Computational and Experimental Simulations in Engineering: Proceedings of ICCES2019 [1st ed. 2020] 978-3-030-27052-0, 978-3-030-27053-7

Table of contents :
Front Matter ....Pages i-xix
Front Matter ....Pages 1-1
Investigation of Systematic Error Characteristics and Error Elimination Method in Digital Image Correlation (Shuichi Arikawa, Yuma Kume, Satoru Yoneyama, Yasuhisa Fujimoto)....Pages 3-9
Simultaneous Identification of Two-Independent Viscoelastic Characteristics with the Virtual Fields Method (Yusuke Hoshino, Yuelin Zheng, Satoru Yoneyama)....Pages 11-20
Optical Phase-Based Method for Dynamic Deflection Measurement of Railroad Bridge (Shien Ri, Qinghua Wang, Hiroshi Tsuda, Hirokazu Shirasaki, Kenichi Kuribayashi)....Pages 21-25
Full-Field Microscale Strain Measurement of Carbon Fiber Reinforced Plastic Using 2-Pixel Sampling Moiré (Qinghua Wang, Shien Ri, Peng Xia, Hiroshi Tsuda)....Pages 27-34
Ascending Order Constraints Sensitivity Optimal Design Method for Steel Structure (Junchen Guo, Xin Zhao)....Pages 35-47
Front Matter ....Pages 49-49
Single Driven Constraint Optimal Structural Design of Tall Buildings Under Period Constraint (Hao Zhang, Jiemin Ding, Xin Zhao, Lang Qin)....Pages 51-60
Numerical Assessment Regarding the Influence of the Stiffness of the Material Used to Build Multi-layer Energy-Absorbing Panels on the Absorption of the Shock Wave Energy (Grzegorz Sławiński, Piotr Malesa, Marek Świerczewski)....Pages 61-79
Front Matter ....Pages 81-81
Computation of Mixed Mode Stress Intensity Factors in 3D Functionally Graded Material Using Tetrahedral Finite Element (Omar Tabaza, Hiroshi Okada, Yasunori Yusa)....Pages 83-90
A Smoothing Gradient-Enhanced Damage Model (Tinh Quoc Bui)....Pages 91-96
Front Matter ....Pages 97-97
Thermal Behavior of Phase Change Material (PCM) Inside a Cavity: Numerical Approach (Md. A. A. Shak, A. M. Bayomy, S. B. Dworkin, J. Wang, M. Z. Saghir)....Pages 99-104
Front Matter ....Pages 105-105
Dynamic Analysis of Stochastic Friction Systems Using the Generalized Cell Mapping Method (Shichao Ma, Xin Ning, Liang Wang)....Pages 107-118
Nonlinear Flight Dynamics of Very Flexible Aircraft (Chen Zhanjun, Fu Zhichao, Lv Jinan, Liu Ziqiang)....Pages 119-123
Front Matter ....Pages 125-125
Hydro-mechanical Properties of Unsaturated Decomposed Granite in Triaxial Compression Test Under Drained-Vented/Undrained-Unvented Condition (X. Xiong, S. Okino, R. Mikami, T. Tsunemoto, X. Y. Qiu, Y. Kurimoto et al.)....Pages 127-133
Effective Foundation Input Motion for Soil-Steel Pipe Sheet Pile (SPSP) Foundation System (Md. Shajib Ullah, Keisuke Kajiwara, Chandra Shekhar Goit, Masato Saitoh)....Pages 135-148
Calculation and Analysis for Fracture Pressure of Deep Water Shallow Formation (Reyu Gao, Jun Li, Kuidong Luo, Hongwei Yang, Qingxin Meng, Wenbao Zhai)....Pages 149-160
The Study on the Influence of Acidification on the Pore and Seepage Field of Carbonate Reservoir (Shuai Cui, Houbin Liu, Teng Liu, Anran Yu, Xu Han)....Pages 161-172
Front Matter ....Pages 173-173
Treatment of Textile Dyes Wastewater Using Electro-Coagulation (Mahmoud A. Elsheikh, Hazem I. Saleh, Hany S. Guirguis, Karim Taha)....Pages 175-187
Stagnation Point Flow and Heat Transfer Over a Permeable Stretching/Shrinking Sheet with Heat Source/Sink (Izyan Syazana Awaludin, Anuar Ishak, Ioan Pop)....Pages 189-199
Heat Transfer Analysis of Icing Process on Metallic Surfaces of Different Wettabilities (Kewei Shi, Xili Duan)....Pages 201-206
Artificial Force Free Boundaries: Particle-Based Fluid Simulation with Implicit Surfaces (Yasutomo Kanetsuki, Susumu Nakata)....Pages 207-222
Droplet-Falling Impact Simulations by Particle-Based Method (Kazuhiko Kakuda, Wataru Okaniwa, Shinichiro Miura)....Pages 223-227
Numerical Modeling of Bridge Piers Scouring Flow Patterns (Mahdi Alemi, João Pedro Pêgo, Rodrigo Maia)....Pages 229-236
Estimation of Electric Field Between the Capillary and Wire-Netting Electrodes During the Electrostatic Atomization from Bio-emulsified Fuel (Chien-hua Fu, Osamu Imamura, Kazuhiro Akihama, Hiroshi Yamasaki)....Pages 237-248
Research on Wind Field of Transmission Tower Under the Complex Terrain (Qiang Shengpei, Qi Fei, Gao Qiang)....Pages 249-255
Front Matter ....Pages 257-257
Application of a New Infinite Element Method for Free Vibration Analysis of Thin Plate with Complicated Shapes (D. S. Liu, Y. W. Chen)....Pages 259-279
Front Matter ....Pages 281-281
Development of Smart-Technology for Forecasting Technical State of Equipment Based on Modified Particle Swarm Algorithms and Immune-Network Modeling (Galina Samigulina, Zhazira Massimkanova)....Pages 283-293
Method of Computer Simulation of Thermal Processes to Ensure the Laser Gyros Stable Operation (Evgenii Kuznetsov, Yuri Kolbas, Yury Kofanov, Nikita Kuznetsov, Tatiana Soloveva)....Pages 295-299
Numerical Approach of Viscous Flow Containing Short Fiber by SPH Method (Nobuki Yamagata, Masakazu Ichimiya)....Pages 301-307
High Viscous Flow Analysis in the 3D Printer by SPH (Masakazu Ichimiya, Nobuki Yamagata)....Pages 309-317
Content Structure for Driving Object Parameters in Contextual Model of Engineering Structure (László Horváth)....Pages 319-333
Quasi Two Dimensional FEM Model for Form Rolling Analysis and Its Application with LS-DYNA (Tomohiko Ariyoshi, Ken-ichi Kawai)....Pages 335-341
The Study of Limit Load and Plastic Collapse Load Under Combined Loads (Ying Zhang, Bin Zheng, Liping Zhang, Zhenyu Liu, Juan Du)....Pages 343-361
Experimental Research on Uniaxial Tensile Behavior and Cyclic Deformation Behavior of TA17 (Juan Du, Xuejiao Shao, Linyuan Kuang, Yuechuan Lu, Minda Yu, Jiang Lu)....Pages 363-373
Study on Uniaxial Tension and Cyclic Deformation Behavior of TA16 at Room Temperature and High Temperature (Xuejiao Shao, Du Juan, Linyuan kuang, Yuechuan Lu, Jiang Lu, Mingda Yu)....Pages 375-386
Front Matter ....Pages 387-387
Implementation of Combined Ohno-Wang Nonlinear Kinematic Hardening Model and Norton-Bailey Creep Model Using Partitioned Stress Integration Technique (Tomoshi Miyamura, Yasunori Yusa, Jun Yin, Kuniaki Koike, Takashi Ikeda, Tomonori Yamada)....Pages 389-396
Cross Domain Recommendations Based on the Application of Fuzzy AHP and Fuzzy Inference Method in Establishing Transdisciplinary Collaborations (Maslina Binti Zolkepli, Teh Noranis Binti Mohd. Aris)....Pages 397-412
Merging Behavior Simulation of Vehicular Platoon (Eisuke Kita, Miichiro Yamada, Daisuke Ishizawa)....Pages 413-418
Front Matter ....Pages 419-419
P-FEM Based on Meshless Trial and Test Functions: Part I-MLS Approximation (Xiang Li, Wei Guo, Xiaoping Chen)....Pages 421-437
Thermodynamic Performance Analysis on Various Configurations of Organic Rankine Cycle Systems (Jun Fen Li, Hang Guo, Biao Lei, Yu Ting Wu, Fang Ye, Chong Fang Ma)....Pages 439-446
Front Matter ....Pages 447-447
Prediction of Rubber Friction on Wet and Dry Rough Surfaces Using Flow Structure Coupling Simulation (Takayoshi Kubota, Yusuke Mizuno, Shun Takahashi, Ryota Asa, Reina Sagara, Yuji Kodama et al.)....Pages 449-463
Zonal Reduced-Order Modeling of Unsteady Flow Field (Takashi Misaka)....Pages 465-474
Real-Time Prediction of Wind and Atmospheric Turbulence Using Aircraft Flight Data (Ryota Kikuchi, Takashi Misaka, Shigeru Obayashi)....Pages 475-487
Front Matter ....Pages 489-489
Stress-Strain-Based Approach to Catastrophic Failure of Steel Structures at Low Temperatures (V. M. Kornev)....Pages 491-510
Cement Failure Caused by Thermal Stresses with Casing Eccentricity During CO2 Injection (Xuelin Dong, Deli Gao, Zhiyin Duan)....Pages 511-524
Optimal Design for Toggle Brace Damper Systems Based on Virtual VD Modal (Bingjie Du, Xin Zhao, Hao Li)....Pages 525-535
Viscous-Tuned Hybrid Structural Vibration Mitigation System: Wind-Induced Response Analyses Method and Case Study (Yue Yang, Xin Zhao, Weixing Shi)....Pages 537-546
Optimal Placement of Friction Dampers in High Rise Buildings Under Seismic Excitation (Apetsi Ampiah, Xin Zhao)....Pages 547-556
Prospect of Using Nano Particles in Compatible Water for EOR Application (M. Al-Samhan, F. Jasim, F. Al-Attar, J. AL-Fadhli)....Pages 557-565
Performance-Based Design Optimization of Steel Braced Frame Using an Efficient Discrete Algorithm (X. Wang, Q. Zhang, X. Qin, Y. Sun)....Pages 567-584
Study on Influence of Mechanical Parameters of Cement Plug on Sealing Integrity of Abandoned Wellbore (Jiwei Jiang, Jun Li, Jiejing Nie, Gonghui Liu, Tao Huang, Wai Li)....Pages 585-595
Front Matter ....Pages 597-597
Hyperelastic Nonlinear Thermal Constitutive Equation of Vulcanized Natural Rubber (Yufei Liao, Chen Li, Weiwei Zhang)....Pages 599-607
Modelling the Edge Crushing Performance of Corrugated Fibreboard Under Different Moisture Content Levels (Aiman Jamsari, Andrew Nevins, Celia Kueh, Eli Gray-Stuart, Karl Dahm, John Bronlund)....Pages 609-620
Multiaxial Stress Based High Cycle Fatigue Model for Adhesive Joint Interfaces (M. A. Eder, S. Semenov, M. Sala)....Pages 621-632
Rating of Polymers for Low-Cost Rapid Manufacturing of Individualized Anatomical Models Used in Presurgical Planning (Magdalena Żukowska, Filip Górski, Adam Hamrol)....Pages 633-644
Front Matter ....Pages 645-645
Research on Annular Pressure Buildup in Deepwater Oil and Gas Well (Xueting Wu, Ling Xiao)....Pages 647-655
Optimal Drilling Parameters Design Based on Single Drilling Depth Indicator in Controlled Gradient Drilling (Jiangshuai Wang, Jun Li, Gonghui Liu, Hongwei Yang, Kuidong Luo)....Pages 657-666
Calculation of Transient Fluctuation Pressure in Deep Water Dual Gradient Drilling (Kuidong Luo, Jun Li, Nan Ma, Reyu Gao, Jiangshuai Wang)....Pages 667-678
Design and Analysis of the Composite Hollow Glass Spheres Separator for Dual-Gradient Drilling in Deep Water (Ruiyao Zhang, Jun Li)....Pages 679-692
Methane Hydrate Generation Model and Software Development Based on P. Englezos Method (Nan Ma, Jun Li, Kuidong Luo, Shujie Liu, Min Wen)....Pages 693-706
Front Matter ....Pages 707-707
Implementation of SPH and DEM for a PEZY-SC Heterogeneous Many-Core System (Natsuki Hosono, Mikito Furuichi)....Pages 709-715
pzqd: PEZY-SC2 Acceleration of Double-Double Precision Arithmetic Library for High-Precision BLAS (Toshiaki Hishinuma, Maho Nakata)....Pages 717-736
Efficient GPU Integration for Multi-loop Feynman Diagrams with Massless Internal Lines (Elise de Doncker, Fukuko Yuasa, Ahmed Almulihi)....Pages 737-747
Front Matter ....Pages 749-749
Structural Damage Detection with Uncertainties Using a Modified Tree Seeds Algorithm (Zhenghao Ding, Jun Li, Hong Hao)....Pages 751-760
Front Matter ....Pages 761-761
Tensile Properties of Carbon Fiber Reinforced Polymer Matrix Composite (Eva Kormanikova, Milan Zmindak, Peter Sabol)....Pages 763-770
Increasing of Fluid Effect on Liquid Storage Laminated Composite Tank During Seismic Excitation (Kamila Kotrasova, Eva Kormanikova)....Pages 771-776
Bending of Piezo-Electric FGM Plates by a Mesh-Free Method (V. Sladek, L. Sator, J. Sladek)....Pages 777-790
Front Matter ....Pages 791-791
Temperature-Dependent Raman Spectroscopy of Graphitic Nanomaterials (Prabhakar Misra, Daniel Casimir, Raul Garcia-Sanchez)....Pages 793-800
Front Matter ....Pages 801-801
A Numerical Mesoscopic Method for Simulating Mechanical Properties of Fiber Reinforced Concrete (Zhimin Zeng, Weizhen Chen, Wenzhao Wang)....Pages 803-812
Front Matter ....Pages 813-813
Simulation of Casting Geometry Effect in Low-Frequency Electromagnetic Casting (Vanja Hatić, Boštjan Mavrič, Božidar Šarler)....Pages 815-825
Prevention Technology of Shrinkage Using Actual Casting Test and Casting Simulation (Atsushi Kishimoto, Yusaku Takagawa, Tomonobu Saito)....Pages 827-837
Front Matter ....Pages 839-839
Large-Scale Serial-Sectioning Observation of 3D Steel Microstructures Based on Efficient Exploring of Etching Conditions Using 3D Internal Structure Microscope (Norio Yamashita, Yuichi Koyanagi, Hiroshi Takemura, Kentaro Asakura, Tadashi Kasuya, Susumu Tsukamoto et al.)....Pages 841-850
Front Matter ....Pages 851-851
Recovering the Initial and Boundary Data in the Two-Dimensional Inverse Heat Conduction Problems Using the Novel Space-Time Collocation Meshfree Approach (Chih-Yu Liu, Cheng-Yu Ku)....Pages 853-859
A Newton’s Second Law Abided Darcy-Brinkman-Forchheimer Framework in Matrix Acidization Simulation (Yuanqing Wu, Maoqing Ye)....Pages 861-866
Calculation of Offshore Platform Internal Force Based on the Secondary Development of ANSYS (Wei Liu, Sheng Dong, Nan Liu)....Pages 867-877
Influence of Salts on Morphology of Structures in Surfactant-Polymer Solutions Explored by Coarse Grained Dynamic Simulation (Dongjie Liu, Fei Liu, Wenjing Zhou, Fei Chen, Jinjia Wei)....Pages 879-884
Dynamic Mesh Technology for the Simulation Study of Single Screw Expander (Lili Shen, Wei Wang, Yuting Wu, Biao Lei, Ruiping Zhi, Chongfang Ma)....Pages 885-889
Flow Pattern, Liquid Holdup and Pressure Drop of Gas-Liquid Two-Phase Flow with Different Liquid Viscosities (Zilong Liu, Ruiquan Liao, Yindi Zhang, Yubin Su, Xiaoya Feng)....Pages 891-905
Theoretical Analysis on the Lifetime of Sessile Droplet Evaporation (Yang Shen, Yongpan Cheng, Jinliang Xu, Kai Zhang)....Pages 907-914
Effect of Swirling Strength on Flow Characteristics of a Heavy-Duty Gas Turbine Annular Combustion Chamber (Zaiguo Fu, Huanhuan Gao, Lingtong Li, Jiang Liu, Zhuoxiong Zeng, Jianxing Ren)....Pages 915-925
PCM-Based Thermal Management of a Wireless Electric Vehicle Charging System (Zaiguo Fu, Lingtong Li, Qunzhi Zhu, Tao Zhang, Zhiyuan Cheng)....Pages 927-932
Study on Gas Channeling Regularity and Anti-channeling Measures of Multi-component Thermal Fluid Huff and Puff for Xinjiang Heavy Oil Reservoirs (Wenwei Wu, Liguo Zhong, Xiaodong Han, Lipeng Tong, Cheng Wang, Caixia Wang et al.)....Pages 933-953
Direct Numerical Simulation on Turbulent Flow over a Truncated Pyramid with a Wavy Surface in an Open Channel (Mitsuhiro Shintani, Mitsuo Matsumoto, Yoshimichi Hagiwara)....Pages 955-968
Recent Progress on Phase Equilibrium Calculation in Subsurface Reservoirs Using Diffuse Interface Models (Tao Zhang, Yiteng Li, Jianchao Cai, Shuyu Sun)....Pages 969-982
Study on Calculation of Mixed Oil-Water Volume During the Commissioning of Large Drop Continuous U-Shaped Liquid Pipelines (Wang Li, Kun Wang, Zhijian Zhang, Sha Chen)....Pages 983-1003
Modelling and the FEM Analysis of the Effects of the Blast Wave on the Side of a Vehicle According to the AEP-55 Vol. 3 Methodology (Piotr Malesa, Grzegorz Sławiński, Marek Świerczewski)....Pages 1005-1014
Simulation Analysis of Thermal Stress of CFST Arch Bridge During Exothermic Hydration Process (Jianyuan Sun, Jinbao Xie, Zhisheng Zhang)....Pages 1015-1021
Front Matter ....Pages 1023-1023
Simulation of Interfacial Sliding Problem of Fiber Reinforced Composites Using Constraint Conditional Finite Element Method (Ryuta Kitamura)....Pages 1025-1031
Front Matter ....Pages 1033-1033
Design of a Wideband Antipodal Vivaldi Antenna with High Gain for Detecting Breast Cancer (Lulu Wang, Huiyong Chen)....Pages 1035-1040
Research on the Fitting Method for QRS Wave Group of ECG Signals Based on Ant Colony Algorithm (Fan Yang, Bo Chen, Kun Zhu, Zhaobi Chu)....Pages 1041-1049
Learning-Based Metal Artifacts Removal in Head CT (Shipeng Xie, Qian Chen)....Pages 1051-1059
Models of Markov Chain with Weights and Its Application in Predicting the Magnetic Field Intensity of Magnetocardiogram Signal (He Jinhao, Chen Bo, Zhu Kun, Huang Kaicheng)....Pages 1061-1067
Sparse-View CT Reconstruction Based on Improved Re-Sidual Network (Yufei Qian, Shipeng Xie, Wenqin Zhuang, Haibo Li)....Pages 1069-1080
The Simulation and Experiment Study of a Wearable, Flexible and Transparent Biomimetic Voltage Source (Zhen Liang, Runhuai Yang, Chenchen Xu, Shaohui Hou, Guoqing Jin, Fuzhou Niu)....Pages 1081-1089
Neural-Network-Based Outcome Classification for Nursing Care (Ning An, Liuqi Jin, Hong Ming, Wenjuan Cheng, Jiaoyun Yang)....Pages 1091-1099
Auto Rickshaw—Pedestrian Head and Neck Impact Injury Mitigation—An Analysis of Impact Material Alternatives, Their Costs and Their Benefits (A. J. Al-Graitti, T. Smith, R. Prabhu, M. D. Jones)....Pages 1101-1123
Modal Analysis of Respiratory Cilia (Qu Jiaqi, Gao Qiang)....Pages 1125-1131
Automated Design of Customized 3D-Printed Wrist Orthoses on the Basis of 3D Scanning (Filip Górski, Przemysław Zawadzki, Radosław Wichniarek, Wiesław Kuczko, Magdalena Żukowska, Izabela Wesołowska et al.)....Pages 1133-1143
Front Matter ....Pages 1145-1145
Research on the Permeability Model of Fractal Fractured Media in 3D Coordinate System (Huan Zhao, Wei Li, Lei Wang, Xin Ling, Dandan Shan, Bing Li et al.)....Pages 1147-1155
Modelling and the FEM Analysis of the Effects of the Blast Wave on the Floor of a Vehicle According to the AEP-55 Vol. 2 Methodology (Marek Świerczewski, Grzegorz Sławiński, Piotr Malesa)....Pages 1157-1168
Front Matter ....Pages 1169-1169
Dynamic Impacts of Catalyst Management on Self-fluidized Pump-Free Ebullated-Bed Reactor (Bo Chen, Zhaohui Meng, Hailong Ge, Jordy Botello, Tao Yang, Xiangchen Fang)....Pages 1171-1185
Front Matter ....Pages 1187-1187
A Fine Simulation Analysis of Rock Fragmentation Mechanism of TBM Disc Cutter with DEM (Yadong Xue, Jie Zhou, Feng Zhao, Hanxiang Zhao)....Pages 1189-1205
Research on Pollution Accumulation Characteristics of Insulators in Natural Pollution Accumulation Experiments (Mao Dong, Gao Qiang)....Pages 1207-1216
Fast Reconstruction of Transient Heat-Flux Distributions in a Laser Heating Process with Time-Space Adaptive Mesh Refinement (Qing-Qing Yang, Jiu Luo, Dong-Chuan Mo, Shu-Shen Lyu, Yi Heng)....Pages 1217-1223
Transfer Learning Approach in Automatic Tropical Wood Recognition System (Rubiyah Yusof, Azlin Ahmad, Anis Salwa Mohd Khairuddin, Uswah Khairuddin, Nik Mohamad Aizuddin Nik Azmi, Nenny Ruthfalydia Rosli)....Pages 1225-1233
Moving Least Squares (MLS) Interpolation Based Post-processing Parametric Study in Finite Element Elastic Problems (Mohd. Ahmed, Mohamed Hechmi El Ouni, Devender Singh, Nabil Ben Kahla)....Pages 1235-1252
Genetically Aerodynamic Optimization of High-Speed Train Based on the Artificial Neural Network Method (Fu Tao, Chen Zhaobo, Wang Zhonglong)....Pages 1253-1270
Front Matter ....Pages 1271-1271
Predicting Structure Dynamic Acceleration Based on Measured Strain (Wang Yuansheng, Lan Chunbo, Qin Weiyang, Yue ZhuFeng)....Pages 1273-1282
Experiment Investigation of Constrained Layer Damping Used for Vibration Suppression of Railway Wheel (Wang Zhonglong, Jiao Yinghou, Chen Zhaobo)....Pages 1283-1290
Free Transverse Vibration of Mindlin Annular and Circular Plate with General Boundary Conditions (Qingjun Hao, Zhaobo Chen, Wenjie Zhai)....Pages 1291-1302
Underdetermined Blind Source Separation for Multi-fault Diagnosis of Planetary Gearbox (H. Li, Q. Zhang, X. R. Qin, Y. T. Sun)....Pages 1303-1316
Study on Sound Transmission Loss of Lightweight FGM Sandwich Plate (C. Li, Z. Chen, Y. Jiao)....Pages 1317-1328
Supervised Learning for Finite Element Analysis of Holes Under Tensile Load (Wai Tuck Chow)....Pages 1329-1339
Back Matter ....Pages 1341-1342

Citation preview

Mechanisms and Machine Science 75

Hiroshi Okada Satya N. Atluri   Editors

Computational and Experimental Simulations in Engineering Proceedings of ICCES2019

Mechanisms and Machine Science Volume 75

Series Editor Marco Ceccarelli Department of Industrial Engineering, University of Rome Tor Vergata, Roma, Italy Editorial Board Alfonso Hernandez Mechanical Engineering, University of the Basque Country, Bilbao, Vizcaya, Spain Tian Huang Department of Mechatronical Engineering, Tianjin University, Tianjin, China Yukio Takeda Mechanical Engineering, Tokyo Institute of Technology, Tokyo, Japan Burkhard Corves Institute of Mechanism Theory, Machine Dynamics and Robotics, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany Sunil Agrawal Department of Mechanical Engineering, Columbia University, New York, NY, USA

This book series establishes a well-defined forum for monographs, edited Books, and proceedings on mechanical engineering with particular emphasis on MMS (Mechanism and Machine Science). The final goal is the publication of research that shows the development of mechanical engineering and particularly MMS in all technical aspects, even in very recent assessments. Published works share an approach by which technical details and formulation are discussed, and discuss modern formalisms with the aim to circulate research and technical achievements for use in professional, research, academic, and teaching activities. This technical approach is an essential characteristic of the series. By discussing technical details and formulations in terms of modern formalisms, the possibility is created not only to show technical developments but also to explain achievements for technical teaching and research activity today and for the future. The book series is intended to collect technical views on developments of the broad field of MMS in a unique frame that can be seen in its totality as an Encyclopaedia of MMS but with the additional purpose of archiving and teaching MMS achievements. Therefore, the book series will be of use not only for researchers and teachers in Mechanical Engineering but also for professionals and students for their formation and future work. The series is promoted under the auspices of International Federation for the Promotion of Mechanism and Machine Science (IFToMM). Prospective authors and editors can contact Mr. Pierpaolo Riva (publishing editor, Springer) at: [email protected] Indexed by SCOPUS and Google Scholar.

More information about this series at http://www.springer.com/series/8779

Hiroshi Okada Satya N. Atluri •

Editors

Computational and Experimental Simulations in Engineering Proceedings of ICCES2019

123

Editors Hiroshi Okada Department of Mechanical Engineering Tokyo University of Science Noda, Chiba, Japan

Satya N. Atluri Department of Mechanical Engineering Texas Tech University Lubbock, TX, USA

ISSN 2211-0984 ISSN 2211-0992 (electronic) Mechanisms and Machine Science ISBN 978-3-030-27052-0 ISBN 978-3-030-27053-7 (eBook) https://doi.org/10.1007/978-3-030-27053-7 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The 25th International Conference on Computational and Experimental Engineering and Sciences (ICCES 2019) was held during March 25–28, 2019 at Tokyo University of Science, Tokyo, Japan. More than 400 distinguished researchers and engineers gathered from various parts of the world and discussed a variety of topics. They include theoretical, analytical, computational, and experimental approaches to the problems of engineering and sciences. It was a memorable event for all the researchers and engineers who attended the conference. The conference series of ICCES started in Tokyo, Japan in 1986 and has been held in various places around the world. It came back to Tokyo for its 25th memorial event. This volume contains articles which were submitted as full papers to ICCES 2019. After their submission, the articles were reviewed rigorously by the volume editors. The volume editors believe that the articles contain the state-of-the-art researches in the field of Computational and Experimental Engineering and Sciences. Prof. Hiroshi Okada Department of Mechanical Engineering Tokyo University of Science Tokyo, Japan Prof. Satya N. Atluri Department of Mechanical Engineering Texas Tech University Lubbock, TX, USA

v

Contents

Part I 1

2

3

4

5

Advances in Experimental Solid Mechanics

Investigation of Systematic Error Characteristics and Error Elimination Method in Digital Image Correlation . . . . . . . . . . . . Shuichi Arikawa, Yuma Kume, Satoru Yoneyama and Yasuhisa Fujimoto Simultaneous Identification of Two-Independent Viscoelastic Characteristics with the Virtual Fields Method . . . . . . . . . . . . . . Yusuke Hoshino, Yuelin Zheng and Satoru Yoneyama Optical Phase-Based Method for Dynamic Deflection Measurement of Railroad Bridge . . . . . . . . . . . . . . . . . . . . . . . . Shien Ri, Qinghua Wang, Hiroshi Tsuda, Hirokazu Shirasaki and Kenichi Kuribayashi

3

11

21

Full-Field Microscale Strain Measurement of Carbon Fiber Reinforced Plastic Using 2-Pixel Sampling Moiré . . . . . . . . . . . . Qinghua Wang, Shien Ri, Peng Xia and Hiroshi Tsuda

27

Ascending Order Constraints Sensitivity Optimal Design Method for Steel Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Junchen Guo and Xin Zhao

35

Part II Advances in Smart and Functional Construction Materials and System for Civil Infrastructure 6

Single Driven Constraint Optimal Structural Design of Tall Buildings Under Period Constraint . . . . . . . . . . . . . . . . . . . . . . . Hao Zhang, Jiemin Ding, Xin Zhao and Lang Qin

51

vii

viii

7

Contents

Numerical Assessment Regarding the Influence of the Stiffness of the Material Used to Build Multi-layer Energy-Absorbing Panels on the Absorption of the Shock Wave Energy . . . . . . . . . Grzegorz Sławiński, Piotr Malesa and Marek Świerczewski

Part III 8

9

10

A Smoothing Gradient-Enhanced Damage Model . . . . . . . . . . . . Tinh Quoc Bui

12

13

14

91

99

Nonlinear Dynamics and Control in Aerospace Engineering

Dynamic Analysis of Stochastic Friction Systems Using the Generalized Cell Mapping Method . . . . . . . . . . . . . . . Shichao Ma, Xin Ning and Liang Wang Nonlinear Flight Dynamics of Very Flexible Aircraft . . . . . . . . . Chen Zhanjun, Fu Zhichao, Lv Jinan and Liu Ziqiang

Part VI

83

Phase Change Material: Numerical and Experimental Results

Thermal Behavior of Phase Change Material (PCM) Inside a Cavity: Numerical Approach . . . . . . . . . . . . . . . . . . . . . . . . . . Md. A. A. Shak, A. M. Bayomy, S. B. Dworkin, J. Wang and M. Z. Saghir

Part V 11

Computational Fracture Modeling on Welded Joints and Advanced Materials

Computation of Mixed Mode Stress Intensity Factors in 3D Functionally Graded Material Using Tetrahedral Finite Element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Omar Tabaza, Hiroshi Okada and Yasunori Yusa

Part IV

61

107 119

Computational and Experimental Methods in Geotechnical and Multidisciplinary Engineering Problems

Hydro-mechanical Properties of Unsaturated Decomposed Granite in Triaxial Compression Test Under Drained-Vented/ Undrained-Unvented Condition . . . . . . . . . . . . . . . . . . . . . . . . . . X. Xiong, S. Okino, R. Mikami, T. Tsunemoto, X. Y. Qiu, Y. Kurimoto and F. Zhang Effective Foundation Input Motion for Soil-Steel Pipe Sheet Pile (SPSP) Foundation System . . . . . . . . . . . . . . . . . . . . . . . . . . Md. Shajib Ullah, Keisuke Kajiwara, Chandra Shekhar Goit and Masato Saitoh

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Contents

15

16

Calculation and Analysis for Fracture Pressure of Deep Water Shallow Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reyu Gao, Jun Li, Kuidong Luo, Hongwei Yang, Qingxin Meng and Wenbao Zhai The Study on the Influence of Acidification on the Pore and Seepage Field of Carbonate Reservoir . . . . . . . . . . . . . . . . . Shuai Cui, Houbin Liu, Teng Liu, Anran Yu and Xu Han

Part VII 17

18

19

20

21

22

23

24

ix

149

161

Computational and Experimental Fluid/Electromagnetic Dynamics and Other Applications

Treatment of Textile Dyes Wastewater Using Electro-Coagulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mahmoud A. Elsheikh, Hazem I. Saleh, Hany S. Guirguis and Karim Taha

175

Stagnation Point Flow and Heat Transfer Over a Permeable Stretching/Shrinking Sheet with Heat Source/Sink . . . . . . . . . . . Izyan Syazana Awaludin, Anuar Ishak and Ioan Pop

189

Heat Transfer Analysis of Icing Process on Metallic Surfaces of Different Wettabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kewei Shi and Xili Duan

201

Artificial Force Free Boundaries: Particle-Based Fluid Simulation with Implicit Surfaces . . . . . . . . . . . . . . . . . . . . . . . . Yasutomo Kanetsuki and Susumu Nakata

207

Droplet-Falling Impact Simulations by Particle-Based Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kazuhiko Kakuda, Wataru Okaniwa and Shinichiro Miura

223

Numerical Modeling of Bridge Piers Scouring Flow Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mahdi Alemi, João Pedro Pêgo and Rodrigo Maia

229

Estimation of Electric Field Between the Capillary and Wire-Netting Electrodes During the Electrostatic Atomization from Bio-emulsified Fuel . . . . . . . . . . . . . . . . . . . . . Chien-hua Fu, Osamu Imamura, Kazuhiro Akihama and Hiroshi Yamasaki

237

Research on Wind Field of Transmission Tower Under the Complex Terrain . . . . . . . . . . . . . . . . . . . . . . . . . . . . Qiang Shengpei, Qi Fei and Gao Qiang

249

x

Contents

Part VIII 25

Application of a New Infinite Element Method for Free Vibration Analysis of Thin Plate with Complicated Shapes . . . . . D. S. Liu and Y. W. Chen

Part IX 26

27

28

Nano/Micro Structures in Application of Computational Mechanics 259

Design, Analysis and Manufacturing Through Simulation

Development of Smart-Technology for Forecasting Technical State of Equipment Based on Modified Particle Swarm Algorithms and Immune-Network Modeling . . . . . . . . . . . . . . . . Galina Samigulina and Zhazira Massimkanova Method of Computer Simulation of Thermal Processes to Ensure the Laser Gyros Stable Operation . . . . . . . . . . . . . . . . Evgenii Kuznetsov, Yuri Kolbas, Yury Kofanov, Nikita Kuznetsov and Tatiana Soloveva Numerical Approach of Viscous Flow Containing Short Fiber by SPH Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nobuki Yamagata and Masakazu Ichimiya

283

295

301 309

29

High Viscous Flow Analysis in the 3D Printer by SPH . . . . . . . . Masakazu Ichimiya and Nobuki Yamagata

30

Content Structure for Driving Object Parameters in Contextual Model of Engineering Structure . . . . . . . . . . . . . . László Horváth

319

Quasi Two Dimensional FEM Model for Form Rolling Analysis and Its Application with LS-DYNA . . . . . . . . . . . . . . . . Tomohiko Ariyoshi and Ken-ichi Kawai

335

The Study of Limit Load and Plastic Collapse Load Under Combined Loads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ying Zhang, Bin Zheng, Liping Zhang, Zhenyu Liu and Juan Du

343

31

32

33

34

Experimental Research on Uniaxial Tensile Behavior and Cyclic Deformation Behavior of TA17 . . . . . . . . . . . . . . . . . Juan Du, Xuejiao Shao, Linyuan Kuang, Yuechuan Lu, Minda Yu and Jiang Lu Study on Uniaxial Tension and Cyclic Deformation Behavior of TA16 at Room Temperature and High Temperature . . . . . . . Xuejiao Shao, Du Juan, Linyuan kuang, Yuechuan Lu, Jiang Lu and Mingda Yu

363

375

Contents

Part X

35

36

37

39

Cross Domain Recommendations Based on the Application of Fuzzy AHP and Fuzzy Inference Method in Establishing Transdisciplinary Collaborations . . . . . . . . . . . . . . . . . . . . . . . . . Maslina Binti Zolkepli and Teh Noranis Binti Mohd. Aris Merging Behavior Simulation of Vehicular Platoon . . . . . . . . . . Eisuke Kita, Miichiro Yamada and Daisuke Ishizawa

389

397 413

Theory and Application of Meshless Methods for the Numerical Solution of Engineering and Scientific Applications

P-FEM Based on Meshless Trial and Test Functions: Part I-MLS Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xiang Li, Wei Guo and Xiaoping Chen Thermodynamic Performance Analysis on Various Configurations of Organic Rankine Cycle Systems . . . . . . . . . . . Jun Fen Li, Hang Guo, Biao Lei, Yu Ting Wu, Fang Ye and Chong Fang Ma

Part XII 40

High-Performance and Intelligent Computing for Real World’s Applications (Celebrating the 60th Birthday of Shinobu Yoshimura)

Implementation of Combined Ohno-Wang Nonlinear Kinematic Hardening Model and Norton-Bailey Creep Model Using Partitioned Stress Integration Technique . . . . . . . . . . . . . . Tomoshi Miyamura, Yasunori Yusa, Jun Yin, Kuniaki Koike, Takashi Ikeda and Tomonori Yamada

Part XI

38

xi

421

439

Data-Driven Estimation and Control of Flow Fields in Engineering Applications

Prediction of Rubber Friction on Wet and Dry Rough Surfaces Using Flow Structure Coupling Simulation . . . . . . . . . . Takayoshi Kubota, Yusuke Mizuno, Shun Takahashi, Ryota Asa, Reina Sagara, Yuji Kodama and Shigeru Obayashi

41

Zonal Reduced-Order Modeling of Unsteady Flow Field . . . . . . . Takashi Misaka

42

Real-Time Prediction of Wind and Atmospheric Turbulence Using Aircraft Flight Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ryota Kikuchi, Takashi Misaka and Shigeru Obayashi

449

465

475

xii

Contents

Part XIII 43

44

45

46

47

48

49

50

Stress-Strain-Based Approach to Catastrophic Failure of Steel Structures at Low Temperatures . . . . . . . . . . . . . . . . . . . . . . . . . V. M. Kornev

491

Cement Failure Caused by Thermal Stresses with Casing Eccentricity During CO2 Injection . . . . . . . . . . . . . . . . . . . . . . . . Xuelin Dong, Deli Gao and Zhiyin Duan

511

Optimal Design for Toggle Brace Damper Systems Based on Virtual VD Modal . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bingjie Du, Xin Zhao and Hao Li

525

Viscous-Tuned Hybrid Structural Vibration Mitigation System: Wind-Induced Response Analyses Method and Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yue Yang, Xin Zhao and Weixing Shi

52

537

Optimal Placement of Friction Dampers in High Rise Buildings Under Seismic Excitation . . . . . . . . . . . . . . . . . . . . . . . Apetsi Ampiah and Xin Zhao

547

Prospect of Using Nano Particles in Compatible Water for EOR Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. Al-Samhan, F. Jasim, F. Al-Attar and J. AL-Fadhli

557

Performance-Based Design Optimization of Steel Braced Frame Using an Efficient Discrete Algorithm . . . . . . . . . . . . . . . X. Wang, Q. Zhang, X. Qin and Y. Sun

567

Study on Influence of Mechanical Parameters of Cement Plug on Sealing Integrity of Abandoned Wellbore . . . . . . . . . . . . . . . . Jiwei Jiang, Jun Li, Jiejing Nie, Gonghui Liu, Tao Huang and Wai Li

Part XIV 51

Structural Damage Identification, Maintenance and Life-Cycle Integrity Management

585

Composite Materials: Modelling, Processing, Design and Application

Hyperelastic Nonlinear Thermal Constitutive Equation of Vulcanized Natural Rubber . . . . . . . . . . . . . . . . . . . . . . . . . . . Yufei Liao, Chen Li and Weiwei Zhang Modelling the Edge Crushing Performance of Corrugated Fibreboard Under Different Moisture Content Levels . . . . . . . . . Aiman Jamsari, Andrew Nevins, Celia Kueh, Eli Gray-Stuart, Karl Dahm and John Bronlund

599

609

Contents

53

54

Multiaxial Stress Based High Cycle Fatigue Model for Adhesive Joint Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. Eder, S. Semenov and M. Sala Rating of Polymers for Low-Cost Rapid Manufacturing of Individualized Anatomical Models Used in Presurgical Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magdalena Żukowska, Filip Górski and Adam Hamrol

Part XV

55

56

57

58

59

61

62

621

633

Computer Simulation of Dynamic Mass and Heat Transfer in Gas-Liquid-Solid 3-Phase Flow in Harvesting Natural Gas from Subsea Gas Hydrate Depositions

Research on Annular Pressure Buildup in Deepwater Oil and Gas Well . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Xueting Wu and Ling Xiao Optimal Drilling Parameters Design Based on Single Drilling Depth Indicator in Controlled Gradient Drilling . . . . . . . . . . . . . Jiangshuai Wang, Jun Li, Gonghui Liu, Hongwei Yang and Kuidong Luo

647

657

Calculation of Transient Fluctuation Pressure in Deep Water Dual Gradient Drilling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kuidong Luo, Jun Li, Nan Ma, Reyu Gao and Jiangshuai Wang

667

Design and Analysis of the Composite Hollow Glass Spheres Separator for Dual-Gradient Drilling in Deep Water . . . . . . . . . Ruiyao Zhang and Jun Li

679

Methane Hydrate Generation Model and Software Development Based on P. Englezos Method . . . . . . . . . . . . . . . . Nan Ma, Jun Li, Kuidong Luo, Shujie Liu and Min Wen

693

Part XVI 60

xiii

New Horizon of Computational Science and Engineering with Heterogeneous Many-Core Processors

Implementation of SPH and DEM for a PEZY-SC Heterogeneous Many-Core System . . . . . . . . . . . . . . . . . . . . . . . Natsuki Hosono and Mikito Furuichi

709

pzqd: PEZY-SC2 Acceleration of Double-Double Precision Arithmetic Library for High-Precision BLAS . . . . . . . . . . . . . . . Toshiaki Hishinuma and Maho Nakata

717

Efficient GPU Integration for Multi-loop Feynman Diagrams with Massless Internal Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . Elise de Doncker, Fukuko Yuasa and Ahmed Almulihi

737

xiv

Contents

Part XVII 63

Structural Uncertainty Quantification and Reliability Analysis

Structural Damage Detection with Uncertainties Using a Modified Tree Seeds Algorithm . . . . . . . . . . . . . . . . . . . Zhenghao Ding, Jun Li and Hong Hao

Part XVIII 64

65

66

763

Increasing of Fluid Effect on Liquid Storage Laminated Composite Tank During Seismic Excitation . . . . . . . . . . . . . . . . . Kamila Kotrasova and Eva Kormanikova

771

Bending of Piezo-Electric FGM Plates by a Mesh-Free Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V. Sladek, L. Sator and J. Sladek

777

70

793

Multi-scale Modeling and Simulation of Functional Materials

A Numerical Mesoscopic Method for Simulating Mechanical Properties of Fiber Reinforced Concrete . . . . . . . . . . . . . . . . . . . Zhimin Zeng, Weizhen Chen and Wenzhao Wang

Part XXI 69

Functional Materials

Temperature-Dependent Raman Spectroscopy of Graphitic Nanomaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prabhakar Misra, Daniel Casimir and Raul Garcia-Sanchez

Part XX 68

Multi-field and Multi-scale Modeling of Advanced Materials

Tensile Properties of Carbon Fiber Reinforced Polymer Matrix Composite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eva Kormanikova, Milan Zmindak and Peter Sabol

Part XIX 67

751

803

Verification and Validation of Metallurgical Processes Simulations

Simulation of Casting Geometry Effect in Low-Frequency Electromagnetic Casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vanja Hatić, Boštjan Mavrič and Božidar Šarler

815

Prevention Technology of Shrinkage Using Actual Casting Test and Casting Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Atsushi Kishimoto, Yusaku Takagawa and Tomonobu Saito

827

Contents

Part XXII 71

73

74

75

76

77

78

Materials Integration-Fusion of Computation and Experiments Through Data Science

Large-Scale Serial-Sectioning Observation of 3D Steel Microstructures Based on Efficient Exploring of Etching Conditions Using 3D Internal Structure Microscope . . . . . . . . . . Norio Yamashita, Yuichi Koyanagi, Hiroshi Takemura, Kentaro Asakura, Tadashi Kasuya, Susumu Tsukamoto and Hideo Yokota

Part XXIII 72

xv

841

Advances in Modeling and Simulation of Heat Transfer and Fluid Flow

Recovering the Initial and Boundary Data in the Two-Dimensional Inverse Heat Conduction Problems Using the Novel Space-Time Collocation Meshfree Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chih-Yu Liu and Cheng-Yu Ku

853

A Newton’s Second Law Abided Darcy-BrinkmanForchheimer Framework in Matrix Acidization Simulation . . . . Yuanqing Wu and Maoqing Ye

861

Calculation of Offshore Platform Internal Force Based on the Secondary Development of ANSYS . . . . . . . . . . . . Wei Liu, Sheng Dong and Nan Liu

867

Influence of Salts on Morphology of Structures in Surfactant-Polymer Solutions Explored by Coarse Grained Dynamic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . Dongjie Liu, Fei Liu, Wenjing Zhou, Fei Chen and Jinjia Wei Dynamic Mesh Technology for the Simulation Study of Single Screw Expander . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lili Shen, Wei Wang, Yuting Wu, Biao Lei, Ruiping Zhi and Chongfang Ma Flow Pattern, Liquid Holdup and Pressure Drop of Gas-Liquid Two-Phase Flow with Different Liquid Viscosities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zilong Liu, Ruiquan Liao, Yindi Zhang, Yubin Su and Xiaoya Feng Theoretical Analysis on the Lifetime of Sessile Droplet Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yang Shen, Yongpan Cheng, Jinliang Xu and Kai Zhang

879

885

891

907

xvi

79

80

81

82

83

84

Contents

Effect of Swirling Strength on Flow Characteristics of a Heavy-Duty Gas Turbine Annular Combustion Chamber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zaiguo Fu, Huanhuan Gao, Lingtong Li, Jiang Liu, Zhuoxiong Zeng and Jianxing Ren PCM-Based Thermal Management of a Wireless Electric Vehicle Charging System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zaiguo Fu, Lingtong Li, Qunzhi Zhu, Tao Zhang and Zhiyuan Cheng Study on Gas Channeling Regularity and Anti-channeling Measures of Multi-component Thermal Fluid Huff and Puff for Xinjiang Heavy Oil Reservoirs . . . . . . . . . . . . . . . . . . . . . . . Wenwei Wu, Liguo Zhong, Xiaodong Han, Lipeng Tong, Cheng Wang, Caixia Wang, Bingyan Liu, Jianbin Liu, Tongchun Hao and Shuang Huang Direct Numerical Simulation on Turbulent Flow over a Truncated Pyramid with a Wavy Surface in an Open Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mitsuhiro Shintani, Mitsuo Matsumoto and Yoshimichi Hagiwara Recent Progress on Phase Equilibrium Calculation in Subsurface Reservoirs Using Diffuse Interface Models . . . . . . . . Tao Zhang, Yiteng Li, Jianchao Cai and Shuyu Sun Study on Calculation of Mixed Oil-Water Volume During the Commissioning of Large Drop Continuous U-Shaped Liquid Pipelines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wang Li, Kun Wang, Zhijian Zhang and Sha Chen

915

927

933

955

969

983

85

Modelling and the FEM Analysis of the Effects of the Blast Wave on the Side of a Vehicle According to the AEP-55 Vol. 3 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005 Piotr Malesa, Grzegorz Sławiński and Marek Świerczewski

86

Simulation Analysis of Thermal Stress of CFST Arch Bridge During Exothermic Hydration Process . . . . . . . . . . . . . . . . . . . . 1015 Jianyuan Sun, Jinbao Xie and Zhisheng Zhang

Contents

Part XXIV

87

xvii

Activities of CAE Advanced Composite Materials and Structures Research Division at Tokyo University of Science

Simulation of Interfacial Sliding Problem of Fiber Reinforced Composites Using Constraint Conditional Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025 Ryuta Kitamura

Part XXV

Computational Modeling in Biomedical Applications

88

Design of a Wideband Antipodal Vivaldi Antenna with High Gain for Detecting Breast Cancer . . . . . . . . . . . . . . . . . . . . . . . . 1035 Lulu Wang and Huiyong Chen

89

Research on the Fitting Method for QRS Wave Group of ECG Signals Based on Ant Colony Algorithm . . . . . . . . . . . . . . . . . . . 1041 Fan Yang, Bo Chen, Kun Zhu and Zhaobi Chu

90

Learning-Based Metal Artifacts Removal in Head CT . . . . . . . . 1051 Shipeng Xie and Qian Chen

91

Models of Markov Chain with Weights and Its Application in Predicting the Magnetic Field Intensity of Magnetocardiogram Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . 1061 He Jinhao, Chen Bo, Zhu Kun and Huang Kaicheng

92

Sparse-View CT Reconstruction Based on Improved Re-Sidual Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1069 Yufei Qian, Shipeng Xie, Wenqin Zhuang and Haibo Li

93

The Simulation and Experiment Study of a Wearable, Flexible and Transparent Biomimetic Voltage Source . . . . . . . . . 1081 Zhen Liang, Runhuai Yang, Chenchen Xu, Shaohui Hou, Guoqing Jin and Fuzhou Niu

94

Neural-Network-Based Outcome Classification for Nursing Care . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1091 Ning An, Liuqi Jin, Hong Ming, Wenjuan Cheng and Jiaoyun Yang

95

Auto Rickshaw—Pedestrian Head and Neck Impact Injury Mitigation—An Analysis of Impact Material Alternatives, Their Costs and Their Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . 1101 A. J. Al-Graitti, T. Smith, R. Prabhu and M. D. Jones

96

Modal Analysis of Respiratory Cilia . . . . . . . . . . . . . . . . . . . . . . 1125 Qu Jiaqi and Gao Qiang

xviii

97

Contents

Automated Design of Customized 3D-Printed Wrist Orthoses on the Basis of 3D Scanning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133 Filip Górski, Przemysław Zawadzki, Radosław Wichniarek, Wiesław Kuczko, Magdalena Żukowska, Izabela Wesołowska and Natalia Wierzbicka

Part XXVI Impact Loading and Failure of Structures and Materials 98

Research on the Permeability Model of Fractal Fractured Media in 3D Coordinate System . . . . . . . . . . . . . . . . . . . . . . . . . 1147 Huan Zhao, Wei Li, Lei Wang, Xin Ling, Dandan Shan, Bing Li and Zhan Su

99

Modelling and the FEM Analysis of the Effects of the Blast Wave on the Floor of a Vehicle According to the AEP-55 Vol. 2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1157 Marek Świerczewski, Grzegorz Sławiński and Piotr Malesa

Part XXVII Modeling and Simulation of Chemical Processes: Optimizations and Applications 100

Dynamic Impacts of Catalyst Management on Self-fluidized Pump-Free Ebullated-Bed Reactor . . . . . . . . . . . . . . . . . . . . . . . 1171 Bo Chen, Zhaohui Meng, Hailong Ge, Jordy Botello, Tao Yang and Xiangchen Fang

Part XXVIII

Theme: Methods for Computer Modeling in Engineering and Sciences

101

A Fine Simulation Analysis of Rock Fragmentation Mechanism of TBM Disc Cutter with DEM . . . . . . . . . . . . . . . . 1189 Yadong Xue, Jie Zhou, Feng Zhao and Hanxiang Zhao

102

Research on Pollution Accumulation Characteristics of Insulators in Natural Pollution Accumulation Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1207 Mao Dong and Gao Qiang

103

Fast Reconstruction of Transient Heat-Flux Distributions in a Laser Heating Process with Time-Space Adaptive Mesh Refinement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1217 Qing-Qing Yang, Jiu Luo, Dong-Chuan Mo, Shu-Shen Lyu and Yi Heng

Contents

xix

104

Transfer Learning Approach in Automatic Tropical Wood Recognition System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1225 Rubiyah Yusof, Azlin Ahmad, Anis Salwa Mohd Khairuddin, Uswah Khairuddin, Nik Mohamad Aizuddin Nik Azmi and Nenny Ruthfalydia Rosli

105

Moving Least Squares (MLS) Interpolation Based Post-processing Parametric Study in Finite Element Elastic Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1235 Mohd. Ahmed, Mohamed Hechmi El Ouni, Devender Singh and Nabil Ben Kahla

106

Genetically Aerodynamic Optimization of High-Speed Train Based on the Artificial Neural Network Method . . . . . . . . . . . . . 1253 Fu Tao, Chen Zhaobo and Wang Zhonglong

Part XXIX

Theme: Sound and Vibration

107

Predicting Structure Dynamic Acceleration Based on Measured Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273 Wang Yuansheng, Lan Chunbo, Qin Weiyang and Yue ZhuFeng

108

Experiment Investigation of Constrained Layer Damping Used for Vibration Suppression of Railway Wheel . . . . . . . . . . . 1283 Wang Zhonglong, Jiao Yinghou and Chen Zhaobo

109

Free Transverse Vibration of Mindlin Annular and Circular Plate with General Boundary Conditions . . . . . . . . . . . . . . . . . . 1291 Qingjun Hao, Zhaobo Chen and Wenjie Zhai

110

Underdetermined Blind Source Separation for Multi-fault Diagnosis of Planetary Gearbox . . . . . . . . . . . . . . . . . . . . . . . . . 1303 H. Li, Q. Zhang, X. R. Qin and Y. T. Sun

111

Study on Sound Transmission Loss of Lightweight FGM Sandwich Plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317 C. Li, Z. Chen and Y. Jiao

112

Supervised Learning for Finite Element Analysis of Holes Under Tensile Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329 Wai Tuck Chow

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1341

Part I

Advances in Experimental Solid Mechanics

Chapter 1

Investigation of Systematic Error Characteristics and Error Elimination Method in Digital Image Correlation Shuichi Arikawa, Yuma Kume, Satoru Yoneyama and Yasuhisa Fujimoto

Abstract In this study, systematic errors and an error elimination method in digital image correlation are investigated for improving accuracy and spatial resolution of deformation measurements. Characteristics of the systematic error of measured displacements in DIC are investigated using an imaging process simulation. The speckle size of the random pattern on the target surface, the gap between each micro lens on the imaging sensor, and the relative position between the micro lenses and the imaged pattern on the micro lenses are considered in the simulation. The error investigation results show that those factors affect the characteristics of the systematic error. Possible error characteristics in actual DIC measurements using random patterns are considered. It is then found that the function of the possible error characteristics shows complex profile and the error becomes zero at integer displacements in pixel. In practical deformation measurement in DIC, the error profile can be obtained using additional measured shift data, if the actual shift amount as the relative translation amount of the target and the camera is known. The function approximation of the error profile and an error elimination method are investigated. Keywords Deformation measurement · Digital image correlation · Systematic error · Error elimination

S. Arikawa (B) Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, Kanagawa 214-8571, Japan e-mail: [email protected] Y. Kume Graduate School of Science and Technology, Meiji University, 1-1-1 Higashimita, Tama-ku, Kawasaki, Kanagawa 214-8571, Japan S. Yoneyama Aoyama Gakuin University, 5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Kanagawa 252-5258, Japan Y. Fujimoto Mitsubishi Electric Corporation, 8-1-1 Tsukaguchi-honmachi, Amagasaki, Hyogo 661-8661, Japan © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_1

3

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1.1 Introduction Recently digital image correlation (DIC) [1] is widely used for various deformation measurements because of its usefulness. The error of measurement results by DIC depends on interpolation functions for the subpixel calculation in the image correlation [2] and is affected by subset shape functions [3]. The measurement accuracy is improved by using high order functions for the subpixel interpolation and by using the large subset area for image correlation. However, using the high order interpolation functions and the large subset area cause degradation of the spatial resolution of the measurement. Therefore, the realization of the coexistence of the high accuracy measurement with the high spatial resolution is still difficult. One of the solutions in a real object scale is using a high resolution camera with optimizing the optical magnification of the imaging lens. However, the image resolution of cameras and the optical resolution of lenses have limitations. On the other hand, if the error that occurs with a small subset area or a low order interpolation function can be eliminated, the coexistence of the high accuracy with the high spatial resolution will be attained. Authors have developed an error elimination method [4, 5]. But some low level errors have remained. To develop a high accuracy error elimination method, the error generation mechanism of DIC measurement should be elucidated more clearly. Also the authors have reported that some physical factors of the imaging process affect the error profile and there is a possibility to develop a high accuracy error elimination method [6]. In this study, imaging process simulations are performed for investigating the effect of various factors such as the arrangement of the micro-lens array on the imaging sensor, the imaged speckle size on the micro-lens array and their relative positions. The effects of various factors for the error profiles are then investigated. Additionally, an error elimination method is considered.

1.2 Imaging Simulation Not only the data processes, but also some physical factors affect the error generation. As shown in Fig. 1.1, the shape and the size of the micro-lens relating the sampling area of light intensities are not the same as those of the pixel. Therefore, some of the light intensity of the imaged pattern enter gaps between the micro-lenses and are lost in the sampling process. This means that the shape and the size of the micro-lens, the imaged speckle size and their relative position affect characteristics of the error in DIC. In the imaging process simulation in this study, the light intensity is calculated from the overlap area of the imaged spackle and the micro-lens array which the shape and the gap between the lenses are considered. Various initial conditions such as the center position distance for the initial state as shown in Fig. 1.1 and various known displacements are given for the simulation images. In this study, grid layout

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Fig. 1.1 Schematic figure of an imaged speckle on a micro-lens array

speckle pattern images and random layout speckle pattern images are generated. The generated images have the size of 500 × 500 pixels with 250 speckles. For the grid layout, each speckle is arranged on each centers of regions of 10 × 10 pixels. For the random layout, each speckle is arranged as random position without their overlaps. When speckle diameter is 5 pixels, the area ratio of the speckles is about 20%. The simulated images are saved as a general data format. The DIC measurements are applied for the images made by the simulation. In the DIC calculation, bi-linear interpolation is used for the sub-pixel calculation. The subset size is set to 31 × 31 pixels. Effects of various factors for the error are then investigated.

1.3 Results and Discussion Error profiles of various center position distance using the grid arranged speckle patterns without the gaps between the micro-lenses are shown in Fig. 1.2a. The center position distance is calculated from 0 to 0.9 pixel with a pitch of 0.1 pixel. Results of 0, 0.2, 0.4, 0.5, 0.6, 0.8 pixel of the center position distance are shown in the figure. These error profiles show various shapes. The averaged profile is also shown in the figure. The averaged profile has relatively small amplitude and becomes zero at the displacements of 0 and 0.5 pixel. In this simulation using the grid arranged speckle pattern, relative positions of the speckles and the micro-lenses in the subset region are the same. Thus each error profile indicates the characteristics of each relative position of the speckle and the micro-lens. The averaged profile means that effects of various initial positions of the speckle on the micro-lens array are averaged. Therefore, if randomly arranged speckle pattern is used, the characteristics of sufficient number of speckles in an enough size of a subset becomes the same as the averaged profile

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(a) Without gaps between micro-lenses

(b) Without gaps, bi-cubic interpolation

(c) With gaps, lens diameter of 0.8 pixel (d) With gaps, square lens of 0.8 × 0.8 pixel Fig. 1.2 Error profiles of various micro-lens conditions

using the grid pattern [6]. The maximum amplitude of the error profiles is 0.051 pixel and that of the averaged profile is 0.021 pixel. The amplitude of the averaged profile is not flat because of the sub-pixel intensity interpolation. If high-order interpolation functions are used, the amplitude can be reduced [2]. Error profiles using bi-cubic polynomial interpolation for the same simulation images are shown in Fig. 1.2b. The maximum amplitude of the error profiles is 0.034 pixel and that of the averaged profile is 0.003 pixel. Especially the amplitude of the averaged profile drastically decreases. Figure 1.2c shows error profiles with the gaps between the micro-lenses. The round shaped micro-lenses with the diameter of 0.8 pixel are used for the simulation. Amplitudes of the error profiles are larger than those without the gaps as shown (a). The maximum amplitude of the error profiles is 0.120 pixel and that of the averaged profile is 0.025 pixel. There is no big difference in the averaged error profile with that without the gaps. Figure 1.2d shows error profiles with the gaps between square shaped micro-lenses. The dimension of the square shaped micro-lenses is 0.8 × 0.8 pixel. Amplitudes of the error profiles slightly decrease from those using the round shaped micro-lenses as shown (c). The maximum amplitude of the error profiles is

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Fig. 1.3 Error profile using randomly arranged pattern

0.097 pixel and that of the averaged profile is 0.027 pixel. The amplitude of the averaged error profile slightly increases from that using the round shaped microlenses. From the above results, it is found that the existence of the gaps between the micro-lenses and the shape of the micro-lenses affect the error profiles. Figure 1.3 shows the error profile using randomly arranged speckle patterns with the gaps between square shaped micro-lenses of 0.8 × 0.8 pixel. About nine speckles are included in the subset region of 31 × 31 pixels. In this situation, the error profile is complex and is not close to the averaged profile using the grid pattern, because the characteristics of each speckle position are not averaged enough. Actually, various patterns of complex error profiles are obtained at the another coordinates. To improve the spatial resolution by reducing the subset size, such error profile must be identified and eliminated.

1.4 Error Elimination Method We propose an error elimination method using Lagrange polynomial with additional shift amounts data. Figure 1.4 shows the relationship between the actual and the measured shift amount on the actual and the measured displacement field at a single coordinate of the measurement field. In this figure, S0 is the target point of the error elimination. S1 to Sn are additional shift points as rigid body translations. At this point, the relationship between the measured shift amount, sm , and the measured displacement, x m , is known. But the relation of the actual shift amount, s, and the actual displacement, x, is unknown. However, the actual shift amount, s, from S0 can be obtained using an additional rigid body rotation measurement [4]. Thus the function including S0 to Sn on s–sm field can be calculated by Lagrange polynomial. The shift points for the polynomial should cover more than one pixel as one cycle of the error profile. Since the error must be zero at all integer displacements in pixel, the relationship between the actual shift amount, s, and the actual displacement, x, can be calculated.

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Fig. 1.4 Relationship between actual shift amount and measured shift amount

Then the actual displacement for the target S0 can be obtained. To apply the proposed error elimination method for actual deformation measurements, image capturing of a single rigid body rotation and some rigid body translations of the relative position of the measurement object and the camera at the initial state is required before the deformation measurement.

1.5 Summary In this study, it was found that the existence of the gaps between the micro-lenses and the shape of the micro-lenses affected the error profiles by using the imaging process simulation. When the subset size was not large enough, complex error profile was obtained. For improving spatial resolution, the error elimination method using Lagrange polynomial with additional shift amount data was proposed by the authors. It is expected that practical verifications and optimizations of the proposed method will be performed and highly accurate displacement measurements with high spatial resolution will be possible.

References 1. Sutton, M.A., Wolters, W.J., Peters, W.H., Ranson, W.H., McNeill, W.R.: Determination of displacements using an improved digital correlation method. Image Vis. Comput. 1(3), 133–139 (1983) 2. Schreier, H.W., Braasch, J.R., Sutton, M.A.: Systematic errors in digital image correlation caused by intensity interpolation. Opt. Eng. 39(11), 2915–2921 (2000) 3. Schreier, H.W., Sutton, M.A.: Systematic errors in digital image correlation due to undermatched subset shape functions. Exp. Mech. 42(3), 303–310 (2002)

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4. Arikawa, S., Yoshida, R., Yoneyama, S., Fujimoto, Y., Omoto, Y.: A method for eliminating periodical error for highly accurate measurement in digital image correlation. In: Proceedings of the International Conference on Advanced Technology in Experimental Mechanics 2015, p. 31 (2015) 5. Arikawa, S., Murata, M., Yoneyama, S., Fujimoto, Y., Omoto, Y.: Elimination of periodical error for bi-directional displacement in digital image correlation method. In: Advancement of Optical Methods in Experimental Mechanics, vol. 3. Conference Proceedings of the Society for Experimental Mechanics Series, pp. 151–155 (2017) 6. Arikawa, S., Kume, Y., Zhang, Y., Yoneyama, S., Fujimoto, Y.: Investigation of error generation mechanism in digital image correlation based on imaging process simulation (in Japanese). J. Jpn. Soc. Exp. Mech. 18(1), 37–42 (2018)

Chapter 2

Simultaneous Identification of Two-Independent Viscoelastic Characteristics with the Virtual Fields Method Yusuke Hoshino, Yuelin Zheng and Satoru Yoneyama Abstract In this study, a method for determining the viscoelastic material properties from displacement fields is proposed. Stress-strain relationship represented by the superposition integral is employed as the viscoelastic constitutive equation. Thus, the unknown properties are relaxation bulk and shear moduli. The virtual fields method based on the principle of virtual work is used as a method for the inverse analysis. The unknown material properties are determined by solving nonlinear simultaneous equations of the virtual work containing the constitutive equations. The validity of the proposed method is demonstrated by identifying the viscoelastic properties of a soft epoxy resin. Results show that the two independent viscoelastic material properties can be identified by proposed method. Keywords Inverse analysis · Virtual fields method · Viscoelasticity · Bulk relaxation modulus · Shear relaxation modulus · Digital image correlation

2.1 Introduction In recent years, the development of polymeric material is a fast-growing, more and more of the polymeric materials are used as machine materials. These materials are used in a wide variety of situations such as vibration control, soundproof and a matrix of carbon fiber reinforced plastics. On the other hand, due to miniaturization and complexity of recent equipment and structures, material properties may differ from the results obtained by a tensile test of a bulk material [1, 2]. Therefore, in order to evaluate the stress and strain, it is necessary to know the material properties. The dynamic viscoelasticity test is frequently performed as a test for determining the viscoelastic material properties. Because viscoelastic material properties are highly dependent on temperature and time, it is possible to obtain a material characteristic by performing experiments under the conditions such as various temperature and frequency in a dynamic viscoelasticity test. Therefore, it must deal with a lot of Y. Hoshino · Y. Zheng · S. Yoneyama (B) Aoyama Gakuin University, Sagamihara 252-5258, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_2

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data in the experiment. The relaxation bulk and shear moduli can be determined from the relation between the relaxation modulus and the viscoelastic Poisson’s ratio [3, 4]. However, viscoelastic Poisson’s ratio has been frequently treated as a constant. In this case, it is difficult to obtain the viscoelastic material properties correctly. There is a method for obtaining the relaxation bulk modulus and the relaxation shear modulus directly, but research to measure them at the same time is not sufficient [5]. Two independent viscoelastic material properties should be obtained simultaneously on the same specimen under the same conditions [6]. This is described by Tschoegl et al. [7] as the standard protocol of the material properties evaluation of viscoelastic solids. In this study, a method for determining the viscoelastic material properties from measured displacement distributions is proposed. Stress-strain relationship represented by convolution integral is employed as the viscoelastic constitutive equation. Thus, the unknown parameters in material properties are relaxation bulk modulus and relaxation shear modulus. The virtual fields method (VFM) based on the principle of virtual work is used as a method for the inverse analysis. The unknown material properties are determined by solving the nonlinear simultaneous equation of virtual work substituted into the constitutive equations. The validity of the proposed method is demonstrated by identifying the viscoelastic properties of soft epoxy resin. Results show that the material properties are identified by proposed method.

2.2 Virtual Fields Method for Linearly Viscoelasticity The principle of virtual work can be written that the work done by the external forces by the virtual displacement is zero when the object in equilibrium under various forces is given arbitrary virtual displacements. The principle of virtual work is expressed by the following equation.  

σi j εi∗j d =

 q

Ti u i∗ d

(2.1)

where σij represents the stress components, ε*ij expresses the virtual strain components, u*i gives the virtual displacement, T i is the traction (the external force components), i and j are the x and y directions,  is the inside of the object, and  is the boundary of the object. As shown in Fig. 2.1,  is composed of Dirichlet boundary  u and Neumann boundary  q .

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Fig. 2.1 Region for analysis, its surface, traction and virtual displacement

When stress and strain tensors are represented as σij and εij , the stress-strain relation can be written as  t dεkk (τ) dτ K (t − τ) σkk (t) = 3K (t)εkk (0) + 3 dτ 0  t dei j (τ) dτ (2.2) si j (t) = 2G(t)ei j (0) + 2 G(t − τ) dτ 0 where σkk (t) expresses the sum of normal stresses, εkk (t) is the sums of normal strains, and sij (t) and eij (t) are the deviatoric stress and strain. The relaxation bulk modulus K(t) and relaxation shear modulus G(t) are expressed by the Prony series as K (t) = K e +

N 

K i e(−t/ρi )

i=1

G(t) = G e +

N 

G i e(−t/ρi )

(2.3)

i=1

where ρi gives the relaxation time. Here, the traction T i , the strain εij , the virtual displacement u*i , and the virtual strain ε*ij are known values because T i and εij can be measured and u*i and ε*ij can be given arbitrarily. Therefore, the unknown parameters are K e , Ge , K i and Gi . Since the stress-strain relationship of the viscoelasticity depends on time, the arbitrary numbers of equation of the principle of virtual work are obtained from the displacement distributions at each time and arbitrary number of virtual displacements. These equations are expressed as t =0

 

σi j εi(n)∗ j d =

 q

Ti u i(n)∗ d

(n)∗ d t = t1  σi j εi(n)∗ j d = q Ti u i .. .. . .   (n)∗ t = tn  σi j εi j d = q Ti u i(n)∗ d

(2.4)

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where the superscript n in parenthesis represents the index of the virtual fields. The stress components are the values calculated from the strains obtained by the measurement. Equation (2.4) shows the system of nonlinear simultaneous equations with unknown material properties. It is possible to determine the unknown parameters by solving Eq. (2.4). In this study, a Newton-Raphson method is used to solve the nonlinear simultaneous equations. Equation (2.4) can be rewritten as  f t0 =



 f t1 =



 f tn =



σi j εi(n)∗ j d − σi j εi(n)∗ j d

 q

 −

σi j εi(n)∗ j d −

q

 q

Ti u i(n)∗ d Ti u i(n)∗ d .. . Ti u i(n)∗ d

(2.5)

Applying the Taylor series expansion and omitting higher order terms than second order, the following equation can be obtained.     ∂ fk ∂ fk K e + G e ( f k )i+1 = ( f k )i + ∂ Ke i ∂G e i         ∂ fk ∂ fk ∂ fk ∂ fk + K 1 + G 1 · · · K n + G n ∂ K1 i ∂G 1 i ∂ Kn i ∂G 1 i (2.6) where K e , Ge , K 1 , G1 , …, K n , and Gn are the correction values of the previous calculated values or the initial values of the unknown parameters. The correction values are obtained so that Eq. (2.6) become (f k )i+1 = 0. Therefore, the following equation can be obtained.    ∂ fk ∂ fk K e + G e ∂ Ke ∂G e         ∂ fk ∂ fk ∂ fk ∂ fk K 1 + G 1 · · · + K n + G n (2.7) + ∂ K1 ∂G 1 ∂ Kn ∂G n 

− fk =

For solving Eq. (2.7) at each time, the following equation is obtained. f = bd

(2.8)

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where ⎡

⎡ ⎢ ⎢ f =⎢ ⎣

⎡ ∂ ft0 ⎤ f t0 ∂K ⎢ ∂ ft1e f t0 ⎥ ⎢ ∂ Ke ⎥ .. ⎥, b = ⎢ ⎢ .. . ⎦ ⎣ . f tn

∂ f t0 ∂G e ∂ f t1 ∂G e

.. .

∂ f t0 ∂ K1 ∂ f t1 ∂ K1

.. .

∂ f tn ∂ f tn ∂ f tn ∂ K e ∂G e ∂ K 1

∂ f t0 ∂G 1 ∂ f t1 ∂G 1

· · · ∂∂ Kft0n · · · ∂∂ Kft1n .. . . .. . . . ∂ f t1 · · · ∂∂ Kftnn ∂G 1

∂ f t0 ∂G n ∂ f t1 ∂G n

⎤

⎥ ⎥ .. ⎥ ⎥, . ⎦

∂ f tn ∂G n

⎤ K e ⎢ G ⎥ e⎥ ⎢ ⎢ ⎥ ⎢ K 1 ⎥ ⎢ ⎥ G ⎥ d=⎢ ⎢ . 1⎥ ⎢ . ⎥ ⎢ . ⎥ ⎢ ⎥ ⎣ K n ⎦ G n

(2.9)

The correction term d can be obtained by the least-squares sense as

−1 d = bT b bT f

(2.10)

The matrix f is determined by substituting the initial values of the unknown parameters into Eq. (2.5). Here, the partial differentiations are not obtained analytically. Therefore, the matrix b is determined by the numerical differentiation. The above computation is repeated until the correction terms become sufficiently close to zero.

2.3 Identification of Viscoelastic Material Properties In order to validate the proposed inverse analysis approach, the viscoelastic material properties are identified from the displacement distribution measured by digital image correlation. A soft epoxy resin (Epikote 828) is used as a test material. This material has long been used as a test material for studying the mechanics of a time-dependent material by one author and coworkers [6, 8]. It is known that this material is linearly viscoelastic and thermorehologically simple. The specimen in Fig. 2.2 is loaded by an electrohydraulic servo fatigue testing machine at various temperatures (T = 248–303 K) in a temperature controlled chamber. The temperature is measured using four thermocouples installed near the specimen and is carefully maintained during Fig. 2.2 Shape of specimen

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the experiment within ±0.2 K. The constant rate displacement load of 0.03 mm/s is applied during the period of 50 s. The load value is measured by a load cell equipped in the test machine whereas the displacement distributions are measured using digital image correlation. To perform the strain measurement with digital image correlation, the specimen surface is painted with black ink on the surface, and white dot pattern is also painted by spray painting, such that the speckle-like pattern is created. It is known that the size of the random pattern should be selected to oversample the intensity pattern by several sensors for accurate measurement. In this study, each random pattern is oversampled by 10–40 pixels. The variation of the speckle-like pattern on the specimen is observed by a monochromatic CMOS camera (2048 × 2048 pixels × 8 bits) equipped with a telephoto lens of the focal length of 200 mm. Various algorithms have been proposed for digital image correlation. In this study, a digital image correlation algorithm that uses the Newton–Raphson method to search for both displacements and displacement gradients is employed [9]. A linear shape function is used for a subset deformation. Therefore, six variables, two displacements and for displacement gradients, in a subset are treaded as unknowns. A bicubic interpolation method is used for obtaining the continuous speckle pattern. The subset size is set to 21 × 21 pixels, and the interval is 10 pixels. The data processing is implemented using software developed by one author. Displacements with the accuracy of approximately 0.02 pixels are obtained using the above procedure. Figure 2.3 shows the finite elements used to define the piecewise virtual fields, which are constructed using 8-noded isoparametric elements. The resultant load values are applied to the upper section of the model as the external load, whereas the load at the lower section is treated as unknown. That is, the virtual displacements equal 0 at the lower section. It is sometimes difficult to obtain appropriate results Fig. 2.3 Finite elements for piecewise virtual fields

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because of measurement errors in the strains, which are obtained by differentiating the displacements. Therefore, the measured displacements are smoothed to minimize the sum of the residuals of the measured and approximated values and the secondorder derivatives of the approximated values. Furthermore, the strains obtained by differentiating the displacements using a finite element mesh are also smoothed in the same manner. After that, the displacements at the nodes are determined to fit both the approximated displacements and the approximated strains because both approximations are performed independently. The procedure for smoothing both the displacements and the strains is found in Ref. [10]. These displacements at the nodes are used as the data input into the algorithm for the virtual fields method. The measured strains and the virtual strains at the integration points of the elements in the model are computed from the displacements and the virtual displacements at the nodes, using a displacement-strain relationship. Next, the integrations in the principle of virtual work are numerically computed. Before applying the virtual fields method to experimental data, the identification algorithm used in this study is validated by applying it to the displacement fields obtained using a finite element method. Because the displacement fields obtained by a finite element method do not contain measurement errors, the material properties are completely obtained without any difficulties. Figure 2.4 shows an example of the image of the specimen. Using digital image correlation, the displacement and strain distributions are obtained as shown in Fig. 2.5. It is observed that the smooth displacement and strain distributions are obtained by the measurement. The load values at various temperatures are shown in Fig. 2.6. The measured displacements in Fig. 2.5 as well as the load values in Fig. 2.6 are used as the input values of the virtual fields method. The bulk and shear moduli at various temperatures obtained using the proposed method are shown in Fig. 2.7. The time and temperature superposition principle is applied to the results in Fig. 2.7 and the master curves are constructed. Figure 2.8 Fig. 2.4 Example of image for digital image correlation measurement

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Fig. 2.5 Displacement and strain components measured using digital image correlation

shows the master curves of the bulk and shear moduli. As shown in this figure, the master curves obtained by the proposed method coincide with those obtained using a dynamic mechanical test. The results show that two independent viscoelastic material properties can be obtained by the proposed method.

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1000 248K 253K 258K 263K 268K 273K 283K 293K 303K

Load, N

100 10 1 0.1 0

10

20

30

40

50

Time, s Fig. 2.6 Load histories at various temperatures

248K 253K 258K 263K 268K 273K 283K 293K 303K

1000

100

10

1 0.1

1

10

Bulk relaxation modulus K(t), MPa

Shear relaxation modulus G(t), MPa

1000

100

10

0.1

Time, s

(a)

1

Time, s (b)

10

Fig. 2.7 Bulk and shear relaxation moduli at each temperature obtained by inverse analysis

Shear and bulk relaxation moduli G(t), K(t), MPa

1000

100

DMA 248K 253k 258k 263K 268K 273K 283K 293K 303K

K(t)

10

G(t) 1 1E-10

Reference temperature TR : 303 K 1E-08 0.000001 0.0001

Time, s Fig. 2.8 Master curves of bulk and shear moduli

0.01

1

100

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2.4 Conclusions An investigation into the use of the virtual fields method is conducted to simultaneously identify two independent viscoelastic material properties. The displacement fields are measured using digital image correlation. Then, the bulk and shear relaxation moduli are identified using the virtual fields method. Because two independent properties can be determined simultaneously, the proposed method fulfills the standard protocol. The measurement example shows that the master curves of the time-dependent bulk and shear moduli for the wide range of the reduced time can be obtained by the proposed procedure. It is expected that the precise stress analysis can be realized by measuring the material properties by the proposed method. Acknowledgements This work was supported by JSPS KAKENHI Grant Number 18K03845.

References 1. Litteken, C., Strohband, R.A., Dauskardt, R.H.: Residual stress effects on plastic deformation and interfacial fracture in thin-film structures. Acta Mater. 53, 1955–1961 (2005) 2. Miura, H.: Materials and mechanics in nano scale (in Japanese). Trans. JSME Ser. A 72, 595–599 (2006) 3. Park, S.W., Schapery, R.A.: Methods of interconversion between linear viscoelastic material functions. Part 1-a numerical method based on Prony series. Int. J. Solids Struct. 36, 1653–1675 (1999) 4. Emri, I., Prodan, T.: A measuring system for bulk and shear characterization of polymers. Exp. Mech. 46, 429–439 (2003) 5. Pierron, F., Grediac, M.: The Virtual Fields Method: Extracting Constitutive Mechanical Parameters from Full-Field Deformation Measurement, pp. 1–93. Springer, Berlin (2012) 6. Hoshino, Y., Tamai, K., Zhang, Y., Yoneyama, S.: Direct measurement and master curve construction of viscoelastic Poisson’s ratio with digital image correlation. Strain 54, e12294 (2018) 7. Tschoegl, N.W., Knauss, W.G., Emri, I.: Poisson’s ratio in linear viscoelasticity a critical review. Mech. Time-Depend. Mater. 6, 3–51 (2002) 8. Yoneyama, S., Ogawa, K., Misawa, A., Takashi, M.: Evaluation of time-dependent fracture mechanics parameters of a moving crack in a viscoelastic strip. JSME Int. J. Ser. A 42, 624–630 (1999) 9. Yoneyama, S.: Basic principle of digital image correlation for in-plane displacement and strain measurement. Adv. Compos. Mater. 25, 105–123 (2016) 10. Yoneyama, S., Koyanagi, J., Arikawa, S.: Measurement of discontinuous displacement/strain using mesh-based digital image correlation. Adv. Compos. Mater. 25, 329–343 (2016)

Chapter 3

Optical Phase-Based Method for Dynamic Deflection Measurement of Railroad Bridge Shien Ri, Qinghua Wang, Hiroshi Tsuda, Hirokazu Shirasaki and Kenichi Kuribayashi Abstract The development of highly reliable structural health monitoring techniques with low-cost is essential to ensure the safety of various aging infrastructures. To measure accurate deflection over the entire structure, in this study, the sampling moiré method is applied to two-dimensional displacement measurement of an old and long metal bridge by utilizing repeated pattern marker. The deflection measured from our method corresponded well with that measured from conventional contact-type ring displacement sensor. In addition, we can measure the deflection of the railroad bridge successfully by use of the repeated rivet patterns without any marker. The digital images of natural repeated patterns are analyzed through the sampling moiré method and the deformation is evaluated. This moiré technique features comparable accuracy to conventional one despite low-cost and easy setup. The aged deterioration of bridges can be diagnosed with deflection measurement at regular intervals. The developed measuring technique is expected to be widely applied to the inspection of various large-scale structures for infrastructure triage. Keywords Optical method · Deformation measurement · Sampling Moiré technique · Repeated pattern

3.1 Introduction Bridge structures are exposed to various external loads such as traffic, earthquakes, and wind loads during their lifetime. As for rail bridges affected by daily train loads, the deflection is increased when soundness of the bridges is decreased. Their structures may get deteriorated along with time in unexpected ways, which may lead to structural damage causing costly repairs and/or heavy loss of human lives. Consequently, structural health monitoring has become an important research topic for S. Ri (B) · Q. Wang · H. Tsuda National Institute of Advanced Science and Technology, Tsukuba 305-8568, Japan e-mail: [email protected] H. Shirasaki · K. Kuribayashi East Japan Railway Co., Saitama 331-8513, Japan © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_3

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continuous assessment and evaluation of structural safety. To measure the deflection of the bridges, several researchers generally use the contact type ring displacement sensor, linear variable differential transformer (LVDT), GPS sensor [1, 2] or digital image correlation (DIC) method [3, 4], as one of the famous optical full-field method. Recently, a sampling moiré method [5] has been developed to measure the small displacement distribution by using a grating pattern, and several practical applications of displacement measurement for large-scale structures have been demonstrated [6, 7]. To broaden the application range, we also developed an accurate displacement distribution measurement technique by utilizing arbitrary repeated pattern [8], and the measurement accuracy of 1/1000 of the pitch can be achieved. In this study, the dynamic displacement measurement of an old metal bridge was carried out by means of the sampling moiré method. The deflection of the bridge was successfully measured by use of the repeated rivet pattern without any artificial marker. This method is innovative, highly cost-effective and easy to implement, and maintains the advantages of dynamic measurement and high resolution with a wide field of view.

3.2 Principle Figure 3.1a illustrates a schematic representation of a periodically repeated pattern, which can be considered the sum of cosines function with multiple frequencies, captured by a digital camera. After performing image processing of down-sampling (the down-sampling pitch is similar to the repeated pattern pitch) and intensity interpolation, multiple phase-shifted moiré fringe patterns can be obtained. The generated

Fig. 3.1 Principle of displacement measurement by utilizing arbitrary repeated pattern

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moiré fringe by image processing with low spatial frequencies can also be considered as the sum of cosines functions with multiple frequencies by a Fourier series as shown in Fig. 3.1b. Then, the phase distribution of the moiré fringe with multiple frequencies can be calculated by phase shifting method using discrete Fourier transform (DFT) algorithm. Finally, the displacement distribution is measured from the phase difference of the moiré fringe with multiple frequencies before and after deformations.

3.3 Experiment Figure 3.2 shows the optical setup for measuring the dynamic deflection of an old railroad bridge using a cinema camera (Canon 1Dc, 4K movie @ 24 fps) and grating marker with a 50 mm pitch. The focus length of the cinema lens was f = 14 mm. The interval between two piers of the bridge was 21 m. Several magnetic grating markers were placed at the bridge, as shown in Fig. 3.2, and the camera was placed at the ground near the pier. The distance from the camera to the target is about 15 m. To compare the accuracy of the deflection measurement, several ring displacement sensors were also placed at a pier and the mid-span for the center point of two piers. Figure 3.3 shows the deflection measurement results of a rail bridge when an express train consisting of 12 cars (JR East, E351) passed through the bridge at a speed of 85 km/h. Figure 3.3a shows the target bridge with steel plate girders. Figure 3.3b–d show the measured results obtained by the conventional ring displacement sensor, the developed method using a grating marker with a 50 mm pitch, and the improved method using the repeated rivet pattern with a 100 mm pitch, respectively. The maximum deflection in the vertical direction measured by a displacement sensor was 5.8 mm. The deflection measured from our method corresponded well with that measured from conventional contact-type ring displacement sensor. In addition, we

Fig. 3.2 Optical setup for measuring deflection measurement of a railroad bridge

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Fig. 3.3 Experimental results of deflection measurement of a railroad bridge: a photography of the target bridge, measured deflection obtained by b conventional ring displacement sensor, c basic sampling moiré method using a grating marker with a 50 mm pitch, and d developed method using a repeated rivet pattern with a 100 mm pitch, in case of an express train passed through at a speed of 85 km/h

can measure the deflection of the rail bridge successfully by use of the repeated rivet patterns without any marker. The difference between the measured deflection between the ring sensor and our developed methods is less than 0.4 mm. This moiré technique features comparable accuracy to conventional sensor despite low-cost and easy setup. Interestingly, compared with other cars (ordinary cars), only the deflection by car No. 9 was 20% smaller because of car No. 9 was a green car (second class car) and it was designed as a light structure for comfortable to ride. Figure 3.4 shows the displacement vector measured by our method during the passage of another train. Unlike a conventional displacement sensor, our approach can easily obtain the displacement data in both x- and y-directions. As a result, fullfield measurement can help us to evaluate the maximum deflection and understand the deformation behavior visually.

3.4 Conclusions In this study, a dynamic displacement measurement method based on recent advanced moiré technique is applied to a rail plate girder bridge with a 21 m span. The applicability and effectiveness of the present method were verified through the practical application on a bridge with steel plate girders. Experimental results demonstrated that the measurement accuracy of our method is the same as the conventional contact type displacement sensor. This method is useful to monitor various infrastructures not only the bridge but also the high building, tunnel, and long rail, etc.

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Fig. 3.4 Displacement vector measured by our method during the passage of a train

References 1. Meng, X., Dodson, A., Roberts, G.: Detecting bridge dynamics with GPS and triaxial accelerometers. Eng. Struct. 29, 3178–3184 (2007) 2. Moschas, F., Stiros, S.: Measurement of the dynamic displacements and of the modal frequencies of a short-span pedestrian bridge using GPS and an accelerometer. Eng. Struct. 33, 10–17 (2011) 3. Lee, J., Shinozuka, M.: Real-time displacement measurement of a flexible bridge using digital image processing techniques. Exp. Mech. 46, 105–114 (2006) 4. Yoneyama, S., Kitagawa, A., Iwata, S., Tani, K., Kikuta, H.: Bridge deflection measurement using digital image correlation. Exp. Tech. 31, 34–40 (2007) 5. Ri, S., Fujigaki, M., Morimoto, Y.: Sampling Moiré method for accurate small deformation distribution measurement. Exp. Mech. 50, 501–508 (2010) 6. Ri, S., Muramatsu, T., Saka, M., Nanbara, K., Kobayashi, D.: Accuracy of the sampling Moiré method and its application to deflection measurements of large-scale structures. Exp. Mech. 52, 331–340 (2012) 7. Ri, S., Saka, M., Nanbara, K., Kobayashi, D.: Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets. Exp. Mech. 53, 1635–1646 (2013) 8. Ri, S., Hayashi, S., Ogihara, S., Tsuda, H.: Accurate full-field optical displacement measurement technique using a digital camera and repeated patterns. Opt. Express 22, 9693–9706 (2014)

Chapter 4

Full-Field Microscale Strain Measurement of Carbon Fiber Reinforced Plastic Using 2-Pixel Sampling Moiré Qinghua Wang, Shien Ri, Peng Xia and Hiroshi Tsuda Abstract Microscale strain distributions are essential parameters for evaluating the mechanical properties and instability behaviors of composite materials. In this study, a recently-developed reconstructed multiplication moiré method from 2-pixel sampling moiré fringes was used to investigate the microscale deformation performance of a carbon fiber reinforced plastic (CFRP) specimen. This method enables wide field of view and high displacement and strain sensitivities for deformation measurement of CFRP. A microgrid as the deformation carrier was fabricated on CFRP by ultraviolet nanoimprint lithography. The in situ full-field distributions of two-dimensional (2D) displacements, normal strains and shear strains of CFRP were quantitatively measured at different three-point bending loads under a laser microscope. The deformation distribution features of CFRP were analyzed, which is helpful to understand its potential damage mechanism. Keywords Deformation measurement · Strain distribution · Image processing · Optical method · Moiré fringe · CFRP

4.1 Introduction Carbon fibre reinforced plastic (CFRP) has been widely used in industrial fields of automobiles, aerospace, railways and infrastructures owing to the advantages of low density and high strength-to-weight ratio. To evaluate its mechanical property and instability behaviour, microscale strain distributions of CFRP under mechanical loading are necessary to be measured experimentally. Optical techniques have drawn great attention in both scientific and engineering fields owing to the merits of non-destructive testing and high sensitivity. In recent years, the commonly used optical methods for microscale deformation measurement include electronic speckle pattern interferometry (ESPI) [1], digital holography [2, Q. Wang (B) · S. Ri · P. Xia · H. Tsuda National Metrology Institute of Japan (NMIJ), National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, Ibaraki 305-8568, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_4

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3], digital image correlation (DIC) [4], geometric phase analysis (GPA) [5] and moiré methods [6]. Each method has its advantages and disadvantages, and is mainly used according to its own appropriate applications. Among these methods, the moiré methods impress the authors due to the advantages of deformation visualization, a large field of view and the strong noise resistance ability. In particular, the microscope scanning moiré method [7] has a wide field of view for microscale deformation measurement, and the sampling moiré method [8, 9] has high deformation measurement accuracy using a spatial phase-shifting technique. Recently a multiplication moiré method from 2-pixel sampling moiré fringes has been developed [10]. This method outperforms the microscope scanning moiré method in terms of doubled deformation sensitivity and simple 2D deformation measurement by digital image processing. In detail, it has the same wide field of view as the microscope scanning moiré method, and both the displacement and strain measurement sensitivities are doubled as in other multiplication moiré methods [11, 12]. 2D strain distributions are simply measurable without rotating the sample stage or the scanning lines, no matter whether the scanning resolution is adjustable or not. Besides, this method can analyze grid images where the grid pitch is only around 2 pixels which is impossible to be recognized by eyes, and has a larger field of view than the sampling moiré method in which the grid pitch is usually greater than 4 pixels. In this study, the multiplication moiré method from 2-pixel sampling moiré is chosen as the research methodology. The full-field microscale displacement and strain distributions of a CFRP specimen are non-destructively investigated under different three-point bending loads.

4.2 Principle of Reconstructed Multiplication Moiré Method The used deformation measurement method was the reconstructed multiplication moiré method with advantages of high deformation sensitivity and a large field of view. Two groups of 2-pixel sampling moiré fringes were first generated from a grid image at each load, and then used to reconstruct multiplication moiré fringes (Fig. 4.1) to double the deformation measurement sensitivity [10]. Both the normal strains in the x and y direction as well as the shear strain were able to be accurately measured without rotating the sample stage or the microscope scanning lines. The detailed implementation procedure is introduced in [9]. To make the field of view as large as possible, the specimen grating pitch py can be as small as around 2 pixels in the recorded image. Since the microscope scanning pitch is 1 pixel, both the specimen grating and the traditional multiplication moiré fringes with frequency of (2/py − 1/pscan_y ) will be recorded [10], where pscan_y represents the scanning pitch in the y direction. The intensity of the recorded image can be expressed as

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Fig. 4.1 Formation principle of 2-pixel sampling moiré and reconstructed multiplication moiré patterns, where m is the fringe order

Irecorded = A cos(2π

y 2 1 ) + B cos[2πy( − )] + C py py pscan_y

(4.1)

where A and B stand for the modulation amplitudes, and C means the background and higher-frequency intensity of the recorded image. Due to the mutual disturbance, both the specimen grating and the multiplication moiré fringes in the recorded image are not distinct, see Fig. 4.1b. If the recorded intensity image is thinned out every 2 pixels followed by linear or high-order intensity interpolation shown in Fig. 4.1c, two 2-pixel down-sampling moiré patterns can be generated with intensities of 1 1 k 2 1 1 k − ) + 2π ] + B cos[2πy( − − ) + 2π ] + C py Ty Ty py pscan_y Ty Ty 1 1 k = A cos[2πy( − ) + 2π ] + D (k = 0, 1 pixel) (4.2) py Ty Ty

Im,y = A cos[2πy(

Because k = 0, 1 pixel and T y = 2 pixels, the intensities of the two sampling moiré patterns can also be represented by 1 1 − )] + D py Ty 1 1 = −A cos[2πy( − )] + D py Ty

Im,y1 = A cos[2πy( Im,y2

(4.3)

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If the microscope magnification remains unchanged and the scanning resolution in the y direction is changed to half, the microscope scanning pitch will be approximately equal to the specimen grating pitch, and the traditional scanning moiré will emerge with intensity of I m,scan_y = A cos[2πy(1/py − 1/T y )] + E. From Eq. (4.3), one of the 2-pixel sampling moiré patterns is the same to the traditional scanning moiré pattern. As the microscope magnification does not change, the field of view of the 2-pixel sampling moiré is also the same to that of the scanning moiré. To achieve higher deformation measurement accuracy by using both of the two sampling moiré patterns, a distinct multiplication moiré pattern can be reconstructed from the multiplicative interference between the two sampling moiré patterns using the following equation [10] Imulti = Im,y1 Im,y2 = −

1 A2 1 A2 cos[4πy( + D2 − )] − 2 py Ty 2

(4.4)

From Eq. (4.4) and Fig. 4.1d, the frequency of the multiplication moiré is twice that of the sampling moiré. Both the displacement and strain measurement sensitivities in the reconstructed multiplication moiré method are twice as high as in the scanning moiré method. Based on the fringe centering method, the displacement of the specimen relative to the sampling pitch in the y direction can be acquired from u y = m y Ty (m y = 0, 0.5, 1, 1.5, 2, 2.5, . . .)

(4.5)

where my indicates the fringe order of the reconstructed multiplication moiré fringes in the y direction. Similarly, the relative displacement in the x direction can be calculated using ux = mx T x , where mx means the fringe order of the reconstructed multiplication moiré fringes in the x direction, and T x is the sampling pitch in the x direction. The x-direction, y-direction and shear strain components are obtainable from the partial differentials of the displacements in the x and y directions. ∂(m x Tx ) ∂u x = ∂x ∂x ∂(m y Ty ) ∂u y εy = = ∂y ∂y ∂u y ∂(m x Tx ) ∂(m y Ty ) ∂u x γx y = + = + ∂y ∂x ∂y ∂x εx =

(4.6)

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Fig. 4.2 a CFRP specimen size, b photo of specimen after bending, and c experimental setup of three-point bending test under a laser scanning microscope

4.3 Experiments and Results 4.3.1 Specimen Preparation and Three-Point Bending Test The CFRP specimen was made up of epoxy resin and K13D carbon fibres with diameters of 10–11 μm [9]. The thickness, the width and the length of the specimen were 1 mm, 4 mm and 22 mm, respectively (Fig. 4.2a). All fiber directions were perpendicular to a 1 × 22 mm2 surface to be observed, which was polished using sand papers and polishing solutions on an automatic polishing machine. A strain gauge was pasted on a 4 × 22 mm2 surface to monitor the maximum tensile strain during the following bending test, as shown in Fig. 4.2b, c. Before testing, a cross grating with pitch of 3 μm was fabricated on the polished surface by ultraviolet nanoimprint lithography (EUN-4200), the process of which is presented in Fig. 4.3a. The used resist was PAK01 and the UV wavelength was 375 nm. The three-point bending test was carried out using a self-developed automatic mechanical loading device under a laser scanning microscope (Lasertec OPTELICS HYBRID). The three-point bending experimental setup is displayed in Fig. 4.2c. The support span was 16 mm, and the span-to-depth ratio was 16 according to the American Society for Testing and Materials (ASTM) standards. During the bending test, a series of grid images were recorded using the microscope.

4.3.2 Displacement and Strain Distributions of CFRP Under different three-point bending loads, the grid images were collected when the magnification of the objective lens was 5× and the image resolution was 1024 × 1024. In this case, the grid pitch was around 2 pixels, and the developed reconstructed

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Fig. 4.3 a Process of UV nanoimprint lithography, b CFRP surface image and 3-μm-pitch grid image, and c 2-pixel sampling moiré patterns and the reconstructed multiplication moiré pattern in the y direction when the strain gauge value is 5330 με

multiplication moiré method was able to be used for deformation measurement in a large field of view. The CFRP surface image, the recorded 3-μm-pitch grid fabricated according to the process in Fig. 4.3a and the region of interest are shown in Fig. 4.3b. The 2-pixel sampling moiré patterns and the reconstructed multiplication moiré pattern in a 1.26 × 0.53 mm2 square region are illustrated in Fig. 4.3c when the strain gauge value is 5330 με. From the grid images when the strain gauge values were 0, 2460, 3500, 4220 and 5330 με, the displacement and strain distributions of CFRP were measured using the reconstructed multiplication moiré method mentioned above. The displacement distributions in the x and y directions under different loads are presented in Fig. 4.4, and the distributions of the normal strains in the x and y directions, and the shear strains under these different loads are listed in Fig. 4.5. From Fig. 4.4, the displacement in the x (axial) direction under each load is positive in the upper-left and bottom-right corners and negative in the other two corners, and the displacement in the y (loading) direction is negative in all the area and minimum along the loading line, both of which agree well with the three-point bending feature. From Fig. 4.5, the strain in the x direction is compressive in the upper area and tensile in the lower area, while the strain in the y direction is tensile in the upper region and compressive in the lower region. The shear strain is negative in the left area, positive in the right area and almost zero in the middle area. With the increase of the bending load, the absolute values of the tensile and compressive (or positive and negative) strains gradually grow. The strain results are useful in finding the weak area on the specimen which is helpful to understand the potential damage mechanism.

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Fig. 4.4 2D displacement distributions of CFRP under different three-point bending loads measured by the multiplication moiré method

Fig. 4.5 Distributions of x-direction strain, y-direction strain and shear strain of CFRP under different three-point bending loads measured by the multiplication moiré method

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4.4 Conclusions In conclusion, the microscale 2D deformations including displacement and strain distributions of CFRP under different three-point bending loads were quantitatively investigated using the reconstructed multiplication moiré method from 2-pixel sampling moiré fringes. In our next work, 2-pixel sampling moiré will be combined with phase analysis to realize full-automatic, high-accuracy, high-sensitivity and largeview-field deformation measurement of materials. Acknowledgements This work was supported by JSPS KAKENHI, grant numbers JP16K17988, JP16K05996 and 18K13665. The authors are also grateful to Dr. Kimiyoshi Naito in National Institute for Materials Science, Japan, for providing the CFRP materials.

References 1. Leendertz, J.A.: Interferometric displacement measurement on scattering surfaces utilizing speckle effect. J. Phys. E: Sci. Instrum. 3(3), 214–218 (1970) 2. Goodman, J.W., Lawrence, R.W.: Digital image formation from electronically detected holograms. Appl. Phys. Lett. 11(3), 77–79 (1967) 3. Xia, P., Wang, Q., Ri., S., Tsuda, H.: Calibrated phase-shifting digital holography based on a dual-camera system. Opt. Lett. 42(23), 4594–4957 (2017) 4. Chu, T.C., Ranson, W.F., Sutton, M.A.: Applications of digital-image-correlation techniques to experimental mechanics. Exp. Mech. 25(3), 232–244 (1985) 5. Liu, Z., Xie, H., Fang, D., Dai, F., Xue, Q., Liu, H., Jia, J.: Residual strain around a step edge of artificial Al/Si(111)-7x7 nanocluster. Appl. Phys. Lett. 87(20), 201908 (2005) 6. Weller, R., Shepard, B.M.: Displacement measurements by mechanical interferometry. Proc. Soc. Exp. Stress. Anal. 6(1), 35–38 (1948) 7. Kishimoto, S., Egashira, M., Shinya, N.: Microcreep deformation measurements by a moiré method using electron-beam lithography and electron-beam scan. Opt. Eng. 32(3), 522–526 (1993) 8. Ri, S., Fujigaki, M., Morimoto, Y.: Sampling moiré method for accurate small deformation distribution measurement. Exp. Mech. 50(4), 501–508 (2010) 9. Wang, Q., Ri, S., Tsuda, H.: Micro/nano-scale strain distribution measurement from sampling moiré fringes. J. Vis. Exp. 123, e55739 (10 pp) (2017) 10. Wang, Q., Ri, S., Tsuda, H.: Digital sampling moiré as a substitute for microscope scanning moiré for high-sensitivity and full-field deformation measurement at micron/nano scales. Appl. Opt. 55(25), 6858–6865 (2016) 11. Li, Y., Xie, H., Chen, P., Zhang, Q.: Theoretical analysis of moiré fringe multiplication under a scanning electron microscope. Meas. Sci. Technol. 22, 025301 (12 pp) (2010) 12. Patorski, K., Wielgus, M., Ekielski, M., Kazmierczak, P.: AFM nanomoiré technique with phase multiplication. Meas. Sci. Technol. 24, 035402 (9 pp) (2013)

Chapter 5

Ascending Order Constraints Sensitivity Optimal Design Method for Steel Structure Junchen Guo and Xin Zhao

Abstract Traditional structural design method usually adopts the process of trial and error methodology to try to search a “feasible solution” that satisfy the structural design requirements. This process is complex and inefficient, and it can’t get the optimal solution. Therefore, integrating computers and advanced optimization ideas into engineering structures to make structural design quickly, accurately, and intelligently is an important in the field of engineering structure design. This paper develops constant incremental sensitivity analysis method that regard sensitivity coefficient as the guide of member cross-sectional optimal design and redistributes structural material under design constraints to improve material efficiency. Based on this analysis method, ascending order constraints sensitivity optimal design method is developed for steel structures. Taking the sensitivity analysis results as a reference standard, the optimal design method can optimize the structural members and minimize the overall structural cost. To verify the accuracy and the optimization ability of the proposed design method, a utility program was developed. At last, a single plane frame structure is exemplified in this paper to discuss the applicability of the optimization design method. The results of program design and manual design are compared and analyzed which confirm the accuracy and effectiveness of the proposed optimal design method. Keywords Ascending order constraints sensitivity · Optimal design · Steel structure

J. C. Guo · X. Zhao (B) Department of Structual Engineering, Tongji University, No. 1239 Siping Road, Shanghai 200092, China e-mail: [email protected] X. Zhao Tongji Architectural Design Group, No. 1239 Siping Road Shanghai 200092, China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_5

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5.1 Introduction Traditional structure design usually adopts the process of modeling-analysiscalculation-artificial modify to try to search a “feasible solution” that satisfied the structural design requirements. This process is complex and inefficient, and it can’t get the most reasonable solution. Therefore, integrating computers and advanced optimization ideas into engineering structures to make structures design quickly, accurately, and intelligently is an advance research in the field of engineering structure design. Intelligent design needs to be based on a certain design method. The design method can be divided into optimal design and compliant design according to the different paths from the initial design model to the optimized design model. Optimal design is based on the experience of selecting the original component size with large redundancy and conservative design. The design method can reduce the size of the component to reduce the design redundancy gradually, and check the design compliance the requirements of the standard. Compliant design is to select the original component size that is negative in design redundancy and initial component size doesn’t satisfy the design constraints. The design method increases the component size to improve the design redundancy gradually, so that the structure just satisfied the requirements of the standard [1, 2]. Based on the constant incremental sensitivity analysis (CISA) method [3, 4], this paper proposes ascending order constraints sensitivity optimal design (AO-CSOD) method and ascending order constraints sensitivity compliant design (AO-CSCD) method. A single plane frame structure is exemplified in this paper to discuss the applicability of the methods AO-CSOD method and AO-CSCD method and the results of the two methods are compared. Finally, the factors that cause the difference between the AO-CSOD method and AO-CSCD method are analyzed and summarized.

5.2 Theoretical Basis 5.2.1 Ascending Order Constraints Constraints can be divided into driven constraints and validation constraints according to the order in which they are used. Drive constraint is a kind of design constraint which needs to consider the influence of design variables in the optimal design, which is considered as the constraint condition in the process of solving design variables. Validation constraint is a design constraint introduced as a verification condition in optimal design [5]. Constraints can be divided into global constraints, assembly constraints, component constraints, sectional constraints and detailing constraints. We can sort constraints according to a certain hierarchy of relationships to our optimization priorities.

5 Ascending Order Constraints Sensitivity Optimal Design …

Detailing Constraint

Sectional Constraint

Component Constraint

37

Assembly Constraint

Global Constraint

Fig. 5.1 Ascending order constraints

First, component constraints, sectional constraints and detailing constraints are optimized as driven constraints, when those constraints of beams or columns were selected as the driven constraints which satisfied the requirements of the standard, Then, we can check other components whether they can satisfy the requirements of the standard, if not, the unsatisfied constraints are further optimized as driven constraints. When the component constraints, sectional constraints and detailing constraints are satisfied (retaining a certain degree of redundancy in the optimal design), the constraints are ascended, and then the assembly constraints and the global constraints are used as validation constraints. If the constraints don’t satisfy the requirements of the standard, the constraints are further optimized as driven constraints until all of the constraints satisfy the requirements of the standard. After a series of optimization with ascending order constraints, the redundancy of the structure will be greatly reduced (Fig. 5.1).

5.2.2 Redundancy Redundancy is a measure of the margin of a performance metric to the design standard limitations and can be defined as formulas (5.1) and (5.2). Rei = (gui − gi )/gui

(5.1)

Rei = (gi − gli )/gli

(5.2)

where Rei is the redundancy of the design constraint ‘I’, gi is design constraint ‘I’, gui is the upper limitation of the design constraint ‘I’, gli is the lower limitation of the design constraint ‘I’, When the constraint limitation is the upper limitation, take the formula (5.1), and when the constraint limitation is the lower limitation, take the formula (5.2). “Over-redundancy” means that the structure has a large optimization space. “Under-redundancy” means that the design indexes of the structure don’t satisfy the requirements of the standard, and the “proper-redundancy” indicates that the design of the structure is satisfactory. In this paper, the proper redundancy is 0–10%, more than 10% is over-redundancy. When all the constraints of the structure satisfy the requirements of the standard, the smaller the redundancy of the structure is, the better the structure is.

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J. C. Guo and X. Zhao AO-CSCD

Over Constrained

AO-CSOD

Over Constrained

Fig. 5.2 The AO-CSCD method and the AO-CSOD method

5.2.3 Optimal Design Method and Compliant Design Method The design method from the primitive design with over-constrained (overredundancy) to the properly-constrained (proper-redundancy) is called optimal design method. At present, the optimal design method is widely used in the optimal design. But the optimal design method also has some shortcomings. The interaction between design constraints in optimal design methods often leads to conservative results. If the designers want to reduce the redundancy under the condition of satisfy the requirements of the standard, the whole process can be inverted, so this paper also proposes a compliant design method. The compliant design optimizes the primitive structure from under-constrained to properly-constrained (Fig. 5.2).

5.2.4 The CISA Method The CISA method is to increase the material volume or material cost of each component group in turn and calculate the change value of design constraints as the sensitivity coefficient of each component group. The formula (5.3) is used to calculate the sensitivity coefficient [6–8]. si,k =gi /νk

(5.3)

where Si,k is the constant incremental sensitivity coefficient of the design constraint ‘I’, gi is the change value of design constraint ‘I’, vk is the change value of design variable k. The CISA method developed in this paper is used in the optimal design and compliant design. We determine the optimal position (beam or column or other component) by the ratio of the increment of structural material (change the section in the standard steel form) to the value of change in the global constraints (story drift). This can better consider the coupling effect between different components such as beams and columns. The redundancy of the structure is reduced effectively and the working efficiency is improved. The CISA method is shown in Fig. 5.3, where the black up arrow indicates an increase in the section size of the component, and the black down arrow indicates a decrease in the section size of the component.

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Sensitive Components Under Constrained Insensitive Components Constraint State Sensitive Components Over Constrained Insensitive Components

Fig. 5.3 The CISA method

5.2.5 The AO-CSCD Method The AO-CSCD method combines the ascending order constraints method with the compliant design method, and further proposes the ascending order constraints sensitivity compliant design method. The flow of AO-CSCD method is shown in Fig. 5.4.

5.2.6 The AO-CSOD Method The AO-CSOD method combines the ascending order constraints method with the optimal design method and applies compliant design method described. Then further proposes the ascending order constraints sensitivity optimal design method. The flow of AO-CSOD method is shown in Fig. 5.5.

5.3 Case Study 5.3.1 A Single-Story Planar Frame Structure A single-story steel frame model with a height of 4 m and a bay of 6 m was selected. Beams and columns adopted the standard H-section of steel table, and the column bottom was hinged to the ground. The material is Q234B steel, the density is 7850 kg/m3 ,

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Primitive Design

The Structural system is determined. The redundancy of the pre-selected component size is under-constrained, and don't satisfy the requirements of the standard.

The compliant design of the components (the redundancy slightly inadequate)

Component constraints Sectional constraints Detailing constraints Normalize constraints Beams and columns can increase the size of the sections at the same time

Satisfy

Ascending order constraints

The compliant design of the assembly (the redundancy slightly inadequate)

Satisfy

CISA

Increase the section size of the components with higher sensitivity coefficient Makes the redundancy slightly inadequate.

Ascending order constraints

The compliant design of the global constraint

Calculate the redundancy Fig. 5.4 The AO-CSCD method

CISA

Check all the constraints Optimize the section size of the Components or assemblies with higher sensitivity coefficient to the global constraints

5 Ascending Order Constraints Sensitivity Optimal Design …

Primitive Design

The Structural system is determined. The redundancy of the pre-selected component size is over-constrained, and satisfy the requirements of the standard.

The compliant design of the components (proper-redundancy)

Component constraints Sectional constraints Detailing constraints Normalize constraints Beams and columns can decrease the size of the sections at the same time

Satisfy

Ascending order constraints

The compliant design of the assembly (the redundancy slightly inadequate)

Compliant design Unsatisfy

41

CISA

Decrease the section size of the components with lower sensitivity coefficients Makes the redundancy proper.

Check the global constraints

Satisfy

Ascending order constraints

The compliant design of the global constraint

CISA

Check all the constraints Optimize the section size of the components or assemblies with lower sensitivity coefficient to the global constraints

Calculate the redundancy

Fig. 5.5 The AO-CSOD method

the elastic modulus is 210 GPa, the Poisson’s ratio is 0.3, the yield strength is 235 MPa and the design values of the strength of material is 215 MPa (Fig. 5.6). As shown in Tables 5.1 and 5.2, select the calculation parameters. The limitation of constraints is shown in Table 5.3. The AO-CSOD method. Primitive design: The redundancy of the component size is over-constrained. Column: 400 × 400 × 13 × 21. Beam: 550 × 200 × 10 × 16. The primitive design parameters of AO-CSOD are shown in Table 5.4. The AO-CSCD method. Primitive design: The redundancy of the component size is under-constrained. Column: 100 × 100 × 6 × 8. Beam: 100 × 50 × 5 × 7.

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Fig. 5.6 A single-story planar frame structure

Table 5.1 Calculation parameters Superimposed dead load (N/m)

Live load (N/m)

Basic wind pressure (kN/m2 )

Shape coefficient

Wind pressure height coefficient

Bay (m)

27,000

12,000

0.75

1.3

1

6

Table 5.2 Calculation parameters Maximum of seismic influence coefficient

Damping ratio

Characteristic site period

Slenderness ratio

0.24

0.02

0.4

100

Table 5.3 Limitation of constraints Slenderness ratio

Beam deflection (live load)

Beam deflection (combination of loads)

Stress ratio (beam)

Stress ratio (column)

Story drift

100

0.0020

0.0025

0.9000

0.9000

0.0040

The primitive design parameters of AO-CSCD are shown in Table 5.5. The design section adopts the international hot-rolled H steel table GB/T 112632010. the section steel table is arranged according to the section area from large to small. In the design process, beams and columns increase and decrease sections in turn according to this list. Table 5.4 The primitive design parameters of AO-CSOD Project

Primitive design

Slenderness ratio

Beam deflection (live load)

Beam deflection (combination of loads)

Stress ratio (beam)

Stress ratio (column)

Story drift

Material dosage (kg)

Value

27.72

1.5 × 10−4

4.9 × 10−4

0.5639

0.3801

0.0020

1926

Redundancy

72.27%

92.69%

80.56%

37.34%

57.77%

50.55%

5 Ascending Order Constraints Sensitivity Optimal Design …

43

Table 5.5 The primitive design parameters of AO-CSCD Project

Primitive design

Slenderness ratio

Beam deflection (live load)

Beam deflection (combination of loads)

Stress ratio (beam)

Stress ratio (column)

Story drift

Material dosage (kg)

Value

112.9

0.0559

0.1821

21.34

10.80

0.1978

191

Redundancy

−12.90%

−2694%

−7183%

−2277%

−1100%

−4845%

5.3.2 The Optimization of Component Constrained It is not necessary to consider incremental sensitivity coefficient under component constraints, but to optimize the structure to proper-redundancy according to the requirements of detailing constraints, sectional constraints and normalize constraints. The constraints used in the case are slenderness ratio, deflection of beam (live load), deflection of beam (characteristic combination of loads), stress ratio of beam, stress ratio of column. Because of the opposite arrangement of sections between the AO-CSOD method and the AO-CSCD method, the section area may change little when the sections increase or decrease, but the parameters such as moment of inertia are quite different from each other. So, they do not converge to the same solution in the calculation. Optimal results under component constraints in AO-CSOD method are shown in the Table 5.6 that the column is 400 × 200 × 8 × 13, the beam is 250 × 255 × 14 × 14. Optimal results under component constraints in AO-CSCD method are shown in the Table 5.7 that the column is 446 × 150 × 7 × 12, and the beam is 250 × 255 × 14 × 14. Table 5.6 Optimal results under component constraints in AO-CSOD method Project

AO-CSOD

Slenderness ratio

Beam deflection (live load)

Beam deflection (combination of loads)

Stress ratio (beam)

Stress ratio (column)

Story drift

Value

45.53

2.1 × 10−4

7.1 × 10−4

0.6668

0.8392

0.0047

Redundancy

54.47%

89.23%

71.52%

25.91%

6.76%

−16.62%

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Table 5.7 Optimal results under component constraints in AO-CSCD method Project

AO-CSCD

Slenderness ratio

Beam deflection (live load)

Beam deflection (combination of loads)

Stress ratio (beam)

Stress ratio (column)

Story drift

Value

45.53

2.6 × 10−4

8.5 × 10−4

0.8263

0.8392

0.0047

Redundancy

54.47%

87.07%

65.93%

8.187%

6.120%

−18.10%

5.3.3 The Optimization of Global Constraints and Sensitivity Analysis Take the AO-CSOD method as an example. It can be seen from the above data that after the component constraint optimization is completed, the story drift didn’t satisfy the design requirements, therefore, execute the global constraints design part in the AO-CSCD method. After executing the global constraint optimization, we carry on the CISA method. In this case, the story drift under wind load and the story drift under seismic action are used as global constrained. The optimal design process contains 2 cycles. The sensitivity coefficients of beams and columns are shown in the Fig. 5.7. Taking the first cycle of compliant design as an example, as we can see in Fig. 5.7, the sensitivity coefficients of the columns under the action of seismic and wind load are higher than the sensitivity coefficients of the beams, so we increase the section size of the columns. If the sensitivity coefficients of the columns under the action of seismic and wind load are lower than the sensitivity coefficients of the beams, then we increase the section size of the beams. If appear the sensitivity coefficient points to the optimized object is inconsistent under two kinds of action, then optimizes section size of the columns and beams at the same time. -3

1

x 10

0.8

0.6

0.4

0.2

0

0

1

Fig. 5.7 The sensitivity coefficients of beams and columns

2

5 Ascending Order Constraints Sensitivity Optimal Design …

45

Optimal results of AO-CSOD method. Column: 300 × 300 × 10 × 15. Beam: 400 × 200 × 8 × 13. Optimal results of AO-CSCD method. Column: 300 × 300 × 10 × 15. Beam: 470 × 150 × 7 × 13.

5.3.4 Comparison and Analysis of Design Results The optimization design results of the AO-CSOD method and the AO-CSCD method are compared with the primitive design results. The results are compared with the manual design results at the same time. Manual design results are shown in the Table 5.8. Column: 338 × 351 × 13 × 13. Beam: 496 × 199 × 9 × 14. The results of the AO-CSOD method are shown in the Table 5.9. The results of the AO-CSCD method are shown in the Table 5.10. Comparing the results of the AO-CSOD method and the AO-CSCD method, we can find that the material dosage of the AO-CSOD method is smaller than that of the AO-CSCD method, and comparing the two design results and the manual design results, it is obvious that both design results are better than the manual design results, and the design efficiency has been greatly improved, from half an hour to just a few dozen seconds. Table 5.8 Manual design results Project

Manual design

Slenderness ratio

Beam deflection (live load)

Beam deflection (combination of loads)

Stress ratio (beam)

Stress ratio (column)

Story drift

Material dosage (kg)

Value

33.21

1.5 × 10−4

5.1 × 10−4

0.6018

0.6447

0.0032

1297

Redundancy

66.78%

92.36%

79.75%

33.13%

28.37%

59.15%

Table 5.9 The results of the AO-CSOD method Project

AOCSOD

Slenderness ratio

Beam deflection (live load)

Beam deflection (combination of loads)

Stress ratio (beam)

Stress ratio (column)

Story drift

Material dosage (kg)

Value

36.84

3.1 × 10−4

0.0010

0.7800

0.7892

0.0039

1127

Redundancy

63.16%

84.50%

59.04%

18.98%

18.98%

1.671%

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Table 5.10 The results of the AO-CSCD method Project

AOCSCD

Slenderness ratio

Beam deflection (live load)

Beam deflection (combination of loads)

Stress ratio (beam)

Stress ratio (column)

Story drift

Material dosage (kg)

Value

36.84

2.6 × 10−4

8.6 × 10−4

0.8182

0.7238

0.0038

1072

Redundancy

63.16%

87.02%

65.78%

9.092%

19.58%

3.995%

5.4 Conclusion This paper proposes the AO-CSOD method and the AO-CSCD method based on the CISA method, The AO-CSOD method and AO-CSCD method are applied to the optimization of a single-story steel frame structure. The results of the two methods are compared and analyzed. The main conclusions are as follows: (1) In the AO-CSOD method and the AO-CSCD method, we can effectively identify the components and their positions that need to be optimized base on the CISA method. The efficiency of optimization has been effectively improved, and the workload of optimal design has been greatly reduced. (2) Due to the characteristics of the algorithm, the optimization effects of the AOCSOD method are not better than that of the AO-CSCD method. However, if a structure system is complicated, it is more reasonable to start the design from the over-redundancy, so we combine the compliance design method in the AOCSOD method, and the AO-CSOD method can also converges to the approximate optimal solution. (3) The AO-CSOD method and the AO-CSCD method can achieve automatic configuration at design progress and have higher design efficiency compared with current manual optimal design. The approximate optimal solutions can be obtained in a short time, and they can save the amount of structural materials and obtain better economic and social benefits.

References 1. Isenberg, J., Pereyra, V., Lawver, D.: Optimal design of steel frame structures. Appl. Numer. Math. 40(1), 59–71 (2002) 2. Tocher, J.L., Karnes, R.N.: The impact of automated structural optimization on actual design. AIAA J. (1971) 3. Adelman, H.M., Haftka, R.T.: Sensitivity analysis of discrete structural systems. AIAA J. 24(5), 823–832 (1986)

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4. Yu, T.Y., Zhao, X.: Virtual work sensitivity method for the optimization design of tall buildings. In: 13th East Asia-Pacific Conference on Structural Engineering and Construction (EASEC-13), Sapporo, Japan (2013) 5. Austin, F.: A rapid optimization procedure for structures subjected to multiple constraints. In: AIAA/ASME/SAE 18th Structures, Structural Dynamics and Materials Conference, San Diego, Calif., pp. 71–79 (1977) 6. Brayton, R.K., Spence, R.: Sensitivity and Optimization. Elsevier, New York (1980) 7. Barrar, C.D.: Structural Optimization Using the Principle of Virtual Work and an Analytical Study on Metal Buildings. Virginia Tech, Virginia (2009) 8. Zhao, X., Dong, Y.M., Yu, T.Y.: Sensitivity analysis of material distribution to structural period for super tall buildings. In: IASS-SLTE 2014 Symposium, pp. 99–104 (2014)

Part II

Advances in Smart and Functional Construction Materials and System for Civil Infrastructure

Chapter 6

Single Driven Constraint Optimal Structural Design of Tall Buildings Under Period Constraint Hao Zhang, Jiemin Ding, Xin Zhao and Lang Qin

Abstract Period constraint is important for the design of super tall buildings. Structural design of numerous super tall buildings is governed by period constraint. Reducing period of tall building may cause excessive stiffness requirement and will induce massive waste of structural member material. Proper vibration period is critical index for the optimal structural design of super tall buildings. The authors developed a series of sensitivity based optimal structural design methods for super tall buildings. The design criteria for tall buildings can be treated as constraints in optimal design, which can be divided into three classes: global constraints, assembly constraints and component constraints. If certain constraint has been considered in solving design variables, it will be defined as driven constraint, otherwise the constraint will be defined as validation constraint. Sensitivity analysis is concerned with the relationship between design variables and the structural response. Based on the sensitivity results, an engineer can decide on the direction of design alternation needed to improve the design effectiveness. In this paper, the formulas for the sensitivity coefficient of vibration period to the structural members are derived, based on which single driven constraint optimal method for structural optimization under period constraint is developed. According to the sensitivity coefficients, cost effective structural design can be achieved by reasonably redistributing materials amongst different sets of structural members. A 468-m tall building project is employed to illustrate the applicability and effectiveness of the proposed optimal design method under period constraint. Keywords Tall buildings · Period constraint · Optimal design · Structural design

H. Zhang · X. Zhao (B) Tongji University, No. 1239 Siping Road, Shanghai, China e-mail: [email protected] J. Ding · X. Zhao Tongji Architectural Design (Group) Co., Ltd., No. 1230 Siping Road, Shanghai, China L. Qin Southwest Electric Power Design Institute Co., Ltd., Chengdu, China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_6

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6.1 Introduction Period is an important parameter for tall building structural design. A tall building structure with high period will cause deformation and comfort problems in normal service situation, especially for those tall buildings in high wind area. In order to reduce the adverse effects caused by high period, structural engineers commonly try to find a proper vibration period for tall buildings to balance structural performance and costs. Goel did a lot of research on the suitable period of tall buildings [1, 2]. In the field of aerospace, structural optimization design with minimum weight as optimization objective had been preliminarily developed, but the initial optimization research was limited to stress constraints and buckling constraints under uniform pressure or shear loads [3]. Sensitivity analysis is concerned with the relationship between design variables and the structural response. Based on the sensitivity results, an engineer can decide on the direction of design change needed to improve the performance measures. Early sensitivity analysis was used to assess the impact of parameter changes on the mathematical model of control systems [4–6]. Yu and Zhao [7] optimized structure cost based on virtual work principle and constraints of inter-story displacement angle under wind load. Zhao and Dong [8] optimized tall buildings cost under period based on virtual work. Zhao and Qin [9] using optimal design method optimized outrigger systems for super tall buildings. Zhao and Qin [10] used optimal method to optimize placement of visco elastic coupling dampers in super tall buildings. Based on virtual work principle, a sensitivity analysis under period constraint is carried out and a single driven constraint optimal method is developed in this paper. According to the sensitivity coefficients, cost effective structural design is achieved by means of reasonably distributing materials among the various components subject to period constraint.

6.2 Basic Principles of Sensitivity Analysis 1. The Relation between Inertial Force Work and Component Cost

W =

P 

k eW

(6.1)

k=1

In the formula (6.1), W represents the virtual work done by inertial force on the first mode of vibration, and the real and virtual modes are the first mode of vibration. K represents the number of components, P represents the total number of components, represents the virtual deformation work corresponding to W of the K component, and the alignment element can be expressed as formula (6.2), and the shell element

6 Single Driven Constraint Optimal Structural Design …

53

can be expressed as formula (6.3): W =

P 

eW,k

(6.2)

k=1

eW,k

 T    2Kk Kk 1 Fϕm,k = Lk Fϕm,k Fϕn,k T K 2K Fϕn,k 6 k k

(6.3)

In the formula (6.3), L is the length of the k-th line element, H is the length of the k-th shell element, D is the height of the k-th shell element, and the internal force of the element under the first mode of vibration, A is the cross-sectional area of the line element, and the shear area of the line element, and B is the torsional and bending moment of inertia of the line element under the inertia force, and B is the thickness of the k-th shell element. Assuming that the change of material cost of one component does not affect the internal virtual work of other components, there are: ∂eW,k ∂W = ∂ck ∂ck

(6.4)

Formula (6.4): ck is the number k material cost for the first component. 2. The Relation between Vibration Mode and Component Cost The relationship between the first-order mode vector and component cost can be converted into the relationship between the mode of each floor and component cost, in which M is the total number of floors. Fi · X i =

P 

ekX i

(6.5)

k=1

e X i ,k

  T   2Kk Kk 1 F X i m,k = Lk Fϕm,k Fϕn,k Kk 2Kk FTX i n,k 6

(6.6)

where F is x-directional unit virtual force acting on the center of mass of story i, as depicted in Fig. 6.3; ekX ji is contribution of member k to F · X ji , and can be expressed for frame element, and contribution for shell element can be obtained similarly. Assuming that the change of material cost of one component does not affect the internal virtual work of other components. ∂e X i ,k ∂ Xi = ∂ck ∂ck

(6.7)

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Fig. 6.1 Sketch of model calculation

3. The relationship between quality and component cost The calculation sketch of the model is shown in Fig. 6.1. The mass of the first layer beam and half of the mass of the first layer column and the second layer column are concentrated in the first layer particle, and the mass of the second layer beam and half of the mass of the second layer column are concentrated in the second layer particle. The relationship between quality matrix M and component cost can be converted into the relationship between quality of each floor and component cost, in which M is the total number of floors. In the process of optimization, the size of component changes, which only affects the quality of adjacent floors. The floors where component K is located are K floors. MK k k Horizontal component: ∂∂co k = ρ /C ∂ ML = 0 (L = K ) ∂cok

(6.8)

M K −1 MK k k Vertical component: ∂ ∂co = ∂∂co k k = 0.5ρ /C ∂ ML = 0 (L = K − 1 and L = K ) ∂cok

(6.9)

In the formula (6.9), the mass of K layer, K − 1 layer and L layer is the mass density of component k, the cost of unit volume of component k, and the cost of component K. 4. The Relation between Periodic Constraints and Component Cost

sTk =

∂T 2π ∂ω π ∂ω2 = − · = − · ∂cok ω2 ∂cok ω3 ∂cok

(6.10)

In the formula (6.10), T and ω is the first-order period of the structure and the first-order circular frequency of the structure respectively.

6 Single Driven Constraint Optimal Structural Design …



∂ φTWMφ π π sTk = − 3 · =− 3 · k ω ∂co ω

=−

π · ω3

∂W ∂cok

55

· φT Mφ − W · ∂φ∂coMφ k  2 φT Mφ M   2

∂ Xi 2 2 2 · φT Mφ − W · 2Mi X i · ∂co · k + X i + Yi + ϕi r i ∂W ∂cok

i=1

T

∂ Mi ∂cok



 2 φT Mφ (6.11)

In the formula (6.11), the horizontal relative displacement of the center of mass of the mode I layer in the X and Y directions, the relative torsion angle of the mode I layer and the rotation radius of the mode I layer can be divided by the square root of the quotient of the mass of the layer. φT Mφ = 1

⎧ ⎡ ⎤⎫ M

∂M ⎬ ∂ekX  ∂T −π ⎨ ∂ekW i ⎦ i k 2 2 2 2 2 ⎣ sT = = 3 · −ω · + X i + Yi + ϕi ri · 2Mi X i · ⎩ ∂cok ω ∂cok ∂cok ∂cok ⎭

(6.12) (6.13)

i=1

Formula (6.13) takes into account the effects of design variables (component cost) on the virtual work W, the first mode and the mass matrix M of the structure of inertial force F on the first mode. The sensitivity coefficients of periodic constraints on component cost can be obtained by extracting the internal force under the first mode of vibration and the virtual unit force, material characteristics, component size, material unit cost, component location and material density of layer I of structural component K.

6.3 Single Driven Constraint Optimal Structural Design Method Equal incremental sensitivity analysis method is to increase the material volume or cost of each component group in turn, calculate the variation of design constraints, and then calculate the sensitivity coefficient. si,k =

gi vk

(6.14)

In the formula (6.14), the sensitivity coefficient of the first design constraint to the design variable k, the change variable of the second design constraint and the change variable of the K design variable are expressed.

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The mathematical model of structural optimization design includes three main elements: design variables, constraints and optimization objectives. In this paper, the objective of optimization is only to consider the cost of construction, which can directly establish the linear function expression of optimization objectives and design variables, as shown below. Ob =

P 

vk

(6.15)

k=1

In the formula (6.15), Ob represents the optimization objective, k represents the component number, P represents the total number of components, and v k represents the design variable. The mathematical model of optimal design can be expressed as: ⎫ Find v k ∈ R n (k = 1, 2, . . . , P) ⎪ ⎪ ⎬ Min Ob ⎪ s.t. gi ≤ Limiti (i = 1, 2, · · · , m) ⎪ ⎭ v k(L) ≤ v k ≤ v k(U ) (k = 1, 2, . . . , P)

(6.16)

In the formula (6.16), gi represents the first design constraint, Limiti represents the i design constraint limit, m represents number of constraints, v k(L) and v k(U ) represents the lower and upper bounds of design variables respectively. Based on the constrained sensitivity optimization design, the sensitivity coefficient is taken as the reference index of component dimension optimization design, and the directional material redistribution of the component is carried out, so that the economic cost of construction of the structure can reach the optimum under the premise of satisfying the design constraints.

6.4 Analysis Principle of Period Rayleigh principle is a principle for calculating approximate natural frequencies of vibration systems, especially the upper bound of minimum natural frequencies. For a conservative system that vibrates near a stable equilibrium position, it is assumed that it is a simple harmonic oscillation with a possible displacement satisfying continuous deformation conditions and displacement boundary conditions. According to the conservation of mechanical energy, the relationship between the maximum potential energy and the maximum kinetic energy of the system can be expressed as following equations. ω2 φT Mφ(Maximum kinetic energy) = φT Kφ(Maximum potential energy) (6.17)

6 Single Driven Constraint Optimal Structural Design …

ω2 =

φT f W φT Kφ = T = T T φ Mφ φ Mφ φ Mφ

57

(6.18)

where M denotes the mass matrix of structures; K is the stiffness matrix of the structure, the mode vector under the inertial force f, f is the inertial force vector, W is the virtual work done by the inertial force F on the mode of vibration. If the first-order circular frequency of the structure is known, the first-order period T of the structure can be calculated by the following formula: T =

2π ω

(6.19)

It can be seen from formula (6.18) and formula (6.19) that the first order natural vibration period of the structure is related to three quantities: the virtual work W of inertial force F on the mode, the first-order mode and the mass matrix M of the structure.

6.5 Case Study A 101-story, 468-m frame-core wall structure is presented to illustrate the optimal design procedure. The model of the structure is shown in Fig. 6.2a. According to Chinese design code, the design characteristic period is 0.5 s and the fortification intensity is 7°. Design spectral response acceleration parameter at short periods is 0.1147g according to evaluation report of site seismic safety. The damping ratio in the frequent earthquake is set to 4%. According to the research of the Chinese Academy of Architectural Sciences, the first-order period of tall building structure Fig. 6.2 a Structural system; b outriggers; c braces; d columns; e core walls

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Fig. 6.3 The elevation view of outriggers

Fig. 6.4 The layout plan

above 250 m was suggested that should be controlled. The first-order period of this case is 8.19 s. The main lateral force-resisting members in the structure are outriggers, braces, SRC column sand core walls, as shown in Fig. 6.2b–e. Three outriggers are set on 23–26 stories (outrigger 1), 47–50 stories (outrigger 2), and 98–100 stories (outrigger 3). The elevation view of outriggers is shown in Fig. 6.3. Several-story braces are set along overall height. The layout plan is shown in Fig. 6.4. Composite columns Z1 and Z2 are separated because of different buried steel. The sensitivity coefficients of period to the material cost of each group of components are shown in Fig. 6.5. As shown in Fig. 6.5, the sensitivity of low-zone members is negative, and the first-order period of low-zone members decreases with the increase of sensitivity, while the sensitivity of high-zone members is positive, while the first-order period of high-zone members increases with the increase of sensitivity. Generally speaking, the cost sensitivity of shear walls is greater than that of stiff steel columns, outrigger trusses and outrigger diagonal braces (Table 6.1).

6.6 Conclusions In this paper, a sensitivity analysis under period constraint is carried out. According to the sensitivity coefficients, cost effective structural design is achieved by means of reasonably distributing materials among the various components subject to period constraint. A tall building is exemplified to illustrate the single driven constraint

6 Single Driven Constraint Optimal Structural Design …

59

Fig. 6.5 Sensitivity coefficients of design constrains to design variables

Table 6.1 Three optimization step period and cost changes Optimization step

Period before optimization(s)(Redundancy)

Period after optimization(s)(Redundancy)

Cost optimization (thousand yuan)

Accumulative Cost optimization (thousand yuan)

1

8.19(3.85%)

7.95(6.93%)

6418.4

6418.4

2

7.95(6.93%)

8.20(3.60%)

16059.5

22477.9

3

8.20(3.60%)

8.50(0.04%)

3726.9

26204.7

optimal method. The result shows that this optimal method is efficient to lower the structural cost. Following conclusions can be obtained. Single driven constraint optimal design method is an efficient optimization design method which provides a good basis for more application about sensitivity analysis based artificial structural optimization method. Using this method can significantly reduce economic cost of tall buildings under period constraint. The formulas for the sensitivity coefficient of vibration period to the structural members are derived. According to the sensitivity coefficients, cost effective structural design can be achieved by reasonably redistributing materials amongst different sets of structural members.

References 1. Goel, R.K., Chopra, A.K.: Period formulas for moment-resisting frame buildings. J. Struct. Eng. 123(11), 1454–1461 (1997) 2. Goel, R.K., Chopra, A.K.: Period formulas for concrete shear wall buildings. J. Struct. Eng. 124(4), 426–433 (1998)

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3. Gerard, G.: Minimum Weight Analysis of Compressive Structures. New York University Press, New York, NY (1956) 4. Tomovic, R.: Sensitivity Analysis of Dynamic Systems. McGraw-Hill Book Co., New York (1963) 5. Gellatly R.A., Gallagher, R.H., Luberacki, W.A.: Development of a Procedure for Automated Synthesis of Minimum Weight Structures. AFFDL-TDR-TR-64-141 (1964) 6. Radanovic, L.: Sensitivity Methods in Control Theory. Pergamon Press, Oxford, England (1966) 7. Yu, T.Y., Zhao, X.: Virtual work sensitivity method for the optimization design of tall buildings. In: 13th East Asia-Pacific Conference on Structural Engineering and Construction (EASEC13), Sapporo, Japan (2013) 8. Zhao, X., Dong, Y.M., Yu, T.Y.: Sensitivity analysis of material distribution to structural period for super tall buildings. In: IASS-SLTE 2014 Symposium, pp. 99–104 (2014) 9. Zhao, X., Qin, L., Dong, Y., et al.: Optimal design of outrigger systems for super tall buildings. In: IABSE Conference Nara 2015. International Association for Bridge and Structural Engineering, Nara (2015) 10. Zhao, X., Qin, L.: Optimal placement of viscoelastic coupling dampers in super tall buildings. In: 8th STESSA Conference Shanghai 2015. International Conference on Behavior of Steel Structures in Seismic Areas, Shanghai (2015)

Chapter 7

Numerical Assessment Regarding the Influence of the Stiffness of the Material Used to Build Multi-layer Energy-Absorbing Panels on the Absorption of the Shock Wave Energy ´ Grzegorz Sławinski, ´ Piotr Malesa and Marek Swierczewski Abstract Mechanical effects of shock waves and waves of relief propagating behind them constitute the main destruction factor of an explosion. The shock wave created as a result of an explosion reaches a destructed object and causes a sudden increase of pressure affecting its surface. The article presents numerical tests regarding the influence of using materials with different stiffness in one of the layers of a multilayer energy-absorbing panel on the reduction of a load coming from the shock wave affecting the model of a light wheeled armoured vehicle. The assumed aim of the tests was to assess the effectiveness of low-density materials with different stiffness used in a layer of additional external multi-layer energy-absorbing panels. Three systems were selected for the initial analysis of the effectiveness of multi-layer energy-absorbing panels. The effectiveness of absorbing the shock wave energy will be assessed based on the analysis of acceleration characteristics in selected construction points of a vehicle which are the carriers of energy-absorbing panels. In order to tests the effectiveness of energy-absorbing panels, numerical calculations were conducted using the finite element method. The numerical model of the problem was created on the basis of respective physical and mathematical models including a range of physical phenomena and mechanisms which are important in terms of the absorption of the air shock wave energy generated during the detonation of explosives. The CONWEP option implemented in the LS-DYNA system was used to simulate the effects of the shock wave. The effectiveness assessment of the energyabsorbing systems was planned as two-stage tests, i.e. a central explosion carried out on the basis of the test methodology included in AEP-55 vol. 2 and a side explosion carried out in a limited scope of the test methodology described in AEP-55 vol. 3. In those two cases, the effectiveness was assessed on the basis of the result comparison of registered accelerations in vehicle’s characteristic construction points. ´ G. Sławi´nski (B) · P. Malesa · M. Swierczewski Military University of Technology, Gen. Witolda Urbanowicza St. 2, 00-908 Warsaw, Poland e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_7

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Keywords FEM analysis · Improvised explosive device · Blast wave · Energy-absorbing panel

7.1 Introduction The current and future battlefield is characterised by an unpredictable nature of actions which can develop from small riots into different directions, which also makes threats unpredictable. Therefore, there is a visible need in different armies around the world to maintain forces which will enable them to participate in operations of various intensity of actions on own territory as well as abroad. The use of such forces is determined by i.a. compliance with the following requirements: – maximising the strategic mobility of military units, which is defined as the ability to quickly provide appropriate forces in the conflict zone; – possibility to gradually intensify the participation of own forces; – maximising the possibilities of constantly exploring the battlefield; – maximising the possibilities of providing fast tactical support. Therefore, light wheeled armoured vehicles are the most frequently used vehicles in many armies around the world. Their aim, when used during patrols, is to protect crews from firing from small arms and explosions of mines and Improvised Explosive Devices (IED), which currently constitute the biggest threat for soldiers [1, 2]. The shock wave created as a result of IED explosion can cause permanent deformations of the vehicle’s chassis and body as well as the perforation of the vehicle’s hull. This, in turn, can hinder further driving or render it completely impossible and expose the crew to firing from small arms, which can result in injuries or even death of crew members [2]. In order to ensure the safety of soldiers performing combat actions, many documents have been created. The aim of those documents is to specify development directions for the military technical equipment which is being designed as well as guidelines how to meet the requirements set for them, which need to be taken into consideration at the stage of designing the military technology. One of such documents is STANAG 4569 applicable in the NATO. The document describes the way of checking the resistance of the supporting structure to the effects of the detonation wave created as a result of a detonation under a wheel or track of a military vehicle or on the side of a military vehicle. The main aim of using STANAG 4569 is to comply with the following requirements of interoperability, i.e.: – completing a mission in the conditions of a specific threat; – providing commanders with the choice of appropriate equipment and NATO member states with planning guidelines to distribute the equipment in a way corresponding to the threats at the theatre of war; – providing member states with the possibility of preparing and modernising their equipment in such a way so that it is appropriate for specific threats.

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The list of protection levels included in the document is based on the probability of ensuring protection to vehicle’s crews in case of a specific threat indicated in AEP-55 [3], which is obligatory when performing acceptance tests. Thus the aim of using equipment of a specified protection level is to strengthen interoperability between countries which participate in multi-nation missions. Taking the above into consideration, the tactical and technical requirements which are being currently formulated in different countries focus mainly on searching for solutions: – which are resistant to piercing with rifle bullets and FSPs mentioned in STANAG 4569 up to protection level 4 inclusive (this level has become a standard for vehicles participating in missions); – which increase resistance to mine explosions. This protection level should be considered together with the design of an armoured vehicle (sometimes level 2 is indicated). Such a multi-variant approach to armouring vehicles renders it necessary to prepare several material solutions with precisely characterised properties. According to that approach, armour plates, which can constitute armour protection, can be divided into two groups: – metal armours (steel, aluminium, titanic); – composite armours which will constitute the subject matter of considerations of the presented numerical analyses.

7.2 Destruction Factors of the Shock Wave The shock wave constitutes the main destruction factor of explosions of open Improvised Explosive Devices (IED), namely devices which are characterised by the complete lack of any casing or a casing which does not produce splinters (paper, fabric, insulating tapes etc.). Also the so-called closed light structures, in which the main explosive is placed in a casing made of plastic or in a thin tin container, generate powerful shock waves [4]. As a result of a sudden chemical change of an explosive, an area of heated gaseous medium, called detonation products, is created [5]. The value of pressure of created detonation products of explosives is even several hundred thousand times higher than the air pressure. Mechanical effects of shock waves and waves of relief propagating behind them constitute the main destruction factor of an explosion. The shock wave, created as a result of an explosion, reaches a destructed object (a person, vehicle, construction) and causes a sudden increase of pressure affecting its surfaces. The set of the shock wave and the wave of relief is called a blast wave. In order to correctly assess the results of blast wave effects on destructed objects, the information

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Fig. 7.1 Characteristic of the course of overpressure P(t) produced by the blast wave in a given point of space

about the course of overpressure P(t), created in a given point of space, is needed. The characteristic of that course is presented in Fig. 7.1. The profile of pressure in the blast wave is characterised by the following values: overpressure on the wavefront (Ps), maximum under pressure (Pd), duration of the positive pulse phase (τ+), duration of the negative pulse phase (τ−). In addition, a value constituting the integral from overpressure after the time in the positive phase of overpressure, called the pulse of the blast wave, is introduced as a characteristic of the blast wave. Such a value can be established from the following dependency: τ+

Is = ∫ P(t)dt

(7.1)

0

If the Ps value of overpressure in the wavefront is known, it is possible to specify the value of overpressure after wave reflection using formulas for the reflected shock wave. If the P(t) function is known, it is also possible to specify changes in time of overpressure affecting the loaded object.

7.3 Multi-layer Protective Systems—Passive Systems Vehicles used during current armed conflicts should ensure such a protection level which corresponds to current threats. The main problem is encountered when developing effective protection of the crew of an armoured vehicle against negative effects of the shock wave. A concept of preparing an effective “anti-explosion” shield should be based on the following assumptions:

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– the shield should be characterised by a tight construction resistant to the effects of external climatic and mechanical factors; – the structure of the shield’s material should be characterised by a step change in mechanical properties and, at the same time, maintenance of high properties in the structure of a given layer; – the shield should have high mechanical properties in terms of very big deformations. When designing a protective structure, multiple factors should be taken into consideration. Factors which might have an influence on the amount of absorbed energy coming from the shock wave definitely include the following: – – – –

the geometry of a panel; the type of the used material; the way of destroying the material; suppression in the material etc.

It has turned out that perfect structures which meet the above-mentioned assumptions are multi-layer structures, which—thanks to the used layers and materials— enable dissipation and absorption of the energy of the shock wave caused by the detonation of explosives, i.a. through appropriate shaping of the protective structure or thanks to the use of modern materials [6]. Desired characteristics of materials used to build energy-absorbing panels include the following: – big relative energy absorption; – small specific weight, fire-resistance; – resistance to temperature and chemical factors. Taking into consideration the number of factors which might have an impact on decreasing the transmission of the shock wave energy to a vehicle and, consequently, to its crew, the article focuses on numerical tests of the influence of using materials with different stiffness in one of the layers of a multi-layer energy-absorbing panel.

7.4 Test Object and Methodology The aim of numerical analyses was to check the capabilities of dissipating the energy of detonated explosives on multi-layer protective panels mounted on the vehicle. The effectiveness of absorbing the shock wave energy will be assessed based on the analysis of acceleration characteristics in selected construction points of a vehicle which are the carriers of energy-absorbing panels. Several systems have been selected for the initial effectiveness analysis of multi-layer energy-absorbing panels: – referential variant—a mass equivalent of energy-absorbing panels, made of armour steel; – variant V1—a multi-layer system according to the configuration presented in Fig. 7.2;

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Fig. 7.2 Protective systems mounted to the floor plate of a military vehicle [Cymat—aluminium foam; variant V1—density ρ = 0.6 (g/mm3 ), variant V2—density ρ = 0.5 (g/mm3 ), variant V2— density ρ = 0.31 (g/mm3 )]

– variant V2—a multi-layer system according to the configuration presented in Fig. 7.2; – variant V3—a multi-layer system according to the configuration presented in Fig. 7.2; The scope of the tests included checking energy-absorbing capabilities of energyabsorbing panels made of components with different stiffness, mounted on the vehicle’s body. Two identical vehicle bodies have been prepared for the tests—a referential body with mass equivalents of energy-absorbing panels made of armour steel and a principal body with different built-up variants of multi-layer energy-absorbing panels. Two types of experimental tests have been conducted and divided into the following stages: 1. explosion under the vehicle (central explosion)—the equivalent of 6 kg TNT detonated from the shooting plate (Fig. 7.3): – the referential system: a body with mass equivalents of energy-absorbing panels made of steel—Fig. 7.4a; – the principal system: a body with mounted energy-absorbing panels— Fig. 7.4a. 2. side explosion—the equivalent of 15 kg TNT (Fig. 7.5): – the referential system: a body with mass equivalents of energy-absorbing panels made of armour steel—Fig. 7.6a; – the principal system: a body with mounted energy-absorbing panels— Fig. 7.6b.

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Fig. 7.3 Way of placing the explosive towards the vehicle—central explosion

Fig. 7.4 The vehicle with the mounted referential system (a) and the mounted protective system (b)

Fig. 7.5 Way of placing the explosive towards the vehicle—side explosion

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Fig. 7.6 The vehicle with the mounted referential system (a) and with the system of multi-layer energy-absorbing panels (b)

The central explosion has been carried out on the basis of the test methodology included in AEP-55 vol. 2, whereas in the case of the side explosion, the test has been conducted in a limited scope of the test methodology described in AEP-55 vol. 3. For comparative reasons, the influence of mass differences of two tested systems mounted to the bottom of the vehicle as well as to its side walls, has been levelled. This has been obtained through the choice of the thickness of the armour plate used in referential systems. The thickness of the armour plate has been chosen in such a way so that its mass corresponds to the mass of multi-layer energy-absorbing panels.

7.5 Numerical Model The numerical model of the problem has been created on the basis of respective physical and mathematical models including a range of physical phenomena and mechanisms which are important in terms of the absorption of the air shock wave energy generated during the detonation of explosives. In order to solve the problem, the finite element method (FEM) with the closedform scheme of integration has been chosen. The FEM integration is available in the commercial LS-Dyna software. The spatial discretisation of the physical system has been built using finite elements and the number of finite elements in particular models is presented in Table 7.1. Loading with the shock wave coming from the explosion has been carried out using the CONWEP model coupled with the traction boundary condition [7]. The material model of a modified Johnson-Cook model has been used to describe the properties of the armoured metal plate (Table 7.2) [8]. The components made of S355 structural steel have been modelled using the linear and plastic model (Table 7.3) [9]. The material model of a Honeycomb model has been used to describe the properties of the aluminium foam and cork [10].

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Table 7.1 Number of elements in numerical models Type element

Variant with the referential metal plate

Variant with the protective panel

Number of elements which the vehicle was made of (shell)

259,320

259,320

Number of elements which the vehicle was made of (solid)

3792

3792

Number of elements which panels were made of (shell)

19,900

9912

Number of elements which panels were made of (solid)

19,016

134,388

302,028

407,412

Total

Table 7.2 Material properties for Armstal 500 steel for the MAT_MODIFIED_JOHNSON_COOK material model [8] Hardness

Yield stress

Strain hardening

Strain rate hardening

Temperature softening

HB

σ0.2

A (MPa)

B (MPa)

n

C

ε˙ 0 (s−1 )

m

488–566

1707

1875

415

0.98

0.001

2 × 10−4

1.0

Table 7.3 Material properties for S355 steel for the MAT_PIECEWISE_LINEAR_PLASTICITY material model [9] ρ (t/mm3 )

E (GPa)

ν (–)

7.8 × 10−9

2.1 × 105

0.3

355

EPS1 (MPa)

EPS2 (MPa)

ES1 (MPa)

ES2 (MPa)

0

0.8

355

550

SIGY (MPa)

The total of 22 contact cards have been described in the numerical models, including the following types: – CONTACT_TIED_NODES_TO_SURFACE_OFFSET, which receives only the translation degrees of freedom in contact points. – CONTACT_TIED_SHELL_EDGE_TO_SURFACE_OFFSET, in which all translation and rotation degrees of freedom are blocked. Measuring points have also been included in the model by introducing measuring elements in them, using the ELEMENT_SEATBELT_ACCELEROMETER card. The measurement has been conducted with the registration frequency of 0.5 μs. The total mass of the object together with energy-absorbing panels corresponding to the real construction has been achieved by using mass elements distributed in the

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front and rear part of the vehicle, using RBE3 elements. This, in turn, has rendered it possible to achieve the total vehicle mass of 11.3 t. Calculations have been stabilised in the initial phase by activating the dynamic relaxation option, which enables a free fall of the vehicle under the pull of gravity. The calculation phase has been established at 200 ms.

7.6 Results of Numerical Analyses As a result of conducted numerical analyses, time characteristics of acceleration changes in characteristic construction points of the vehicle have been obtained. Their analysis and comparison regarding two variants of loading the vehicle and different variants of materials used to build multi-layer energy-absorbing panels, have been used to test energy absorption and indicate the best design and material solution of the protective system. All registered courses from the measuring points have been filtered using the SAE 1000 filter.

7.6.1 Explosion Under the Vehicle—Analysis of Results The location and description of the measuring points are presented in Fig. 7.7 and in Table 7.4. Figures 7.8, 7.9 and 7.10 present the diagrams of acceleration courses for the tests carried out using the explosive placed next to the vehicle for the referential system as well as the system with energy-absorbing panels in three variants.

Fig. 7.7 Location and orientation of coordinate systems in the measuring points in the vehicle

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Table 7.4 Description of the measuring points for the central explosion (the referential system and the system with the energy-absorbing panel) Measuring point number

Measuring point description

Remarks

1

Floor, centre

See Fig. 7.7

2

Mounting of the middle seat

See Fig. 7.8

3

Mounting of the side seat

See Fig. 7.9

Fig. 7.8 Characteristic of changes in acceleration on the vehicle’s floor

Tables 7.5 and 7.6 present the maximum and minimum acceleration values, registered in the measuring points for the referential system as well as the system with energy-absorbing panels.

7.6.2 Explosion Next to the Vehicle—Analysis of Results The location and description of the measuring points are presented in Fig. 7.11 and in Table 7.7. Description of the measuring points for the central explosion (the referential system and the system with the energy-absorbing panel). Figures 7.12, 7.13 and 7.14 present the diagrams of acceleration courses for the tests carried out using the explosive placed next to the vehicle for the referential system as well as the system with energy-absorbing panels in three variants.

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Fig. 7.9 Characteristic of changes in acceleration in the mounting point of the middle seat

Fig. 7.10 Characteristic of changes in acceleration in the mounting point of the side seat

7 Numerical Assessment Regarding the Influence of the Stiffness … Table 7.5 Values of maximum and minimum accelerations in particular measuring points for the variant with the referential metal plate

73

Measuring point

Acceleration amax (g)

Acceleration amin (g)

Floor, centre, ax

10763.56

−10544.47

Floor, centre, ay

1590.10

−2829.17

Floor, centre, az

1202.61

−1320.39

Mounting of the middle seat, ax

2048.39

−1959.50

Mounting of the middle seat, ay

1951.42

−1907.37

Mounting of the middle seat, az

2693.78

−2564.79

Mounting of the side seat, ax

1183.80

−1114.07

Mounting of the side seat, ay

2035.58

−2652.05

Mounting of the side seat, az

619.62

−671.95

Tables 7.8 and 7.9 present the maximum and minimum acceleration values registered in the measuring points for the referential system as well as the system with energy-absorbing panels. The numerical analyses included in the article constitute the continuation of the previous works in the field of determining the effects of the shock wave generated by small explosives (not exceeding 4 kg) on a vehicle and its crew. When analysing the results for the side explosion, the reduction of the maximum/minimum values by 40% (between referential variants and V1) and by 50% (between referential variants and V3) has been observed. In the case of the explosion under the vehicle resulted in the growth of 30% (between referential variants and V1) and 40% (between referential variants and V3). It is worth noting that the decrease in the maximum values of accelerations and the courses of acceleration and time characteristics in selected points has been observed in each of the analysed variants of the energy-absorbing panel (for different levels of stiffness of the material in one of the layers). This is more visible in the case of the effects of the shock wave on the vehicle’s side, which undoubtedly results from the distance of the detonation place from the loaded hull’s surface. In the case of the explosion under the vehicle, the reduction of those values is more visible in the mounting points of seats (the middle and side seats) than on the vehicle’s floor. The main conclusion drawn for the analyses is that it is possible to use different materials for energy-absorbing layers depending on the place where they are mounted. The use of materials which are less stiff (and, at the same time, have a smaller mass) seems more beneficial for the protection of side surfaces of the vehicle. On the other hand, medium stiff materials are better for the protection of the vehicle’s bottom, where the pressure of the shock wave has definitely higher values.

9504.89

1771.10

1479.13

2204.53

2924.81

1790.34

1577.16

2880.91

1571.11

Floor, centre, ay

Floor, centre, az

Mounting of the middle seat, ax

Mounting of the middle seat, ay

Mounting of the middle seat, az

Mounting of the side seat, ax

Mounting of the side seat, ay

Mounting of the side seat, az

Variant V1

Acceleration amax (g)

Floor, centre, ax

Measuring point

10458.69 1652.71 1777.78 2183.71 2573.73 1762.44 1743.56 2948.55 1515.93

−1897.32 −1733.01 −2145.17 −2894.11 −1925.03 −1598.02 −2683.29 −1661.90

Variant V2

Acceleration amax (g)

−9589.84

Acceleration amin (g)

−1636.21

−2627.15

−1568.45

−1872.59

−2843.07

−2148.66

−1451.04

−1719.17

−13619.16

Acceleration amin (g)

1323.50

2051.01

1134.20

1742.24

2766.78

1883.21

1594.17

1690.66

7650.06

Variant V3

Acceleration amax (g)

Table 7.6 Values of maximum and minimum accelerations in particular measuring points for the variant with the protective panel

−1362.40

−2611.84

−1122.83

−1755.85

−3152.79

−2016.98

−1545.58

−1671.46

−9492.41

Acceleration amin (g)

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Fig. 7.11 Location and orientation of coordinate systems in the measuring points in the vehicle

Table 7.7 Description of the measuring points for the central explosion (the referential system and the system with the energy-absorbing panel) Measuring point number

Measuring point description

Remarks

1

Mounting of the side seat

See Fig. 7.11

2

Mounting of the middle seat

See Fig. 7.12

3

Floor, centre

See Fig. 7.13

The obtained significant consistency of the numerical and experimental results allows for the statement that the identical assumptions and conditions for modelling also in current analyses will render it possible to obtain correct results which will be used to develop a methodology of experimental tests regarding the real vehicle— energy-absorbing panel system.

7.7 Conclusions No vehicle is universal, which means that no vehicle protects the crew against all possible threats. The complete protection of vehicles, e.g. against IED explosions, is not possible due to indefinable level and type of threats.

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Fig. 7.12 Characteristic of changes in acceleration in the mounting of the side seat

Fig. 7.13 Characteristic of changes in acceleration in the mounting of the middle seat

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Fig. 7.14 Characteristic of changes in acceleration of the sensor on the vehicle’s floor

Table 7.8 Values of maximum and minimum accelerations in particular measuring points for the variant with the referential metal plate

Measuring point

Acceleration amax (g)

Acceleration amin (g)

Side seat, ax

1481.47

−1446.27

Side seat, ay

2698.54

−2418.03

775.81

−895.88

Middle seat, ax

2450.77

−2404.76

Middle seat, ay

2906.33

−3665.17

Side seat, az

2168.47

−2139.93

Floor, ax

16119.33

−8699.58

Floor, ay

2097.00

−2706.65

Floor, az

1873.59

−1937.41

Middle seat, az

Ideas of design solutions of armours and shields usually lead to the creation of very heavy vehicles with limited mobility. A low level of movement safety expressed by the stability and steerability of the vehicle, poor movement dynamics and other poor functional properties.

1986.79

1987.36

1937.16

11660.98

1470.73

1173.75

Middle seat, az

Floor, ax

Floor, ay

Floor, az

681.28

Side seat, az

Middle seat, ay

1849.81

Middle seat, ax

1109.45

Side seat, ay

Variant V1

Acceleration amax (g)

Side seat, ax

Measuring point

1051.45 1917.75 611.61 1862.60 1856.34 1320.68 12842.65 1522.66 1440.85

−855.62

−1136.93

−1956.71

−2451.02

−2243.85

−7735.71

−3526.98

−1452.05

Variant V2

Acceleration amax (g)

−1698.66

Acceleration amin (g)

−1708.37

−3828.10

−7896.72

−1477.63

−2378.11

−1774.00

−1023.20

−1848.09

−793.88

Acceleration amin (g)

1492.92

1581.34

9868.39

1326.79

1926.13

1984.02

729.07

1393.39

1097.65

Variant V3

Acceleration amax (g)

Table 7.9 Values of maximum and minimum accelerations in particular measuring points for the variant with the protective panel

−1293.26

−2181.97

−6785.54

−1543.58

−2608.89

−2043.79

−1173.52

−1261.18

−838.31

Acceleration amin (g)

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79

Providing protection to people depends not only on the used technical means but also on the assumed tactics. Acknowledgements The research was carried out within Project No. DOBRBIO4/022/13149/2013 “Improving the Safety and Protection of Soldiers on Missions Through Research and Development in Military Medical and Technical Areas”. Supported and co-financed by NCR&D. Poland.

References 1. Kowalkowski, S.: Improwizowane urz˛adzenia wybuchowe – definicje. PWL no. 6/2010 2. Motrycz, G.: Cases of using improvised explosive devices. Szybkobie˙zne Pojazdy G˛asiennicowe (44) nr 2 (2017) 3. AEP-55, Vol. 2, Edn. 1, Procedures for Evaluating the Protection Levels of Logistic and Light Armoured Vehicle Occupants for Grenade and Blast Mine Threats Level, NATO/PFP Unclassified (2005) 4. Ciszewski, T., Kamyk, Z.: Zagro˙zenie IED we współczesnych konfliktach zbrojnych, Mi˛edzynarodowa Konferencja Naukowo-Techniczna „Problemy detekcji i utylizacji materiałów niebezpiecznych”, pp. 45–54. Military Institute of Technical Engineering in Wrocław, Ko´scierzyna (2010) 5. Włodarczyk, E.: Wst˛ep do mechaniki wybuchu. Wydawnictwo Naukowe PWN, Warszawa (1994) ´ 6. Kciuk, S., M˛ez˙ yk, A., Swito´ nski, E.: Najnowsze tendencje w projektowaniu pojazdów specjalnych, Szybkobie˙zne Pojazdy G˛asienicowe (48/49) nr 2/3 (2018) 7. Tabatadaei, Z., Volz, J.: A comparison between three different blast method in LS-Dyna: LBE, MMALE, Coupling of LBE and mm-ALE. In: 12th International LS-Dyna Users Conference, Detroit (2012) 8. Tria, D.E., Tr˛ebi´nski, R.: Dynamic characterization and constitutive modelling of ARMSTAL 500 steel. Probl. Mechatron. Armament Aviat. Saf. Eng. 6, 3(21), 19–40 (2015) ´ 9. Sławi´nski, G., Swierczewski, M., Malesa, P.: Risk assessment regarding the injuries of the lower limbs of the driver of a military vehicle in the case of an explosion under the vehicle. AISC 831, 1–15 (2019). ISSN 2194-5357 ´ 10. Sławi´nski, G., Swierczewski, M., Stanisławek, S., Dziewulski, P., Malesa, P.: Zastosowanie zaawansowanych technik komputerowych do opracowania systemów ochronnych dla pojazdów wojskowych przed wybuchem miny lub IED, Bezpiecze´nstwo w aspekcie zagro˙ze´n technicznych i medycznych wynikaj˛acych z u˙zycia improwizowanych urz˛adze´n wybuchowych (IED), Red.: Andrzej Chciałowski, Grzegorz Gielerak, Jerzy Małachowski, Wojskowy Instytut Medyczny, Warszawa (2017)

Part III

Computational Fracture Modeling on Welded Joints and Advanced Materials

Chapter 8

Computation of Mixed Mode Stress Intensity Factors in 3D Functionally Graded Material Using Tetrahedral Finite Element Omar Tabaza, Hiroshi Okada and Yasunori Yusa Abstract In this paper, a formulation of interaction integral method for the calculation of Stress Intensity Factor (SIF) for three-dimensional cracks in functionally graded material (FGM) is discussed. The SIF evaluations were carried out using somewhat rough FEM model around the crack front while maintaining the path independent property of the interaction integral. The proposed method was examined on both homogeneous material and FGM, for the accuracies of evaluated SIFs under mode I and mixed mode loadings. The results indicate that the present interaction integral method maintain its path independent property. Keywords Stress intensity factor · Interaction integral method · Domain integral method · Fracture mechanics · Finite element method

8.1 Introduction In the evaluations of residual strengths and lives of structures, it is important to accurately evaluate the fracture parameters, such as stress intensity factors (SIFs) and energy release rate. For fatigue and stress corrosion cracking problems, the SIFs play a key role [1]. Engineering structures generally have complex configurations, which lead to the initiation of cracks in the locations of high stress concentration [2, 3]. The stress analysis is usually carried out using the finite element method (FEM). The three-dimensional model is created using a computer aided design (CAD) program and then the FEM model is generated using an automatic-mesh generation program. The most time-consuming process in FEM analysis with a crack is in its model generation. It generally requires some manual operations by a CAD operator. Okada et al. [4] presented a method to generate a FEM model with a crack consisting of the quadratic tetrahedral element. Then an in-house software was developed and applied to the three-dimensional crack problems (see for example Daimon and Okada O. Tabaza · H. Okada (B) · Y. Yusa Department of Mechanical Engineering, Faculty of Science and Technology, Tokyo University of Science, 2641 Yamazaki, Noda, Chiba 278-8510, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_8

83

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[5]). When tetrahedral elements were used, a large integral domain was required to maintain the accuracy as pointed out by Okada and Ohata [6]. A problem with the path independency appears when large integral domain is used. To overcome this problem a correction term is used [5]. In this paper, a methodology to accurately calculate the stress intensity factor in three-dimensional functionally graded material using tetrahedral elements only, is proposed. This methodology is based on the domain integral method developed by Okada and Ohata [6]. In Sect. 8.2, a brief review of the domain integral method and the interaction integral method is presented. In Sect. 8.3, some numerical examples are presented to demonstrate the accuracy of this method, conclusions and some remarks are shown in Sect. 8.4.

8.2 Formulation of J-Integral and Interaction Integral Method The stress strain relationship for linear elasto-static body that are used in this paper is expressed by, σi j = Di jkl (x)εkl .

(8.1)

Here, σ ij and εkl represent the stresses and the strains, Dijkl (x) are the Cartesian components of a forth order tensor that expresses the generalized Hook’s law. Dijkl (x) ti p are the functions of the spatial coordinates. Where Di jkl (x) = Di jkl at the crack tip. The material at the crack tip is assumed to be elastic and isotropic. J-integral is defined as an integral on an infinitesimally small surface surrounding the crack front.    1 ∂u i 3D ε ε W n 1 − n i σi j d Sε (8.2) J = lim  ∂ x1 ε → 0  Sε →0 Here, n iε are the components of unit outward normal vector on the integral surface S , ε and  are the radios and the length of the integral surface. W is the strain energy density and u i are the displacements. W is given by, ε

W =

1 1 σi j εi j = Di jkl εi j εkl 2 2

(8.3)

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8.2.1 Interaction Integral Method The interaction integral is based on the superposition of two independent solutions that satisfy the equilibrium. Their superposition can generate another equilibrium state. The superscripts (1) and (2) designate the two different solutions. The J-integral as defined in Eq. (8.2) can be rewritten as 

(1)

KI

 (2) 2

+ KI



 (2) 2



(1)

 (2) 2

KI I I + KI I I

+ E 2G 

   ∂u (1) 1 ∂u (2) (1) (2) 3D (1) (2) =J = lim + W + W + W (1,2) n ε1 − n iε σi j + σi j d Sε ∂ x1 ∂ x1 ε →0   Sε →0 E

+

(1)

KI I + KI I

(8.4) E where E  = 1−v 2 or E for the plane strain or plane stress condition, respectively. E, G and v are respectively the Young’s modulus, the shear modulus and Poisson’s ratio. By splitting the contributions of the states (1), (2) and their interaction terms, we can establish the formulation of interaction integral method. We let the state (1) be the solution of the boundary value problem, which is the FEM solution for the given elastic boundary value problem. Solution (2) is the auxiliary solution, and it is set be the asymptotic solution with respect to mode I, II or III crack deformation of linear isotropic solid.

 1 2  (1) (2) (1) (2) K + K I(1) K + K K K (2) I I I I I I E µ II III   ∂u (2) ∂u (1) 1 j j ε (1) ε (2) (1,2) ε = lim n 1 − n i σi j − n i σi j W d Sε  ∂ x ∂ x ε 1 1 → 0 Sε  →0

(8.5)

(2) By setting K I(2) = 1 and K I(2) I = K I I I = 0 in Eq. (8.5) we can extract mode I SIF (1) of the actual problem (K I ). The same operation is carried out to find mode II or III SIF.

8.2.2 Domain Integral Method To perform the numerical computations for the interaction integral method the domain integral method is adopted. The function of virtual crack propagation q = q(x), is introduced to develop the domain integral formulation. q at the crack-front

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expresses the virtual crack extension. q(x) is a continuous and piecewise differentiable function over the domain of integration. The value of q is set to be zero at the outer surface of the volume V. By applying Gauss divergence theorem and integrating by parts,  1 2  (1) (2) (2) (2) K I K I + K I(1) + K I(1) I KI I I I KI I I  E G ⎤ ⎡   (2) (1) ∂u ∂u ∂q 1 ⎣ j j =− − σi(2) dV ⎦ W (1,2) δ1i − σi(1) j j A ∂ x1 ∂ x1 ∂ xi V ⎤ ⎡ 



  (2) (1) (1,2) ∂u ∂u W 1 ⎣ ∂ ∂ j j − − σ (1) − σ (2) qd V ⎦ A ∂ x1 ∂ xi i j ∂ x1 ∂ xi i j ∂ x1

(8.6)

V

(1) (2) the crack (u i(2) ) with W (1,2) = 21 Di(2) jkl εi j εkl . By setting the asymptotic solution of   (2) ∂u (2) ti p ti p (2) (2) j 1 ∂u i . The = D ε and ε = + the elastic constants Di jkl , where σi(2) j ij i jkl kl 2 ∂x j ∂ xi ti p

relation Di jkl = Di jkl (x) is assumed to hold at the crack tip, and the equilibrium equation

∂σi(2) j ∂ xi

= 0 is satisfied. Applying to Eq. (8.4), we can rewrite it as

2 2 (1) (2) (2) KI I I KI I I (K (1) K I(2) + K I(1) I KI I ) + E I G ⎤ ⎡   (2) (1) ∂u ∂u 1 ⎣ ∂q j j =− − σi(2) dV ⎦ W (1,2) δ1i − σi(1) j j A ∂ x1 ∂ x1 ∂ xi V ⎤ ⎡   (2)  1 ⎣ ti p (1) ∂εi j − Di jkl − Di jkl (x) εkl qd V ⎦ A ∂ x1

(8.7)

V

  is a crack front segment which is assumed to be locally straight. ΔA = Δ qdx 3 , is the area of the virtual crack propagation defined by integrating q over the length .

8.3 Numerical Examples In this section, some results of numerical analyses are presented. Because of the lack of analytical solutions of 3D FGM crack problems, the accuracy of the proposed interaction integral method was tested for the problems of homogeneous material with available analytical solutions.

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First, the problems of mode I crack in a plate were considered. Next mixed mode problems were solved by introducing an inclined crack in a plate subjected to tension. The material was assumed to be homogenous material or FGM. The results for the SIFs were normalized by substituting over the analytical mode I SIF homogenous solution for each case.

8.3.1 In-plane Crack in Homogenous Material and FGM Here the crack face was set to be perpendicular to the direction of tensile stress. Young’s modulus and Poisson’s ratio of homogenous material were set to be E = 21 GPa, v = 0.3. For FGM Young’s modulus was set to change from 21 to 2100 GPa as shown in Fig. 8.1b. The boundary conditions were set to emulate a circular crack in an infinite body. The height H, width W and thickness t of the cube were set to be 10 mm, 10 mm and 5 mm respectively. The radius r of the circular crack was 1 mm. The radius is much smaller than the height, width and thickness. A uniformly distributed force (P = 100 MPa), were applied on both top and bottom surfaces. The same boundary conditions were applied for all cases. Three different sizes of integral domain were set to examine the path-independent property. Their radiuses were crack × 8, crack × 12 and crack × 16, where crack = (a × 0.0255), a was the radius of the penny shaped crack. The integral domains are shown in Fig. 8.2. In the graphs the notations (KI_8, KI_12 and KI_16) represent the radius of the domain integral for (crack × 8, crack × 12 and crack × 16) respectively. K_th represents the theoretical solution (Fig. 8.3). The results of SIF in homogenous material are compared with the analytical solution provided in the stress intensity factors handbook by Murakami et al. [7].

(a)

(b)

(c)

Fig. 8.1 The statements of the boundary value problem. a Boundary conditions. b Young’s modulus change pattern. c Crack front angle

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(a)

(b)

(c)

Fig. 8.2 The three integration domains. a crack × 8, b crack × 12, c crack × 16

(a)

(b)

Fig. 8.3 a Homogenous material deformation, b FGM material deformation

The evaluated SIFs are summarized in Fig. 8.4.

(a)

(b)

Fig. 8.4 Normalized SIF results for a homogenous material, b FGM

8 Computation of Mixed Mode Stress Intensity Factors …

(a)

89

(b)

Fig. 8.5 Mixed mode normalized SIF results for a homogenous material, b FGM

The homogenous solution was in a very good agreement with the theoretical solution. In the FGM the value of KI increased by increasing the value of Young’s modulus.

8.3.2 Inclined Crack with Homogenous Material and FGM The results of SIFs for the mixed mode problem are shown in Fig. 8.5. They are the solution of the inclined crack. For the homogenous material, a very good agreement was found with theoretical solution for the three modes of SIF. In the case of the FGM the values of the three modes SIF were larger than the homogenous material. The zero and maximum values for mode II and III respectively deviated from θ = 90°.

8.4 Dissections and Conclusions Dissections and conclusions are listed as follows. • The results of the crack problems for the homogenous material agree very well with the corresponding theoretical solution. The maximum error was less than 1.3%. • We can clearly find that there is very small or no difference between the three domains. Therefore, the so-called path-independency property holds. • The FGM effect on the SIF is that its value increases along with the Young’s modulus. This happens because of the moment that is caused by the difference in deformation between the two sides of the crack, this non-equilibrium state in deformation will cause; first the applied stress to diverse from its direction, and second cause the mentioned before moment.

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Acknowledgements A part of present research performed by Hiroshi Okada was supported by JSPS (Japan Society for Promotion of Science) Grand-in Aid for scientific Research (c) No. 16K05988. The support is gratefully acknowledged.

References 1. Atluri, S.N., Sampath, S.G., Tong, P.: Structural Integrity of Aging Airplanes. Springer, Berlin, Heidelberg (1991) 2. Nakamura, T., Taniguchi, K., Hirano, S., Narita, M., Sato, T.: Stress corrosion cracking in welds of reactor vessel nozzle at OHI-3 and of other vessel’s nozzle at Japan’s PWR plants. ASME PVP: 2009-77344 (2009) 3. Qian, X., Swaddiwudhipong, S., Nguyen, C.T., Petchdemaneengam, Y., Marshall, P., Ou, Z.: Overload effect on the fatigue crack propagation in large-scale tubular joints. Fatigue Fract. Eng. Mater. Struct. (2013). https://doi.org/10.1111/ffe.12013 4. Okada, H., Kawai, H., Tokuda, T., Fukui, Y.: Development of automated crack propagation analysis system (2nd report, the crack propagation analysis system and finite element model generation for the crack propagation). Trans. Jpn. Soc. Mech. Eng. Ser. A 76(772), 1681–1688 (2010) 5. Ryutaro, D., Okada, H.: Mixed-mode stress intensity factor evaluation by interaction integral method for quadratic tetrahedral finite element with correction terms. Eng. Fract. Mech. 115, 22–42 (2014) 6. Okada, H., Ohata, S.: Three-dimensional J-integral evaluation for cracks with arbitrary curvatures and kinks based on domain integral method for quadratic tetrahedral finite element. Eng. Fract. Mech. 109, 58–77 (2013) 7. Murakami, Y., et al.: Stress Intensity Factors Handbook Volume 2, 1st edn. Committee on Fracture Mechanics. The Society of Materials Science, Japan (1987)

Chapter 9

A Smoothing Gradient-Enhanced Damage Model Tinh Quoc Bui

Abstract An implicit gradient-enhanced damage model with the characteristic length which is a stress level dependent parameter is developed. Both displacements and nonlocal equivalent strain fields are approximated using low-order elements. The spurious damage zones and stress oscillation induced by the conventional strain-based gradient damage can be eliminated. Numerical examples are considered to show the accuracy and performance of the developed method. Keywords Gradient damage · Fracture · FEM · Quasi-brittle

9.1 Introduction Conventional local damage approaches suffer several drawbacks including the meshdependent problems and spurious damage zones. To alleviate the problems, the nonlocal damage models have introduced by providing an internal length scale to the governing equations. The implicit gradient-enhanced models are proved to be effective for computational implementation, where the internal length scale is introduced through gradient terms of an equivalent strain field [1]. Compared to the conventional non-local models, approaches that incorporate with an evolving length scale can produce more physical damage zones at high levels of deformation [2]. The length scale here can be formulated as a function of local equivalent strain, non-local equivalent strain, stress, or damage parameters. A stress-based length scale model incorporated with the second order gradientenhanced damage model is presented in this paper. The implementation of linear basic functions for both the displacement and the coupled non-local equivalent strain fields results in an oscillation of the stress profile, as stated in [3]. Such oscillation of the stress field is self-compensated and therefore does not affect the calculation of the internal force, in which the numerical integration performs over the whole element. Nevertheless, in approach which exploits the stress at quadrature points to estimate T. Q. Bui (B) Department of Civil and Environmental Engineering, Tokyo Institute of Technology, Tokyo, Japan e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_9

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the stress-based evolving interaction zones, the ill stress profile is problematic. This paper proposes a technique to smooth out the oscillation of the stress profile based on non-local weighted averaging, in which the low-order functions is maintained whilst exploiting the stress field without stress oscillation. All these features of the developed model will be verified through numerical examples [4].

9.2 Constitutive Model The degradation of isotropic quasi-brittle materials is represented by introducing a scalar damage parameter D into the constitutive equation. The stress-strain relation is written as σ = (1 − D) C ε

(9.1)

where σ is the Cauchy stress tensor, C the elasticity matrix and ε the strain vector and D denotes the scalar damage variable, 0 ≤ D ≤ 1, where D = 0 indicates virgin material and D = 1 totally damaged material. The damage evolution law is defined by D = g(κ)

(9.2)

In non-local approaches, the damage evolution is driven by the non-local equivalent strain ε¯ eq instead of a local equivalent strain εeq , and a loading function f = ε¯ eq − κ

(9.3)

where κ is the maximum value of ε¯ eq in loading history. The damaged zone propagates ˙ ≥ 0, f D˙ = 0. when satisfying the Kuhn-Tucker conditions f (¯εeq , κ) ≤ 0, D(κ) The scalar function of εeq may take various forms, in case of quasi-brittle materials, the modified von Mises equivalent strain is given by Bui et al. [5] εeq

k−1 1 I1 + = 2k(1 − 2ν) 2k



(k − 1)2 2 12k I + J2 (1 − 2ν)2 1 (1 + ν)2

(9.4)

where ν is the Poisson’s ration, I1 is the first invariant and J2 is the second invariants of the deviatoric strain tensor, and k is defined as the ratio between compressive strength and tensile strength.

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9.3 Smoothing Gradient-Enhanced Damage Model In [1], the non-local equivalent strain is expressed through the implicit diffusion equation ε¯ eq − c∇ 2 ε¯ eq = εeq

(9.5)

where ∇ 2 is the Laplacian and the gradient coefficient c is defined with respect to the internal length scale l as c = 0.5 l 2 . A homogeneous Neumann boundary condition must be introduced to obtain a unique solution of ε¯ eq ∇ ε¯ eq . n = 0 on 

(9.6)

The equilibrium equation and boundary conditions are ∇σ + b = 0 in ; u = u¯ on u ; σ.n = t¯ on t

(9.7)

The use of a constant internal length scale leads to incorrect prediction of shear bands in compression test [3] and spurious damage at high levels of deformation. The stress-based nonlocal damage model introduced in [2] is to alleviate this problem. The shape and radius of the interaction domain is modified according to the stress state at each material points. An anisotropic gradient activity coefficient c is incorporated into (9.5) instead of a constant c in order to capture the anisotropic diffusion. The diffusion equation is reformulated in principal coordinate (x1 , x2 ) at each material point, which rotates an angle α from the global coordinate. Taking into account the anisotropy of the diffusion term gives [4, 5]: ε¯ eq (x1 , x2 ) − c1 σ2

∂ 2 ε¯ eq (x1 , x2 ) ∂ 2 ε¯ eq (x1 , x2 ) − c2 = εeq (x1 , x2 ) 2 ∂ x1 ∂ x22 σ2

(9.8)

where c1 = f 12 c and c2 = f 22 c. c1 , c2 are principal stresses and f t is the ultimate t t strength at the material point x.   c1 0 . Anisotropic gradient activity coefficient c is derived as c = 0 c2 Stress oscillation occurs in case of low-order element as shown in [3]. The oscillation magnifies as the mesh is refined. In the zones where damage parameter changes sharply, i.e., the transition zones between damaged elements and healthy elements, the stresses extracted at Gauss points are not adequate to use to calculate the stressbased internal length scale. Hence, the smoothing of stress is crucial in determination of stress principal directions. The stress at material point x is replaced by its non-local counterpart:

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σ(x) =

1

(x, ξ)

 ψ(x, ξ)σ(ξ)d

(9.9)



where ψ(x, ξ) is the weight function (e.g., Gaussian distribution).

9.4 Numerical Results A square concrete plate (H = 60 mm) under compression is conducted. The displacement control Newton-Raphson method is implemented for tracing the equilibrium curves of the structures. The plane strain condition is assumed. The boundary condition and the uniform load applied on the top edge of the plate via prescribed displacement as shown in Fig. 9.1. The damage is triggered by a small weakened zone at the bottom-right corner. Geometry dimensions and parameters are: the Young’s modulus E = 20 GPa; the Poisson’s ratio is ν = 0.2; and the length scale parameters lc = 2 mm and κi = 0.0001. The local equivalent strain is calculate based on the modified von-Mises rule (9.4) with k = 1, and the damage evolution law follows the exponential formula [4, 5] D =1−

 κi  1 − α + α e−β(κ−κi ) κ

(9.10)

For this particular compressive example, α = 0.92, β = 300 is taken. The plate discretized with 80 × 80 linear quadrilateral (Q4) elements is used for the analysis. The equivalent strain and damage evolution representing the shear band appear to be static. This result is similar to that of the non-local integral approach [2] and the static nature of shear bands. Figure 9.2 shows the evolution of damage bands in the plate for constant length scale and stress-based evolving length scale approach. The force-displacement curve of the structure is represented in Fig. 9.3 in comparison with those derived from the constant length scale approach and the non-local Fig. 9.1 Geometry and boundary conditions of the plate

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Fig. 9.2 Damage evolution for constant length scale model (a, c, e, g) and stress-based model (b, d, f, h), subjected to prescribed displacement d = 0.006, 0.01, 0.03 and 0.05 mm, respectively

Fig. 9.3 Force-displacement curve

integral model. The stress-based approaches generate steeper curves in the strain softening phase, compared with standard model. However, the peak value calculated by the developed model is in good agreement with references.

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References 1. Peerlings, R.H.J., de Borst, R., Brekelmans, W.A.M., de Vree, J.H.P.: Gradient-enhanced damage for quasi-brittle materials. Int. J. Numer. Methods Eng. 39, 937–953 (1996) 2. Giry, C., Dufour, F., Mazars, J.: Stress-based nonlocal damage model. Int. J. Solids Struct. 48(25–26), 3431–3443 (2011) 3. Simone, A., Asker, H., Peerlings, R.H.J., Sluys, L.J.: Interpolation requirements for implicit gradient-enhanced continuum damage models. Commun. Numer. Methods Eng. 19, 563–572 (2003) 4. Nguyen, H.A.T., Bui, Q.T., Hirose, S.: Smoothing gradient damage model with evolving anisotropic nonlocal interactions tailored to low-order finite elements. Comput. Meth. Appl. Mech. Eng. 328, 498–541 (2018) 5. Bui, Q.T., Nguyen, H.A.T., Doan, H.D., Hirose, S.: Numerical failure simulation of quasibrittle materials using a second-order implicit theory. In: Proceedings of the 4th International Conference on Engineering Mechanics and Automation (2016)

Part IV

Phase Change Material: Numerical and Experimental Results

Chapter 10

Thermal Behavior of Phase Change Material (PCM) Inside a Cavity: Numerical Approach Md. A. A. Shak, A. M. Bayomy, S. B. Dworkin, J. Wang and M. Z. Saghir

Abstract In this study, the melting process and thermal behaviour of a PCM in the presence of an embedded U-shaped heat source was numerically investigated using the COMSOL-3D Multiphysics software. Two different cases were analyzed in this study. In the first case, a rectangular cavity filled with a paraffin phase change material and an embedded U-shaped heating source was analyzed. Paraffin was used because of its low melting point temperature which is approximately 32 °C. It was observed that the melting rate was around the heating source due to the low conductivity of the paraffin. In the second case, Bentonite and paraffin were used in four different combinations. It was observed that the heat transfer rate was higher for Bentonite than for paraffin. This is because Bentonite has higher thermal conductivity and a low heat capacity. Keywords Paraffin · PCM · Bentonite · Thermal conductivity · Heat capacity · Heat storage

10.1 Introduction North America faces great challenges during the winter season, making it critical to find different ways to store energy. In the current study, some concepts involving storing thermal energy using PCM or Bentonite or a combination of PCM and Bentonite in a cavity were analyzed. PCM can absorb, store and release large amounts of latent heat [1]. Bashar and Siddiqui [1] conducted a study to investigate the melting process and thermal behavior of Phase Change Material (PCM) due to heat transfer convection by a U-shaped tube. The results revealed the importance of using paraffin Md. A. A. Shak · S. B. Dworkin · M. Z. Saghir (B) Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON, Canada e-mail: [email protected] A. M. Bayomy Canadian Nuclear Laboratories, 286 Plant Rd, Chalk River, ON, Canada J. Wang McClymont & Rak Engineers Inc, 1271 Denison St, Markham, ON, Canada © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_10

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as a means of storing energy. Touatia et al. [2] focused their study in extracting heat from a solar system and storing it in a phase change material. Plotze et al. [3] conducted a study describing the thermal properties, heat conductivity, heat capacity, and the thermal diffusivity of Bentonite. The aim of this paper is to investigate the effectiveness of using PCM as heat storage media.

10.2 Model Description A three-dimensional rectangular cavity filled with PCM (paraffin wax) and/or Bentonite was used in this investigation. The length of the cavity was 149.86 mm, the height was 115.22 mm, and the depth was 13 mm. A U-shaped copper tube was used inside then cavity with a thickness 0.2 mm, an outer diameter of 4.76 mm, and a vertical length of 113.22 mm. This U shape tube was used as a heat source. A clearance of 14.95 mm was maintained between the two legs of the tube. Hot water (39 °C) was flowing through the tube at a flow rate of 0.37 l/min for six hours. The inlet temperature of the heat transfer fluid (HTF) selected was 39 °C, because this temperature generally comes from the solar collector during the day time.

10.2.1 Governing Equations and Boundary Conditions The governing equation used in this simulation are; The energy equation: ρc p

∂T + ρc p u . ∇T + ∇ . q = Q ∂t

(10.1)

Navier-Stokes equation: ρ

∂u + ρ(u . ∇)u = ∇ . [−P I + μ(∇u + ∇u)T ] + F ∂t

(10.2)

The continuity equation: ρ∇ . (u) = 0

(10.3)

No slip boundary condition was used in the rectangular cavity. Open boundary condition was used in inlet, and outlet of U-shaped copper tube.

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Fig. 10.1 Normal element mesh

10.2.2 Mesh Sensitivity Analysis and Convergence Criteria In the present study, a tetrahedral element is used to perform the numerical model. To observe the grid dependency, the calculation of maximum and minimum temperature on the surface for heat transfer fluid flow through the U-shaped copper tube was performed for different numbers of domain, as shown in Fig. 10.1. The convergence criteria depend on the temperature, the velocity, the pressure, and the number of iterations. The convergence is deemed acceptable when the change in variable between two consecutive iteration was less than 1e–5.

10.3 Results and Discussions In the first case the cavity was filled with paraffin and flow is circulating inside the U shape pipe. The flow duration was for six hours. It is clear from Fig. 10.2a, b and c that by increasing the circulation time, the surface temperature inside the cavity increased significantly. This is happened because, heating source U-shaped copper tube transfer heat to the PCM continuously during the simulation time and PCM absorbed the heat. A horizontal temperature distribution in the cavity at different times is shown in Fig. 10.3. This time difference showed the heat absorbed by the PCM toward the two sides of the cavity. One can see that the temperature of the PCM inside the heating source was approximately 31 °C, and the temperature inside the U-shaped tube was approximately 39 °C for all three-time steps, respectively. However, outside the heating source, the temperature decreased gradually. Near the two side walls, the

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(a) t = 2 hours

(b) t = 4 hours

(c) t = 6 hours

Fig. 10.2 Temperature contours at different times 70 Inflow T inside the tube

Two hours Four hours Six hours

60

T (degC)

50 40

T outside the heating source

30 T inside the two legs of the heating source

20 Outflow T inside the tube

10 0

-0.1

-0.08

-0.06

-0.04

-0.02

0 x (m)

0.02

0.04

0.06

0.08

0.1

Fig. 10.3 Temperature distributions at y = 0 m of the cavity along x-axis

temperature was approximately 24 °C after two hours and increased to 26 °C after six hours. This is further proof of the low conductivity of the PCM. The other model consisted of embedding the heat source with Bentonite then paraffin outside the bentonite and vice versa. The last case involved Bentonite only. A horizontal temperature distribution of all the models is shown in Fig. 10.4. The model consisted of three hours of heat added to the cavity and three hours of heat removed from the cavity. Figure 10.4a, b show the surface temperature distributions of all four models under heating and cooling conditions inside the rectangular cavity, respectively. The results revealed that the heat transfer rate during the heating period was higher for Bentonite than paraffin. Figure 10.4b also shows that the heat transfer rate during cooling was higher for Bentonite than PCM. This is because Bentonite has a higher thermal conductivity than the PCM. The heat transfer rate (Q) of the cavity is related to the mass flow rate (m), ˙ heat capacity (cp ) and temperature difference (T). The heat transfer rate was calculated using the following equation: Q = mc ˙ p T

(10.4)

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40

40

35 30 T (deg C)

20 15

Model A three hours Model B three hours Model C three hours Model D three hours

10 5 0 -0.1

-0.05

0 x (m)

0.05

0.1

T(deg C)

30

25

20 10 0

-0.1

(a) t = 3 hours

-0.05

Model A six hours Model B six hours Model C six hours Model D six hours

0 0.05 x (m) (b) t = 6 hours

0.1

Fig. 10.4 Temperature distributions at y = 0 m of the cavity along x-axis

Table 10.1 Total heat transfer analysis

Model

Heat supplied QS (W/m2 )

Heat Extracted QA (W/m2 )

Efficiency QA ∗100% η=Q S

Model A Model B

10472.48 127845.9

7130.73 70916.4

68 55

Model C

6246.63

5159.19

83

Model D

4991.65

1665.72

33

The ratio of the energy supplied (QS ) to the energy absorbed (QA ) after six hours in all four models are presented in Table 10.1. In model A, the entire cavity is filled with paraffin, so it stored large amounts of heat and released large amounts of heat when the heat was extracted. In model C, the heat source was surrounded by paraffin and Bentonite was placed on the outside region. During heat absorption, all of the heat was released to the paraffin because the heat source and paraffin were too close to each other. That is why the efficiency of model C is higher than any other models. In models B and D, the efficiency is low because model B is filled with Bentonite only and in model D, the heat source was surrounded by bentonite and on the outside is the PCM.

10.4 Conclusion This numerical study was conducted to investigate a time-dependent heat transfer process in the presence of a phase change material and bentonite inside a rectangular cavity. The results revealed that the geometry of the heating source has a great impact on the melting behaviour of the PCM as well as the melting period. It is also clear from the results that the position of the heating source has a significant effect on

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the heat transfer rate in a thermally insulated cavity. It is clear from the results that bentonite is more sensitive to temperature variations in a cavity than PCM because of higher thermal conductivity. Acknowledgements Acknowledge the financial support of Natural Sciences & Engineering Research Council of Canada (NSERC), Ryerson University, and McClymont & Rak Engineers Inc. for funding this project.

References 1. Bashar, M., Siddiqui, K.: Investigation of heat transfer during melting of a PCM by a U-shaped heat source. Energy Res. 41, 2091–2107 (2017) 2. Touatia, B., Kerroumia, N., Virgoneb, J.: Solar thermal energy discharging from a multiple phase change materials storage tank. Appl. Solar Energy 53, 185–189 (2017) 3. Plotze, M., Scharli, U., Koch, A., Weber, H.: Thermophysical properties of Bentonite. In: International Meeting, Little, France Clays in Natural & Engineering Barriers for Radioactive Waste Confinement, vol. P/THME/19, pp. 579–580 (2007)

Part V

Nonlinear Dynamics and Control in Aerospace Engineering

Chapter 11

Dynamic Analysis of Stochastic Friction Systems Using the Generalized Cell Mapping Method Shichao Ma, Xin Ning and Liang Wang

Abstract Friction systems are a kind of typical non-smooth systems in the actual engineering and often generates complicated dynamics. It is difficult to handle this systems by conventional analysis methods directly. In this context, we investigate the stochastic responses of friction systems using the generalized cell mapping method under random excitation in this paper. To verify the accuracy and validate the applicability of the suggested approach, we present two classical nonlinear friction systems, i.e., Coulomb force model and Dahl force model as examples. Meanwhile, this method is in good agreement with Monte Carlo simulation method and the computation time is greatly reduced. In addition, further discussion finds that the adjustable parameters can induce the stochastic P-bifurcation in the two examples, respectively. Keywords Friction systems · Dynamic analysis · Stochastic responses · Cell mapping

11.1 Introduction Spacecrafts are complex mechanical systems which contain a wealth of nonlinear dynamic behavior with each component. Among them, the instability of systems that may result from the non-smooth features require designers to develop the most stringent solution to ensure the overall system security. As a typical non-smooth S. Ma (B) · X. Ning (B) Northwestern Polytechnical University School of Astronautics, Xi’an 710072, People’s Republic of China e-mail: [email protected] X. Ning e-mail: [email protected] National Key Laboratory of Aerospace Flight Dynamics, Xi’an 710072, People’s Republic of China L. Wang Northwestern Polytechnical University School of Science, Xi’an 710129, People’s Republic of China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_11

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factor, friction is a very complicated phenomenon arising at contact surfaces in the space manipulator [1]. And many researchers have paid attention to the problems of friction systems in the actual production. Some interesting results have been proposed [2]. And dynamic friction models, such as Dahl, Bliman-Sorine, LuGre, Coulomb as well as atomic scale and fractal models, have been developed to reasonably describe some special phenomena [3–5]. On the other hand, the uncertainly caused by random forces increases the complexity of studying such systems. And some researchers used the various analysis methods to obtain the stochastic responses of friction systems [6–11]. Among these methods, the generalized cell mapping (GCM) method has been demonstrated to be a very efficient tool due to its ability of global analysis of the strongly nonlinear systems, especially for the stochastic systems. Recently, researchers do many research based on the cell mapping method which was first proposed by Hsu [12]. Various improvements were applied to analyze the dynamical phenomena, such as stochastic response, crisis, first-passage problem, global analysis [13–19]. And we can see that cell mapping method is a fast and effective numerical method. However, existing methods are to approximate discontinuities to smooth functions for friction systems. In this paper, we research the nonlinear dynamics of non-smooth friction systems by using stochastic generalized cell mapping (SGCM) method. The purpose is to preserve the non-smooth properties of the friction systems. And this paper is arranged as follows. In Sect. 11.2, we introduce the friction system model and the SGCM method. In Sect. 11.3, we present the responses analysis of two different friction force models by this method. And Monte Carlo (MC) method is used to verify the effectiveness. Finally, conclusions are drawn in Sect. 11.4.

11.2 Friction System Model and the SGCM Method Consider a mechanical model of a single-degree-of-freedom Duffing friction oscillator (Fig. 11.1) subjected to Gaussian white noise excitation ˙ = ξ(t). x¨ + α x˙ + κ x + μx 3 + F(x, x)

(11.1)

Where x, x, ˙ x¨ are displacement, velocity and acceleration, respectively. The dot · represents the differentiation with respect to the time t. α is the damping coefficient, κ is the linear stiffness coefficient and μ is the nonlinear stiffness coefficient representing the intensity of stiffness nonlinearity, and F(x, x) ˙ is the discontinuous friction model. ξ(t) is the Gaussian white noise excitation which satisfies the following condition [20]: E[ξ(t)] = 0,

E[ξ(t)ξ(t + τ )] = 2σ δ(τ ).

(11.2)

And the 2σ represents the intensity of Gaussian white noise, δ(·) is the Dirac delta function.

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Fig. 11.1 Schematic of friction system with random excitation

For the friction system described above, we will introduce the stochastic generalized cell mapping method. Suppose that S is the interested space of the system and then divide  evenly into Nc intervals of uniform size s, each interval is named a cell. And these cells are labeled with integers N = {1, 2, . . . , Nc } (Fig. 11.2). Then, we establish a one-step transition probability matrix. Each cell generates s¯ random sample trajectories for these N cells. And if the cell i(1 ≤ i ≤ N ) has Si sample points falling from the cell j (1 ≤ j ≤ N ), then the probability  one step transition  from cell i to j is assigned to be pi j = Si /S with i Si = S and i pi j = 1. Here, P represents the one step transition probability matrix with the element pi j . Finally, according to the C-K equation [16] Fig. 11.2 Division of interested space 

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 p(x, m t) =

p(x, t|x0 , 0) p(x0 , (m − 1) t)dx0 ,

(11.3)

we can obtain the probability distribution vector p(m) after m cycles with the initial vector p(0). Among, p(m + 1) = P · p(m) or p(m) = P m · p(0).

(11.4)

11.3 Dynamic Responses Analysis of the Friction Systems In this section, we directly utilize the SGCM method to obtain the stochastic responses by the marginal and joint probability densities. Two classical nonlinear Duffing systems with different non-smooth friction force models are presented as examples. One is the Coulomb friction force, and the other one is the Dahl friction force. In addition, we also find the stochastic P-bifurcation with variable stiffness. And we compare the results obtained by SGCM methods with those from MC method and evaluate the effectiveness and the applicability of the proposed procedure for different friction force models.

11.3.1 Example 1: Coulomb Friction Force Now, we consider this Duffing system with Coulomb friction excited by Gaussian white noise in which

Or can write it as 

˙ = ξ(t). x¨ + α x˙ + κ x + μx 3 + f c sgn(x)

(11.5)

x˙ = y y˙ = −αy − κ x − μx 3 − f c sgn(y) + ξ(t),

(11.6)

which f c is the amplitude of friction and sgn(.) represents the signum function. The parameter values of the system are initially chosen as α = 1, κ = 0.01, μ = 0.01. And in this case, the intensity of Gaussian white noise 2σ is chosen as 0.01. In order to compute the one-step transition probability matrix, the interesting domain is chosen as  = {−3 ≤ x ≤ 3, −3 ≤ x˙ ≤ 3}. And the selected domain is divided into 50 × 50 cells, 2000 random sample trajectories are generated from each cell. Therefore, 5,000,000 random sample trajectories in total are used to construct the one-step transition probability matrix. MC method takes the same number of samples. Hence, using the method illustrated in Sect. 11.2, the steady-state probability

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density functions (PDFs) of displacement x and velocity y(y = x) ˙ are shown in Figs. 11.3 and 11.4 with the adjustable coefficient f c . In Fig. 11.3a, we can see that the marginal PDF of displacement x presents a bimodal state (red solid line) with f c = −0.02. As f c increases, the topological structure of marginal PDFs of x changes from bimodal state to unimodal state. And when f c = 0.01, the topological structure of marginal PDF is a unimodal state (brown solid line) obviously. The marginal PDFs of velocity x˙ have the similar trends with the friction coefficient f c increasing, which are shown in Fig. 11.3b. In Fig. 11.4, the joint PDFs of displacement x and velocity x˙ are displayed with the adjustable coefficient f c . Among them, the results of SGCM method is in the left side of Fig. 11.4a–d when f c = −0.02, −0.01, 0, 0.01, respectively. The corresponding results of MC method are in the right side. With the increasing of f c , the steady-state joint PDFs vary from a “crater” into one “peak”. By combining Figs. 11.3 and 11.4, we can find that the value of the friction coefficient f c will induce changes in the topological structure of the stochastic system, thus affecting the stability of the system. This phenomenon is called stochastic P-bifurcation [17]. In addition, it can be seen that the results of SGCM method is in good agreement with the results of MC method. And Table 11.1 is the time comparison of the SGCM method and MC method, which shows the high efficiency of SGCM obviously. Therefore, this method is an efficient approach to analyze the response of friction system with noise fluctuation.

11.3.2 Example 2: Dahl Friction Force The second example considers the following Duffing system with Dahl friction subjected to Gaussian white noise. The equations of motion of the system are written as x¨ + α x˙ + κ x + μx 3 + λ f D = ξ(t),

(11.7)

λ|x| ˙ x. f˙D = x˙ − fC

(11.8)

Where λ f D is the Dahl friction force, suppose that f D = z. Rewrite the system as ⎧ ⎪ ⎨ x˙ = y y˙ = −αy − κ x − μx 3 − λz + ξ(t) ⎪ ⎩ z˙ = y − λ|y| x. fc

(11.9)

For Eq. (11.9), the effects of different values of α are considered under the fixed parameters α = 0.02, μ = 1, f c = 0.05 and the Dahl friction force parameter λ = 0.06. The intensity of Gaussian white noise 2σ is chosen as 0.02. The interested domain of the SGCM method is chosen as  = {−2 ≤ x ≤ 2, −2 ≤ y ≤ 2} with a

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(a)

(b)

Fig. 11.3 The steady-state marginal PDFs of system (6) for x and y with f c = −0.02, −0.01, 0, 0.01. a The marginal PDFs of x; b the marginal PDFs of y. Solid lines: the SGCM method results; symbols: MC method results (color figure online)

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Fig. 11.4 The steady-state joint PDFs of system (6) for x and y with f c = −0.02, −0.01, 0, 0.01. Left sides: the SGCM method results; right sides: MC method results. a f c = −0.02; b f c = −0.01; c f c = 0; d f c = 0.01

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Table 11.1 The calculation time comparison of two methods for system (6) Methods

Values (fc ) −0.02 s

SGCM MC

−0.01 s

0s

0.01 s

249.7 s

252.4 s

248.2 s

260.9 s

54208.2 s

54036.7 s

54912.7 s

55610.5 s

cell structure of 50 × 50. And 2000 random sample trajectories are also generalized from each cell. Hence, out of cells, there are a total of 5,000,000 sample trajectories to construct the one-step transition probability matrix. For comparison, the MC method is used. In Figs. 11.5 and 11.6, the steady-state PDFs of displacement x and velocity y of system Eq. (11.9) are respectively plotted with the adjustable damping coefficient α. In Fig. 11.5, blue line, green line, pink line and orange line represent the steadystate responses with α = 0.5, 0, −0.5, −1, respectively. As shown in Fig. 11.5a, the topological structure of marginal PDF of displacement x is unimodal shape when α = 0.5. But as α decreases, the peaks of the marginal PDFs start to go down and it can be seen that the shape of the marginal PDFs of x changes from one peak to two peaks. However, there is no significant change of the topological structure velocity y with the α decreasing in Fig. 11.5b. The marginal PDFs of velocity always maintain the shape of single peak no matter how α changes. Figure 11.6 show the joint PDFs of displacement and velocity when α changes. Among them, the results of SGCM method is in the left side of Fig. 11.6a–d when α = 0.5, 0, −0.5, −1, respectively. The corresponding results of MC method are in the right side. The joint PDFs of x and y display the change from one peak to two peaks. These figures demonstrate the occurrence of the stochastic P-bifurcation when the damping coefficient α decreases from 0.5 to −1. Of course, the results of the SGCM method are well coincident with the MC simulations, which verify the effectiveness of the method. The calculate time comparison of the SGCM method and MC method are shown in Table 11.2, it can be seen that the SGCM method has advantage of time obviously.

11.4 Conclusions In this paper, we investigate the stochastic responses of the nonlinear friction systems with adjustable coefficient property under Gaussian white noise excitation. Using SGCM method, we can obtain the steady-state probability density functions of systems displacement and velocity with different friction force model. To verify the accuracy and validate the applicability of the suggested approach, we present two classical nonlinear friction systems with adjustable coefficient, i.e., Coulomb force model and Dahl force model as examples. By comparing with the Monte Carlo method, it is proved that the high efficiency of the SGCM method under two examples. In addition, both adjustable coefficients f c and α can induce stochastic

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(a)

(b)

Fig. 11.5 The steady-state marginal PDFs of system (9) for x and y with α = 0.5, 0, −0.5, −1. a The marginal PDFs of x; b The marginal PDFs of y. Solid lines: the SGCM method results; symbols: MC method results (color figure online)

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Fig. 11.6 The steady-state joint PDFs of system (9) for x and y with α = 0.5, 0, −0.5, −1. Left sides: the SGCM method results; right sides: MC method results. a α = 0.5; b α = 0; c α = −0.5; dα=1

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Table 11.2 The calculation time comparison of two methods for system (9) Methods

Values (α) 0.5 s

SGCM MC

0s

−0.5 s

−1 s

248.4 s

246.0 s

240.5 s

243.2 s

80670.1 s

78120.2 s

77960.4 s

77420.8 s

P-bifurcation in the two systems, respectively, due to affecting the stability of the systems. Without any approximation, the non-smooth of the friction systems still retain, we conclude that the proposed method is more advanced than the common analytic method. We are currently working in addressing for engineering interests, the dynamic analysis of stochastic response and bifurcation is of great significance to practical engineering, we will report results of more complicated problems in future publications. Acknowledgements This work was supported by the National Science Foundation of China through the Grants (11872306, 11772256), the Central University Fundamental Research Fund (3102018zy043).

References 1. Awrejcewicz, J., Olejnik, P.: Analysis of dynamic systems with various friction laws. ASME Appl. Mech. Rev. 58, 389–411 (2005) 2. Xu, W., Wang, L., Feng, J.Q., Qiao, Y., Han, P.: Some new advance on the research of stochastic non-smooth systems. Chin. Phys. B. 27, 110503-110501-110506 (2018) 3. Astrom, K.J., de Wit, C.C.: Revisiting the LuGre friction model, stick-slip motion and friction dependence. IEEE Control Syst. Mag. 28, 101–114 (2008) 4. Berger, E.J.: Friction modeling for dynamic system simulation. ASME Appl. Mech. Rev. 55, 535–577 (2002) 5. Olsson, H., Astrom, K.J., de Wit, C.C., Gafvert, M., Lischinsky, P.: Friction models and friction compensation. Eur. J. Control. 4, 176–195 (1998) 6. Baule, A., Touchette, H., Cohen, E.G.D.: Stick-slip motion of solids with dry friction subject to random vibrations and an external field. Nonlinearity 24, 351–372 (2010) 7. Kumar, P., Narayanan, S., Gupta, S.: Stochastic bifurcation analysis of a Duffing oscillator with Coulomb friction excited by Poisson white noise. Procedia Eng. 144, 998–1006 (2016) 8. Guerine, A., El Hami, A., Walha, L., Fakhfakh, T., Haddar, M.: Dynamic response of a Spur gear system with uncertain friction coefficient. Adv. Eng. Softw. 000, 1–10 (2016) 9. Fang, Y.N., Liang, X.H., Zuo, M.J.: Effects of friction and stochastic load on transient characteristics of a spur gear pair. Nonlinear Dyn. 93, 599–609 (2018) 10. Sun, J.J., Xu, W., Lin, Z.F.: Research on the reliability of friction system under combined additive and multiplicative random excitations. Commun. Nonlinear Sci. Numer. Simul. 54, 1–12 (2018) 11. Jin, X.L., Wang, Y., Huang, Z.L.: Approximately analytical technique for random response of LuGre friction system. Int. J. Non-Linear Mech. 104, 1–7 (2018) 12. Hsu, C.S.: Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear System. Springer, New York (1987)

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13. Hong, L., Xu, J.X.: Crises and chaotic transients studied by the generalized cell mapping digraph method. Phys. Lett. A 262, 361–375 (1999) 14. Tongue, B.H., Gu, K.Q.: A theory basis for interpolated cell mapping. SIAM J. Appl. Math. 48(5), 1206–1214 (1988) 15. Sun, J.Q.: Random vibration analysis of a non-linear system with dry friction damping by the short-time Gaussian cell mapping method. J. Sound Vib. 180, 785–795 (1995) 16. Yue, X.L., Xu, W., Jia, W.T., Wang, L.: Stochastic response of a ϕ6 oscillator subjected to combined harmonic and Poisson white noise excitations. Phys. A 392, 2988–2998 (2013) 17. Wang, L., Xue, L.L., Xu, W., Yue, X.L.: Stochastic P-bifurcation analysis of a fractional smooth and discontinuous oscillator via the generalized cell mapping method. Int. J. Non-linear Mech. 96, 56–63 (2017) 18. Wang, L., Ma, S.C., Sun, C.Y., Jia, W.T., Xu, W.: Stochastic response of a class of impact systems calculated by a new strategy based on generalized cell mapping method. ASME J. Appl. Mech. 85, 054502 (2018) 19. Yue, X.L., Xu, Y., Xu, W., Sun, J.Q.: Probabilistic response of dynamicsl systems based on the global attractor with the compatible cell mapping method. Phys. A 516, 509–519 (2019) 20. Zhu, W.Q., Cai, G.Q.: Introduction to Stochastic dynamics. Science and Technology Press (2017)

Chapter 12

Nonlinear Flight Dynamics of Very Flexible Aircraft Chen Zhanjun, Fu Zhichao, Lv Jinan and Liu Ziqiang

Keywords Very flexible aircraft · Nonlinear · Couple · Flight dynamics · Dynamic stall · Stability In order to get more solar energy and better flight performance, HALE aircraft are normally built with relatively long and slender high-aspect ratio wings, so the wings of HALE aircraft are very flexible, with large wing deformations during flight. The natural frequencies are low enough so it tends to exhibit an overlap with the rigidbody flight dynamic frequencies [1] and the coupling is harmful. The geometrically nonlinearities following the high deformation and overlap between structure and flight dynamic mode lead to difficulties in modeling, loads analysis, simulation and control of HALE configuration aircraft. One of the recommendations from the Helios mishap report was to “develop more advanced, multidisciplinary (structures, aeroelastic, aerodynamics, atmospheric, materials, propulsion, controls, etc.) time-domain analysis methods appropriate to highly-flexible, morphing vehicles.” Researchers have built plenty of useful tools on the subject with frequency and time-domain analysis by coupling nonlinear aerodynamic, structure and flight dynamics since 2003, including UM/NAST by Cesnik and Brown et al. [2, 3], NATASHA by Patil and Hodges et al. [4, 5], SHARP by Palacios and Murua et al. [6, 7] and so on.

C. Zhanjun (B) · F. Zhichao · L. Jinan · L. Ziqiang China Academy of Aerospace Aerodynamics, Beijing 100074, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_12

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Fig. 12.1 Geometry values of vehicle models

This paper develops couple tool constituted by geometrically-exact nonlinear mixed-formulation couple with unsteady aerodynamic model. A conventional layout HALE configuration aircraft is trimmed first. The effect of couple between structure and flight dynamics is explored preliminarily by stability analysis. And then the modulated excitation signals are adopted to evaluating the flight quality via timedomain simulations. It is very important to get static/unsteady airloads accompanied with the deformation of wings with little calculated amount. Strip theory and vortex lattice method are used to get unsteady airloads in this paper. The former combined with the dynamic stall model can simulate the aerodynamic force after the wing stall, but cannot give the three-dimensional aerodynamic effect of the wing. The latter is opposite. In this paper, two methods coupled geometric nonlinear beam model are established to establish the time domain analysis method of flexible wing respectively, to study the effect of dynamic stall and three-dimensional effect on the dynamic response of flexible wing. The geometry of the aircraft model is shown in Fig. 12.1, while the parameters of wings can be found in Table 12.1. The main wing is flexible while the other parts are modelled as rigid parts. The aircraft are trimmed with three inputs, angle of attack, thrust, and elevator deflection for different flexibility of the main wing in Fig. 12.2. The solution of ONERA and UVLM are illustrated compared to Murua solutions [6]. Because the out-of-plane deformation is too large, the normal component of lift is reduced. As the flexibility increases, the trim angle increases sharply. The ONERA aerodynamic model cannot simulate the three-dimensional aerodynamic effect. The aerodynamic force near the tip of the wing is larger, which result in greater out-of-plane deformation, so the larger angle of attack is required. Figure 12.3 shows the root loci of the eigenvalue analysis with ONERA model. The short period frequency is relatively large and it is easy to overlap with the structural modal frequency. The long-period mode in the right picture is close to the imaginary axis, indicating that the long-term stability margin of the aircraft is small. The eigenvalue of the spiral mode is always a real root. As the flexibility increases, the eigenvalue moves from the left side of the imaginary axis to the right side, not far from the imaginary axis, which means that the this modal diverge slowly. The Dutch rolling mode has the same trend as the long-period mode, and the frequency is lower. The difference is that as the flexibility increases, the oscillation damping decreases slightly. The structural flexibility has an obvious influence on the rolling

12 Nonlinear Flight Dynamics of Very Flexible Aircraft Table 12.1 The parameters of wings

Item

121 Value

Main wing (flexible) chord/m

1.0

Semi-span/m

16.0

Ref. axis location (from L.E.)

50% chord

Center of gravity (from L.E.)

50% chord

Mass per unit span (kg/m)

0.75

Rotational inertia (Ixx)

0.1

Torsional rigidity (GJ)

1.0e4

Flat bend inertia (Iyy)

2.0e4

In-plane bend inertia (Izz)

4.0e6

Fuselage (rigid) Length/m

10

Mass per unit span (kg/m)

0.08

Rotational inertia (Ixx)

0.01

Elevator (rigid) chord/m

0.5

span/m

5.0

Center of gravity (from L.E.)

50% chord

Mass per unit span (kg/m)

0.08

Rotational inertia (Ixx)

0.01

Vertical tail (rigid) chord/m

0.5

span/m

2.5

Center of gravity (from L.E.)

50% chord

Mass per unit span (kg/m)

0.08

Rotational inertia (Ixx)

0.01

mode. As the flexibility increases, the damping increase rapidly, which causes the root degenerate into the negative real root. For the very flexible aircraft, it is necessary to carry out nonlinear simulations to evaluate the stability and verify the theoretical analysis. The simulations of the very flexible model, s1.0 for short, medium flexibility model, s2.0 for short, and rigid model accompany by frozen models s1.0_fz, s2.0_fz are executed disturbed by ramp signals initially. Figure 12.4 shows that the pitch rate response of vehicles disturbed by ramp signals added to the elevator. All of the models are stability after few periods of oscillation. But the amplitude of flexible model is largest.

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(a) incidence

(b) elevator angle

(d) thrust

(c) displacement in the tip

Fig. 12.2 The trim characteristics at V = 25 m/s following the wing flexibility

Fig. 12.3 Root loci of the eigenvalue analysis at V = 25 m/s. Only the imaginary part and the pure real root information are displayed, the right picture is enlarged in the dotted line on the left picture, and the arrow starts from the rigid mode

Figure 12.5 shows that the rolling angular rate response of vehicles disturbed by ramp signals added to the ailerons differentially. According to the result of eigenvalue analysis, the damping of roll modal is so large that the aircraft roll oscillation cannot be excited in simulations. Compared the different models, it can be found that the model with larger amplitude of out-of-plane static deformation deviate largely from the equilibrium position after the disturbance disappears.

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Fig. 12.4 Pitch rate response of vehicles disturbed by ramp signals added to the elevator. The aerodynamic model of left picture is ONERA, while UVLM for the right

Fig. 12.5 Rolling angular rate response of vehicles disturbed by ramp signals added to the ailerons differentially. The aerodynamic model of left picture is ONERA, while UVLM for the right

References 1. Su, W., Cesnik, C.E.S.: Dynamic response of highly flexible flying wings. AIAA J. 49(2), 324–339 (2011) 2. Cesnik, C.E.S., Brown E.L.: Modeling of high aspect ratio active flexible wings for roll control. In: 43rd AIAA Structures, Structural Dynamics, and Materials Conference, AIAA 2002-1719 3. Su, W.: Coupled nonlinear aeroelasticity and flight dynamics of fully flexible aircraft. Ph.D. thesis, The University of Michigan, UK (2008) 4. Patil, M.J., Hodges, D.H.: Flight dynamics of highly flexible flying wings. J. Aircr. 6(43), 1790–1798 (2006) 5. Patil, M.J.: Nonlinear gust response of highly flexible aircraft. In: 48th AIAA / ASME / ASCE / AHS / ASC Structures, Structural Dynamics, and Materials Conference, AIAA-2007-2103 6. Murua, J.: Flexible aircraft dynamics with a geometrically-nonlinear description of the unsteady aerodynamics. Ph.D. thesis, Department of Aeronautics, Imperial College London, UK (2012) 7. Murua, J., Palacios, R., Graham, R.: Application of the unsteady vortex-lattice method in aircraft aeroelasticity and flight dynamics. Prog. Aerosp. Sci. 55, 46–72 (2012)

Part VI

Computational and Experimental Methods in Geotechnical and Multidisciplinary Engineering Problems

Chapter 13

Hydro-mechanical Properties of Unsaturated Decomposed Granite in Triaxial Compression Test Under Drained-Vented/Undrained-Unvented Condition X. Xiong, S. Okino, R. Mikami, T. Tsunemoto, X. Y. Qiu, Y. Kurimoto and F. Zhang Abstract Generally speaking, most of the geomaterials in surface ground are in unsaturated state. The mechanical and hydraulic properties of unsaturated soil are much more complicated than those of saturated soil. To investigate these properties, it is important to conduct systematic experiments under various conditions. In this paper, triaxial compression tests on decomposed granite under drained/undrained condition were conducted. Two kinds of special loading conditions, that is, constant suction and constant degree of saturation were adopted during shear stage. Based on the test results, it is found that the drainage of Masado specimens influence its hydro-mechanical behaviors. It is also understood that the variation of suction and degree of saturation under drained condition cannot be described by existing water retention curve models, in which the influence of deformation was not considered. The test results are also helpful for the understanding of the moisture characteristics of the decomposed granite. Keywords Unsaturated soil · Triaxial test · Drained-undrained condition · Degree of saturation

X. Xiong · S. Okino · R. Mikami · F. Zhang (B) Nagoya Institute of Technology, Gokiso, Nagoya 466-8555, Japan e-mail: [email protected] T. Tsunemoto Kyoto, Kyoto Prefecture 602-8570, Japan X. Y. Qiu SGIDI Engineering Consulting Co., Ltd., 388, Yixian Road, Shanghai, China Y. Kurimoto Nagoya Institute of Technology, Shimizu Corporation, 3-4-17, Etchujima, Tokyo 135-8530, Japan © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_13

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13.1 Introduction Geomaterials in many cases are in unsaturated state. Comparing with saturated soil, the mechanical and hydraulic properties of unsaturated soil are complex. Therefore, it is undoubtedly important to study the mechanical and hydraulic properties of saturated-unsaturated soil. In early unsaturated soil constitutive models, van Genuchten model [4], VG model) is often used, which cannot consider the influence of void ratio/soil deformation on water retention curve (WRC). As shown in many isotropic loading tests [2, 3], the degree of saturation will keep on increasing even when the suction is remained constant. In other words, WRC is dependent on the deformation of unsaturated soil. Therefore, it is important to study how stress-strain relation of the unsaturated soil couples with its water retention characteristics under various conditions. Most of the researchers conducted undrained unsaturated soil triaxial tests under vented condition [1], and only few focused on undrained-unvented condition. In this paper, a kind of decomposed granite, called as Masado with high permeability, was tested with triaxial compression tests under drained-vented and undrained-unvented conditions, in order to investigate its hydro-mechanical properties at different loading paths.

13.2 Sample Material and Test Apparatus 13.2.1 Test Sample Material Masado is a typical decomposed granitic that is widely distributed in western Japan. In construction, it is often used as pavement materials of roads. Furthermore, compared to silty clay commonly used in unsaturated tests, the permeability of Masado is much larger, which could significantly shorten the testing time required for unsaturated tests. Therefore, Masado was selected as the test material for the unsaturated tests. Figure 13.1 is the grain grading curve of Masado. Some pre-tests were conducted in according to the JIS A 1205 and JIS A 1202 and the physical material properties of Masado were obtained as shown in Table 13.1. Moreover, as shown in Fig. 13.2, 100

Weight percent P(d) [%]

Fig. 13.1 Grain grading curve of Masado

80 60 40 20 0 0.001

0.01

0.1

Perticle size d [mm]

1

10

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Table 13.1 Material properties of Masado Liquid limit wL (%)

Plasticity index Ip

Specific gravity Gs

Maximum dry density ρd (g/cm3 )

Optimum water content wopt (%)

Non-plastic

Non-plastic

2.66

1.85

13.7

2.00

Dry density, ρd [g/cm 3]

Fig. 13.2 Compaction curve of Masado

Ze

1.90

ro

1.80

air

vo

id

cu

1.70

rv

e

1.60 1.50 5.0

10.0

15.0

20.0

Water content, w [%]

25.0

compaction test (JIS A 1210) also conducted to find the optimum moisture content for sample preparation. In this test, only part of Masado with particle diameter less than 2 mm was used.

13.2.2 Triaxial Apparatuses The unsaturated triaxial test apparatus shown in Fig. 13.3 was used in the compression tests, which utilized axis-translation method to control suction. The main feature of triaxial apparatus is that both the porous stone (pore air pressure) and the ceramic disk (pore water pressure) are embedded in the axial cap, and the test time is shortened under double-end-face drainage condition. To obtain the volume change of unsaturated samples, differential manometer is utilized to measure the pressure difference between the burette with the standard water surface and the inner Fig. 13.3 Outline of unsaturated triaxial test apparatus

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chamber. Unsaturated triaxial apparatus can control four kinds of pressure including axial pressure, confining pressure, pore air pressure and pore water pressure by air pressure.

13.3 Triaxial Compression Test 13.3.1 Test Method The samples for triaxial apparatus, with the same initial moisture content w0 = 15%, was 10.0 cm in height and 5.0 cm in diameter. The specimens were prepared by static compaction method for three layers and the target void ratio was 0.65. In all cases, the specimens were consolidated under drained-vented condition, and then were sheared under drained-vented or undrained-unvented conditions. During shear stage, suction and degree of saturation were controlled to be constant respectively under drained-vented condition, while the water content was constant under undrained-unvented condition. Figures 13.4 shows the stress paths of the unsaturated specimens reached the initial stress state of triaxial compression tests under constant-suction (CS), constant-degree-of-saturation (CDS) and constantwater-content (CWC) conditions respectively. The shear rate was 0.0025% min and the maximum deviatoric strain was 15%. The confining stress σ net 3 was kept constant during the shear stage.

13.3.2 Test Results Tests results of unsaturated Masado are shown in following figures. As shown in Fig. 13.5, the degree of saturation decreased gradually during the shear stage under CS condition, and it is known from Fig. 13.7 that the suction changed during the shearing A B, C, D

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stage under CDS condition, which means that finite deformation has influence on WRC. These influences cannot be described by existing model [5]. Therefore, proper modification should be done on the existing model in order to consider properly the influence of finite deformation on both the variations of the saturation and the suction (Figs. 13.6, 13.7, 13.8 and 13.9). In all loading paths, the peak strength and dilatancy of specimens under the same net confining stress increased with the increase of initial suction at the beginning of shear stage. This increment was lager under drained condition than that under 400

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undrained condition. In addition, comparing Fig. 13.7 with Fig. 13.10, the dilatancy of specimens with the same confining stress was smaller under CWC condition. As shown in Figs. 13.6 and 13.8, with the increase of the confining pressure, the peak strength increased, and the volumetric strain changed from dilation to contraction. Comparing the stress-strain relation of the unsaturated Masado with its water retention characteristics, it is found that the changes of degree of saturation under CS condition and the variation of suction have the same tendency with the dilatancy. However, under CWC condition, the changes of degree of saturation are totally the 400

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same with the dilatancy, while the variation of suction has reverse tendency with the dilatancy. Therefore, it could be concluded that the drainage of Masado specimens influence its hydro-mechanical behaviors. To model hydro-mechanical properties of unsaturated soil under undrained-unvented condition, the compressibility of air should be taken into consideration.

13.4 Conclusion In this paper, to investigate hydro-mechanical properties of unsaturated Masado at different loading paths under drained/undrained condition, a series of triaxial compression tests under CS, CDS and CWC conditions were conducted. The main conclusions can be made as: (1) The changes of degree of saturation and suction under CS and CDS conditions imply that finite deformation has influence on WRC. These influences cannot be described by existing model [5]; (2) In all loading paths, the peak strength and dilatancy of specimens under the same net confining stress increased with the increase of initial suction at the beginning of shear stage; (3) The drainage of Masado specimens influences its hydro-mechanical behaviors. To model hydromechanical properties of unsaturated soil under undrained-unvented condition, the compressibility of air should be taken into consideration.

References 1. Rahardjo, H., Heng, O.B., Choon, L.E.: Shear strength of a compacted residual soil from consolidated drained and constant water content triaxial tests. Can. Geotech. J. 41(3), 421–436 (2004) 2. Sharma, R.S.: Mechanical behaviour of unsaturated highly expansive clays. Doctoral dissertation, University of Oxford, England (1998) 3. Sun, D.A., Sheng, D., Sloan, S.W.: Elasto-plastic modelling of hydraulic and stress–strain behaviour of unsaturated soils. Mech. Mater. 39(3), 212–221 (2007) 4. Van Genuchten, M.T.: A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44(5), 892–898 (1980) 5. Zhang, F., Ikariya, T.: A new model for unsaturated soil using skeleton and degree of saturation as state variables. Soils Found. 51(1), 67–81 (2011)

Chapter 14

Effective Foundation Input Motion for Soil-Steel Pipe Sheet Pile (SPSP) Foundation System Md. Shajib Ullah, Keisuke Kajiwara, Chandra Shekhar Goit and Masato Saitoh Abstract An experimental investigation to evaluate the effective foundation input motion (EFIM) of a soil-steel pipe sheet pile (SPSP) foundation system under 1g conditions was carried out through scaled model testing on a shaking table. The scaled model with 20 piles interlocked together to form a circular assembly was embedded in dry cohesionless Gifu sand housed in a laminated shear box. Tips of all the piles were rigidly bolted at the bottom of the shear box while the heads were rigidly connected to a footing. Amplitude and frequency dependent EFIM at the footing level of SPSP foundation model is obtained under dynamic ground excitations. Carried out experiments encompass a range of low-to-high amplitude of lateral harmonic ground excitation, covering elastic-to-inelastic behaviour of soil. Results show that the amplitude of EFIM at the SPSP footing level and the resonant frequency of the soil-SPSP foundation system decreases with the increase in the amplitude of excitation due to the fact that the increase in the loading amplitude results in the increase in the soil strain and thus decreases the soil stiffness. Results are also obtained in the form of kinematic interaction factors (KIF). The KIF is found approximately equal to unity up to the resonant frequency of the soil-SPSP foundation system, however, decreases above the resonant frequency reflecting the filtering effect of soil-SPSP kinematic interaction. For higher amplitude of excitations, an increase in the KIF (more than unity) is observed for a considerable number of frequencies, particularly around the lower frequency region. This can be attributed to the nonlinearity induced in the soil due to the higher amplitude of excitation.

Md. S. Ullah (B) · K. Kajiwara · C. S. Goit · M. Saitoh Graduate School of Science and Engineering, 255 Shimo-Okubo, Saitama, Japan e-mail: [email protected] K. Kajiwara e-mail: [email protected] C. S. Goit e-mail: [email protected] M. Saitoh e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_14

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Keywords Steel pipe sheet pile (SPSP) foundation · Kinematic response · Scaled model · Shaking table · Frequency domain

14.1 Introduction Steel pipe sheet pile (SPSP) foundation is a useful way to support large-scale structures such as bridge structures spanning over a river or a bay where the surface ground is very weak. Steel pipes with couplings welded on the sides of the steel pipes are sequentially driven into firm bearing layer to form an enclosure of different shapes (e.g. circular, oval, rectangular, etc.) wherein the couplings are interlocked to join the driven steel pipes with each other [1]. The interlocking parts are thoroughly washed by jets of water and filled with concrete mortar grouting. Finally, the interlocked steel pipes are integrated by connecting the top of each pipes to a rigid footing to form the SPSP foundation system [1]. This type of foundation is becoming a popular choice for large scale bridge projects (e.g. Nhat tan bridge in Vietnam, Tokyo Gate bridge in Japan, 2nd Meghna bridge in Bangladesh, etc.) in various parts of the world as it offers strong rigidity and large vertical bearing capacity. Furthermore, the use of SPSP foundation can reduce construction cost and time since no temporary cofferdam is needed because of its applicability as a cofferdam. The static mechanical behaviour of this kind of foundation based on model experiments are available in the literature [2, 3]. Furthermore, vertical shear behaviour of SPSP joint and other fundamental design and construction details are also available [4–9] based on field data, experimental investigations, and FEM simulation. The dynamic response analysis of a soil-foundation-structure system is most commonly carried out through well-known sub-structuring technique [10, 11]. It is a common practice to assume that the foundation motion is equal to the free field motion for convenience [12]. However, the soil-foundation kinematic interaction may substantially modify the foundation input motion from the free field motion due to the difference in stiffness between the soil and the foundation and the inertia of the foundation [13]. Accordingly, SPSP foundation also interacts with soil under dynamic excitation as seen for other types of foundation structures (pile foundation, caisson etc.) as studies [5] suggest that the SPSP foundation exhibits an intermediate behaviour between the caisson and ordinary pile group foundations. Therefore, it is desirable to consider the soil-foundation interaction effects in the dynamic response analysis of foundation and structure supported on it to avoid inaccurate estimation of the response [14]. A few studies focusing on the dynamic response of soil-SPSP foundation system adopting FEM simulation technique are available [15–17]. Experimental investigation on the dynamic response of soil-SPSP foundation is also available in the literature based on model testing [18, 19], however, the law of similarity between the model and the prototype has not been considered in these model experiment and the experiments were conducted for a very small amplitude of ground excitation. Considering such aforesaid facts, it is very important to investigate the dynamic response at the footing level of the SPSP foundation in response to ground excitation to understand the

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soil-SPSP foundation kinematic interaction behaviour and for proper assessment of the dynamic response of structure founded on SPSP foundation. Furthermore, the nonlinear behaviour of soil under strong ground motion and the nonlinear vertical shear behaviour of SPSP joint add sources of additional complexity in understanding the dynamic response of soil-SPSP foundation and such behaviour is still largely unknown. In response to such, the current study investigates the effective foundation input motion (EFIM) induced by soil-SPSP foundation interaction for low-to-high amplitude of ground excitation through scaled model testing.

14.2 Scaled Model Experiment A physical experimental model of soil-SPSP foundation system was prepared and experiment was conducted on a shaking table under 1g conditions. A detailed description of the scaling relationships between the model and the prototype, experimental setup (i.e. soil-SPSP foundation model, shake table, shear box), loading cases, data recording, and data processing that are employed in this scaled model vibration test is presented in the subsequent sections.

14.2.1 Scaling Relationships The law of similitude derived by Kokusho and Iwatate [20] for the dynamic model testing of soil under 1g conditions is adopted in the current experimental framework. This scaling law considers the ratio of forces acting on model and prototype, providing a loading frequency relationship between the model and the prototype as ωm = η−1/4 λ−3/4 ωp

(14.1)

where, ωm is the cyclic loading frequency on the model and ωp is the cyclic loading frequency on the prototype. Subscripts m and p refer to the model and the prototype, respectively, for all the equations in this section. In Eq. (14.1), η is the density scaling ratio of the model to the prototype and λ is the geometric scaling ratio of the model to the prototype. η=

lm ρm and λ = ρp lp

In addition, the ratio of the dynamic strain between the model and the prototype is given by

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γm = η1/2 λ1/2 γp

(14.2)

where, γ m is the dynamic strain in the model and γ p is the dynamic strain in the prototype. Based on these relations, other scaling parameters (e.g. force, stress, young’s modulus, shear wave velocity, frequency, etc.) between the model and the prototype was derived. For the present experimental program, λ was adopted as 1/16.5, i.e., the prototype is 16.5 times larger than the model, and η was adopted as 0.35.

14.2.2 Experimental Setup A one degree of freedom shaking table owned by Saitama University was used for the experiment. The size of the top plate of the shaking table is 1800 mm × 1800 mm with a maximum stroke of ±200 mm. A specially designed laminar shear box of inner dimension of 1200 mm × 800 mm × 1035 mm was used to house the soil-SPSP foundation model on the shaking table as shown in Fig. 14.2. The laminar shear box consists of a set of rectangular metallic frames stacked on the top of each other with ball bearings in between each frame to minimize the shear resistance of the housing, i.e., enabling the shear box to move freely in horizontal shear. Dry cohesionless Gifu sand was used to model soil. The standard properties of Gifu sand are available in the literature [21]. 20 hollow circular standard aluminium pipes of outer diameter 30 mm and inner diameter 26 mm were arranged in a manner to form a circular assembly of diameter 316.5 mm for the SPSP model. Length of each pipe was 960 mm. These pipes were interlocked to each other by a type of joint as shown in Fig. 14.1, where the centre of rotation of the joint was located at the middle of jointed part. A similar type of joint model with the centre of rotation at one end of the jointed part has been utilized by Kimura et al. [2]. This joint provides vertical shear resistance through friction. The lateral rigidity of the joint, on the other hand, is derived from the hoop stress generated due to the pressure exerted from the soil located on both inside and outside of the circular assembly (see Fig. 14.2). The length of the jointed part is 950 mm and provides a clear space of 5 mm both at the top and the bottom to allow the pipe to move vertically. All the aluminium pipes were connected rigidly at the top by means of a rigid footing made of a circular plate with diameter 350 mm and thickness 8.5 mm. To maintain fixed tip condition during the experiment, the bottom part of all the pipes were rigidly connected by means of another circular plate of same dimension as the top plate. This circular plate was further rigidly connected to the bottom part of the shear box by a rectangular steel plate of thickness 5 mm. Polyurethane foam was sprayed on the open joint parts along the whole joint length to seal against the possible movement of sand inside the joint.

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Fig. 14.1 Schematic layout of the soil-SPSP foundation model for EFIM estimation (all dimensions in mm)

14.2.3 Loading, Data Recording, and Processing The experiments were carried out under kinematic loading condition (i.e., ground excitation) to estimate the EFIM of the soil-SPSP foundation system. Lateral harmonic accelerations of amplitudes of 0.5, 1, 2, 3, 4, and 5 m/s2 were applied at the base of the laminar shear box in the frequency range of 6–35 Hz to induce low-to-high levels of strain in the soil for encasing elastic-to-inelastic behaviour of soil. The wide frequency range was adopted to encompass typical structural and soil natural periods. Translational response in terms of acceleration in the direction of loading was measured by the accelerometer placed at central position of the circular footing (accelerometer #2 as in Fig. 14.1). Rocking response was measured by the accelerometers placed at the outer peripheral points on the circular footing (accelerometers #1 and #3 in Fig. 14.1). Response at soil surface was measured by an assembly of three accelerometers (accelerometers #5, #6, and #7) placed on soil 75 mm apart from each other as shown in Fig. 14.1. All the accelerometers used in the experiment had the measuring capacity of ±50 m/s2 with maximum frequency range up to 130 Hz. As for the data recording, data were recorded in time domain. Fast Fourier Transform (FFT) technique was employed to convert the recorded time domain data into frequency domain. Furthermore, to eliminate possible effects of

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Fig. 14.2 Experimental photo

noise from the recorded data, measured data were passed through a band pass filter. A band pass range of 0.8–1.2 times the frequency was used. One aluminium pipe (as indicated in Fig. 14.1), was instrumented with strain gauge at different depths (0, 75, 150, 225, 300, 450, 600 mm) to measure the induced strain in the loading direction.

14.3 Experimental Result and Discussion The EFIM of the soil-SPSP foundation system (Au ) is evaluated as the amplification ratio of motion at the footing with respect to the input motion at the base of the laminar shear box, i.e. Au = u¨ eff /¨ug , where u¨ eff and u¨ g are the recorded acceleration at the footing and input harmonic excitation at the base, respectively. The corresponding phase difference between the aforesaid motions is Φu . The amplification of motion at soil surface is also evaluated as (As = u¨ sur /¨ug ), where u¨ sur is the recorded acceleration at the soil surface. The corresponding phase difference soil surface motion and input motion at base of laminar shear box is Φs . These defined

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response quantities are presented in Fig. 14.3 through Fig. 14.8 for all the amplitude and frequency of lateral harmonic accelerations mentioned in Sect. 14.2.3. It is clearly seen from the presented figures, that the resonant frequency and the maximum amplification ratio of the soil-SPSP foundation system decreases with the increase in input loading amplitude. For input loading amplitude of 0.5 m/s2 , the resonant frequency is 17 Hz and corresponding amplification ratio is 6.1 while the resonant frequency and amplification ratio decreases to 8 Hz and 2.2, respectively for loading amplitude of 5 m/s2 . This decrease in amplification ratio corresponds to a respective increase in damping of the soil-foundation system. This decreasing trend of resonant frequency and maximum amplification ratio with the increasing loading amplitude is presented in Fig. 14.9. This behaviour is understandably attributed to the decreasing stiffness of the soil-foundation system accompanied by the increase in strain level in soil due to increasing loading amplitude. The mean shear strain in soil (μs ) at resonance is found as 2.48 × 10−4 for input loading of amplitude 0.5 m/s2 , while −3 2 it increases  × 10 for 5 m/s ; the mean shear strain in soil is obtained as:  to 1.73 μs = [ usur − ug /H ], where H is the height of the soil layer used in the current experiment i.e. 1035 mm. From the amplification ratio (Au , As ) results, it has been observed that the SPSP footing follows the soil surface motion at lower frequency until the resonant frequency. And above the resonant frequency, the SPSP footing motion is smaller than soil surface motion suggesting that the soil-SPSP foundation kinematic interaction effect filter out the high frequency components of the soil surface motion. However, for higher amplitude of input loading (see Figs. 14.6, 14.7 and 14.8), SPSP footing response is higher than soil surface response around lower frequency region indicating amplification of low frequency components. And a clear difference in phase between Φu and Φs of these two responses has been observed

Fig. 14.3 Amplification ratio with corresponding phase at footing (Au , Φu ) and at soil surface (As , Φs ) in response to lateral harmonic excitation of amplitude 0.5 m/s2

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for such higher amplitude of input loading, whereas the phase results shows same pattern for lower amplitude of input loading (see Figs. 14.3, 14.4, and 14.5). The modification in SPSP footing motion from the soil surface motion is reflected in the horizontal kinematic interaction factor (KIF), evaluated as the ratio of the footing motion to the soil surface motion (Iu = u¨ eff /¨usur ). The KIF results for all loading amplitude and frequency range considered in this study are presented in Fig. 14.10. The results show that for lower amplitude of input excitation (0.5–2 m/s2 ), the horizontal KIF is approximately equal to unity around the lower frequency region (up

Fig. 14.4 Amplification ratio with corresponding phase at footing (Au , Φu ) and at soil surface (As , Φs ) in response to lateral harmonic excitation of amplitude 1.0 m/s2

Fig. 14.5 Amplification ratio with corresponding phase at footing (Au , Φu ) and at soil surface (As , s ) in response to lateral harmonic excitation of amplitude 2.0 m/s2

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to resonant frequency of the soil-SPSP foundation system) while the KIF decreases (less than unity) in the higher frequency region above the resonant frequency showing the filtering of high frequency component of soil surface motion by soil-SPSP kinematic interaction discussed above. Such behaviour is similar to that of the KIF of pile group foundation which offers flexural behaviour under kinematic loadings [22, 23]. However, for higher input loading amplitude (3−5 m/s2 ), the KIF values found larger than unity for a number of frequency particularly around the lower frequency region shows that the foundation input motion gets amplified compared to the soil surface motion (Figs. 14.6, 14.7 and 14.8). The reason for this increase in KIF (more than unity) is the different phases of the footing motion and soil surface motion (see Fig. 14.6, 14.7 and 14.8). Different phases indicate a delay in soil surface movement compared to footing movement and this is caused by the nonlinear behaviour of soil induced by strong excitation (for example, μs is 1.32 × 10−3 for loading amplitude of 5.0 m/s2 at 9 Hz). For better understanding of the effect of the nonlinear behaviour of soil-SPSP foundation system on the KIF, the experimentally recorded bending strain of the instrumented aluminium pipe (as detailed in Sect. 14.2.3) for all input loading amplitude at lower frequency of 7 and 9 Hz are plotted in Figs. 14.11 and 14.12, respectively. In Fig. 14.12 for lower amplitude of loading (0.5–2 m/s2 ), the bending strain pattern indicates that the top of the pipe movement is restrained by adjacent soil while for higher loading amplitude (3–5 m/s2 ) the top of the pipe is almost free to move (i.e. no restriction from adjacent soil). Similar change in bending strain pattern (i.e. change in boundary condition at the top of the pipe) has also been observed at 7 Hz (see Fig. 14.11) for higher amplitude of loading (4–5 m/s2 ). For 3 m/s2 at 7 Hz, the bending pattern is similar to that of the lower amplitude (0.5–2 m/s2 ) pattern since the mean shear strain in soil is smaller (μs = 7.45 × 10−4 ). The changes in boundary

Fig. 14.6 Amplification ratio with corresponding phase at footing (Au , u ) and at soil surface (As , Φs ) in response to lateral harmonic excitation of amplitude 3.0 m/s2

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Fig. 14.7 Amplification ratio with corresponding phase at footing (Au , Φu ) and at soil surface (As , Φs ) in response to lateral harmonic excitation of amplitude 4.0 m/s2

Fig. 14.8 Amplification ratio with corresponding phase at footing (Au , Φu ) and at soil surface (As , Φs ) in response to lateral harmonic excitation of amplitude 5.0 m/s2

condition at the pile head observed through the bending strain pattern is an indication of possible induced separation at the interface between soil and foundation structure at lower depth due to the induced nonlinearity in soil with increased amplitude of loading. This nonlinear behaviour of soil for higher amplitude of loading resulting in higher foundation motion compared to soil surface motion as seen in Fig. 14.10. This kind of increase in KIF (i.e. higher foundation motion compared to soil surface motion) around the lower frequency region has also been observed for single pile

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Fig. 14.9 Resonant frequency and amplification ratio at footing for different loading amplitude

Fig. 14.10 Horizontal Kinematic interaction factor for different input loading amplitude

and pile group due to the similar change of boundary condition at pile head as in [24]. Along with such nonlinear behaviour of soil, the sliding of joint between the pipes could possibly contribute to the increase in foundation motion compared to soil surface motion. The rotational KIF is not reported in this paper since it is found negligibly insignificant compared to the horizontal KIF. This is because of the dominance of flexural behaviour of the SPSP foundation system under kinematic loading condition.

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Fig. 14.11 Bending strain profile for different loading amplitude for 7 Hz

Fig. 14.12 Bending strain profile for different loading amplitude for 9 Hz

14.4 Conclusion In this study, the effective foundation input motion (EFIM) of a soil-steel pipe sheet pile (SPSP) foundation system is evaluated through shaking table experiment. The experiment was carried out with a scaled model of soil-SPSP foundation system under lateral dynamic loading condition. The experimentally measured loading amplitude

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and frequency dependent EFIM at the footing level of the SPSP foundation is presented in this paper and these quantities can be directly employed to substructure method for total dynamic response calculation of such soil-SPSP foundation system. A wide range of loading amplitude and frequency is employed in the experimental program which covers soil behaviour from elastic-to-inelastic range. The EFIM results shows that the resonant frequency and the amplification ratio of motion of the soil-SPSP foundation system decreases with increasing input loading amplitude. This decreasing pattern is accompanied by the increase in shear strain in soil with the increase in loading amplitude. The kinematic interaction factor (KIF) of the soil-SPSP foundation system is also reported which shows higher filtering effect on foundation motion around higher frequency range, particularly above the resonant frequency of soil-SPSP foundation system. Such behaviour attributed to the soil-SPSP foundation kinematic interaction under dynamic loading. The KIF is approximately equal to unity up to resonant frequency of the soil-SPSP foundation system for lower amplitude of lateral dynamic excitation indicating almost no filtering effect. However, for higher amplitude of excitation an increase in KIF (more than unity) around the lower frequency has been observed. The reason for such observed increase in KIF is the nonlinear behaviour of soil induced by the higher loading amplitude at lower frequency. Acknowledgements This work was supported by JSPS KAKENHI Grant number 17K14713. Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the authors and do not necessarily reflect the views of the sponsors.

References 1. Katayama, T.: Recent technology development of steel pipe sheet pile foundation in Japan. In: Proceedings of the International Workshop on Recent Advances in Deep Foundations, pp. 335–339. Port and Airport Research Institute, Yokosuka, Japan (2007) 2. Kimura, M., Inazumi, S., Too, J.K.A., Isobe, K., Mitsuda, Y., Nishiyama, Y.: Development and application of h-joint steel pipe sheet piles in construction of foundations for structures. Soils Found. 47(2), 237–251 (2007) 3. Kimura, M., Isobe, K., Nishiyama, Y.: Development of three-dimensional frame analysis method for H-joint steel pipe sheet pile foundation system. In: Proceedings of the International Workshop on Recent Advances in Deep Foundations, pp. 341–348. Port and Airport Research Institute, Yokosuka, Japan (2007) 4. Kawakami, K., Okubo, T., Komada, K., Okahara, M.: Sheet pile foundation and its structural characteristics against horizontal loads. In: Proceedings of the 7th Joint Panel Conference of the US-Japan Cooperative Program in Natural Resources, pp. VII7–VII60. Tokyo, Japan (1975) 5. Asama, T., Shioi, Y., Okahara, M., Mitsuie, Y.: Dynamic analysis of sheet pile foundations. In: Proceedings of the 12th Joint Panel Conference of the US-Japan Cooperative Program in Natural Resources, pp. 235–242. Gaithersburg, Maryland (1980) 6. Kunihiko, O., Kiyosi, Y., Hiroya, O.: Development of new foundation method using steel pipe sheet pile with strength pipe-junction “Hyper-Well SP”. JFE Tech. Rep. 8, 44–50 (2006) 7. Kiyomiya, O.: Tokyo Gate bridge-design and construction of steel pipe sheet pile foundation. Steel Constr. Today Tomorrow 31, 14–18 (2011)

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8. Specification for Highway Bridges, Part IV: Substructures. Japan Road Association (2012) 9. Positive and negative alternating shear test of steel pipe pile foundation joint. JASPP (Japanese Technical Association for Steel Pipe Piles and Sheet Piles) Technical Report (2016). (in Japanese) 10. Kausel, E., Whitman, R.V., Morray, J.P., Elsabee, F.: The spring method for embedded foundation. Nucl. Eng. Des. 48, 377–392 (1978) 11. Mylonakis, G., Nikolaou, A., Gazetas, G.: Soil-pile-bridge seismic interaction: kinematic and inertial effects part I: soft soil. Earthq. Eng. Struct. Dyn. 26(3), 337–359 (1997) 12. Laura, R.D., Grossi, Y., Sanctis, L.D., Viggiani, G.M.B.: An analytical solution for the rotational component of the Foundation Input Motion induced by a pile group. Soil. Dyn. Earthq. Eng. 97, 424–438 (2017) 13. Gazetas, G.: Seismic response of end-bearing single piles. Int. J. Soil. Dyn. Earthq. Eng. 3(2), 82–93 (1984) 14. Safak, E.: Detection and identification of soil-structure interaction in buildings from vibration recordings. J. Struct. Eng. 121(5), 899–906 (1995) 15. Tongxiang, A., Kiyomiya, O., Trung, T.: Seismic performance of steel pipe sheet pile foundation on soft ground. In: Proceedings of the 15th World Conference on Earthquake Engineering, Lisbon, Portugal (2012) 16. Trung, N.T., Kiyomiya, O.: Response analysis of steel sheet pipe pile foundation by three simple types of models. In: Proceedings of the 39th Annual Conference of JSCE, Yokohama, Japan, Kanto Branch (2012) 17. Trung, N.T., Kiyomiya, O., An, T.: Comparison of three seismic analysis models of steel sheet pile bridge foundation. In: Proceedings of the 15th Symposium on Performance-based Seismic Design Method for Bridges of JSCE, Tokyo, Japan (2012) 18. Uno, K., Aso, T., Tsutusmi, H., Kitagawa, S.: Dynamic characteristics of steel pipe piled well foundations. In: Proceedings of the 10th World Conference on Earthquake Engineering, Balkema, Rotterdam (1992). ISBN 90 5410 060 19. Aso, T., Uno, K., Kitagawa, S., Morikawa, T.: A dynamic model test and analysis of a steel pipe piled well foundation. In: Proceedings of the 11th World Conference on Earthquake Engineering, Paper No. 1085, Acapulco, Mexico, (1996). ISBN 0 08 042822 3 20. Kokusho, T., Iwatate, T.: Scaled model tests and numerical analyses on nonlinear dynamic response of soft grounds. Proc. Jpn. Soc. Civ. Eng. 285, 57–67 (1979) 21. Ishida, T., Watanabe, H., Ito, H., Kitahara, Y., Matsumoto, M.: Static and dynamic mechanical properties of sandy materials for model tests of slope failure under the condition of low confined stress (Research report no. 380045). Central Research Institute of Electrical Power Industry, Japan (1981) 22. Tazoh, T., Wakahara, T., Shimizu, K., Matsuzaki, M.: Effective motion of group pile foundations. In: Proceedings of the 9th World Conference on Earthquake Engineering, vol. 3, pp. 587–592, Tokyo-Kyoto, Japan (1988) 23. Laora, R.D., Sanctis, L.D.: Pile induced filtering effect on the foundation input motion. Soil Dyn. Earthq. Eng. 46, 52–63 (2013) 24. Kaynia, A.M.: Dynamic stiffness and seismic response of pile groups. Ph.D. dissertation, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, MA (1982)

Chapter 15

Calculation and Analysis for Fracture Pressure of Deep Water Shallow Formation Reyu Gao, Jun Li, Kuidong Luo, Hongwei Yang, Qingxin Meng and Wenbao Zhai Abstract The deep water shallow strata has the characteristics of low overburden pressure, poor cementation properties and low fracture pressure of the formation. Therefore, the wall rupture and collapse are prone to occur. Currently, there is no perfect theoretical model for shallow formation pressure prediction. This paper assumes that the shallow strata are composed of homogeneous and isotropic ideal elastoplastic materials. An elastoplastic mechanical model of a circular wellbore is established. By introducing the theory of excess pore pressure and rock damage degree, the theoretical formula of formation fracture pressure under uniform ground stress is corrected. Verification by using actual log data. Furthermore, the law of the rupture pressure of the borehole wall with the change of the super-pore pressure coefficient and the degree of fracture coefficient is further analyzed. The results show that the well wall will be damaged to a certain extent during the drilling process, and the drilling fluid column pressure will produce squeezing effect on the well wall to cause the super-pore pressure in the shallow stratum. The wall rupture pressure decreases with the increase of the super-pore pressure coefficient and the rock fragmentation coefficient. Compared with the actual results of the field engineering, the calculation results under the model are closer to the actual situation than the traditional model calculation results, which explains the fracture mechanism of shallow water formations more reasonably. Therefore, when considering the problem of the shallow strata rupture mechanism, the relevant theory in geotechnical mechanics should be introduced for a more comprehensive analysis. Keywords Deep water shallow strata · Fracture pressure · Ideally elastoplastic · Excess pore pressure · Rock fragmentation coefficient

R. Gao · J. Li (B) · K. Luo · H. Yang · Q. Meng · W. Zhai China University of Petroleum, Changping, Beijing, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_15

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Nomenclature σh σv ϕ C σr σθ σz p po pp pw rw rp p A Bd Kv

Horizontal horizontal stress Overburden pressure Rock internal friction angle Cohesion Radial stress Circumferential stress Axial stress Formation pore pressure Wellbore pore pressure after drilling Drilling liquid column pressure Well radius Interface elastoplastic zone interface radius Excess pore pressure Excess pore pressure coefficient Rock fragmentation coefficient Rock integrity factor

15.1 Preface With the deepening of oil and gas exploration and development, the number of discoveries of large-scale onshore oil and gas resources in the world has decreased significantly. According to statistics, the world’s offshore oil resources account for 34% of the world’s total oil resources, of which the proven reserves exceed 100 billion tons, and the proportion is still rising. Deep water is the future direction of petroleum exploration and development. Deep water exploration and development is the development trend of world petroleum industry. But in the deepwater drilling process, the cost is very high, the average daily cost is more than one million dollars. In addition to the general problems of marine operations such as harsh environment, frequent typhoons, shallow geological hazards, equipment reliability, etc., the conditions of marine geology are more complex than those of land, and the development of deepwater drilling is more difficult and challenging. Because of the existence of sea water, the rock cementation in shallow strata is weak, and the overlying strata have low pressure relative to the same depth of land. The strata are mainly saturated soils, and have not yet formed well-consolidated rocks. The stratum is mainly composed of saturated soil, and no well-consolidated rock has been formed. The problem of narrow density window in shallow formation is particularly obvious in deepwater drilling process, This part of the formation is more prone to rupture and collapse, resulting in increased non-production time, more cost losses, seriously affecting the development of deep-sea oil and gas resources. These seriously affect the development of deep-sea oil and gas resources [1].

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The traditional fracture pressure calculation model is mainly based on elasticity [2, 3], such as Eaton Method, Stephen Method, Anderson Method and Huang Rongzun Method and some modified empirical models based on elasticity, etc. But controlled by consolidation theory of saturated soil, the wellbore will enter the plastic yield state before it breaks down during drilling in shallow formation. So it doesn’t apply to these models. Aadnay et al. [4, 5] constructed an elastic-plastic model for calculating fracture pressure, taking into account the plastic zone around the wellbore. But he thinks that the starting point of wellbore fracture is located on the cementation surface of elastic-plastic zone, and the calculated fracture pressure is much higher than that calculated by elastic model, which is not consistent with the lower fracture pressure in deep water shallow formation. Sun et al. [6] proposed a new calculation model of formation homogeneous fracture pressure in view of the drawbacks of Aadnay’s model, which improved the accuracy of prediction of formation fracture pressure, but the result he chosed is the pressure when vertical fracture occurs in wellbore. This is inconsistent with Wojtanowicz et al. [7] argued that when a wellbore is fractured, it can only form horizontal fractures but not vertical ones in shallow formations. The deviation of fracture pressure calculated by Sun for horizontal fracture is large. Therefore, on the basis of previous studies, considering the influence of excess pore pressure in Geomechanics theory, and considering the fragmentation characteristics of shallow formation in deep water, this paper introduces the fragmentation degree coefficient to modify the formula of borehole stress. Then, the degree of variation of the influence of various coefficients on formation fracture pressure in the model is compared and analyzed.

15.2 Introduction of Modified Coefficient of Mechanical Model and Calculation of Fracture Pressure 15.2.1 Elastic-Plastic Model of Circular Borehole Under Uniform in Situ Stress in Shallow Strata Deep water shallow formation has short sedimentation time, low cementation strength and large Poisson’s ratio. The difference between horizontal in situ stresses is small due to less tectonic movement, which can be approximately considered to be affected by uniform horizontal in situ stresses. Therefore, it is assumed that shallow strata are composed of homogeneous and isotropic ideal elastic-plastic materials. The formation is elastic before drilling and receives the effect of overlying strata pressure σv and uniform horizontal in situ stress σh . After drilling, there is a plastic zone around the wellbore, and the plastic yield obeys Mohr-Coulomb strength criterion. The force model is shown in Fig. 15.1. According to Mohr-Coulomb strength criterion, the yield function of shallow strata can be expressed as

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Fig. 15.1 Borehole stress model under uniform in situ stress in deep water shallow strata

σ1 =N σ3 +σ0

(15.1)

When drilling fluid column pressure A exceeds horizontal in situ stress B excessively, the wellbore around the wellbore will enter a plastic state. According to Andrew K. Wojtanowicz et al., when wellbore is fractured in deep shallow formation, only horizontal fractures can be formed on the wellbore. The stress condition on the borehole wall satisfies σr > σθ > σz , the yield function is [8] σr = N σz + σ0

(15.2)

From the plane strain theory, the increment of circumferential stress in the plastic zone around the well can be expressed as [9] σz = (σr + σθ )/2

(15.3)

The axial stress on the borehole wall can be obtained as follows. σz = σv +

(σr − σh ) + (σθ − σh ) 2

(15.4)

Simultaneous (15.2), (15.4) and boundary conditions σr = pw , (r = rw ) show that the plastic stress on the borehole wall is ⎧ ⎨ σr = pw σ = [(2 − N ) pw − 2σo ]/N − 2σv + 2σh ⎩ θ σz = ( pw − σo )/N

(15.5)

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In the plane strain problem under axisymmetric condition, the stress equilibrium equation is σr − σθ dσr + =0 dr r

(15.6)

The distribution of stress field in plastic zone obtained by simultaneous formulas (15.2), (15.4), (15.6) and boundary conditions σr |r =rw = pw is as follows ⎧ ⎪ ⎪ ⎨ σr = σθ = ⎪ ⎪ ⎩ σz =



N σh N −1  N σh N −1 1 N −1 (σh







   2(N −1)

1 σ + Pw − NN−1 σh − σv − N1 σo rrw N N o       2(N −1) − σv − N1 σo + 2−N Pw − NN−1 σh − σv − N1 σo rrw N N      2(N −1) − σv − σo ) + N1 Pw − NN−1 σh − σv − N1 σo rrw N

− σv −

(15.7)

15.2.2 Calculation of Excess Pore Pressure During drilling, due to the invasion of drilling fluid into surrounding strata, a dense mud cake will be formed on the borehole wall. Mud cake prevents fluid from seeping into each other in well and formation. This variable of pore pressure caused by external loads is called excess pore pressure, record as p: p = p p − p po

(15.8)

In the loading process, the external pressure of the saturated system is borne by the two phases of water and particle skeleton respectively. Among them, the water pressure is called neutral pressure, and the rock skeleton pressure is called effective pressure. This neutral pressure caused by additional stress is different from the pore water pressure caused by hydrostatic pressure in soil. It is called residual pore water pressure or excess pore water pressure [10]. The formula for calculating the excess pore pressure from Skemption [11] is as follows: p = B[σ3 + A(σ1 − σ3 )]

(15.9)

The B in formula 15.9 is the coefficient of pore pressure under the action of isotropic stress and deviating stress. For saturated soils, B = 1.0; A is the coefficient of pore pressure under biased stress, it can be solved experimentally or experientially (Table 15.1). Increments of maximum and minimum principal stresses for σ1 and σ3 , respectively.

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Table 15.1 Empirical values of A [12]

Soil type (saturation)

A

Soft fine sand

2.00–3.00

Sensitive clay

0.75–1.50

Normally consolidated clay

0.50–1.00

Mild overconsolidated clay

0–0.50

Severely overconsolidated clay

−0.50–0

According to formula 15.5, wellbore stress increment caused by drilling is

σ1 = σr = σr − σh = pw − σh σ3 = σz = σz − σv = ( pwN−σo ) − σv

(15.10)

The simultaneous formulas (15.9), (15.10) can be obtained:  ( pw − σo ) ( pw − σo ) p = A pw − σh − + σv + − σv N N

(15.11)

15.2.3 Calculation of Rock Fragmentation Coefficient and Fracture Pressure The characteristics of shallow stratum are short sedimentation time, weak soil cementation and unconsolidated diagenesis. Influenced by drilling operation, the structural plane becomes weaker, which further reduces the wellbore stability. Therefore, the formation fragmentation degree coefficient Bd is introduced to describe the effect of shallow soil fragmentation degree on the equivalent stress state of the surrounding rock of the shaft wall.  = e0.5Bd σz,r,θ σz,r,θ

(15.12)

Where Bd satisfies 0 ≤ Bd ≤ 1, the larger the Bd value is, the more serious the rock fragmentation is. Bd can be calculated by formula

(15.13)

In geotechnical engineering, the integrality coefficient A of rock mass can be expressed as the square ratio of P-wave velocity of rock mass to rock. Quantitative classification of stratigraphic rock integrity is made according to the development degree of structural planes in Engineering Rock Mass Classification Standard (GB50218-94). The specific categories of rock mass integrity factor K v are shown in Table 15.2.

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Table 15.2 Corresponding relationship between K v and rock mass integrity degree of qualitative Division Kv

>0.75

0.75–0.55

0.55–0.35

0.35–0.15

6.5) is known as adsorption. At higher pH values above 9.0, Al(OH)−4 is also present in the system. Freshly formed amorphous Al(OH)3 “sweep flocs” has big surface areas, which is important for a rapid adsorption of soluble organic compounds and accumulation of colloidal particles [13]. Both hydrogen and oxygen gas are evolved near the anode and cathode as each gas bubble nucleates. These bubbles carried the contaminants (dye particle) to the liquid surface as shown in Fig. 17.1 [3]. The difference between the two electrolytes could be illustrated by the scientific truth that NaCl solution provides large conductivity to the solution and the formed chemicals are good oxidants to degrade organic compounds, therefor increasing the efficiency of operation. Average levels of chloride ions in wastewater minimize the passivation layer on the anodes surface, improving the removal efficiency of pollutants in the electrocoagulation method [5]. When the chlorides were presented in the samples the products from anodic discharge of chlorides were Cl2 and OCl− . The OCl− is a strong oxidant, which capable to oxidize organic compounds presented in the wastewater [14]. When adding NaCl in solution there are three equations: 2Cl− − 2e → Cl2

(17.5)

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Fig. 17.1 Electrocoagulation mechanism

Cl2 + H2 O → HOCl + Cl− + H+

(17.6)

HOCl ↔ OCl− + H+

(17.7)

The magnitude of electrode material consumed or dissolved during the electrocoagulation mechanism depended significantly on the intensity of current as explained by the Faraday law as follows (17.4), (17.6). m=

MIt n F vol

where: m is the mass of dissolved metal (gm/l); M is the molecular weight (g/mol) (MAl = 26.98 g/mol); I is the current intensity (Ampere); t is the contact time (second); n is the number of electrons involved in the oxidation reduction reaction (nAl = 3); and F is the Faraday constant (96,485 C/mol) and vol is the sample volume (liter).

17.3 Materials and Methods 17.3.1 Experimental Set-Up Experiments were implemented in a batch electrochemical circular glass cell shown in Fig. 17.2 that had the following dimensions: 15 cm diameter and 20 cm height. The total volume of 2 l of wastewater was treated in the electrochemical cell with 15 cm wetted depth and 5 cm free board. A rotating anode in the center of electrocoagulation cell and composed of rod with four shafts. Every shaft had the following dimensions: 2.5 cm width, 15 cm height and 0.3 cm thickness. The rotating anode was attached

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Fig. 17.2 Schematic diagram: (1) DC power supply (2) electrical wires (3) electrocoagulation cell (4) cathode (5) rotating anode (6) DC motor (7) speed regulator

to a motor with adjustable speed in order to maintain the anode rotations. The motor is DC electrical type and supplies various steady state speeds (20, 40 and 80 rpm). The cathode was a cylindrical roller with 11 cm diameter and 17 cm height. The gap between cathode and reactor 1.5 cm whereas it was 3 cm between cathode and anode. These electrodes were made of aluminum and immersed height was 12 cm and there is clearance of 3 cm between electrodes and the bottom of reactor. The metal electrodes were dropped to the wastewater sample and connected to digital multi meter, KEW SNAP model-2012, for measurement the current and the potential between the electrodes. The D.C. power supply output had three different current conditions: 0.4 A, 0.7 A and 1 A with the volts of 13 V, 17 V and 21 V, respectively.

17.3.2 Synthetic Wastewater Synthetic wastewater was set by adding dosages of dye (reactive blue 19) and 1 gm/l of NACL to 1 l of tap water and mixed for 3 min. The mixture showed a uniform blue color. The initial pH ranged from 5.9 to 8.2.

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17.3.3 Experimental Method First EC-cell (batch system) was filled with 2 l of synthetic wastewater at room temperature 25 °C. Electrodes were submerged and then the current was passed by the regulated DC power supply. The overall efficiency of the reactor was tested using three main variables; processing time, current intensity and the anode’s overall rotational speed. The electrolysis time (RT) was maintained in the range of 0–20 min. Three main current densities (CD); 0.4, 0.7, 1 A with various steady-state anode rotational speeds of 0, 20, 40 and 80 rpm were examined. The reaction was timed, beginning when the D.C. power supply and D.C. motor were switched on. Samples of 15 ml of wastewater were withdrawn from the depth of 5 cm below the free surface of wastewater at regular time intervals of 5 min. The effect of the electrochemical treatment was determined by measuring concentration of dye at the regular time intervals of 5 min. COD was measured at the beginning and at the end of the run. After each run the electrodes were cleaned and rinsed with HCl (10% concentration) to remove the oxides formed at the anode surface and then dried. The EC batch rounds were executed 12 times.

17.3.4 Analytical Measurement The experimental parameters measured were dye concentration, COD, conductivity, TDS and pH. Analysis was carried out by the standard method for the examination of water and wastewater (22nd edition, 2012) [15]. The COD was measured by the closed reflux, titration method and the colour was also estimated using a 4802 uv/vis a double beam spectrophotometer (λ → 585) [16]. The removal efficiency of dye and COD was determined as (A0 − A)/A0 and (C0 − C)/C0, respectively. In order to accomplish the aims of these study three sets of runs were planned using concentration of dye 200 mg/l. The three sets of runs were done using 21 V, 17 V and 13 V, respectively. Each set of them contained four experiments with different rotating anodes (0, 20, 40, 80 rpm.).

17.4 Result and Discussion 17.4.1 Effect of Current Intensity In all electrocoagulation operations, current intensity is the most important factor that control the interaction rate in the electrocoagulation cell [1, 10, 17]. It was noted that the removal efficiencies of dyes were from 90.3% to 96%, from 79.5% to 93.5% and from 72% to 88.9% for 21 V, 17 V and 13 V, respectively at initial dye concentration 200 mg/l after 20 min as shown in Figs. 17.3, 17.4 and 17.5.

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100% 90% 80% 70% 60%

no rotaƟon

50%

20 r.p.m

40%

40 r.p.m

30%

80 r.p.m

20% 10% 0% 1

2

3

4

5

Fig. 17.3 Effect of anode rotation on dye removal (200 mg/l–21 V–1A)

100% 90% 80% 70% 60%

no rotaƟon 20 r.p.m 40 r.p.m 80 r.p.m

50% 40% 30% 20% 10% 0% 1

2

3

4

5

Fig. 17.4 Effect of anode rotation on dye removal (200 mg/l–17 V–0.7A)

At initial dye concentration of 200 mg/l, initial COD concentration of 280 mg/l, after 20 min contact time and rotating speed of 20 rpm, the maximum removal of dye and COD were 96%–92.1%, 93.5%–90.3%and 88.9%–82.4% for 1 A, 0.7 A and 0.4 A, respectively as shown in Figs. 17.3, 17.4 and 17.5. A current intensity is applied between the metallic electrodes that are submerged in the wastewater, which encourage the electrodes dissolution that is being responsible for the creation of coagulant species [5, 10, 15].

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100% 90% 80% 70% 60%

no rotaƟon

50%

20 r.p.m

40% 30%

40 r.p.m

20%

80 r.p.m

10% 0% 1

2

3

4

5

Fig. 17.5 Effect of anode rotation on dye removal (200 mg/l–13 V–0.4A)

Increasing current intensity encourages the generation of hydrogen gas bubbles and reduces its size which leads to increasing of the upward flow and removal of contaminants by flotation [5, 9, 16–18].

17.4.2 Effect of Rotational Speed of Anode The rotational speed is an effective parameter to transfer the coagulant substance that is formed and affects the dissolution rate of electrodes in the reactor. It is responsible for the homogeneity of the reactor’s components including the temperature and pH [3]. On the other hand, a high rotational speed can potentially break or damage flocs that are formed in the reactor, and convert to smaller flocs that are difficult to separate from the water [3]. The maximum removals of dye and COD after 20 min reaction time were 96% and 92.1% at 20 rpm for 200 mg/l initial dye concentration and 21 V whereas the removal of dye decreased to 94%, 91.5% and 90.3% in case of no rotation, 40 rpm and 80 rpm, respectively as shown in Fig. 17.3.at the same condition but for 17 V and 13 V the maximum removal of dye were 93.5% and 88.9% in addition the removal of dye decreased to 90%–85%, 84.2%–77.4% and 79.5–72%, respectively as shown in Figs. 17.4 and 17.5.

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17.4.3 Effect of Contact Time Increasing reaction time will lead more discharging of metal ions and then enhanced high removal efficiencies of dyes and COD [19]. When the reaction time increased, more hydrogen bubbles were generated at the cathode; these bubbles increased the mixing capacity and improved the flotation ability of the EC cell [14]. At a small interaction time, the removal ratios are low. This was due to the fact that, the total quantity of produced coagulant was not enough to eliminate all the particles of dye in the wastewater and it required more reaction time to reach higher efficiency of dye removal [20]. The concentration of dye was measured every 5 min to know the effect of contact time. It was found that the removals of dye were 59%, 85%, 90% and 96% at 5 min, 10 min, 15 min, 20 min, respectively as shown in Fig. 17.4 for 200 mg/l initial dye concentration, 1 A current intensity and 200 rpm rotational speed of anode. The removals of dye after 5 min were from 50 to 60% at static and 20 rpm state whereas there were less than 50% at 40 and 80 rpm state In the case of three different currents as shown in Figs. 17.3, 17.4 and 17.5.

17.4.4 Sludge Production Sludge layers were formed at the top of the reactor and were analyzed separately during the electrocoagulation process [20]. The quantity of sludge produced was related significantly to the quantity of contaminants removed [21]. It was observed that the volume of sludge increased with increasing the current intensity and rotational speed of anode. The maximum sludge was 618.5 cm3 at 80 rpm state and for 1 A. The minimum sludge was 70.7 cm3 at static state and for 0.4 A. Initial PH was from 7.4 to 7.58 and final PH was from 8.55 to 8.79 due to the reactions occurring at the cathode. At this cathode, water molecules dissociate into hydrogen bubbles and hydroxyl ions that leads to increasing the pH medium [1, 7, 16]. The final pH was always higher than the initial pH 7 [22]. Total cost Economic cost plays an important role in the implementation of any system of treatment. Cost including electrical energy and metal consumption and electrical energy was calculated from the following equation: E=

(V)(I)(t) vol

where E is the energy consumption (kwh/m3 ); v is the current voltage (volt); I is the current intensity (ampere); t is the contact time (hour) and vol is the sample volume (litre).

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ET = Ee + Em Ee → energy consumption for electrodes Em → energy consumption for motor Total Cost (LE/m3 ) = ET(kwh/m3 ) × Price (LE/kwh) + m(kg/m3 ) × metal Price (LE/kg). Where price of kwh of electricity equals 0.94 LE; price of kg aluminum equals 32 LE. Total cost was approximately 5 LE for treating one cubic meter of synthetic wastewater with initial concentration of dye 200 mg/l using 21 V current voltage and 1 A current intensity at deferent rotational speed of anode. At this condition, removal efficiencies of dye and COD ranges from 90.3 to 96% and from 87.5 to 92.1%, respectively as shown in Table 17.1. Total cost was approximately 3 LE for treating on cubic meter of synthetic wastewater with initial concentration of dye 200 mg/l using 17 V current voltage and 0.7 A current intensity at deferent rotational speed of anode. At this condition, removal efficiencies of dye and COD ranges from 79.5 to 93.5% and from 74.6 to 90.3%, respectively as shown in Table 17.2. Total cost was approximately 1.5 LE for treating one cubic meter of synthetic wastewater with initial concentration of dye 200 mg/l using 13 V current voltage and 0.4 A current intensity at deferent rotational speed of anode. At this condition, removal efficiencies of dye and COD ranges from 72.3 to 88.9% and from 70.2 to 82.4%, respectively as shown in Table 17.3. Total cost was 1.524 LE for treating one cubic meter of synthetic wastewater for initial concentration of dye 200 mg/l using 13 V and 0.4 A at 20 rpm rotational speed of anode. At this condition, the removal efficiency of dye and COD were 88.9% and 82.4%, respectively. COD values reduced from 280 to 49.28 mg/l. Based on Table 17.1 Total cost of treatment for dye 200 mg/l using 21 V, 1 A

Table 17.2 Total cost of treatment of dye 200 mg/l using 17 V, 0.7 A

State of motor (rpm)

Total cost (lE/m3 )

% removal Dye

COD

0

5.0800

94.0

91.2

20

5.0843

96.0

92.1

40

5.0867

91.5

89.0

80

5.0890

90.3

87.5

State of motor (rpm)

Total cost (lE/m3 )

% removal Dye

COD

0

3.14

90

88

20

3.1435

93.5

90.3

40

3.1450

84.2

80

80

3.148

79.5

74.6

17 Treatment of Textile Dyes Wastewater … Table 17.3 Total cost of treatment of dye 200 mg/l using 13 V, 0.4 A

185

State of motor (rpm)

Total cost (lE/m3 )

% removal Dye

COD

0

1.5218

85

78.3

20

1.524

88.9

82.4

40

1.5265

77.4

73

80

1.528

72.3

70.21

300 250 200

outlet COD inlet COD

150 100 50 0 1

Fig. 17.6 COD values for synthetic wastewater at 13 V and 20 rpm for 200 mg/l initial dye concentration

COD values; it is possible to discharge the waste water directly on the water bodies according to the Egyptian law as shown in Fig. 17.6 and Table 17.3.

17.5 Conclusions 1. Current intensity is the most important factor that controls the interaction rate in the electrocoagulation cell and initial concentration of dye has no effect on COD and dye removal efficiencies. 2. The best performance was obtained at 20 rpm rotational speed. The maximum removals of dye and COD after 20 min reaction time were 96%–92.1%, 93.5%–90.3% and 88.9%–82.4% at 21 V, 17 V and 13 V, respectively using 20 rpm for 200 mg/l initial dye concentration.

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3. The total cost was 1.524 LE for treating one cubic meter of synthetic textile waste water using 13 V and 0.4 A at 20 rpm rotational speed of anode after 20 min. At this condition, the removal efficiency of dye and COD were 88.75% and 79%, respectively.

References 1. Kobya, M., Demirbas, E.: Evaluations of operating parameters on treatment of can manufacturing wastewater by electrocoagulation. J. Water Process Eng 8, 64–74 (2015) 2. Saxena, S., Raja, A.: Natural dyes: sources, chemistry, application and sustainability issues. Roadmap to Sustainable Textiles and Clothing, pp. 37–80. Springer, Singapore (2014) 3. Naje, A.S., Chelliapan, S., Zakaria, Z., Ajeel, M.A., Sopian, K., Hasan, H.A.: Electrocoagulation by solar energy feed for textile wastewater treatment including mechanism and hydrogen production using a novel reactor design with a rotating anode. RSC Adv. 6, 10192–10204 (2016) 4. Pajootan, E., Arami, M., Mahmoodi, N.M.: Binary system dye removal by electrocoagulation from synthetic and real colored wastewaters. J. Taiwan Inst. Chem. Eng. 43, 282–290 (2012) 5. Fajardo, A.S., Rodrigues, R.F., Martins, R.C., Castro, L.M., Quinta-Ferreira, R.M.: Phenolic wastewaters treatment by electrocoagulation process using Zn anode. Chem. Eng. J. 331–341 (2015) 6. Kim, T.H., Park, C., Shin, E.B., Kim, S.: Decolorization of disperse and reactive dyes by continuous electrocoagulation process. Desalination 150, 165–175 (2002) 7. Kim, T.H., Park, C., Shin, E.B., Kim, S.: Decolorization of disperse and reactive dye solutions using ferric chloride. Desalination 161, 49–58 (2004) 8. Paz, A., Carballo, J., Pérez, M.J., Domínguez, J.M.: Biological treatment of model dyes and textile wastewaters. Chemosphere 181, 168–177 (2017) 9. Fajardo, A.S., Martins, R.C., Silva, D.R., Martínez-Huitle, C.A., Quinta-Ferreira, R.M.: Dye wastewaters treatment using batch and recirculation flow electrocoagulation systems. J. Electroanal. Chem. 801, 30–37 (2017) 10. Kumar, B., Patel, S.: Effects of operational parameters on the removal of brilliant green dye from aqueous solutions by electrocoagulation. Arab. J. Chem. (2013) 11. Aswathy, P., Gandhimathi, R., Ramesh, S.T., Nidheesh, P.V.: Removal of organics from bilge water by batch electrocoagulation process. Separation and Purification Technol. 159, 108–115 (2016) 12. Khorram, A., Fallah, N.: Treatment of textile dyeing factory wastewater by electrocoagulation with low sludge settling time: optimization of operating parameters by RSM. J. Environ. Chem. Eng. 6, 635–642 (2018) 13. Kobya, M., Ciftci, C., Bayramoglu, M., Sensoy, M.T.: Study on the treatment of waste metal cutting fluids using electrocoagulation. Sep. Purif. Technol. 60, 285–291 (2008) 14. Kil, H., Min, S.: Optimization of color and COD removal from livestock wastewater by electrocoagulation process: application of Box–Behnken design (BBD). J. Ind. Eng. Chem. (2015) 15. APHA (2012) Standard Methods For The Examination of Water and Wastewater, 22nd edn. American Public Health Association, Washington, D.C. 16. Palahouane, B., Drouiche, N., Aoudj, S., Bensadok, K.: Cost-effective electrocoagulation process for the remediation of fluoride from pretreated photovoltaic wastewater. J. Ind. Eng. Chem. 127–131 (2015) 17. Elabbas, S., Ouazzani, N., Mandi, L., Berrekhis, F., Perdicakis, M., Pontvianne, S., Leclerc, J.P.: Treatment of highly concentrated tannery wastewater using electrocoagulation: influence of the quality of aluminium used for the electrode. J. Hazard. Mater. 319, 69–77 (2016)

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18. Yang, T., Qiao, B., Li, G.C., Yang, Q.Y.: Improving performance of dynamic membrane assisted by electrocoagulation for treatment of oily wastewater: effect of electrolytic conditions. Desalination 363, 134–143 (2015) 19. Choudharya, A., Sanjay, D.: Performance evaluation of non rotating and rotating anode reactor in electro coagulation process. J. Mater. Sci. Eng. (2017) 20. Naje, A.S., Chelliapan, S., Zakaria, Z., Abbas, S.A.: Electrocoagulation using a rotated anode: a novel reactor design for textile wastewater treatment. J. Environ. Manag. 176, 34–44 (2016) 21. Kobya, M., Demirbas, E.: Evaluations of operating parameters on treatment of can manufacturing wastewater by electrocoagulation. J. Water Process Eng. 8, 64–74 (2015) 22. Aswathy, P., Gandhimathi, R., Ramesh, S.T., Nidheesh, P.V.: Removal of organics from bilge water by batch electrocoagulation process. Sep. Purif. Technol. 108–115 (2016)

Chapter 18

Stagnation Point Flow and Heat Transfer Over a Permeable Stretching/Shrinking Sheet with Heat Source/Sink Izyan Syazana Awaludin, Anuar Ishak and Ioan Pop

Abstract The present study highlights the problem of magnetohydrodynamic (MHD) stagnation point flow and heat transfer with heat source/sink over a permeable stretching/shrinking sheet. The partial differential equations are reduced to a set of ordinary differential equations using the similarity transformations. The ordinary differential equations are then solved numerically by employing the bvp4c function available in the MATLAB software. The graphical illustrations show the effects of several parameters on the skin friction coefficient, local Nusselt number and the local Sherwood number. The stability analysis is carried out to determine the stability of the solutions in a long run. Keywords Shrinking · Dual solutions · Stability · Stagnation flow

18.1 Introduction In recent decades, studies pertaining to the boundary layer flow and heat transfer over a stretching or shrinking sheet have attracted considerable attention among the researchers due to its extensive applications in the industry which include the production of papers, fine-fiber mattes, roofing shingles, and insulting materials [6]. In regard to this matter, it is also important to note the significance of the movements that occur in the fluid caused by the stretching/shrinking sheet which is parallel to it in order to control the drag and heat flux for the purpose of ensuring the best quality of the final product. The problem of the stretching sheet with movable velocity from the slit was first introduced by Crane [5]. Meanwhile, Miklavˇciˇc and Wang [16] I. S. Awaludin Faculty of Management, Multimedia University, 63100 Cyberjaya, Selangor, Malaysia I. S. Awaludin · A. Ishak (B) Faculty of Science and Technology, School of Mathematical Sciences, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia e-mail: [email protected] I. Pop Department of Mathematics, Babes, -Bolyai University, 400084 Cluj-Napoca, Romania © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_18

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were among the early researchers that investigated the flow over a shrinking sheet with the velocity moves towards the slit. Following this, the issue is becoming a common trend among the researchers which further emphasized the need to extend the problem of flow and heat transfer over a stretching/shrinking sheet [1, 8, 10, 14, 17, 18]. The study of magnetohydrodynamic (MHD) flow is still of current interest due to its wide applications in the industry. We note that the study of the incompressible viscous flow in the presence of MHD was first conducted by Pavlov [19]. The rate of the heat transfer in the cooling process of the drawing strips can be controlled by the presence of the magnetic field in order to obtain an excellent quality of the final product [4]. Apart from that, the application of strong external magnetic field is able to reduce the shrinking effect in the sheet [11]. The scarcity in the study of fluid flow and heat transfer with chemical reaction has attracted the interest of the researchers to investigate this effect in different physical aspects. Bhattacharyya and Layek [2] investigated the steady MHD flow and chemically reactive solution using quasi-linearization method. Following it, Bhattacharyya et al. [3] decided to merge their study for the shrinking sheet case using the same method. The study of the chemical reactions is very important in the industry, especially in an oxidation of solid materials and ceramics manufacturing. As reported by Mabood et al. [12], the flow field can be affected by the appearance of the chemical reaction and magnetic field that appear on the stretching sheet. The problem of MHD stagnation-point flow over the stretching/shrinking sheet in the presence of heat generation/absorption and chemical reaction effects was solved by Freidoonimehr et al. [7]. Motivated by the previous-mentioned literature, the present study aims to extend the analysis of Freidoonimehr et al. [7] by emphasizing on the existence of dual solutions and the temporal stability of the solutions in a long run.

18.2 Basic Equations The current study considers a steady two-dimensional MHD stagnation point flow of an incompressible viscous and electrically conducting fluid that is driven by a permeable stretching/shrinking sheet. It is assumed that the velocity of the external flow is given by u e (x) = ax, while the velocity of the stretching/shrinking sheet is given by u w (x) = bx where a and b are the constants with b > 0 for stretching and b < 0 for shrinking. It is assumed that the temperature and the concentration at the surface are constants, denoted by Tw and Cw , respectively. The ambient (inviscid) temperature and the ambient concentration are also constants and represented by T∞ and C∞ , with an assumption that Tw > T∞ and Cw > C∞ . Apart from that, a uniform magnetic field of strength B0 is applied in the positive direction of the y-axis. In addition, it is also assumed that the magnetic Reynolds number and the induced magnetic field are small and negligible.

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Under these assumptions, the governing equations can be written as [7]:

u

∂v ∂u + =0 ∂x ∂y

(18.1)

∂u ∂ 2u du e ∂u σ B02 +v = ue +ν 2 − (u − u e ) ∂x ∂y dx ∂y ρ

(18.2)

∂T k ∂2T ∂T Q0 +v = + (T − T∞ ) ∂x ∂y ρc p ∂ y 2 ρc p

(18.3)

∂C ∂ 2C ∂C +v = D 2 − R(C − C∞ ). ∂x ∂y ∂y

(18.4)

u

u

subjected to the following boundary conditions: u = u w (x), v = vw , T = Tw , C = Cw at y = 0, u → u e (x), T → T∞ , C → C∞ as y → ∞.

(18.5)

where ν is the kinematic viscosity, σ is the electrical conductivity, ρ represents the fluid density, T is the fluid temperature, k is the thermal conductivity of the fluid, c p is the specific heat capacity at constant pressure, Q 0 represents the volumetric rate of heat generation or absorption, C is the concentration, D is the diffusion coefficient, and R denotes the reaction rate of the solute. On another note, the following similarity transformations are employed in order to solve Eqs. (18.1)–(18.4) along with the boundary conditions (18.5):  η=y

√ a T − T∞ C − C∞ , ψ = x aν f (η), θ (η) = , φ(η) = , (18.6) ν Tw − T∞ Cw − C∞

where η is the similarity variable, while ψ represents the stream function that is defined as u = ∂ψ/∂ y and v = −∂ψ/∂ x which identically satisfies Eq. (18.1). Using (18.6), Eqs. (18.2)–(18.4) are respectively transformed to f  + f f  − f 2 + 1 − M f  + M = 0

(18.7)

1  θ + εθ + f θ  = 0 Pr

(18.8)

1  φ − βφ + f φ  = 0 Sc

(18.9)

subjected to the transformed boundary conditions f (0) = s,

f  (0) = λ, θ (0) = 1, φ(0) = 1,

f  (η) → 1, θ (η) → 0, φ(η) → 0 as η → ∞,

(18.10)

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where λ represents the constant stretching/shrinking parameter, in which λ > 0 denotes the stretching, while λ < 0 refers to the shrinking sheet. In Eqs. (18.7)–(18.10), s indicates the constant mass flux parameter where s > 0 corresponds to suction and s < 0 for injection, M is the magnetic parameter, Pr is the Prandtl number, ε > 0 refers to the heat source parameter whereas ε < 0 is for the counterpart heat sink. Finally, Sc describes the Schmidt number and β denotes the reaction rate parameter. They are defined as: −vw b , s=√ , a aν ν R Sc = , β = . D a

λ=

M=

ν Q0 σ B02 , Pr = , ε = , aρ α aρc p (18.11)

The physical quantities of interest in the present study include the skin friction coefficient C f , the local Nusselt number N u x and the local Sherwood number Sh x which can be expressed as Cf =

τw , ρu 2e (x)

N ux =

xqw xqm , Sh x = k(Tw − T∞ ) D(Cw − C∞ )

(18.12)

where τw is the surface shear stress, while qw and qm respectively refer to the heat flux and the mass flux from the surface of the sheet which are respectively defined as:       ∂u ∂T ∂C , qw = −k , qm = −D . (18.13) τw = μ ∂ y y=0 ∂ y y=0 ∂ y y=0 Substituting Eq. (18.6) into Eq. (18.13) produce  −1/2 Re1/2 N u x = −θ  (0), Re−1/2 Sh x = −φ  (0) x C f = f (0), Rex x

(18.14)

where Rex = u e (x)x/ν is the local Reynolds number.

18.3 Flow Stability In the current study, the numerical results show the presence of dual solutions for Eqs. (18.7)–(18.9) which are subjected to the boundary conditions (18.10). Since two solutions exist for a single value of parameter, the temporal stability analysis is important to be conducted in order to reveal the solutions that are physically stable despite the fact that the solutions manage to fulfil the far field boundary conditions asymptotically. Generally, stability analysis can be attained by applying the disturbance over the time as well as recognizing the development of disturbance through the

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smallest eigenvalues. In this case, it is convenient to introduce the following dimensionless time variables, τ , by following the pioneered work performed by Merkin [13] and Merrill et al. [15]: √ √ u = ax f  (η, τ ), v = − aν f (η, τ ), ψ = x aν f (η, τ ),  a C − C∞ T − T∞ , φ(η, τ ) = , η=y θ (η, τ ) = , τ = at Tw − T∞ Cw − C∞ ν

(18.15)

Equations (18.7)–(18.9) for the unsteady case can be written as follows: ∂2 f ∂3 f + f − ∂η3 ∂η2



∂f ∂η

2 +1−M

∂2 f ∂f +M− =0 ∂η ∂η∂τ

(18.16)

1 ∂ 2θ ∂θ ∂θ − =0 + εθ + f 2 Pr ∂η ∂η ∂τ

(18.17)

1 ∂ 2φ ∂φ ∂φ − =0 − βφ + f 2 Sc ∂η ∂η ∂τ

(18.18)

which are subjected to the following boundary conditions: f (0, τ ) = s,

∂f (0, τ ) = λ, θ (0, τ ) = 1, φ(0, τ ) = 1, ∂η

∂f (η, τ ) → 1, θ (η, τ ) → 0, φ(η, τ ) → 0 as η → ∞. ∂η

(18.19)

According to Weidman et al. [21], the basic flow of the solutions f (η) = f 0 (η), θ (η) = θ0 (η), and φ(η) = φ0 (η) with the disturbances need to be perturbed in order to determine the stability of the solutions: f (η, τ ) = f 0 (η) + e−γ τ F(η), θ (η, τ ) = θ0 (η) + e−γ τ G(η), φ(η, τ ) = φ0 (η) + e−γ τ H (η)

(18.20)

where γ represents the eigenvalue, while F(η), G(η), and H (η) are small relative to f 0 (η), θ0 (η), and φ0 (η). Hence, the following system of linearized eigenvalue problems is obtained as a result of substituting Eq (18.20) into Eqs. (18.16)–(18.18):   F  + f 0 F  + f 0 F + γ − 2 f 0 − M F  = 0

(18.21)

1  G + f 0 G  + Fθ0 + (ε + γ )G = 0 Pr

(18.22)

1  H + f 0 H  + Fφ0 + (γ − β)H = 0 Sc

(18.23)

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along with the boundary conditions F(0) = 0, F  (0) = 0, G(0) = 0, H (0) = 0, F  (η) → 0, G(η) → 0, H (η) → 0 as η → ∞.

(18.24)

The stability of the system is determined by the smallest eigenvalue γ , say γ1 . Following Harris et al. [9] and Merrill et al. [15], without loss of generality, we set F  (0) = 1 to obtain the values of γ1 .

18.4 Results and Discussion The present study comprehensively performed the numerical computations for several values of magnetic parameter M, Prandtl number Pr, heat source/sink parameter ε, Schmidt number Sc, reaction rate parameter β, and suction parameter s on the velocity profiles, temperature profiles, concentration profiles, skin friction coefficient, local Nusselt number, and local Sherwood number. Equations (18.7)–(18.9) subject to the boundary conditions (18.10) were solved numerically using bvp4c function available in the MATLAB software as proposed by Shampine et al. [20]. In addition, all numerical computations are generated by setting the relative error tolerance to 10−7 with the configured convergence criterion in order to achieve accuracy up to six decimal places. Figures 18.1, 18.2 and 18.3 depict the variation of the skin friction coefficient f  (0), the local Nusselt number −θ  (0) and the local Sherwood number −φ  (0) versus λ for various values of M, which are M = 0.3, 0.5, and 0.8 at constant values: Pr = 1, ε = 0.5, Sc = 0.3, β = 0.4, and s = 2. As can be observed in those figures,

Fig. 18.1 Variation of the skin friction coefficient, f  (0) versus λ for various values of M when s=2

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Fig. 18.2 Variation of the local Nusselt number −θ  (0) versus λ for various values of M when Pr = 1, ε = 0.5, Sc = 0.3, β = 0.4 and s = 2

Fig. 18.3 Variation of the local Sherwood number −φ  (0) versus λ for various values of M when Pr = 1, ε = 0.5, Sc = 0.3, β = 0.4 and s = 2

dual solutions are possible when the sheet is shrunk, λ < 0, whereas the solution is unique when the sheet is stretched, λ > 0. Figure 18.1 shows the increment in magnetic parameter that contributes to the increment of the skin friction coefficient f  (0) due to the interaction between magnetic field and the electrically conducting fluid that forms a drag-like force known as Lorentz force. It is important to note that this force produces the retardation effect that can increase the velocity gradient, thus allowing it to be applied in the electromagnetic coating of wires. Figures 18.2 and 18.3 demonstrate the variations of the local Nusselt number −θ  (0) and the local Sherwood number −φ  (0) with λ for various values of M. As observed from these figures, two solutions are obtained for a certain range of the shrinking strength, λ < 0. On the other hand, the solution is unique for the stretching

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case, λ > 0. Solutions are possible up to λ = λc , which depends on M. These values of λc are given in Figs. 18.1 to 18.3. The range of λ for which the solution exists increases as the magnetic parameter M increases. On the other hand, Fig. 18.4 shows the velocity boundary layer thickness for the first solution is thinner compared to that of the second solution, which satisfies the far field boundary conditions asymptotically. Apart from that, Fig. 18.5 exhibits the concentration profiles for several values of Sc when Pr = 1, s = 2, ε = 0.5, β = 0.4, λ = −2.6, and M = 0.3. The finding revealed that the boundary layer thickness is reduced as the value of Schmidt number increases.

Fig. 18.4 Velocity profiles f  (η) for several values of s when λ = −2.6 and M = 0.3

Fig. 18.5 Concentration profiles φ(η) for several values of Sc when Pr = 1, s = 2,ε = 0.5, β = 0.4, λ = −2.6 and M = 0.3

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Fig. 18.6 Plot of the lowest eigenvalues γ1 versus λ for Pr = 1, ε = 0.5, Sc = 0.3, β = 0.4, s = 2 and M = 0.3

As previously discussed, dual solutions exist in this study for a certain range of the shrinking strength. In regard to this matter, a temporal stability analysis is performed by solving the linearized eigenvalue problems Eqs. (18.21)–(18.24) in order to identify the stable solution. Referring to Fig. 18.6, the first solution is found to be stable and physically reliable, whereas the second solution is revealed to be unstable as time passes, thus impractical in the real world applications. The stability of the solutions can be determined by finding the smallest eigenvalues γ in (18.20). The positive smallest eigenvalues signify the presence of an initial decay of disturbance, thus indicating that the flow is stable. On the other hand, the negative smallest eigenvalue represents a growth of disturbance which causes the flow to be unstable in a long run.

18.5 Conclusions The present study carried out an analysis of the MHD stagnation-point flow over a stretching/shrinking sheet in the presence of heat source/sink and chemical reaction effects. The governing nonlinear partial differential equations were reduced into a system of ordinary differential equations using similarity transformations, which were then numerically solved by utilizing the bvp4c function in the MATLAB software. The effects of the involved parameters on the skin friction coefficient, local Nusselt number, and the local Sherwood number were also discussed. The findings revealed that the increment of magnetic parameter and suction tended to result in the increment of the skin friction coefficient. However, the opposite trend occurred on the local Nusselt number and the local Sherwood number. In addition, it is worth to note that dual solutions were found to exist for a certain range of the shrinking

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strength. The temporal stability analysis revealed that only one of the solutions is stable and physically reliable in a long run. Acknowledgements The financial support received from the Universiti Kebangsaan Malaysia is gratefully acknowledged.

References 1. Bachok, N., Ishak, A., Pop, I.: Melting heat transfer in boundary layer stagnation-point flow towards a stretching/shrinking sheet. Phys. Lett. A 374, 4075–4079 (2010) 2. Bhattacharyya, K., Layek, G.C.: Chemically reactive solute distribution in MHD boundary layer flow over a permeable stretching sheet with suction or blowing. Chem. Eng. Commun. 197, 1527–1540 (2010) 3. Bhattacharyya, K., Arif, M.G., Pramanik, W.A.: MHD boundary layer stagnation-point flow and mass transfer over a permeable shrinking sheet with suction/blowing and chemical reaction. Acta Tech. 57, 1–15 (2012) 4. Chakrabarti, A., Gupta, A.S.: Hydromagnetic flow and heat transfer over a stretching sheet. Q. Appl. Math. 37, 73–78 (1979) 5. Crane, L.J.: Flow past a stretching plate. J. Appl. Math. Phys. (ZAMP) 21, 645–647 (1970) 6. Fisher, E.G.: Extrusion of Plastics. Wiley, New York (1967) 7. Freidoonimehr, N., Rashidi, M.M., Jalilpour, B.: MHD stagnation-point flow past a stretching/shrinking sheet in the presence of heat generation/absorption and chemical reaction effects. J. Braz. Soc. Mech. Sci. Eng. 38, 1999–2008 (2016) 8. Gupta, P.S., Gupta, A.S.: Heat and mass transfer on a stretching sheet with suction and blowing. Can. J. Chem. Eng. 55, 744–746 (1977) 9. Harris, S.D., Ingham, D.B., Pop, I.: Mixed convection boundary-layer flow near the stagnation point on a vertical surface in a porous medium: Brinkman model with slip. Transp. Porous Media 77, 267–285 (2009) 10. Jahan, S., Sakidin, H., Nazar, R., Pop, I.: Unsteady flow and heat transfer past a permeable stretching/shrinking sheet in a nanofluid: a revised model with stability and regression analyses. J. Mol. Liq. 261, 550–564 (2018) 11. Lok, Y.Y., Ishak, A., Pop, I.: MHD stagnation-point flow towards a shrinking sheet. Int. J. Numer. Methods Heat Fluid Flow 21, 61–72 (2011) 12. Mabood, F., Khan, W.A., Ismail, A.I.M.: MHD stagnation point flow and heat transfer impinging on stretching sheet with chemical reaction and transpiration. Chem. Eng. J. 273, 430–437 (2015) 13. Merkin, J.H.: On dual solutions occurring in mixed convection in a porous medium. J. Eng. Math. 20, 171–179 (1985) 14. Merkin, J.H., Najib, N., Bachok, N., Ishak, A., Pop, I.: Stagnation-point flow and heat transfer over an exponentially stretching/shrinking cylinder. J. Taiwan Inst. Chem. Eng. 74, 65–72 (2017) 15. Merrill, K., Beauchesne, M., Previte, J., Paullet, J., Weidman, P.: Final steady flow near a stagnation point on a vertical surface in a porous medium. Int. J. Heat Mass Transf. 49, 4681–4686 (2006) 16. Miklavcic, M., Wang, C.Y.: Viscous flow due to a shrinking sheet. Q. Appl. Math. 64, 283–290 (2006) 17. Mohd Nasir, N.A.A., Ishak, A., Pop, I.: Stagnation-point flow and heat transfer past a permeable quadratically stretching/shrinking sheet. Chin. J. Phys. 55, 2081–2091 (2017) 18. Othman, N.A., Yacob, N.A., Bachok, N., Ishak, A., Pop, I.: Mixed convection boundary-layer stagnation point flow past a vertical stretching/shrinking surface in a nanofluid. Appl. Therm. Eng. 115, 1412–1417 (2017)

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19. Pavlov, K.B.: Magnetohydrodynamic flow of an incompressible viscous fluid caused by the deformation of a plane surface. Magnitnaya Gidrodinamika 4, 146–147 (1974) 20. Shampine, L., Kierzenka, J., Reichelt, M.: Solving boundary value problems for ordinary differential equations in MATLAB with BVP4C. https://classes.engineering.wustl.edu/che512/ bvp_paper.pdf (2000) 21. Weidman, P.D., Kubitschek, D.G., Davis, A.M.J.: The effect of transpiration on self-similar boundary layer flow over moving surfaces. Int. J. Eng. Sci. 44, 730–737 (2006)

Chapter 19

Heat Transfer Analysis of Icing Process on Metallic Surfaces of Different Wettabilities Kewei Shi and Xili Duan

Abstract Superhydrophobic surfaces are promising in delaying ice formation and reducing ice accumulation, which can promote the safety and integrity of structures and human safety in harsh environments. This paper investigates heat transfer during the icing process on stainless steel surfaces of different wettabilities. The results demonstrate that a poorer wetting condition, i.e., (super)hydrophobic surfaces, or a smaller impact velocity leads to a smaller final contact area Ac between a water droplet and the cold surface. Also, the average cooling rate q is proportional to the final contact area Ac . Therefore, surface wettability can affect the changes on final contact area of water droplet, and the average cooling rate is influenced by the final contact area. In another word, the water droplet icing process can be delayed when it impacts on hydrophobic or superhydrophobic surfaces. Keywords Superhydrophobic surface · Icing · Heat transfer

19.1 Introduction Ice formation and accretion (accumulation) can cause many engineering problems. For example, it can diminish the stability of marine and offshore structures, leads to slipping threat, and contributes to breakdown of infrastructures and operational equipment [1, 2, 4, 11–13]. Many ice protection techniques have been investigated by researchers. One promising passive anti-icing techniques is designing a surface that has low surface energy which can repell water and prevent icing [5, 7, 8, 12]. However, hydrophobic surfaces are not always icephobic [3, 9]. The relationship between surface wettability and icephobicity is still unclear. The coupled droplet dynamics and heat transfer process from a water droplet to the surface has not

K. Shi · X. Duan (B) Faculty of Engineering and Applied Science, Memorial University of Newfoundland, 240 Prince Phillip Drive, St. John’s, NL A1B 3X5, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_19

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been fully understood [1, 2, 10]. This paper represents a simplified heat transfer analysis of low temperature water droplets impacting on stainless steel surfaces with different wettabilities, which can help better understand why the icing process has been delayed on hydrophobic surfaces.

19.2 Experimental Setup and Methods 19.2.1 The Experimental Setup and Procedure The experimental setup for droplet icing on surfaces of different wettabilities contains: a droplet generation system, a droplet temperature control system, a target surface temperature control system and a high-speed imagine system (Fig. 19.1). For details, please refer to [10]. The total icing time, t icing , [10] has been recorded via the high-speed imagine system. Five runs are conducted at each condition to minimize experimental uncertainties. After each run, the sample surface was washed by deionized water and completely dried before the next run.

Syringe Pump Drop generator

high-speed camera

Sample surface on adjustable base

PC

Fig. 19.1 Experimental setup

Temperature controller

LED Light and diffuser

19 Heat Transfer Analysis of Icing Process on Metallic Surfaces … Table 19.1 Uncertainies

Parameters

Uncertainty

T i and T w

±0.1 °C and ±1 °C

Di and t icing

±10 µm and ±0.5 ms

We and Wei

±0.5% and ±4%

QV and Q

±0.9% and ±1.9%

q and D*c

±1.9% and ±0.3%

203

19.2.2 Sample Surfaces of Various Wettabilities Seven machined or coated stainless steel (17-4PH) surfaces with various wettabilities have been tested. The details of surface fabrication, dimensions, static and dynamic contact angles of these samples can be found in [10].

19.2.3 Uncertainties Analysis The uncertainties include precision errors obtained from equipment manuals and random errors within multiple measurements. The method of Kline and McClintock [6] has been used to determine the uncertainties of average heat transfer rate, heat transfer coefficient and other dimensionless parameters. Table 19.1 represents the uncertainties in these experiments and heat transfer analysis.

19.3 Heat Transfer Analysis of the Icing Process Icing experiments on horizontal surfaces were conducted under different conditions. The temperatures of the sample surfaces were controlled at T w = −10 °C. The droplet initial diameters Di were 1.80 mm. The droplet impact speeds varied from 0.77, 0.99, to 1.17 m/s. The droplet initial temperature was kept at T i = 5 °C. All experiments were conducted under approximately the same room air temperature (15 °C) and relative humidity (30%). The cooling rate of a droplet on a surface, q, can be calculated by Eq. 19.1: q=

dQ = −U Ac (Tw − Td ) dt

(19.1)

where q is the freezing heat transfer rate, U is heat transfer coefficient, Ac is interfacial area, and T w and T d are the wall temperature of the substrates and the temperature of a droplet. The total heat transferred per volume of the droplet in the whole freezing process can be calculated by:

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QV =

Q = −ρC p (T f − Ti ) + ρ Hls Vd

(19.2)

where QV is the heat transferred per volume, V d is the volume of the droplet, ρ is the density of water, C p is specific heat capacity, H ls is the latent heat of freezing, and T f and T i are the final temperature and initial temperature of the droplet. Therefore, the average cooling rate q can be calculated by: q=

Q Q V · Vd = t t

(19.3)

where t is the total icing time of the droplet. The Weber number (We) indicates whether the kinetic or the surface tension energy is dominant. If the surface wettabilities are taken into consideration, the impact Weber number (Wei ) is introduced [10]. W ei =

ρv2 l We = 0.5(1 − cos θa ) 0.5σ (1 − cos θa )

(19.4)

where ρ is the density of the fluid (kg/m3 ), v is its velocity (m/s), l is its characteristic length, typically the droplet diameter (m), and σ is the surface tension (N/m), θ a is the advancing contact angle of the droplet on a surface. The final contact area Ac can be calculated by the final contact diameter, Dc . Therefore, the dimensionless final contact diameter D*c is introduced as follow: Dc∗ =

Dc Di

(19.5)

where Di is the initial diameter of a droplet before it impacts on a surface. Figure 19.2 shows the average cooling rates q varied with the final contact areas Ac of droplets on different target surfaces. q tends to increase while the contact areas Fig. 19.2 The average cooling rates of droplets icing changed with final contact areas

0.30 0.25

q

—

0.20 0.15 0.10 0.05

6

8

10

12

-6

Ac x10

14

16

19 Heat Transfer Analysis of Icing Process on Metallic Surfaces …

(a)

(b)

2.4

205

2.2 2.0

*

2.0 Dc

Dc

*

2.2 1.8 1.6 1.4 1.2 1.2

We=14.0 We=23.3 We=32.7

1.6 2.0

1.6

1.8

SCA(rad)

2.4

1.4 10

20

30 Wei

40

50

Fig. 19.3 a D*c of water droplets with different impact speeds changed with SAC (rad); b D*c of water droplets varied with Wei number

Ac is rising. This indicates that a higher contact area leads to a higher cooling rate, which is also indicated from Eq. 19.1. This contact area is the key for the cooling process of droplets icing on the cold surfaces, and it relates to the surface wettabilities and impact velocities of droplets. Figure 19.3 shows the variations of D*c with (a) the SCA (static contact angle of water droplets on target surfaces) and (b) the Wei number. D*c decreases when the SCA increases, i.e., a better wetting condition will lead to a larger final contact area. In another word, a water droplet will extend wider when it impacts on a surface with a better wettability, and the final contact area of a water droplet will be smaller when it impacts on a more hydrophobic surface. Also, higher Wei number leads to a larger D*c of water droplets. In addition, lower Wei means a lower impact speed on a same target surface or a droplet impact under same speed on a poor wetting surface. Figure 19.3b demonstrates that a lower Wei leads to a smaller D*c of water droplets, which means that a droplet impacts under same speed on a poor wetting surface will result in smaller final contact area. The experimental results demonstrate that hydrophobic or superhydrophobic surfaces can delay the icing process of a droplet, because the final contact area Ac of a water droplet on the surface decreases, which contributes to a relative lower cooling rate for these surfaces.

19.4 Conclusion The average cooling rate q is proportional to the final droplet-surface contact area Ac . A better wetting condition contributes to a larger final contact area Ac . This means hydrophobic or super-hydrophobic surfaces will have smaller contact areas and delay the cooling process of a droplet on it. Lower impact Weber number, due to lower impact velocity or a poor wetting surface, results in smaller D*c of water droplets, i.e.,

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smaller final contact area Ac . So, for droplets with the same impact velocity, a poor wetting surface will lead to smaller final contact area Ac and lower average cooling rate q, resulting in icing delay. Acknowledgements Financial support for this research from Petroleum Research Newfoundland & Labrador (PRNL) is gratefully appreciated.

References 1. Alizadeh, A., Bahadur, V., Zhong, S., Shang, W., Li, R., Ruud, J., Yamada, M., Ge, L., Dhinojwala, A., Sohal, M.: Temperature dependent droplet impact dynamics on flat and textured surfaces. Cit. Appl. Phys. Lett. 100, 111601 (2012) 2. Bahadur, V., Mishchenko, L., Hatton, B., Taylor, J.A., Aizenberg, J., Krupenkin, T.: Predictive model for ice formation on superhydrophobic surfaces. Langmuir 27(23), 14143–14150 (2011). https://doi.org/10.1021/la200816f 3. Hejazi, V., Sobolev, K., Nosonovsky, M.: From superhydrophobicity to icephobicity: forces and interaction analysis. Sci. Rep. 3(1), 2194 (2013). https://doi.org/10.1038/srep02194 4. Heyun, L., Xiaosong, G., Wenbin, T.: Icing and anti-icing of railway contact wires. Reliability and safety in railway, pp. 295–314 (2012) 5. Johnson, R.E., Dettre, R.H.: Wetting of low-energy surfaces. Marcel Dekker, New York (1993) 6. Kline, S.J., McClintock, F.A.: Describing uncertainties in single-sample experiments. J. Mech. Eng. 75(1), 3–8 (1953). https://doi.org/10.1016/j.chaos.2005.11.046 7. Kulinich, S.A., Farhadi, S., Nose, K., Du, X.W.: Superhydrophobic surfaces: are they really ice-repellent? Langmuir (2011). https://doi.org/10.1021/la104277q 8. Mishchenko, L., Hatton, B., Bahadur, V., Taylor, J.A., Krupenkin, T., Aizenberg, J.: Design of ice-free nanostructured surfaces based on repulsion of impacting water droplets. ACS Nano 4(12), 7699–7707 (2010). https://doi.org/10.1021/nn102557p 9. Nosonovsky, M., Hejazi, V.: Why superhydrophobic surfaces are not always icephobic. ACS Nano 6(10), 8488–8491 (2012). https://doi.org/10.1021/nn302138r 10. Pan, Y., Shi, K., Duan, X., Naterer, G.F.: Experimental investigation of water droplet impact and freezing on micropatterned stainless steel surfaces with varying wettabilities. Int. J. Heat Mass Transf. 129, 953–964 (2019). https://doi.org/10.1016/J.IJHEATMASSTRANSFER. 2018.10.032 11. Ryerson, C.C.: Ice protection of offshore platforms. Cold Reg. Sci. Technol. 65(1), 97–110 (2011). https://doi.org/10.1016/j.coldregions.2010.02.006 12. Sojoudi, H., Wang, M., Boscher, N.D., McKinley, G.H., Gleason, K.K.: Durable and scalable icephobic surfaces: Similarities and distinctions from superhydrophobic surfaces. Soft Matter (2016). https://doi.org/10.1039/c5sm02295a 13. Yamada, Y., Ikuta, T., Nishiyama, T., Takahashi, K., Takata, Y.: Droplet nucleation on a welldefined hydrophilic–hydrophobic surface of 10 nm order resolution (2014). https://doi.org/10. 1021/la503615a

Chapter 20

Artificial Force Free Boundaries: Particle-Based Fluid Simulation with Implicit Surfaces Yasutomo Kanetsuki and Susumu Nakata

Abstract We develop improved boundary conditions for particle-based fluid simulation with implicit surfaces. The implicit surfaces are well suited to represent complex obstacles, specifically, when the shape of the obstacles is smooth or varies with time. In existing particle-based fluid simulation with implicit surfaces, an artificially defined force is applied. Although this artificial force successfully enforces nonpenetration boundary conditions to the obstacles, some stability and low-accuracy issues remain. The instability of stacking and clustering, and low-accuracy pressure fields occur because of the artificially defined force and incompatibility of the effective radius when using particle methods. This study presents an improved boundary treatment for implicit surfaces. The proposed boundary conditions follow only the formulation of particle-based methods and do not require any artificial force. When we apply the discretization techniques of particle-based methods, the governing equations are also considered on implicit surfaces. Given that the proposed treatment of boundary conditions is applicable to any particle-based method, we adapted our formulation to both explicit and implicit schemes. The results of the test show that our method does not suffer from instability near the boundary in comparison with existing methods. Moreover, the proposed method can also produce better pressure fields than the existing ones. Keywords Particle-based method · Smoothed particle hydrodynamics · Moving particle semi-implicit · Fluid simulation · Implicit surfaces

Y. Kanetsuki (B) Graduate School of Information Science and Engineering, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, Shiga 525-8577, Japan e-mail: [email protected] S. Nakata College of Information Science and Engineering, Ritsumeikan University, 1-1-1 Noji-Higashi, Kusatsu, Shiga 525-8577, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_20

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20.1 Introduction We develop improved boundary conditions for particle-based fluid simulation in this study. In particle-based fluid simulation methods, both fluids and obstacles are generally represented by particles. Although fluids are represented by particles in the proposed simulation, we employ implicit surfaces to represent obstacles. Implicit surfaces [1] are one of the representation techniques of objects used in computer graphics. When we employ particles to represent obstacles, forming even a simple shape is not easy because of the particle patterns. [2] developed a method to address this issue; however, this method is only applicable to planar boundaries. Meanwhile, implicit surfaces can accurately generate smooth shapes from point sets [3], polygons [4], and user interactions [5]. Thus, these surfaces do not suffer from particle patterns. In addition, because implicit surfaces can simply change its shape, time-varying boundaries are also easily applicable. Implicit surfaces were adopted in particle-based fluid simulation by [6]. In this technique, the same artificial force as polygon models [7] was employed. Although the existing method [6] successfully simulates fluid motion with complex obstacles, some stability and accuracy issues remain. In [6, 7], the artificial force moves the fluid particles to a certain distance to the obstacles when the particles approached the obstacles. This force can enforce non-penetration boundary conditions. However, the definition is unphysical, and the effective radius is incompatible to particle methods. We propose boundary conditions following the discretization of particle methods. The proposed boundary conditions do not require any artificial force and are applicable to any particle-based simulation method. To verify the flexibility of our method, we apply the proposed technique to both explicit and implicit formulations. As the explicit method, we use smoothed particle hydrodynamics (SPH) [8]. Meanwhile, we utilize moving particle semi-implicit (MPS) [9] as the implicit method. The test results show that our method can improve the stability and accuracy compared with the existing method.

20.2 Boundary Conditions with Artificial Forces in the Existing Method Here, we explain the existing boundary conditions for SPH and MPS. The artificial force is defined to ensure that the fluid particles do not penetrate to the obstacles. In both methods, we assume that fluids are discretized by particles as x t1 , x t2 , . . . , x tN at time t. We can simulate the motion of fluids by solving the Navier-Stokes equations. Dv 1 = − ∇ p + ν∇ 2 v + g, Dt ρ

(20.1)

20 Artificial Force Free Boundaries …

209

where v is velocity, ρ is density, p is pressure, ν = μ/ρ is the viscosity coefficient, and g is gravity acceleration. Meanwhile, obstacles are represented by implicit surfaces in d-dimensional domains as f (x) = 0, x ∈ Rd .

20.2.1 SPH for Implicit Surfaces with Artificial Forces In SPH with implicit surfaces, Eq. 20.1 can be discretized as  v it+t

=

v it

+ t

   2  1 − ∇ p + ν∇ v i + g , ρ i

x it+t = x it + tv it+t , where t is the time interval. Nakata and Sakamoto [6] divided the pressure and viscosity terms into the contributions from the fluids and obstacles. 

 1 = + − ∇p , ρ i i



  2  2  2v , ν∇ v i = ν∇ v + ν∇

1 − ∇p ρ











− ρ1 ∇ p i

(20.2)



i







− ρ1 ∇ p i



(20.3)

i







1  and ν∇ v are the contributions of the fluids and − ∇p ρ 2



and where i i

  2 v are the contributions of the obstacles. The terms affected by the fluids can ν∇ i be simply estimated using the SPH as 

1 − ρ1 ∇ p = − t i ρi









ν∇ 2 v = i

μ ρit

mj

j∈ifluid , j=i

j∈ifluid , j=i

mj

v tji ρ tj

pit + p tj 2ρ tj

∇Wijt ,

∇ 2 Wijt ,

where ifluid is the index set of the fluid particles within the effective radius of x it , m j

is mass, Wijt = W x it − x tj is the kernel function, and v tji = v tj − v it . We employ the cubic spline kernel described in [10]. For the effect of the obstacles, given that we represent obstacles as implicit surfaces, no particles are present on and inside the obstacles. To address this issue, [6] employed an artificial force for the pressure term and assumed the ghost particles whose physical quantities are the same in the effective region.

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Fig. 20.1 Ghost particles arranged along the tangential plane



 1 − ∇p ρ



i

  l −  x it − x inear,t  x t − x inear,t  it , = ·  x − x near,t  t 2 i i

 μm W v it 1 2  t  2v = − ν∇ ∇ W x i − x˜ j , t i ρi ρ0 ghost

(20.4)

(20.5)

j∈i

where l is the initial spacing, x inear,t is the nearest ghost from the fluid particle x it ghost (Fig. 20.1), m W is the mass of the ghost, i is the index set of the ghost particles x˜ j within the effective radius, and ρ0 is the rest density. Herein, we assume that the obstacles are static, the density of the obstacles ρ0 is constant, and the ghost particles are arranged along the tangential at  x inear,t .  t planenear,t  < l, The pressure term (Eq. 20.4) is affected if the distance satisfies  x i − x i whereas the viscosity term (Eq. 20.5) has effects within the effective radius of SPH. For the quantities that include x inear,t in their definitions, [6] utilized the characteristics of implicit surfaces. By applying the technique described by [11], the distance to the closest point on the obstacles can be approximated using the following equation:   t  f x    t  x − x near,t  =   i  . i i ∇ f x t  i

(20.6)

Given that the gradients of the function are  perpendicular  to the tangential plane of the boundary, the direction x it − x inear,t / x it − x inear,t  can be approximated as follows:     ∇ f x it f x it x it − x inear,t  t  =   t  ·   t  . (20.7)  f x  ∇ f x   x − x near,t  i i i i

20 Artificial Force Free Boundaries …

211

To evaluate the density ρit , [6] also divided the summation of SPH into two. ρit = ρˆit + ρ˜it , where ρˆit is the contribution of the fluids, and ρ˜it is the contribution of the obstacles. ρˆit can be simply evaluated with SPH as ρˆit =

m j Witj

j∈ifluid

and ρ˜it can be obtained using the ghost particles as follows: ρ˜it = m W

  W x it − x˜ j .

(20.8)

ghost

j∈i

The pressure pit is evaluated using the equation of state in SPH as the following:     pit = max Cs2 ρit − ρ0 , 0 ,

(20.9)

where Cs is the sound speed. The sound speed in SPH is set slower than the actual one, and we determine Cs in accordance with [12]. ghost in Eqs. 20.5 and 20.8 is Given that the summationof the kernelfunction for i near,t  t  , the values are efficiently obtained using the function of the distance x i − x i precomputed lookup tables constructed using the ghost particles arranged along the flat plane [6]. Therefore, the ghost particles are not required in the simulation process; they are only used during precomputation.

20.2.2 MPS for Implicit Surfaces with Artificial Forces In MPS with implicit surfaces, Eq. 20.1 can be discretized as    v i∗ = v it + t ν∇ 2 v i + g , x i∗ = x it + tv i∗ ,   1 v it+t = v i∗ + t − ∇ p , ρ i   1 t+t ∗ 2 xi = x i + t − ∇ p . ρ i When we divide the pressure and viscosity terms into the contributions from the fluids and obstacles similar to Eqs. 20.2 and 20.3, the contributions of the fluids are simply evaluated using the MPS as

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− ρ1 ∇ p i

d =− ρ0 n 0

2d ν∇ 2 v = ν i λn 0



∗

j∈ifluid , j=i

p ∗j − pi ∗ ∗  2 x ji Wij ,  ∗ x ij 

v tji Witj ,

j∈ifluid , j=i ∗

where n 0 is the rest particle number density, pi = min



j∈ifluid

 p ∗j , and λ is the Laplacian

coefficient. In MPS, a different kernel function described by [9] is employed. For the effect of the obstacles, the pressure term is artificially defined similar to Eq. 20.4, and the viscosity term is approximated as follows: 

   l −  x i∗ − x inear,∗  x ∗ − x inear,∗ 1  i∗ , − ∇p = ·  x − x near,∗  ρ t 2 i i i 

  2d  2 v = −νv t ν∇ W x it − x˜ j . i i λn 0 ghost

(20.10) (20.11)

j∈i

the pressure term (Eq. 20.10) is also affected if the distance satisfies  ∗Here,near,∗  < l, while the viscosity term (Eq. 20.11) is affected within the effecx − x i i tive radius of MPS. To evaluate the pressure, the following pressure Poisson equation is used.   ρ0 n i∗ − n 0 1 ∗ ∇ pi =− 2 − p . t n0 ρ0 Cs2 i





2

(20.12)

The Laplacian of the pressure is also divided into two as 

 

 ∇2 p i = ∇2 p +  ∇2 p .

i

i

The contribution of the fluids is simply discretized using MPS as 

2d ∇2 p = i λn 0



pji∗ Wij∗ ,

j∈ifluid , j=i

where pji∗ = p ∗j − pi∗ . For the contribution of the obstacles, the following equation can be derived from Eq. 20.10.  ∗   

  x − x near,∗   x ∗ − x near,∗  l − 2ρ 0 i i i i  . ∇2 p = i t 2 λ

(20.13)

20 Artificial Force Free Boundaries …

213

The particle number density n i∗ is computed in the same manner as the other values. We divide the contributions into two as n i∗ = nˆ i∗ + n˜ i∗ , where the term from the fluids is discretized as

Wi∗j . nˆ i∗ = j∈ifluid , j=i

Meanwhile, the one from the obstacles is discretized as follows:

  W x i∗ − x˜ j . n˜ i∗ =

(20.14)

ghost j∈i

ghost

in Eqs. 20.11 and Given that the summations of thekernel function  for i 20.14 are the function of the distance  x it − x inear,t  or  x i∗ − x inear,∗ , the values are efficiently computed using the precomputed lookup tables constructed by employing the ghost particles arranged along the flat plane in the same manner as the SPH. Therefore, the ghost particles are not required in the simulation process; they are only used during precomputation.

20.3 Improvement of Boundary Conditions Without Artificial Forces Here, we develop an improved boundary condition for implicit surfaces. In the previous section, we explained the existing methods for implicit surfaces that employ SPH and MPS. Although the existing methods successfully combine particle methods with complex obstacles using implicit surfaces, some stability and low-accuracy issues remain. The problems of the existing artificial force are as follows: • The artificial pressure term is not physical because the fluid particles lose their energy when they approach the obstacles, the force is determined to be independent on the state around the fluid particle, and the force includes t in its definition. This condition causes the stacking and clustering problems along the boundaries. • The effective radius of the artificial pressure term is not compatible with that of particle methods, thereby deteriorating the pressure distribution. To address these problems, we propose the formulation to estimate the physical quantities in accordance with the particle methods. This new formulation does not involve the artificially defined force. Furthermore, the proposed formulation can be derived using fewer assumptions than the existing method.

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The artificial pressure term is defined in the existing methods because the pressure of obstacles cannot not be easily estimated. Hence, we develop a technique to estimate the pressure on the obstacle. The evaluated obstacle pressure is applied to the original pressure term.

20.3.1 Improved SPH for Implicit Surfaces To avoid using the artificial pressure term (Eq. 20.4), we develop a technique to estimate the pressure of obstacles in SPH. In our method, the same pressure term to that of fluids is also used for the obstacles. 

  p t + pinear,t ∇W x it − x˜ j , − ρ1 ∇ p = −m W i t near,t i 2ρi ρi ghost



(20.15)

j∈i

where pinear,t and ρinear,t are the pressure and density of the obstacles, respectively. In the existing method, because pinear,t and ρinear,t cannot be simply evaluated, the artificial pressure term is defined. Although the viscosity term (Eq. 20.5) assumes that ρinear,t = ρ0 in the existing method, we use the evaluated density ρinear,t in the viscosity term for consistency. 

 μm W v it 2  t  2v = − ∇ W x i − x˜ j . ν∇ near,t t i ρi ρi ghost

(20.16)

j∈i

In Eqs. 20.15 and 20.16, we assume that the physical quantities of the obstacles are constant in the effective region of each fluid particle at each time step, similar to the existing method. To evaluate pinear,t and ρinear,t , the particles on and inside the obstacles are required. Because we represent the obstacles with implicit surfaces, no particles are necessarily present there. To address this problem, we formulate the evaluation method of pinear,t and ρinear,t at x inear,t . Then we use the same values, pinear,t and ρinear,t , for other ghost particles within the effective radius of x it . First, we estimate x inear,t using   x t − x inear,t . x inear,t = x it −  x it − x inear,t   it  x − x near,t  i i By substituting Eqs. 20.6 and 20.7 into the above equation, we can derive x inear,t

=

x it

   t  ∇ f x it − f x i   2 . ∇ f x t  i

20 Artificial Force Free Boundaries …

215

Second, we estimate ρinear,t at this point. To evaluate ρinear,t , we divide the value into the contributions from the fluids and obstacles. ρinear,t = ρˆinear,t + ρ˜inear,t where the effect of the fluids can be estimated with the fluid particles around x inear,t . ρˆinear,t =

  m j W x inear,t − x tj ,

ifluid j∈

ifluid indicates the set of the indices of the fluid particles within the effective where  radius of x inear,t . For the computation of ρ˜inear,t , we assume that the ghost particles are arranged along the flat plane. From this assumption, ρ˜inear,t can be expressed as follows:

  W x inear,t − x˜ j , (20.17) ρ˜inear,t = m W i j∈

ghost

i indicates the set of the indices of the ghost particles within the effective where  radius of x inear,t . By substituting ρinear,t into Eq. 20.9, we can evaluate pinear,t . ghost ighost in Eqs. 20.8, and  The summation of the kernel function for i 20.15–20.17 can be precomputed using the ghost particles, similar to the existing methods.During actual  simulation, these values can be evaluated as functions of the distance  x it − x inear,t  or constant. ghost

  ⎧ ρ˜it = m W A  x it − x inear,t  ⎪ ⎪

 near,t  x t −x near,t ⎪  t ⎪ i ⎨ − 1 ∇ p = m W pi +t pnear,t B  x it − x inear,t   x it −x inear,t  ρ 2ρ ρ i i i

, i i  W t  2 v = − μm v i C  x t − x near,t  ⎪ ν∇ ⎪ near,t t i i ⎪ ρ ρ ⎪ i i i ⎩ ρ˜inear,t = m W D

where ⎧      ⎪ A x it − x inear,t  = W x inear,t − x˜ j ⎪ ⎪ ghost ⎪ j∈i ⎪ ⎪  t      ⎪ near,t ⎪ = ∇W x near,t − x˜ j  ⎪ B xi − xi ⎪ i ⎨ ghost i  t  j∈  .  near,t = ⎪ C  xi − xi ∇ 2 W x inear,t − x˜ j ⎪ ⎪ ghost ⎪ j∈i ⎪ ⎪   near,t  ⎪ ⎪ ⎪ D= W xi − x˜ j ⎪ ⎩ ghost j∈i

Here, we divide the length and direction of the gradient summation as described by [13].

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20.3.2 Improved MPS for Implicit Surfaces To demonstrate the flexibility of our proposed method, we also apply the proposed technique to MPS. Instead of utilizing the artificial pressure term (Eq. 20.10), we use the pressure term discretized by MPS as follows: 

 1 − ∇p ρ

∗

where pi = min





∗

=− i

 near,∗  pinear,∗ − pi ∗ ∗ x n˜ i , − x  ∗  i i 2 ρ0 n 0  x i − x inear,∗ 

 p ∗j , pinear,∗ . Similar to SPH, we obtain the pressure pinear,∗ at the

j∈ifluid point x inear,∗ ,

boundary and then set pinear,∗ for all x˜ j . To consistently evaluate pi∗ and pinear,∗ , we construct pressure Poisson equations at both x i∗ and x inear,∗ . The Poisson equation of the fluid x i∗ can be constructed in the same technique as Eq. 20.12. However, we use

  2d  near,∗  − pi∗ n˜ i∗ pi ∇2 p = i λn 0 instead of Eq. 20.13 in accordance with the formulation of MPS. The Poisson equation of the obstacle can be similarly constructed as follows:   near ρ0 n inear,∗ − n 0 1 near,∗ , − p ∇2 p i = − 2 t n0 ρ0 Cs2 i



 near where ∇ 2 p i is the Laplacian of the pressure at x inear,∗ and n inear,∗ is the particle number density of x inear,∗ . The Laplacian can be divided into the contributions of the fluids and obstacles as

near  2 near  2 near  +  ∇2 p . ∇ pi = ∇ p

i

i

We discretize both terms following the MPS formulation. Because we assume

near  2 = 0. Then, we obtain the that the pressure of all ghost particles is equal, ∇ p i

following equation. 

   near 2d  ∗ ∇2 p i = p − pinear,∗ W x inear,∗ − x ∗j . λn 0 fluid j j∈i

In summary, the pressure is computed using the following equations. 

 2d ∗ 1 2d pi∗ − ni + 2 2 λn 0 t Cs λn 0

j∈ifluid , j=i

Wij∗ p ∗j −

n∗ − n0 2d ∗ near,∗ n˜ i pi = ρ0 i 2 , λn 0 t n 0

20 Artificial Force Free Boundaries …



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   n near,∗ − n 0 2d near,∗ 1 2d pinear,∗ − nˆ i + W x inear,∗ − x ∗j p ∗j = ρ0 i 2 . 2 2 λn 0 t Cs λn 0 fluid t n 0 j∈i

In the above equations, we rearrange the pressure at the left-hand sides. The particle number density n inear,∗ can be evaluated in the same manner as ρinear,t in SPH. n inear,∗ =



    W x inear,∗ − x ∗j + W x inear,∗ − x˜ j .

ifluid j∈

i j∈

ghost

i in the above equation is constant The summation of the kernel function for  because we assume that the ghost particles are arranged along the flat plane. ghost

20.4 Results We confirm the performance of our proposed methods. For all of the tests, we used the neighbor search algorithm described by [14]. First, we compare the results of the existing method (explained in Sect. 20.2.1) and proposed method (developed in Sect. 20.3.1) in SPH. Figure 20.2 shows the shape of the obstacles used in our SPH tests. We employed f (x) = 0.45 −



x 2 + y2 + z2

for implicit surfaces in Test 1, where x = (x, y, z) ∈ R3 , and then used f (x) = min(x, 1 − x, y, z, 0.3 − z) in Test 2. Both tests were implemented in three dimensions. The fluid particles were located inside the surfaces, and the numbers of fluid particles were 95,442 and 42,000

Fig. 20.2 Obstacles used in our test cases with SPH. The one at the left is used in Test 1, and the one at the right is used in Test 2

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Fig. 20.3 Results of Test 1 using the existing method (left column) and proposed method (right column). The bottom row shows the enlarged view of the top row

in Test 1 and 2, respectively. The particles were arranged with a distance of 0.01, and the effective radius was set to 0.0133. We used 0.001 for both m i and m W , and μ = 1. Figure 20.3 shows the results of Test 1. When using the existing method, we can observe that the fluid particles stack along the boundary. This stacking problem is caused by the artificially defined force that simply moves the particles to the distance l from the boundary. Owing to the definition of the artificial force, the fluid particles cannot escape from the boundary. Meanwhile, the proposed method produced the natural free surface of the fluid particles. Because we eliminated the artificial force and use the formulation in accordance with SPH, the fluid particles can be naturally pushed by the obstacles. Figure 20.4 illustrates the results of Test 2. When using the existing method, the fluid particles are violent around the corner of the boundary. Meanwhile, the proposed method generates smooth motion of the fluid particles at the same place. Thus, we can confirm that the proposed method remarkably improves the stability. Subsequently, we compared the performance of the existing method (explained in Sect. 20.2.2) and the proposed method (developed in Sect. 20.3.2) in MPS. We used Bi-CGStab [15] with diagonal scaling to solve the Poisson equations, and the matrix was stored in ELLpack [16]. Figure 20.5 shows the comparison of the results of the existing and proposed methods in one dimension. The test results demonstrate that the fluid particle loses all momentum energy when it approaches the boundary owing to the artificial force. This enforces the fluid particles not to escape from the

20 Artificial Force Free Boundaries …

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Fig. 20.4 Results of Test 2 using the existing method (left column) and proposed method (right column). The bottom row shows the enlarged view of the top row

boundary and causes the stacking problems. However, the proposed method naturally reflects the fluid particle. Moreover, we compared the particle distribution and pressure distribution of both methods. We apply a similar condition to Test 1. However, the simulation was implemented in two dimensions. The obstacle is defined as f (x) = 0.512 − x 2 − y 2 , where x = (x, y) ∈ R2 . Figure 20.6 depicts the particle distribution in both methods. A total of 1907 fluid particles were used for each method. According to the results of the existing method, we can confirm that the particles stacked along the boundary. Moreover, the fluid particles caused the clustering around the boundary and generate the gap between the first and second layers of the fluid particles from the boundary. However, these instability problems were addressed when the proposed method was used. Figure 20.7 illustrates the comparison of the pressure fields obtained using the existing and the proposed methods. In this test case, 3867 fluid particles filled half in the same obstacle were simulated. The entire fluid region of the existing method contained non-negligible noise, whereas the proposed method eliminated the noise

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Fig. 20.5 Results of the one-dimensional free-fall test. The top row shows the results when using the existing method, whereas the bottom row illustrates the results when using the proposed method. The blue particle is the fluid particle, the green line is the distance l from the obstacle, and the brown line is the obstacle boundary. The simulation is increasing from left to right

Fig. 20.6 Comparison of the existing and the proposed methods in two-dimensional test. The left figure was obtained using the existing method, and the right was obtained using the proposed method

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Fig. 20.7 Comparison of the pressure fields obtained from the existing method (left) and the proposed method (right). The pressure values are colored with red (high) to blue (low)

in most parts. On the basis of these results, we can confirm that the proposed method successfully improves the accuracy. Some noise can be observed around the boundary even in the proposed method. This noise is caused by the assumption of the flat plane, and we believe that the method proposed by [17] resolves this problem by considering the actual shape of the boundary.

20.5 Conclusion We have developed improved boundary conditions for particle-based fluid simulation without an artificial force. The existing methods used for implicit surfaces suffer from instability and low-accuracy issues because of the artificially defined force. We derived the formulations that follow only the discretization of SPH and MPS, and then successfully eliminate the artificial force. During the derivation of our formulation, we did not add new assumptions; we only applied the assumptions used in the existing methods. The test results showed that our proposed technique can resolve the instability issues that occurred in the existing methods. In addition, we demonstrated that the proposed method remarkably eliminated the noise in the fluid. These results indicate that our method improved stability and accuracy during the particle-based fluid simulation. Acknowledgements This work was supported by JSPS KAKENHI Grant Numbers JP00351320 and JP17J00443.

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References 1. Berger, M., Levine, J.A., Nonato, L.G., Taubin, G., Silva, C.T.: A benchmark for surface reconstruction. ACM Trans. Graph. 32(2), Article No. 20 (2013) 2. Band, S., Gissler, C., Teschner, M.: Moving least squares boundaries for SPH fluids. In: Jaillet, F., Zara, F. (eds.) Workshop on Virtual Reality Interaction and Physical Simulation. The Eurographics Association, Geneva (2017) 3. Kazhdan, M., Hoppe, H.: Screened poisson surface reconstruction. ACM Trans. Graph. 32(3), Article No.29 (2013) 4. Yngve, G., Turk, G.: Robust creation of implicit surfaces from polygonal meshes. IEEE Trans. Visual Comput. Graph. 8(4), 346–359 (2002) 5. Schmidt, R., Wyvill, B., Sousa, M.C., Jorge, J.A.: ShapeShop: sketch-based solid modeling with BlobTrees. In: ACM SIGGRAPH 2007 Courses, Article No. 43. ACM, San Diego (2007) 6. Nakata, S., Sakamoto, Y.: Particle-based parallel fluid simulation in three-dimensional scene with implicit surfaces. J. Supercomput. 71(5), 1766–1775 (2015) 7. Harada, T., Koshizuka, S., Kawaguchi, Y.: Improvement in the boundary conditions of smoothed particle hydrodynamics. Comput. Graph Geom. 9(3), 2–15 (2007) 8. Müller, M., Charypar, D., Gross, M.: Particle-based fluid simulation for interactive applications. In: Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 154–159. Eurographics Association, San Diego (2003) 9. Koshizuka, S., Oka, Y.: Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl. Sci. Eng. 123(3), 421–434 (1996) 10. Monaghan, J.J.: Smoothed particle hydrodynamics. Rep. Prog. Phys. 68(8), 1703–1759 (2005) 11. Taubin, G.: Distance approximations for rasterizing implicit curves. ACM Trans. Graph. 13(1), 3–42 (1994) 12. Becker, M., Teschner, M.: Weakly compressible SPH for free surface flows. In: Proceedings of the 2007 ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 209–217. Eurographics Association, San Diego (2007) 13. Bender, J., Koschier, D.: Divergence-free smoothed particle hydrodynamics. In: Proceedings of the 14th ACM SIGGRAPH/Eurographics Symposium on Computer Animation, pp. 147–155. ACM, Los Angeles (2015) 14. Hoetzlein, R.C.: Fast fixed-radius nearest neighbors: interactive million-particle fluids. In: GPU Technology Conference 2014, Santa Clara (2014) 15. van der Vorst, H.A.: Bi-CGStab: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992) 16. Bell, N., Garland, M.: Efficient sparse matrix-vector multiplication on CUDA. NVIDIA Technical Report NVR-2008-004 (2008) 17. Kanetsuki, Y., Nakata, S.: Improved wall boundary conditions with implicitly defined walls for particle based fluid simulation. In: ECCM ECFD 2018, Glasgow (2018)

Chapter 21

Droplet-Falling Impact Simulations by Particle-Based Method Kazuhiko Kakuda, Wataru Okaniwa and Shinichiro Miura

Abstract In this paper, we present the application of the GPU-based particle simulations to droplet-falling impact problem including free surfaces with surface tension and drag force. The particle approach is based on the MPS (Moving Particle Semiimplicit) method using logarithmic weighting functions. We adopt the inter-particle potential force model with a potential coefficient as a surface tension model. The GPU-implementation consists mainly of the search for neighboring particles in the locally uniform grid cell using hash function and solving the Poisson equation with respect to the pressure fields. Numerical results demonstrate the workability and validity of the present approach through the droplet-falling impact with surface tension and drag force. Keywords Particle method · MPS · GPU · Droplet-falling impact · Surface tension · Drag force

21.1 Introduction In the simulation-based fields of science and engineering, there are various gridless/meshless-based particle methods, such as SPH (Smoothed Particle Hydrodynamics) method [2, 10], and MPS (Moving Particle Semi-implicit) one [6, 8] to simulate effectively complicated fluid flow problems. Recently, the physics-based computer simulations on the GPU (Graphics Processing Units) [3, 4] have increasingly become an important strategy for solving efficiently various problems, such as fluid dynamics, solid dynamics, and so forth. In our previous work, we have presented a GPU-based MPS scheme using logarithmic weighting function for solving effectively 2D/3D problems of incompressible fluid flow [5]. The purpose of this paper is to present the application of the GPU-based MPS method to 3D complicated fluid flow problem, namely, the droplet-falling impact K. Kakuda (B) · W. Okaniwa · S. Miura Nihon University, Narashino, Chiba 275-8575, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_21

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[1, 9] with surface tension and drag force. As the surface tension model, we adopt the inter-particle potential force with a potential coefficient [7]. The GPUimplementation consists mainly of the search for neighboring particles in the locally uniform grid cell using hash function [3] and solving the Poisson equation with respect to the pressure fields. The workability and validity of the present approach are compared with experimental data [9].

21.2 MPS Formulation The particle interaction models of the MPS as illustrated in Fig. 21.1a are prepared with respect to differential operators, namely, gradient, divergence and Laplacian [8]. The incompressible viscous fluid flow is calculated by a semi-implicit algorithm, such as SMAC (Simplified MAC) scheme. For the standard MPS formulation, the selection of a weighting function is a key factor in the particle-based framework. If the distance r between the coordinates r i and r j is close, then there is a possibility that the computation fails suddenly with unphysical numerical oscillations. Therefore, in order to stabilize such spurious oscillations generated by the standard MPS strategy, we consider the following logarithmic weighting function as shown in Fig. 21.1b.  w(r ) =

log 0

 re  r

(0 < r < re ) (re ≤ r )

(21.1)

It is important to take into account the surface tension and the drag forces in order to solve accurately the MPS simulation, such as droplet-falling impact with the free surface of a liquid fluid. As the surface tension, we use conveniently the following inter-particle force at particle i:

(a) Particle interaction models (3D)

(b) Profiles of weighting functions

Fig. 21.1 Particle interaction models and weighting functions

21 Droplet-Falling Impact Simulations by Particle-Based Method

f isurf =

225

 C(r − r0 )(r − h) r ij mi r i= j

(21.2)

  where r =  r ij  is the distance between two particles i and j, r0 is the initial distance between two particles, m i is the mass at particle i, and the potential force coefficient C is explicitly given in the literature [7]. On the other hand, the drag force vector at particle i is aerodynamically expressed as drag

fi

=

mi 1 ρi C D Si ui ui  ρi 2

(21.3)

where Si is the projection area of particle i, and C D is the drag coefficient.

21.3 Numerical Example In this section, we present numerical results obtained from applications of the abovementioned MPS approach to the droplet-falling impact involving surface tension and drag force. The initial velocities in this problem are assumed to be zero everywhere in the interior domain. We set also the gravity of 9.8 m/s2 .

21.3.1 Droplet-Falling Impact Simulation Table 21.1 gives the summary of the parameters for the droplet-falling impact problem. The droplet-falling impact on liquid surface has been investigated experimentally, theoretically and computationally by many researchers [1, 9]. It is well-known Table 21.1 A summary of the parameters (droplet-falling impact)

Liquid surface area

6.25 × 10−4 m2

Liquid depth

0.007 m

Droplet radius

0.001035 m

Droplet initial height

0.2 m

Number of total particles

8,86,432

Number of fluid particles

5,38,742

Initial distance of two particles

0.0002 m

Density

1032 kg/m3

Viscosity coefficient

0.0025 Pa s

Surface tension coefficient

0.002361 N/m

Time increment (Δt)

0.00005 s

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(a) t = 0.20441[s]

(c) t = 0.25691[s]

(b) t = 0.21441[s]

(d) t = 0.21716[s] (side-view)

Fig. 21.2 Droplet-falling impact simulation (droplet initial height: 0.20 m, 1.93 m/s)

that such impact leads to the forming of crown-like structure so-called milk-crown phenomena and so forth, under some conditions. In order to reproduce such phenomena, the surface tension is dominant, and there are also the influence of viscosity, impact velocity and liquid depth. Figure 21.2 shows the geometrical configuration and the particle behaviors at different time and viewpoint for the droplet initial height of 0.20 m. For the maximum cavity depth of the droplet, the relative error between our result, 4.280 mm, as shown in Fig. 21.2d and the experimental data [9], 4.115 mm, is 4.01%.

21.4 Conclusions We have presented the GPU-based MPS approach for solving numerically 3D complicated fluid flow problem, namely, the droplet-falling impact with surface tension and drag force. As a surface tension model, we have adopted the inter-particle potential force model with a potential coefficient. The agreement between the present result and the experimental data appears also satisfactory.

References 1. Fedorchenko, A.I., Wang, A.-B.: On some common features of drop impact on liquid surfaces. Phys. Fluids 16(4), 1349–1365 (2004) 2. Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181, 375–389 (1977)

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3. Green, S.: Particle simulation using CUDA, NVIDIA Corporation (2010) 4. Harada, T., Masaie, I., Koshizuka, S., Kawaguchi, Y.: Accelerating particle-based simulations utilizing spatial locality on the GPU. Trans. JSCES (in Japanese), Paper No. 20080016 (2008) 5. Kakuda, K., Obara, S., Toyotani, J., Meguro, M., Furuichi, M.: Fluid flow simulation using particle method and its physics-based computer graphics. CMES Comput. Model Eng. Sci. 83(1), 57–72 (2012) 6. Khayyer, A., Gotoh, H.: Modified moving particle semi-implicit methods for the prediction of 2d wave impact pressure. Coast. Eng. 56, 419–440 (2009) 7. Kondo, M., Koshizuka, S., Suzuki, K., Takimoto, M.: Surface tension model using inter-particle force in particle method. In: Proceedings of the 5th Joint ASME/JSME Fluids Engineering Conference (FEDSM2007), FEDSM-37215 (2007) 8. Koshizuka, S., Oka, Y.: Moving-particle semi-implicit method for fragmentation of incompressible fluid. Nucl. Sci. Eng. 123, 421–434 (1996) 9. Liow, J.L.: Splash formation by spherical drops. J. Fluid Mech. 427, 73–105 (2001) 10. Lucy, L.B.: A numerical approach to the testing of the fission hypothesis. Astron. J. 82(12), 1013–1024 (1977)

Chapter 22

Numerical Modeling of Bridge Piers Scouring Flow Patterns Mahdi Alemi, João Pedro Pêgo and Rodrigo Maia

Abstract Numerical models, compared to the physical models, provide real-scale predictions and are cost-effective. However, the reliability of the numerical results depends on how the governing equations are mathematically solved. This paper summarizes details of a numerical model, developed under the scope of the study of bridge piers local scour, which can be applied in different river engineering applications. The paper reports applications of the numerical model, namely the prediction of the turbulent flow behavior around (i) a single bridge pier, (ii) a compound pier (a circular pier on a circular foundation) and (iii) a complex bridge pier (a pile-supported pier) on a scoured bed: case (i) and (iii) at the equilibrium stage and case (ii) at an intermediate stage of the local scour hole development process. The corresponding numerical results are in good agreement with those obtained from referenced experimental and numerical data. Further, concerning the complex bridge pier case, the study results go beyond the background knowledge available for this case. Keywords Numerical modeling · River engineering · Bridge piers · Scoured bed

22.1 Introduction Bridges are important and vital structures which are exposed to many natural hazards due to the river environment, namely local scouring. In general, the local pier scour occurs as a result of the flow-sediment-pier interaction. Therefore, a good understanding of the flow mechanism is a fundamental step to estimate the accurate scour depth around piers what is of major importance in river engineering. In recent years, numerical modeling has been increasingly used to investigate bridge pier related flow and riverbed disturbances. In fact, compared to the classical physical models, by means of current powerful and affordable computers, numerical models are able to present further results and at lower costs. Moreover, numerical modeling avoids M. Alemi · J. P. Pêgo · R. Maia (B) Faculdade de Engenharia, Departmento de Engenharia Civil, Universidade do Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_22

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issues related to scaling of experimental results. Hence, this study was aimed to contribute to the numerical investigation of the flow features around bridge piers. However, it is noteworthy to mention that numerical models have limitations associated to the domain size, the corresponding grid cells number and the simulation time. It is well known that the turbulent flow structure around bridge piers is complex, including flow separation, reattachment, and vortices. Moreover, the flow has irregular boundaries with smooth/rough surfaces. Therefore, selecting adequate solution methods is essential to simulate the flow features properly. For the bridge pier flow predictions, an in-house numerical model was developed that solves the set of the flow equations in a simple and accurate way. Details of the developed model and the corresponding numerical results are hereafter summarized. The work presented here will serve as a basis for future broader analysis of the local scour hole dimensional and flow features around complex bridge piers. Further, the developed model will be improved to study different river engineering problems, namely open channel flow confluences.

22.2 Numerical Model Details The required key features for the numerical model were: (i) accurate prediction of the flow structure; (ii) easy generation of the grid for complex geometry problems; and (iii) reasonable computational time for the numerical predictions without employing a large number of computational resources. In summary, the developed numerical model solves the space-filtered NavierStokes and continuity equations (LES Smagorinsky model) in the Cartesian grid system using a fractional-step method. In addition, a wall function was incorporated into the model that provides the approximate wall boundary conditions, helping to reduce the computational cost compared to the sole use of the LES model. Since the numerical model uses a Cartesian grid, the pier and bed geometries were described by means of the Fractional-Area-Volume-Obstacle-Representation method [1]. The implicit pressure equation was solved with the successive over-relaxation method and the parallelization of the calculations was achieved by using the FORTRAN OpenMP library. Temporal discretization was performed by the second-order Adams-Bashforth scheme. Concerning the spatial terms, the convection terms were approximated by the QUICK scheme (for stability reasons) and all remaining spatial terms were approximated by the second-order central difference scheme. Overall, the numerical model is second-order accurate in both space and time.

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22.3 Applications 22.3.1 Single Bridge Pier on a Scoured Bed The turbulent flow around a circular pier at the equilibrium stage of the local scour hole process was predicted for the pier geometry and flow conditions (ReD = 67,500 and Reh = 81,000) of the experiment conducted by Graf and Istiarto [2], helping to assess the capability of the developed numerical model to capture the main flow features around a single bridge pier, which are already well known and documented. The corresponding time-averaged flow structure at the vertical symmetry plane, upstream and downstream of the pier, is shown in Fig. 22.1. The predicted main flow features are the down-flow in front of the pier, the horse-shoe vortex system at the pier’s base and the wake flow behind the pier. According to the present results, the approach flow is vertically diverted in front of the pier. The corresponding down-flow is deflected upstream and downstream at the scour hole front bed bottom, forming a large and strong vortex V2 with a small junction counter-rotating vortex V1. The incoming flow is also separated at the upstream rim of the scour hole due to the change in the bed surface slope, that resulting in the formation of a small clockwise vortex (V3). A counter-clockwise vortex (V4) attached to the bottom was predicted just downstream of V3. These two last vortices are weak and may not support the sediment transport as strongly as V2 or V1 does. Similar physical observations and numerical predictions were reported by Graf and Istiarto [2] and by Kirkil et al. [3]. Downstream of the pier, a reverse and upward flow was predicted just behind the pier, as expected, its horizontal length being maximum at the top domain boundary. After that, the flow is mostly directed upwards what fosters sediment transport downstream of the scour hole. Reverse flow length at top boundary

Flow

Fig. 22.1 Time-averaged flow structure at the vertical symmetry plane, single bridge pier

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22.3.2 Compound Pier on a Scoured Bed The turbulent flow around a compound pier with scoured bed, developed after 7 h from the beginning of the scour hole process (intermediate stage), was numerically predicted for the same pier geometry and flow conditions (Reh = 44,800) of the experiment conducted by Kumar and Kothyari [4]. The compound pier model consists of a circular pier with diameter 0.114 m supported by a circular foundation with diameter 0.21 m for a configuration where the top surface of the foundation was located below the initial channel bed level. For this case, numerically predicted velocity profiles and velocity components fluctuations are compared with the ones obtained from the reference study at different longitudinal positions on the symmetry plane. Figure 22.2 pictures the corresponding numerical and experimental profiles of the time-averaged longitudinal velocity component (u-velocity) ¯ at six different longitudinal positions on the symmetry plane. From it, one can clearly see that, overall, the present numerical results are in agreement with those obtained from the reference study, although u-velocity ¯ values at x = 0.14 m are under-predicted over the full water depth. The difference is maximum at a region close to the bed and is minimized when moving towards the top domain boundary. In other words, the longitudinal length of the reverse flow (negative u-velocity) ¯ region close to the bed was numerically predicted larger than at the experimental study. Nevertheless, the predicted size at the top domain boundary is

Fig. 22.2 Comparison of the present numerical results (u-velocity ¯ values) with the experimental results at different positions x = −0.17, −0.14, −0.1, 0.14, 0.25 and 0.4 m (measured from the pier center), compound pier case

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Fig. 22.3 Comparison of the present numerical results (w-velocities ¯ and root-mean-square values of the velocity components fluctuations) with the experimental results at different positions downstream of the compound pier

in good agreement with the physical observation of Kumar and Kothyari [4]. To note that corresponding experimental and numerical w-velocity ¯ (time-averaged vertical velocity component) values and root-mean-square values of the velocity components fluctuations at x = 0.14 m, as well as at x = 0.25 m are in good agreement, as shown in Fig. 22.3.

22.3.3 Complex Bridge Pier on a Scoured Bed Nowadays, most modern bridges are usually built on piers with foundations of complex geometries (e.g., pile-supported piers). In this case, the flow structure is largely affected by the interaction between the pier elements. In this study, the flow structure around a complex bridge pier on the scoured bed (developed after 72 h from the beginning of the scouring process) was numerically predicted for the same conditions of the experiment conducted by Beheshti and Ataie-Ashtiani [5]. The pier case consists of a column and a pile cap supported by a group of piles, a common geometry of the complex piers, for a situation where all pier elements are exposed to the approaching flow. Overall, the present numerical study provided further details about the flow field structure compared to the reference experimental study, helping to better understand the flow mechanism around the selected complex pier on the scoured bed. The time-averaged 3-D streamlines around the selected complex pier are shown in Fig. 22.4a. Moreover, the time-averaged streamlines at the vertical plane that

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(a) 5

3 4 4 2

1: Horse-shoe vortex 2: Recirculation region 3: Vertical side-vortex 4: Horizontal tube vortex 5: Wake vortex

1

(b)

Pile cap

Fig. 22.4 Time-averaged streamlines: a around the complex bridge pier; b at the vertical plane containing the four in-line piles axes

contains the four in-line piles axes, are presented in Fig. 22.4b. According to the present results, above the pile cap, the incoming flow is deflected vertically and sideways by the column and a vertical side-vortex is formed immediately after the upstream side edges of the column. Downstream of that vortex, on top of the pile cap surface, a horizontal vortex is observed along the column side. Concerning the flow deflection in front of the column, the corresponding down-flow interacts with the pile cap, resulting in the formation of a small vortex above the pile cap. The pile cap also deflects the incoming flow vertically and sideways. The flow separates at the lower upstream edge of the pile cap and a flow recirculation occurs immediately after that edge below the pile cap. The flow also separates at the upstream side edges of the pile cap and a very small vortex is formed immediately after those edges at each side of the pile cap. Downstream of it, a horizontal tube vortex was predicted along the pile cap sides. In fact, due to the formation of the above referred

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vertical side-vortex along the column sides, the incoming flow directly above the pile cap is diverted towards the pile cap side faces, resulting in the formation of the mentioned tube vortex. Below the pile cap, the flow also separates at the piles surface and reverse flow regions with variable sizes are formed behind the piles. Inside the scour hole, the horse-shoe vortex system was numerically predicted as a large vortex associating several smaller vortices. Downstream of the complex pier, the wake flow is detected behind the column. Concerning the pile cap, two counter-rotating vortices are observed just downstream of it at the vertical plane considered, which are the result of the interaction between the outflow from the pile group zone and the flow above the pile cap.

22.4 Conclusions A numerical model was developed to solve the flow equations set in domains with complex geometry boundaries in a simple and accurate way. The numerical model was applied to predict the turbulent flow around single, compound and complex bridge piers on the scoured bed. The model enabled to characterize the most relevant flow features of the studied cases and the corresponding results were in good agreement with referenced experimental and numerical results. The numerical study results encompass zones where experimental measurements are difficult or impossible, what is much relevant for the complex pier case as, to the authors’ knowledge, there is no numerical study of the flow field on such a complex pier on the scoured bed. Overall, the study demonstrated the high suitability of the present numerical model to similar investigations on complex geometries on any bed shape. Acknowledgements The authors thank FCT—Fundação para a Ciência e a Tecnologia, Portugal for the funding to this research with project reference UID/Multi/04423/2019.

References 1. Hirt, C.W., Sicilian, J.M.: A porosity technique for the definition of obstacles in rectangular cell meshes. In: 4th International Conference on Numerical Ship Hydrodynamics. Washington, DC (1985) 2. Graf, W.H., Istiarto, I.: Flow pattern in the scour hole around a cylinder. J. Hydraul. Res. 40, 13–20 (2002). https://doi.org/10.1080/00221680209499869 3. Kirkil, G., Constantinescu, S.G., Ettema, R.: Coherent structures in the flow field around a circular cylinder with scour hole. J. Hydraul. Eng. 134, 572–587 (2008). https://doi.org/10. 1061/(ASCE)0733-9429(2008)134:5(572)

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4. Kumar, A., Kothyari, U.C.: Three-Dimensional flow characteristics within the scour hole around circular uniform and compound piers. J. Hydraul. Eng. 138, 420–429 (2012). https://doi.org/10. 1061/(ASCE)HY.1943-7900 5. Beheshti, A.A., Ataie-Ashtiani, B.: Scour hole influence on turbulent flow field around complex bridge piers. Flow Turbul. Combust. 97, 451–474 (2016). https://doi.org/10.1007/s10494-0169707-8

Chapter 23

Estimation of Electric Field Between the Capillary and Wire-Netting Electrodes During the Electrostatic Atomization from Bio-emulsified Fuel Chien-hua Fu , Osamu Imamura , Kazuhiro Akihama and Hiroshi Yamasaki Abstract The electrospray is an injection system which is using the high electric potential to cause the liquid atomization without traditional pressure force. This investigation reports the difference of electrode distances between the capillary and wire-netting effect on electrostatic atomization from bio-emulsified fuel. The wellknown biofuel, such as biodiesel fuel (BDF), which is expected as carbon neutral fuel and contained water remained in the production phase, thus, making the biodiesel as emulsified fuel can take advantage of water fractions to provide low temperature condition (LTC) and result in the reduction of nitrogen oxide (NOx) emission. In addition, the viscosity of emulsified fuel is depending on the class of surface-active agent, nevertheless the emulsified fuel has the large electric conductivity which can be used in electric potential supplied to control and atomize into several particles. In this paper, the breakup and atomization process of bio-emulsified fuel was investigated with the different electrode distances to estimate the effects of electric field. The atomization mode and injection angle is reported in deal. Furthermore, as the electric field distribustion is estimated with the droplet movement and effect of electric field on the atomization mode is dicussed in this paper. Keywords Electrostatic atomization · Emulsion · Biodiesel fuel

23.1 Introduction In the last few decades, the fossil fuel is supporting our social infrastructure and economy. However, the fossil fuel produces a large amount of carbon dioxide (CO2 ) to make global warming occurred. As the result, the global warming which rises in average temperature [1] of earth and makes the climate changed. To satisfy our C. Fu Graduate School of Industrial Technology, Nihon University, Chiba 275-8575, Japan O. Imamura (B) · K. Akihama · H. Yamasaki College of Industrial Technology, Nihon University, Chiba 275-8575, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_23

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energy needs, the Kyoto protocol and Paris agreement provide the platform to solve and communicate the issue with each other between the major industrialized and developing countries. In the current system of alternative energy supplied, there are solar, wind, ocean current, biomass, biofuels and nuclear plant, especially the nuclear plant is efficiency energy provider and be a CO2 free technology [2]. Nevertheless, the nuclear plant still cannot be a favorable energy provider to humankind, especially after the 2011 Fukushima disaster [3]. Alternative fuel, such as biodiesel fuel (BDF), which has already used in transprotation sector and can classified as first generation, second generation and third generation [4]. In the combustion area, the biodiesel is be a high cetane number and oxygen content to improve the thermal efficiency [5, 6] and it can also result the reduction of carbon monoxide (CO), hydrocarbon (HC), particular matter (PM) and Polycyclic aromatic hydrocarbons (PAHs). However, it will produce the higher emission of nitrogen oxides (NOx) than the conventional diesel fuel [7, 8] because the property of high oxygen content which can result in high combustion temperature [9–14]. Furthermore, comparing to the fossil diesel, the biodiesel still has some shortcomings need to be solved, such as lower energy content, cold start problem and high viscosity etc. The water diesel emulsion has been introduced around 1970s [15] and it further version, biodiesel blends which is composed of fatty acid methyl/ethyl esters [7], had some advantages on combustion in diesel engine which can trade-off the emission between reductions of CO and PM, but it will lead to increase NOx emissions [16–18]. The major mechanism of water emulsions results in micro-explosion phenomena, whereby it can control the overall boiling temperature during the combustion and create a vapor expansion break up the droplets into smaller droplets which is called secondary atomization. The major benefit of secondary atomization is that enhances air and fuel mixing to improve combustion efficiency and lead the emission reduction. According to Ithnin et al. [19] water in diesel emulsions can reduce NOx and PM concurrently as well as induce improved combustion efficiency without engine modifications. In addition, the finer emulsion droplets, the greater HC, CO reductions with even better engine performances, and it leads to comparatively greater reduction in NOx emissions which is reported by Attia et al. [20]. To mixture the water and diesel in emulsions fuel, the surfactants must be added in the emulsions to minimize the coalescence mechanism of the water phase as well as reducing the surface tension between the diesel and water phases which is reported by Ghannam and Selim [21]. In the recent papers, the results show that NOx and PM which have been proven to be responsible for respiratory and lung problems in people [22]. Concerning to the NOx emission, there are two major control techniques, which is called exhaust gas recirculation (EGR) and low temperature combustion (LTC). Regarding to the LTC, the main strategy is that control the timing of injection for large quantities of exhaust gases [23, 24], hence lower cylinder temperature and lengthens the ignition delay will be occurred, which translates to improved fuel evaporation and homogeneities. Consequently, the LTC can suppress NOx and soot formation [25–28] by improving the mixture between fuel and air during the combustion in diesel engine. This concept

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is like the characteristics of water-diesel emulsifications. The water emulsions create the heat sink effect and lead to reduction of NOx formation rates which is reported by Fahd et al. [29]. According to the Lin et al., they founded that adding the water into diesel fuel can reduce PM emission greatly [30, 31] and lower cylinder temperature will be occurred, which translating less pyrolysis reactions to form cenospheres. Additionally, the emulsified fuel has a potential to be a NOx-PM trade off fuel, which is observed by Chang et al. and Huo et al. [32, 33], then Subramanian [34] reported that adding water into diesel can also reduce NOx and PM emissions. At the earlier of 20th century, the electrostatic atomization theory was described by Zeleny in 1915 [35, 36]. However, the first, who calculated and found the drop is no longer be a stable under the electric force, is Lord Rayleigh in the end of 19th century [37]. An electrospray is an injection system which is using high electric potential to cause liquid atomization without any traditional pressure forces. As shown in Fig. 23.1, the high voltage power supply is set between the capillary and wire-netting electrodes to make the electric force, which is called field intensity, then the electrostatic field will be established between the capillary and wire-netting. Typically, a capillary is charged at a high electric potential and liquid is passed through it. Through electrophoresis, the positive ions in the solution are driven toward a region of lower potential, hence the liquid in the capillary will break into instability and several fine droplets, which contains the positive charge, down to the wirenetting electrode (negative electrode), while the force of positive ion is high enough to overcome the energy barrier to form the liquid stream between the liquid and surrounding environment. Increasing the high electric potential, the spray mode will be changed to establish different jet morphologies. Once the electric potential applied, the forces due to the fluid inertia meet or exceed the force due to surface tension, then additional ligaments break off from the original jet. From this backgorund, the paper aims to the use of Water/Biodiesel fuel for the combustion and application of electricstatic atomization to Water/Biodiesel emulsion is considerded. In this paper, fundameneal characteristics of electrostatic atomization

Fig. 23.1 The mechanism of electrospray

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of model Water/Biodiesel emulsion is investigated experimentally. As the electrostatic atomiozation of Water/Biodiesel, which is non-newtonian fluid, has not known well, the characteristic of electrostatic atomiozation is reproted in detail. Since the electric field estimation during the electric atomization is considered to be an important factor, the estimation of electic field is dicussed in this paper.

23.2 Experimental 23.2.1 Apparatus and Water-Biodiesel Fuel Emulsification The schematic diagram of the experiment is shown in Fig. 23.2. The electrospray is formed by stainless steel capillary, nitrogen gas container and high electric potential supply. The capillary was commercially purchased with an inner diameter of 170 µ and connected to high electric potential supply which was charged by GS50 N Model (Green-Techno Co. Ltd.). As shown in Fig. 23.2, the steel mesh (negative electrode) was set upward the capillary (positive electrode), whereby, the mechanism of electrospray results in the electrostatic field between the steel mesh and capillary. The nitrogen gas container was giving the 0.05 MPa pressure to push the emulsion fuel towards the capillary through the small tube. During the spraying, the Phantom high-speed camera was used to capture the atomization behavior under the maximum 120000fps with 128 × 128 resolution. The still images and high definition movies were taken by Nikon D7100 camera which is capable of 24 megapixels (6000 × 3348) of resolution, Hi2.0 ISO (= ISO25600), and the maximum 8000fps with the Nikon lens (AF-S DX NIKKOR 18-300 mm f/3.5-6.3G ED VR) throughout the experiment.

Fig. 23.2 The schematics of experimental apparatus

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To mixture the water and diesel fuel blend in emulsified fuel, the surfactant must be added in the emulsion to minimize the coalescence mechanism of the water phase as well as reducing the surface tension between the diesel fuel and water phases. In this study, the Sorbitan Monooleate (C24 H44 O6 ), which is also called Rheodol, was added to the biofuel and pure water as a surface-active agent. Then, using the small dispersing instrument (IKA T10 basic ULTRA-TURRAX) to make it emulsified. The component ratio of emulsified fuel is Oleic Methyl Ester blend 70%, pure water 27% and Sorbitan Monooleate 3%. In addition, Oleic Acid Methyl Ester is a major blend of BDF in water addition to result in the emulsified fuel.

23.3 Results and Discussions 23.3.1 Definition and Motion of Spray Mode Before discussing the properties of electrospray, the modes of spray must be defined. In this study, three different modes can be identified as mode A, B and C which is shown in Figs. 23.3, 23.4 and 23.5. These typical modes show the shape of electrostatic atomization for Water/Biodiesel emulsion at various applied voltages. For instance, the mode A was occurred at lower electric potential supplied such as 4.5–7 kV and mode B was occurred at a little higher electric potential supplied such as 8–15 kV with 30 mm electrode distance. Moreover, the mode C was occurred at high electric potential supplied such as 16–24 kV with 30 mm electrode distance. The experimental fuel is Water/Biodiesel emulsion which has the original physical properties than the other well-known fuels and it gives some difficulties to define the modes as other papers, thus, the definition of address on modes may differ with Fig. 23.3 Mode A (dripping)

Fig. 23.4 Mode B (oscillating jet)

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Fig. 23.5 Mode C (multi-jet)

some reports in this study. The mode A is also called as dripping mode here, which its particle was rotating during spraying, and mode B can also have called as Oscillating jet mode here, which the top of liquid thread was oscillating to cut off and separate the liquid thread into particles. Differ from the mode A and B, the mode C has two jets during the spraying and each jet was oscillating to release the particles which is also called as multi-jet mode here in this study. To figure out the physical properties of three modes of Water/Biodiesel emulsion, the motion of secondary breakup in each mode is roughly estimated. In this study, the secondary breakup is defined as that the cut off liquid thread atomized into several particles. One of typical movements of secondary breakup in each mode is shown in Figs. 23.6, 23.7 and 23.8 with 60 mm electrode distance. In the condition of 60 mm electrode distance, the mode A was presented from 4.5 kV to 7 kV and its liquid thread did not become thinner and it took around 0.004167 s to form the large and several small particles in secondary breakup process at 7 kV which is shown in Fig. 23.6. At 9 kV, the jet of electrospray was changed into mode B and its liquid thread starts swinging and become thinner and took around 0.00139 s to separating the liquid thread into particles as shown in Fig. 23.7.

Fig. 23.6 The motion of secondary breakup at 7 kV (60 mm) (Mode A)

Fig. 23.7 The motion of secondary breakup at 9 kV (60 mm) (Mode B)

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Fig. 23.8 The motion of secondary breakup at 13 kV (60 mm) (Mode C)

Moreover, charging further electric potential supplied at 13 kV, the jet was changed into mode C and its length of liquid thread becomes shorter and shit into two unstable jets. At 13 kV, there has two jets occurred at the same time, but it has an order to break the liquid thread into particles. For example, in Fig. 23.8, right-hand side of jets has a longer length and the breakup occur from the thinner tip of liquid thread with swinging to the other side. However, the left-hand side of jet has the same coordinate in next breakup process. As the result, the mode C has an order to atomize particles in each jet. Referring to the physical motion of mode C, in Fig. 23.8, each jet took around 0.0008 s to form the particles. From these images, the breakup into particles process seems to depend on the water droplet diameter in emulsion and the time of secondary breakup becomes shorter with the increase in applied voltages which is shown in Fig. 23.13.

23.3.2 Injection Angle of Electrospray Figure 23.9 shows the spray angle of Water/Biodiesel emulsion under the electrostatic force. As shown in Fig. 23.9, the liquid thread is spraying at right-hand side and breakup in secondary at the tip of liquid thread. In this study, the spray angles of Fig. 23.9 The mechanism of electrospray

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Fig. 23.10 The spray angle in various electrodes

various electrode distances are estimated to understand the electrospray’s behaviors ◦ which is shown in Fig. 23.10. Referring to Fig. 23.10, the spray angle of θ1 is 15 which was resulted in 30 mm electrode distance at 10 kV. However, the electrospray has the behavior that the spray will swing as the circle to make the symmetrical angle, θ2 , occurred at next step of breakup process. For example, in Fig. 23.7, the motion of spray is breakup and prepared next breakup process at right-hand side first around 0.03056 s. Then, its movement is getting to central at 0.03139 s and breakup at left-hand side at the end around 0.03195 s. Moreover, in Fig. 23.9, the particles are atomizing upward in swinging which are following the blue heavy line in the image. The dashed blue line illustrate that the behavior of particles will be resulted in next step of breakup process. In the study, this behavior of electrospray can correspond to each electric potential supplied at various electrode distances and the relationship between each electrode distance is shown in Fig. 23.10. From this point of view, the area of atomized particles and spray direction can be forecasted and has a potential for use on valuable controlling combustion system. The shape of electrospray will be particularly as a circular cone in the case of Water/Biodiesel emulsion. Figure 23.10 illustrates the relationship between the spray angles, θ1 , and various electrode distances from Water/Biodiesel emulsion. As shown in Fig. 23.10, each electrode distance has similar electrospray characteristics which results in the magnitude of spray angles increase from mode A to mode B. Then, the spray angles will be the unstable curve at mode C.

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Fig. 23.11 Effect of electric field on Water/Biodiesel emulsion

23.3.3 Effect of Electric Field Between the Electrodes Figure 23.11 shows the modes of electrospray in each electric potential supplied with various electrode distances. The spray mode A, B, and C shown in Figs. 23.3, 23.4 and 23.5 are summarized as the function of applied voltage V and electrode distance D. From Fig. 23.11, spray mode seems not to be the function of V /D but to be the function V 2 /D. This indicates that the electric field driven by the flying droplets should be considered because V /D corresponds to representative electric field strength without movement of electric charges between the electrodes. For the droplet, it is assumed to balance ween electric force qE and Stokes’ Law of resistance and to be 1-D movement, then, q E = 3π ηdu where q is the electric charge of droplet, E is the electric field strength, η is the viscosity coefficient of the gas, d is droplet diameter, and u is the velocity of droplet. If the number density of droplet, n, is introduced, the current density, j, is j =u

1 n q = ρe2 E n 3π η d

where ρ e is the electric charge density. If j is integrated in plane parallel to the negative electrode, the electric current is obtained. Here we used the assumption that the electric current is a function of ρ 2e E even though particle size distribution is different. The electric potential between the electrodes is calculated using the Poisson’s equation under the condition that the electric current is uniform between

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the electrodes. Figure 23.12 shows the calculation result of electric potential and electric field strength with and without droplet movement. If there is no droplet movement the electric field is uniform because of 1D model, but the electric field increases with the distance from the anode if the droplet movement is considered. Figure 23.13 shows the comparison of electric current and electric field strength in the condition of V 2 /D = 4.9 MV2 /m. The comparison case is V = 14 kV when D = 40 mm and V = 17.1 kV when D = 60 mm. The values at 2 mm from the anode is used for comparison. Electric charge and current is normalized the case of Fig. 23.12. Figure 23.13 shows that electric field strength is different in the D = 40 and 60 mm condition. However the electric current is almost the same if the electric charge density is small enough. If the electric charge density depends on the flow rate, charge density is expected to be independent of electrode distance. From this result, it is expected that the electric current is dominant factor for spray mode observed in this study.

23.4 Conclusions The fundameneal characteristics of electrostatic atomization of model Water/Biodiesel emulsion is investigated experimaltally. The characteristic of electrostatic atomization is investigated in detail and the estimation of electic field is dicussed. The key conclusions are summarized as follows:

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1. For Water/Biodiesel emulsion, the electrostatic atomization mode can be divided into 3 modes in this study. The first mode is similar to dripping mode and the other modes have long liquid thread. The latter is divided to two modes from their behaviour of liquid. 2. The injection angle increases with the increase in the electric potential. This is related to the change of the mode. 3. From the experimental results with various electrode distances, spray mode seems not to be the function of V /D but to be the function V 2 /D. From the calculation results of electric field with the droplet movement, not the electric field but the electric current seems to be the key factor to determine the atomization mode.

References 1. IPCC: Climate change 2007: synthesis report. In: Pachauri, R.K., Reisinger, A. (eds.) Core Writing Team. IPCC, Geneva, Switzerland (2007) 2. Baruch, J.J.: Combating global warming while enhancing the future. Technol. Soc. 30, 111–121 (2008) 3. Mori, S., Miyaji, K., Kamegai, K.: CCS, nuclear power and biomass an assessment of option triangle under global warming mitigation policy by an integrated assessment model MARIA23. Energy Proc. 37, 7474–7483 (2013) 4. No, S.-Y.: Inedible vegetable oils and their derivatives for alternative diesel fuels in CI engines: a review. Renew. Sustain. Energy Rev. 15, 131–149 (2011) 5. Rakopoulos, D.C., Rakopoulos, C.D., Fiakoumis, E.G., Dimaratos, A.M., Kyritsis, D.C.: Effects of butanol-diesel fuel blends on the performance and emissions of a high-speed DI diesel engine. Energy Convers. Manage. 51, 1989–1997 (2010) 6. Demirbas, A.: Importance of biodiesel as transportation fuel. Energy Policy 35, 4661–4670 (2007) 7. Wang, H., Deneys Reitz, R., Yao, M., Yang, B., Jiao, Q., Qiu, L.: Development of an n-heptanen-butanol-PAH mechanism and its application for combustion and soot prediction. Combust. Flame 160, 504–519 (2013) 8. Chang, Y.-C., Lee, W.-J., Wang, L.-C., Yang, H.-H., Cheng, M.-T., Lu, J.-H., et al.: Effects of waste cooking oil-based biodiesel on the toxic organic pollutant emissions from a diesel engine. Appl. Energy 113, 631–638 (2014) 9. Mwangi, J.K., Lee, W.J., Whang, L.M., Wu, T.S., Chen, W.H., Chang, J.S., et al.: Microalgae oil: algae cultivation and harvest, algae residue Torre faction and diesel engine emission test. Aerosol. Air Qual. Res. (2015) 10. Lee, W.-J., Liu, Y.-C., Mwangi, F.K., Chen, W.-H., Lin, S.-L., Fukushima, Y., et al.: Assessment of energy performance and air pollutant emissions in a diesel engine generator fueled with water-containing ethanol-biodiesel-diesel blend of fuels. Energy 36, 5591–5599 (2011) 11. Wang, X., Cheung, C., Di, Y., Huang, Z.: Diesel engine gaseous and particle emissions fueled with diesel-oxygenate blends. Fuel 94, 317–323 (2012) 12. Mwangi, J.K., Lee, W.-J., Tsai, J.-H., Wu, T.S.: Emission reductions of nitrogen oxides, particulate matter and polycyclic aromatic hydrocarbons by using microalgae biodiesel, butanal and water in diesel engine. Aerosol. Air Qual. Res. 15, 901–914 (2015) 13. Lapuerta, M., Armas, O., Rodriguez-Fernandez, J.: Effect of biodiesel fuels on diesel engine emissions. Prog. Energy Combust. Sci. 34, 198–223 (2008) 14. Lim, C., Lee, J., Hong, J., Song, C., Han, J., Cha, J.-S.: Evaluation of regulated and unregulated emissions from a diesel-powered vehicle fueled with diesel/biodiesel blends in Korea. Energy 77, 533–541 (2014)

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Chapter 24

Research on Wind Field of Transmission Tower Under the Complex Terrain Qiang Shengpei, Qi Fei and Gao Qiang

Abstract In order to explore the influence of complex terrain on the wind field around the transmission tower, this paper numerically simulates the complex terrain of the location of the collapsed tower in Zhenjiang, China. Based on the Computational Fluid Dynamics method, the large eddy simulation turbulence model is built to solve the fluid domain, and the flow field distribution of complex terrain is analyzed. The results show that wind velocity suddenly increased at the location of the collapsed tower under the complex terrain, and the velocity is generally higher than the original gradient wind velocity. The sudden spurt in wind velocity caused by terrain cannot be ignored. Keywords Complex wind field · Geographic information processing system · Computational fluid dynamics · Large eddy simulation

24.1 Introduction The transmission tower is mainly built in a mountainous area with complex terrain, and the terrain leads to the increase of the wind velocity. Therefore the transmission line is subjected to a larger wind load than it on the flat ground, which is the main reason that the transmission tower may be damaged by the wind field. In order to avoid the collapse of transmission towers, it is important to study and master the characteristics and laws of wind field under complex terrain. It is of great significance to the safe operation of transmission lines in mountainous areas. The earlier simulations were performed by Uchida T and Ohya Y using large eddy simulation to study the unsteady three-dimensional flow of complex terrain [1]. The simulation results demonstrate the influence of terrain effect on the wind field, and simulate the local acceleration effect and flow separation. Guo Wenxing established a Computational Fluid Dynamics (CFD) model for the typical mountain topography, and selected different calculation parameters to numerically simulate the wind field. Q. Shengpei · Q. Fei · G. Qiang (B) School of Mechanical and Electric Engineering, Soochow University, Suzhou 215021, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_24

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At the same time, the numerical simulation results were compared with the actual wind tunnel test, and a set of better calculation parameters were obtained [2]. Zhou Zhiyong et al. carried out numerical simulation of wind field on a wide range of complex terrain and provided a scheme for reference, in which the method of reverse modeling was used for the construction of complex terrain surface [3]. Tang Hongxu et al. carried out complex terrain modeling under Gambit [4]. In contrast, the reverse modeling method is faster and more efficient. Liu Zhenqing et al. used large eddy simulation to analyze the characterization of turbulence through 3-D hills and 2-D ridges. By extracting the velocity spectrum, it is found that the 2-D ridge is very sensitive to the conditions of turbulence, and the 3-D hill is less affected by the turbulence conditions [5]. In the wind-resistance analysis of transmission tower structures, the acquisition of wind environment data is generally derived from statistical data of various regions, without considering the influence of complex terrain on wind velocity. Aiming at solving this problem, this paper numerically simulates the complex terrain wind field in the collapsed tower area in Zhenjiang on the basis of the CFD method, and studies the influence of complex terrain on the wind field around the transmission tower.

24.2 Numerical Simulation of the Wind Field in the Collapsed Tower Area in Zhenjiang 24.2.1 Numerical Simulation Test Method In this paper, the results of complex terrain wind field are obtained based on the numerical simulation test method (see Fig. 24.1).

24.2.2 Solid Modeling of Complex Terrain The red dot is marked as the approximate location of the collapsed tower in Zhenjiang (see Fig. 24.2a).The digital elevation model of Zhenjiang collapsed transmission tower area is obtained by using geographic data cloud platform. Geographic Information System (ArcGIS) is used to intercept the parts to be studied and draw the topographic contours [6]. At the same time, the geographic coordinates are transformed into projection coordinates. Elevation grid points are extracted by contour lines and point cloud coordinate files are output. The point cloud coordinate file is imported into the reverse modeling software, and the terrain surface is modeled inversely to obtain the model file of the terrain surface (see Fig. 24.2b). The average deviation between the fitted terrain surface and the original point cloud is about

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Terrain raster data of the collapsed tower area in Zhenjiang The file of terrain point cloud Obtain terrain elevation data based on ArcGIS platform IMAGEWARE Reverse Modeling Atmospheric boundary layer wind characteristics ICEM CFD meshing Fluent recognizable grid files

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(b)

Fig. 24.2 The sketch map of real terrain and simulated (a an actual topographic map, b the fitted complex terrain surface)

±0.36 m. This suggests that the model can meet the simulation requirements, which is in accordance with Jiang Baosong [7].

24.2.3 Numerical Wind Tunnel Before the numerical simulation, we first establish a numerical wind tunnel based on the terrain surface. Its size is closely related to the calculation accuracy. Too small

252 Fig. 24.3 Fluid domain partitioning

Q. Shengpei et al.

Velocity Inlet

Fluid domain 2

Symmetry

Fluid domain 1 Wall

Outflow

Table 24.1 The element type and the number of grids in each domain Fluid domain 1

Fluid domain 2

Total

Mesh type

Tetrahedral meshes

Hexahedral meshes

–

Number of grids

12021504

972312

12993816

a model can result in an excessive blocking rate, while too large which will make the computational overhead unbearable. Therefore, the proposed recommendation makes the basin’s blocking ratio not more than 3% [8]. Then the size of the flow field is 1200 m × 9000 m × 1000 m. The fluid domain is meshed by using ICEM CFD software. In the meshing, considering the calculation accuracy and efficiency, the calculation model is divided into blocks (see Fig. 24.3). The tetrahedral grid is used in the near mountain area and the hexahedral grid is used in the high altitude area. At the same time, the mesh of mountain area is refined. The element type and the number of grids in each area is shown in Table 24.1. It is shown that the number of hexahedral meshes generated by fluid domain 2 takes up only 7.5% of the total number of grids, which greatly reduces the computational cost while ensuring accuracy.

24.2.4 Selection of Boundary Conditions The air velocity in the atmospheric boundary layer is relatively low and can be considered as an incompressible gas flow. The boundary conditions are shown in the figure above (see Fig. 24.3). Define the flow velocity as v, turbulent dynamic energy as k, and turbulent dissipation as ε at the boundary. The landform of Zhenjiang belongs to Class B landforms. Class B landforms are townships or suburbs with sparse houses [9]. The inlet boundary calculation parameters are shown in Table 24.2.

24 Research on Wind Field of Transmission Tower …

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Table 24.2 List of entry boundary calculation parameters Inlet wind velocity v

Turbulent dynamic energy k

v(z)  = 21.65(z/10)0.16 z ≤ ZG

k=  1.5(u¯ × I(z))2 z ≤ ZG

38.24

z > ZG

z > ZG

0

I (z) = 0.14(z/10)−0.16 u¯ = v(z)

Turbulent dissipation ε  3 3 0.09 4 k 2 /l z ≤ ZG ε= 0 z > ZG 1

l = 100(z/30) 2

Note z is the height from the ground; z G is the height of the atmospheric boundary layer; I is the turbulence intensity; l is the turbulent integral scale

24.2.5 Turbulence Model and Calculation Parameters In this paper, the large eddy simulation (LES) is used for numerical simulation, and the WALE sub-lattice model is selected in the CFD calculation software Fluent. The flow field is solved by the SIMPLE algorithm. In Fluent, the default of limit viscous viscosity is limited to 105 , and the turbulent viscosity ratio limit should be greater than the maximum value in the atmospheric boundary μt /μ, so the limit value is modified to 108 when calculating for this model [10]. The calculation and test time step is adopted as 0.01 s.

24.3 Analysis of the Results This paper mainly considers the influence of complex terrain on the wind field, so it is important to consider the influence of the terrain on the wind field at the position of the transmission tower behind the mountain (see Fig. 24.4). Through the post-processing of the numerical simulation results, the velocity cloud map of the mountain interface can be obtained. It can be seen that vortices of Fig. 24.4 The sectional view of the main mountain

Main mountain peak Section Collapse position

Terrain

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different scale appear in the wake of the mountain (see Fig. 24.5). The flow separation occurs when the fluid passes through the mountain, and the acceleration effect of the mountain wind is formed when flowing through the top of the mountain. The figure shows the comparison between the wind velocity at the collapsed tower at a certain point after convergence and the original gradient wind velocity (see Fig. 24.6). When encountering the mountain body, part of the airflow climbs over the mountain. The wind velocity above the windward surface and the mountain body is obviously accelerated, and this disturbance may lead to an increase in the wind velocity near the wake of the mountain. The results show that the maximum wind velocity increment is 50% at 3.36 m from the ground, and the wind velocity increment also keeps above 40% within 20 m away from the ground. Thus it can be seen that such statistics can be used for areas with flat terrain, but if the tower is built at a complex terrain, the specific terrain

Fig. 24.5 The section of the velocity cloud

Simulated wind speed at the inverted tower

Fig. 24.6 Changes of wind velocity values along the height

0.39

3.17

4.43

5.85

8.04

Original gradient wind speed

24 Research on Wind Field of Transmission Tower …

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must be considered to obtain the correct information about the wind velocity and wind pressure distribution at the tower, so as to provide reference for the building standards of the transmission tower.

24.4 Conclusion This paper numerically simulates the complex terrain wind field at the location of the collapsed tower in Zhenjiang, and calculates the percentage of wind velocity increment at the location of the collapsed tower. (1) Large eddy simulation can better simulate the wind velocity of the boundary layer near the ground. (2) Within the height of 20 m from the ground, the maximum increment of the wind velocity keeps at 40% or more. (3) Increased wind velocity makes transmission towers in complex terrain withstand greater wind loads, thus the construction of transmission towers in complex terrain requires considering the effects of the complex terrain. Acknowledgements This work was supported by the National Natural Science Foundation of China (No. 11772211).

References 1. Uchida, T., Ohya, Y.: Large-eddy simulation of turbulent airflow over complex terrain. J. Wind Eng. Ind. Aerodyn. 91(1–2), 219–229 (2003) 2. Guo, W.: CFD Multi-scale simulation of wind flow over hilly terrain. Harbin Institute of Technology (2010) 3. Zhou, Z., et al.: Numerical simulation study of wind environment for the flow around large region with complex terrain. Q. J. Mech. 1, 101–107 (2010) 4. Tang, H., et al.: Three-dimensional terrain model based on GAMBIT. J. Geol. Hazards Environ. Preserv. 1, 61–65 (2013) 5. Liu, Z., Ishihara, T., Tanaka, T., et al.: LES study of turbulent flow fields over a smooth 3-D hill and a smooth 2-D ridge. J. Wind Eng. Ind. Aerodyn. 153, 1–12 (2016) 6. Ouyang, X.: ArcGIS Geographic Information System. Science Press, Beijing (2010) 7. Jiang, B.: Simulation of complex terrain wind field based on GIS platform. School of Civil Engineering, Tongji University (2008) 8. Wang, T.: Large eddy simulation of atmospheric boundary layer flow based on FLUENT. Chin. J. Comput. Mech. 29(5), 734–739 (2012) 9. GB 50009—2012 Building structure load specification. China Building Industry Press, Beijing (2012) 10. Liang, S., et al.: Numerical simulation method for wind flow around complex terrain. J. Eng. Thermophys. 6, 945–948 (2011)

Part VIII

Nano/Micro Structures in Application of Computational Mechanics

Chapter 25

Application of a New Infinite Element Method for Free Vibration Analysis of Thin Plate with Complicated Shapes D. S. Liu and Y. W. Chen

Abstract A novel infinite element method (IEM) is presented in this paper for solving plate vibration problems. In the proposed IEM, the substructure domain is partitioned into multiple layers of geometrically similar finite elements which use only the data of the boundary nodes. A convergence criterion based on the trace of the mass matrix is used to determine the number of layers in the IE model partitioning process. Furthermore, in implementing the Craig-Bampton (CB) reduction method, the inversion of the global stiffness matrix is calculated using only the stiffness matrix of the first element layer. The validity and performance of the proposed method are investigated by means of four illustrative problems. The first example considers the case of a simple clamped rectangular plate. It is observed that the IEM results are in good agreement with the theoretical results for all six natural frequencies. The second example considers the frequency response of a clamped rectangular plate with a crack. The main feature of IEM is that a very fine and good quality virtual mesh can be created around the crack tip. The third and fourth examples consider the natural frequency of a multiple point supported plate and a perforated plate, respectively. The results are obtained just need to adjust the reference point or boundary nodes. The parametric analyses for various geometric profiles are easy to be conducted using these numerical techniques. In general, the results presented in this study confirm that the proposed IEM algorithm provides a fast, direct and accurate means of simulating the dynamic response of various plate structures. Keywords Infinite element method · Craig-Bampton method · Plate vibration · Eigenvalue problem

D. S. Liu · Y. W. Chen (B) Department of Mechanical Engineering and Advanced Institute of Manufacturing with High-Tech Innovations, National Chung Cheng University, Chiayi 621, Taiwan, Republic of China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_25

259

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25.1 Introduction The classical finite element method solving large structural or singularity problems usually requires the construction of a large number of elements, resulting in insufficient computer memory or slow computation. Therefore, many scholars have proposed methods to reduce the degree of freedom of the model (such as Irons-Guyan reduction method, modal synthesis method, etc.) or other numerical methods (such as boundary element method, meshless method, etc.) to solve the above problems. The similarity of elements is also one of the numerical methods. Thatcher [1] combines the concepts of finite element methods and similarity to create many refined triangle elements near a singularity to approximate the solution. For the problem of cracked structural, Ying [2] produces a number of similar triangular elements near the crack tip and combines them into a single element, was called infinite similar element method (ISEM). Han and Ying [3, 4] continue to study the derivation and application of the ISEM, the many elements are reduced into a superelement using elemental similarity, and consider the inertial effect, was called infinite element method (IEM). Ying [5] proved that there exists a transformation matrix to relate the nodal displacement vector between the inner and outer layers, therefore the total stiffness matrix could reduce to form a combined stiffness matrix related only to boundary nodes. Notably, it was shown that the combined stiffness matrix could converge to a certain constant quantity as the number of layers approached to infinity. Liu has been working on the study of the IEM since the 2000s. Liu and Chiou [6, 7] proposed a hybrid IEM/FEM scheme for analyzing various types of elastic and singularity problems. Using this efficient numerical technique, a very fine mesh pattern can be established the singularities without increasing the degree of freedom of the global FEM solution. Moreover, it could be easily allowed to conduct parametric analyses for various crack sizes without changing the FE mesh. Liu et al. [8] extended the IEM formalism to the problem of moisture diffusion in 2D and 3D systems. In a later study, Liu et al. [9] proposed a Plate IEM (PIEM) algorithm based on Mindlin-Reissner theory to investigate the effects of the hole size, hole position and hole profile/area on the bending strength of plates with through-thickness holes. The fractal finite element method (FFEM) also makes use of self-similar finite elements. Leung and Su [10–12] developed FFEM to model the crack region combined with a system stiffness matrix. Only a few degrees of freedom are required to model the singular behavior ahead of the crack tip. Many studies have shown that the use of IEM to reduce the problem to one concerning the boundary nodes only yields a significant reduction in the computational cost. However, the accuracy of the solutions obtained for higher-order natural frequencies is rather poor. Liu and Lin [13] integrated Craig-Bampton method with a dynamic infinite element (DIE) formulation in order to improvement the efficiency of IEM without sacrificing its accuracy, where the Craig-Bampton method (CBM) [14] is a well-known technique for reducing the size of finite element (FE) models; particularly when these models involve two or more connected subsystems. However, Liu and Lin [13] did not discuss the convergence of the element layer, and the

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computational efficiency needs to be improved. Accordingly, this paper proposes a robust convergence criterion for the DIE formulation based on the trace of the mass matrix. Furthermore, the efficiency of the CBM reduction process is improved by calculating the symmetric block-tridiagonal stiffness matrix using the method proposed by Jain et al. [15]. This integrated scheme is called the similar finite element method (IEM).

25.2 Infinte Element Mothed 25.2.1 Finite Element Formulation of Mindlin-Reissner Plate The equation of motion of the plate is established as: ¨ + [K ]{δ} = 0 [M]{δ}

(25.1)

with mass matrix [M], stiffness matrix [K ] and the displacement vector {δ} . From  T have the form Mindlin-Reissner plate theory, the displacement {δ} = u, v, w given in Eq. (25.2). ⎧  ∂w ⎪ ⎨ u = −zθx (x, y) = −z ∂ x − γx z  − γ yz v = −zθ y (x, y) = −z ∂w ∂y ⎪ ⎩ w = w(x, y)

(25.2)

where x and y are in-plane axes located at the mid-plane of the plate, and z is along the direction of plate thickness as seen in Fig. 25.1. The θx and θ y are the rotations of the mid-plane about the y and x axes, respectively; and γ is the angle caused by the transverse shear deformation. Mapping from the physical coordinates to the natural coordinates, the rotate and transverse displacements can be expressed as ⎧ n ⎪ ⎪ θ = Hi (ζ, η)(θx )i ⎪ x ⎪ ⎪ i=1 ⎪ ⎨ n  θy = Hi (ζ, η) θ y i ⎪ i=1 ⎪ ⎪ n ⎪ ⎪ ⎪ Hi (ζ, η)wi ⎩w =

(25.3)

i=1

 where H i are the 4-node plate finite element shape functions, and ζ, η is represented as natural coordinates. From that the element matrices are established as

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z, w y

x, u

h x

dx y, v Fig. 25.1 Coordinate frame of the plate element ⎧    e e e ⎪ ⎨ K = K b + K s 1 1 1 h3 1 = 12 −1 −1 [Bb ]T [Db ][Bb ] det[J ]dζ dη + κh −1 −1 [Bs ]T [Ds ][Bs ] det[J ]dζ dη ⎪  

 ⎩ 1 1 M e = ρ −1 −1 [M]T [I ][N ] det[J ]dζ dη

(25.4)



 where the K be is the bending stiffness matrix and K se is the shear stiffness matrix, in which h is the thickness of the plate, κ is the shear energy correction factor, and [J ] is the Jacobian matrix:  [J ] =

∂x ∂ζ ∂x ∂η

∂y ∂ζ ∂y ∂η

 (25.5)

[Bb ], [Bs ] and [N ] consist of the shape functions as listed in Eqs. (25.6), (25.7) and (25.8), respectively. In addition, [Db ], [Ds ] and [I ] are related to dimension or material properties of model as listed in Eqs. (25.9), (25.10) and (25.11), respectively. ⎡ ∂H

⎤ 0 0 ∂∂Hx2 0 0 · · · ∂∂Hx4 0 0 ⎢ ⎥ ∂H ∂H ∂H [Bb ] = ⎣ 0 ∂ y1 0 0 ∂ y2 0 · · · 0 ∂ y4 0 ⎦ ∂ H1 ∂ H1 0 ∂∂Hy2 ∂∂Hx2 0 · · · ∂∂Hy4 ∂∂Hx4 0 ∂y ∂x 1

∂x

 [Bs ] =

−H1 0 0 −H1

∂ H1 ∂x ∂ H1 ∂y

−H2 0 0 −H2

∂ H2 ∂x ∂ H2 ∂y

· · · −H4 0 · · · 0 −H4

∂ H4 ∂x ∂ H4 ∂y

(25.6)  (25.7)

25 Application of a New Infinite Element Method for Free Vibration …

263

⎡

⎤ H1 0 0 H2 0 0 · · · H4 0 0 [N ] = ⎣ 0 H1 0 0 H2 0 · · · 0 H4 0 ⎦ 0 0 H1 0 0 H2 · · · 0 0 H4 ⎡ ⎤ 1ν 0 E ⎣ [Db ] = ν1 0 ⎦ 1 − ν2 0 0 1−ν 2   E 10 [Ds ] = 2(1 − ν) 0 1 ⎡ 3 ⎤ h 0 0 ⎢ 12 h 3 ⎥ [I ] = ⎣ 0 12 0⎦ 0 0 h

(25.8)

(25.9)

(25.10)

(25.11)

25.2.2 Infinite Element Model The basic concept of IEM involves partitioning the computational domain into multiple layers of geometrically-similar elements. As shown in Fig. 25.2, for element I in the upper-most layer, the local nodes i are numbered 1, 2, 3, and 4, respectively, in the counterclockwise direction. Furthermore, for each node i, the global coordinates are denoted as (xiI , yiI ). Taking the global origin O as the center of similarity, element II in the second layer can be obtained using a simple proportionality ratio c. More particularly, the global coordinates of elements I and II are related as follows: 

 xiI I , yiI I = cxiI , cyiI

(25.12)

where c is a proportionality ratio. From Eq. (25.12), and recalling Eq. (25.5), the determinants of the Jacobian matrices of the first and second element layers are related as follows: Fig. 25.2 Schematic representation of geometrically-similar 2D elements in IEM formulation

Y

2 3 I II

1 4

(x , y ) (x , y ) II i

O

I i

I i

II i

X

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det[J ] I I = c2 det[J ] I

(25.13)

Similarly, from Eq. (25.12), and recalling Eq. (25.6), it can be shown that: [Bb ] I I =

1 [Bb ] I c

(25.14)

Therefore, the matrices of the first and second element layers are related as shown in Eq. (25.15), where the mass matrix [M] and bending stiffness matrix [K b ] of the first element layer have the forms shown in Eq. (25.16). Note that the mass matrix is simplified as a diagonal matrix by a lumped-mass formulation. Lumped-mass is preferred for computational speed in dynamic analysis. 

[M] I I = c2 [M] I [K b ] I I = [K b ] I     ⎧ L1 0 M1 −D ⎪ I ⎪ = ⎨ [M] = −D T M2 0 L2   ⎪ K 1 −A ⎪ ⎩ [K b ] I = −A T K 2

(25.15)

(25.16)

From Eq. (25.15), it is seen that quadrilateral elements I and II can both be mapped using the same square-shaped master element. In other words, if the coordinate node values of an element are similar to those of other elements, these elements are designated as similar elements. Combining the matrices from first to n-th element layer, the mass and bending stiffness matrices can be established as in Eqs. (25.17) and (25.18), where the R = L 1 + c2 L 2 and Q = K 1b + K 2b . ⎡

⎤ L1 0 0 0 0 0 ⎢ 0 R 0 0 ⎥ 0 0 ⎢ ⎥ ⎢ ⎥ 0 0 ⎢ 0 0 c2 R 0 ⎥ ⎢ ⎥ [M] S E F = ⎢ .. ⎥ . ⎢ 0 0 0 ⎥ 0 0 ⎢ ⎥ ⎣ 0 0 0 0 c2(n−2) R ⎦ 0 2(n−1) L2 0 0 0 0 0 c ⎡ ⎤ K 1b −A 0 0 0 0 ⎢ −A T Q −A 0 0 0 ⎥ ⎢ ⎥ ⎢ ⎥ T 0 ⎥ ⎢ 0 −A Q −A 0 ⎥ [K b ] S E F = ⎢ .. .. .. ⎢ ⎥ . . . 0 ⎥ ⎢ 0 0 ⎢ ⎥ ⎣ 0 0 0 −A T Q −A ⎦ 0 0 0 0 −A T K 2b

(25.17)

(25.18)

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For shear stiffness, let the matrix [Bs ] be partitioned into two submatrices: 

 [Bs ] = Bs∗ + Bs∗∗  0 0 ∂∂Hx1 0 0 = 0 0 ∂∂Hy1 0 0

∂ H2 ∂x ∂ H2 ∂y

   ··· −H1 0 0 −H2 0 0 · · · + ··· 0 −H1 0 0 −H2 0 · · ·

(25.19)

Substituting Eq. (25.19) into Eq. (25.4), the shear stiffness is re-represented as follow:  

 [K s ] = K s∗ + K s∗∗ + K s∗∗∗

(25.20)

where ⎧   1  1 ∗ T

∗ ∗ ⎪ ]dζ dη ⎪ ⎨ K s = κh −1 −1 B s [Ds ] B s det[J  ∗∗ T

 1 1 ∗∗ ∗ T ∗∗ K s = κh −1 −1 Bs [Ds ] Bs + Bs [Ds ] Bs∗ det[J ]dζ dη ⎪ ⎪ ⎩ K ∗∗∗  = κh  1  1 B ∗∗ T [D ] B ∗∗  det[J ]dζ dη s

−1 −1

s

s

s

(25.21) According to geometric similarity, the relationship of the shear stiffness matrix between the first and second element layers is: I

I

I

[K s ] I I = K s∗ + c K s∗∗ + c2 K s∗∗∗

(25.22)

where the submatrices of the first element layer are shown as: ⎧   ∗

∗I ⎪ K 1 −A∗T ⎪ ⎪ Ks = ⎪ ⎪ −A∗ K ∗ ⎪ ⎪  ∗∗ 2 ∗∗T  ⎨

∗∗  I K 1 −A = Ks ∗∗ ⎪ −A K ∗∗ ⎪  ∗∗∗ 2 ∗∗∗T  ⎪ ⎪

∗∗∗  I ⎪ K 1 −A ⎪ ⎪ = ⎩ Ks −A∗∗∗ K 2∗∗∗

(25.23)

Similarly, combining the shear stiffness matrices from first to n-th element layer, can be established as: ⎡ ⎤ K 1s −A1 0 0 0 0 ⎢ −A T Q −A 0 0 0 ⎥ ⎢ ⎥ 2 2 1 ⎢ ⎥ T 0 0 ⎥ ⎢ 0 −A2 Q 3 −A3 ⎥ (25.24) [K s ] S E F = ⎢ .. .. .. ⎢ ⎥ . . . ⎢ 0 0 0 ⎥ ⎢ ⎥ T ⎣ 0 0 0 −An−1 Q n −An ⎦ 0 0 0 0 −AnT K 2s

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where ⎧ s K = K 1∗ + K 1∗∗ + K 1∗∗∗ ⎪ ⎪ ⎨ 1s K 2 = K 2∗ + c(n−1) K 2∗∗ + c2(n−1) K 2∗∗∗ ⎪ A = A∗ + c(i−1) A∗∗ + c2(i−1) A∗∗∗ , i = 1, 2, . . . , n ⎪ ⎩ i Q i = Q ∗ + c(i−2) Q ∗∗ + c2(i−2) Q ∗∗∗ , i = 2, 3, . . . , n

(25.25)

Recalling Eq. (25.1), the eigenvalue formulation can be obtained by assembling the aforementioned equations follows:  [K ] S E F − ω2 [M] S E F {u} S E F = 0

(25.26)

[K ] S E F = [K b ] S E F + [K s ] S E F

(25.27)

T  {u} S F E = δ0 δ1 δ2 · · · δn−1 δ n

(25.28)

where

Before applying Eq. (25.26) for further numerical analysis, a detailed convergence study involving the mass matrix must first be performed. In practice, the convergence of the mass matrix depends primarily on the number of element layers selected in the IE formulation. Recalling Eq. (25.17), the trace of the mass matrix has the form:  T r [M](n) SFE

= T r L 1 + c2(n−1) L 2 +

n 

 c2(N −2) R

(25.29)

N =2

if c < 1, then lim T r [M](n) S F E = T r [M] S F E

n→∞

(25.30)

where n denotes the number of chosen element layers, and T r [M] S F E converges to a certain constant quantity as the number of element layers approaches infinity. The critical number of element-layers is determined when the desired accuracy criterion, Eq. (25.31), is satisfied. Notably, Eq. (25.29) shows that the convergence analysis involves only the mass matrix of the first element layer.

ε=

(i) T r [M](i+1) S F E − T r [M] S F E

T r [M](i) SFE

× 100%

(25.31)

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25.2.3 Model Reduction In this study, the Craig-Bampton procedure is adopted to reduce the infinite element model. The mass and stiffness matrices and displacement vector in the Eq. (25.26) can be partitioned respectively as:

(25.32)

(25.33)

(25.34) where the superscripts B and I refer to boundary and interior nodes, respectively. The vectors uB and uI are physical displacements of boundary and interior points, respectively. According to the Craig-Bampton reduction method, the coordinate transformation equation relating the final coordinates of the substructure to the initial coordinates is given as:

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 {u} S F E =

uB uI

!



I 0 = φC φ N



pB pN

! = [φ]{ p} S F E , p B ≡ u B

(25.35)



 where matrices φ C and φ N are the desired matrices of the constraint modes and normal modes, respectively. The constraint modes are defined as the mode shapes of the interior freedoms due to successive unit displacements of the boundary points, with all other boundary points being totally constrained. To determine the constraint modes, the forces acting at all the interior freedoms are set equal to zero, i.e.     K I B u B + K I I u I = {0}

(25.36)

−1 I B  B  C  B   I K u ≡ φ u u = − K II

(25.37)

or

The normal modes are defined as the normal modes of the substructure given a totally constrained boundary. In other words, they represent the eigenvectors of the eigenvalue problem:

−1 I I  I   I   I  iωt

M φ =0 u = φ e , [I ] − ω2 K I I

(25.38)

N These eigenvectors form the respective columns

I I of matrix φ . is a symmetric block-tridiagonal As can be seen from Eq. (25.33), the K matrix. Consequently, the inversion of K I I can be written as the following a semiseparable matrix: ⎡

K

 I I −1

U1 V1T U1 V2T U1 V3T ⎢ V2 U1T U2 V2T U2 V3T ⎢ T T T ⎢ = ⎢ V3 U1 V3 U2 U3 V3 ⎢ . . . .. .. ⎣ .. Vn U1T Vn U2T Vn U3T

· · · U1 VnT · · · U2 VnT · · · U3 VnT . .. . ..

⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

(25.39)

· · · Un VnT

where matrices [Ui ] and [Vi ] are computed as: ⎧ U1 = I ⎪ ⎪ ⎪ ⎪  ⎪ U2 = A−1 ⎪ 2 Q2 ⎪ ⎨ U = A−1  Q  U − A T  U , i = 2, . . . , n − 1 i+1 i−1 i  i+1 i+1T i −1 T ⎪ V = K U − A U ⎪ n n n−1 2 n ⎪ ⎪ T T  −1 ⎪ V = V K A ⎪ n n n−1 2 ⎪ −1  T  ⎩ T T Ai+1 , i = n − 2, . . . , 1 ViT = Vi+1 Q i+2 − Vi+2 Ai+2

(25.40)

25 Application of a New Infinite Element Method for Free Vibration …

269

−1

According to Jain et al. [15], the diagonal blocks of K I I denoted as Di , can be obtained recursively as: ⎧  −1 ⎪ ⎨ D1 =  Q 2 − A2 S1T    T −1 T Di = Q i+1 − A S D S I + A , i = 2, . . . , n − 1 i−1 i−1 i+1 i i ⎪ ⎩ D = K −1  I + A T D S n n−1 n−1 n 2

(25.41)

where matrices [Si ] are defined as follows: T = ViT Si Vi+1

(25.42)

In the present study, [Si ] are computed using the following numerically stable recursive procedure: "

Sn−1 = An K 2−1    T  −1 Q i+2 − Si+1 Ai+2 Si = Ai+1 , i = n − 2, . . . , 1

(25.43)

The remaining block entries are then computed in a numerically stable manner as follows: ⎡ ⎢ ⎢ # $−1 ⎢ ⎢ I I =⎢ K ⎢ ⎢ ⎣

D1 S1 D1 S1 S2 D1 D2 D2 S2 (D1 S1 )T D3 (D1 S1 S2 )T (D2 S2 )T . . . . . . . . .  T  T  T D1 S1 . . . Sn−1 D2 S2 . . . Sn−1 D3 S3 . . . Sn−1

⎤ · · · D1 S1 . . . Sn−1 · · · D2 S2 . . . Sn−1 ⎥ ⎥ ⎥ · · · D3 S3 . . . Sn−1 ⎥ ⎥ ⎥ . .. ⎥ . ⎦ . . ··· Dn

(25.44)

−1

As can be seen from the aforementioned equations, K I I is calculated using only the stiffness matrix of the first element layer, that is greatly improved the efficiency of computational work. Finally, the eigenvalue formulation of the IEM can be obtained by substituting the coordinate transformation relation, i.e., Eq. (25.35), into Eq. (25.26) and then pre-multiplying by the transpose of [φ], i.e.,  T [φ] [K ] S F E [φ] − ω2 [φ]T [M] S F E [φ] { p} S F E = 0

(25.45)

25.3 Case Studies 25.3.1 Rectangular Plate Consider a rectangular plate shown in Fig. 25.3a. Let the plate dimensions and properties be defined as follows: a = 1.2 m, b = 1.0 m, thickness h = 0.01 m, mass

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b

X O

a

(a) Schematic illustration

(b) Virtual mesh configuration

Fig. 25.3 Schematic illustration and virtual mesh configuration of the clamped rectangular plate

density ρ = 2330 kg/m3 , Young’s modulus E = 200 GPa, and Poisson’s ratio ν = 0.3. Furthermore, assume that the plate is clamped on all four sides, the natural frequency ωm,n are estimated respectively as: ωm,n =

  m 2 a

+

n 2  b

% Dπ 4 , m, n = 1, 2, . . . 2ρh

(25.46)

In implementing the IE model, 88 nodes are deployed at the boundary, and the virtual mesh pattern is shown in Fig. 25.3b. For an ideal element, the aspect ratio should have a value of 1.0, so the ratio should be equal to 0.875 in this case. A number of required c element layer s = 40 is obtained by the convergence criterion Eq. (25.31). The convergence process is shown in Fig. 25.4. Overall, the results presented above indicate that in solving the clamped plate vibration problem, the proportionality ratio should be set as c = 0.875, the number of virtual element layer as s = 40. The first six mode shapes of the rectangular plate are depicted in Fig. 25.5. The results for the six lowest natural frequencies are shown in Table 25.1 with the theoretical results. Comparing the sets of natural frequencies reveals, the IEM results are in satisfactory agreement with the theoretical results.

25.3.2 Cracked Plate The second example considers a rectangular plate containing a through-thickness crack, as shown in Fig. 25.6a. The dimensions of the plate are assigned as follows: a = 0.5 m, b = 1.0 m, thickness h = 0.01 m. The same material properties are assigned as previous example: mass density ρ = 2330 kg/m3 , Young’s modulus E = 200 GPa, and Poisson’s ratio ν = 0.3. The plate is assumed be clamped on all four sides and

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Fig. 25.4 The convergence process of the required virtual element layer

1st mode

2nd mode

3rd mode

4th mode

5th mode

6th mode

Fig. 25.5 First six mode shapes of clamped rectangular plate

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Table 25.1 First six natural frequencies of clamped rectangular plate (Hz) Methods

Mode no. 1

2

3

4

5

6

Exact solution

40.78

90.93

113.00

163.14

174.51

233.35

IEM

40.65

90.73

112.90

162.91

174.94

233.99

0.32

0.22

0.09

0.14

0.25

0.27

Error (%)

Y

crack

b

l

X a (a) Schematic illustration

(b) Virtual mesh configuration

Fig. 25.6 Schematic illustration and virtual mesh configuration of the cracked plate

the crack has a length l. The crack length l is assigned various values in the range of l = 0.1 ~ 0.4 m in this study. The proportionality ratio is set as c = 0.875, and number of virtual element layer s = 40 as in the previous example. Figure 25.6b shows the virtual mesh configuration and virtual node arrangement of the IE model. Given various crack lengths, the results obtained from the proposed IEM for the six lowest natural frequencies as shown in Fig. 25.7. The results show that the fourth and fifth frequencies are highly sensitive to the crack length. Figure 25.8 shows the first six mode shapes given crack lengths of l = 0.1, 0.2, 0.3, and 0.4 m, respectively. Using these efficient numerical techniques, a very fine and good quality grid pattern can be created around each crack tip without many freedoms. Furthermore, the results of obtaining IEM only need to adjust the reference point. One is easily allowed to conduct parametric analysis for various crack sizes without changing the boundary nodes.

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600

Natural frequencies (Hz)

500 1st mode

400

2nd mode 3rd mode

300

4th mode 200

5th mode 6th mode

100 0 0.1

0.15

0.2

0.25

0.3

0.35

0.4

Crack sizes (m) Fig. 25.7 First six natural frequencies of cracked plate (Hz)

25.3.3 Multiple Point Supported Plate The third example considers a multiple point supported plate shown in Fig. 25.9a, a square plate is locked on the four symmetrically point supports. The dimensions of the plate are assigned as follows: a = 1 m and thickness h = 0.01 m, and the same material properties as first example. Figure 25.9b shows the virtual mesh configuration and virtual node arrangement of the corresponding model. The model consists of four subdomains, each of which is an IE model. As shown, the reference points are set at the support points, respectively. The proportionality ratio c and number of virtual element layer s are same as the first example. Figure 25.10 shows the six lowest natural frequencies obtained from IEM, and consider the support position is shifted form x 0 = 0.1 m to x 0 = 0.4 m. The maximum fundamental frequency is achieved in the range around x 0 = 0.175 ~ 0.275. Li et al. [16] have confirmed this finding. Figure 25.11 shows the first six mode shapes given support positions of x 0 = 0.1, 0.2, 0.3, and 0.4 m, respectively. As in the second example, the results of obtaining IEM only need to adjust the reference point. It is easily allowed to conduct parametric analysis for various support position without changing the boundary nodes. Furthermore, this example demonstrates the feasibility of combining IEM subdomains.

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(a) l = 0.1 m

(b) l = 0.2 m

(c) l = 0.3 m

(d) l = 0.4 m

Fig. 25.8 First six mode shapes of cracked plate

25.3.4 Perforated Plate The third example considers a perforated plate shown in Fig. 25.12a. The dimensions of the plate are assigned as follows: a = 0.8 m and thickness h = 0.01 m, and the same material properties as first example. The plate is assumed be clamped on all four sides and the circle holes have a radius r. The circle hole radius r is assigned various values in the range of r = 0.03 ~ 0.07 m in this study. Figure 25.12b shows the virtual mesh configuration and virtual node arrangement of the corresponding model. The model consists of sixteen subdomains, each of which is the same IE

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Y

a x0 x0 X a (a) Schematic illustration

(b) Virtual mesh configuration

Fig. 25.9 Schematic illustration and virtual mesh configuration of the multiple supported plate

180

Natural frequencies (Hz)

160 140 120

1st mode

100

2nd mode 3rd mode

80

4th mode

60

5th mode

40

6th mode

20 0

Support positions (m) Fig. 25.10 First six natural frequencies of the multiple point supported plate (Hz)

model. The proportionality ratio c and number of virtual element layer s are same as the first example. Figure 25.13 shows the six lowest natural frequencies given various circle holes radius. Figure 25.14 shows the first six mode shapes given holes radius of r = 0.03, 0.04, 0.05, 0.06, and 0.07 m, respectively. This is another case to show the convenience of IEM. The results are obtained just need to adjust the boundary nodes. The parametric analyses for various circle hole radius are easy to be conducted using these numerical techniques. Furthermore,

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(a) x0=0.1 m

(b) x0=0.2 m

(c) x0=0.3 m

(d) x0=0.4 m

Fig. 25.11 First six mode shapes of the multiple point supported plate

r

a

a (a) Schematic illustration

(b) Virtual mesh configuration

Fig. 25.12 Schematic illustration and virtual mesh configuration of the perforated plate

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400 350 300

1st mode

250

2nd mode

200

3rd mode 4th mode

150

5th mode

100

6th mode

50 0

0.03 0.035 0.04 0.045 0.05 0.055 0.06 0.065 0.07

Hole sizes (m) Fig. 25.13 First six natural frequencies of the perforated plate (Hz)

this example not only demonstrates the feasibility of combining IEM subdomains, but also copying.

25.4 Conclusions The infinite element method (IEM) is presented for solving plate vibration problems in this study. In the proposed method, the substructure domain is partitioned into multiple layers of geometrically similar finite elements, which use only the data of the boundary nodes, and the degree of freedom of the model is reduced using CraigBampton (CB) method. In implementing the proposed IE model, the number of element layers has been determined using a convergence criterion based on the trace of the mass matrix. The convergence analysis involves only the mass matrix of the first element layer. Furthermore, in Craig-Bampton (CB) reduction process, the inversion of the symmetric block-tridiagonal global stiffness matrix has been computed using the method proposed by Jain et al. [15] such that only the stiffness matrix of the first element layer need be considered. The validity of the proposed IEM has been demonstrated by means of four illustrative examples involving a rectangular plate, a rectangular plate containing a through-thickness crack, a rectangular with multiple point supports, and a perforated plate. The first example has demonstrated the general feasibility and accuracy of the proposed method, while the second example has demonstrated the applicability of the proposed method to geometric singularity problems (e.g., cracks). Finally, the third and fourth examples show the advantages

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(a) r=0.03 m

(b) r=0.04 m

(c) r=0.05 m

(d) r=0.06 m

(e) x0=0.07 m

Fig. 25.14 First six mode shapes of the perforated plate

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of IEM for parametric analysis. In general, the results presented in this study have shown that the proposed IEM algorithm provides a fast, direct and accurate tool for simulating the dynamic characteristics of plate structures. Acknowledgements This work was partially supported by the Advanced Institute of Manufacturing with High-tech Innovations (AIM-HI) from The Featured Areas Research Center Program within the framework of the Higher Education Sprout Project by the Ministry of Education (MOE) in Taiwan. This research was also supported by R.O.C. MOST Foundation Contract No. MOST1083017-F-194 -001.

References 1. Thatcher, R.W.: Singularities in the solution of laplace’s equation in two dimensions. J. Inst. Math. Appl. 16, 303–319 (1975) 2. Ying, L.A.: The infinite similar element method for calculating stress intensity factors. Sci. Sinica 21, 19–43 (1978) 3. Han, H.D., Ying, L.A.: An iterative method in the finite element. Math. Numer. Sinica 1, 91–99 (1979) 4. Han, H.D., Ying, L.A.: The infinite element method for eigenvalue problems. Numer. Math. 1, 39–50 (1982) 5. Ying, L.A.: An introduction to the infinite element method. Math. Pract. Theory 2, 69–78 (1992) 6. Liu, D.S., Chiou, D.Y.: A coupled iem/fem approach for solving the elastic problems with multiple cracks. Int. J. Solids Struct. 40, 1973–1993 (2003) 7. Liu, D.S., Chiou, D.Y.: 3D IEM formulation with an IEM/FEM coupling scheme for solving elastostatic problems. Adv. Eng. Softw. 34, 309–320 (2003) 8. Liu, D.S., Zhuang, Z.W., Chung, C.L.: Modeling of moisture diffusion in heterogeneous epoxy resin containing multiple randomly distributed particles using hybrid moisture element method. CMC: Comput. Mater. Continua 13, 89–113 (2009) 9. Liu, D.S., Tu, C.Y., Chung, C.L.: Coupled PIEM/FEM algorithm based on mindlin-reissner plate theory for bending analysis of plates with through-thickness hole. CMES: Comput. Model. Eng. Sci. 92, 573–594 (2013) 10. Leung, A.Y.T., Su, R.K.L.: Mode I crack problems by fractal two level finite element methods. Eng. Fracture Mech. 48, 847–859 (1994) 11. Leung, A.Y.T., Su, R.K.L.: Mixed-mode two-dimensional crack problem by fractal two level finite element method. Eng. Fracture Mech. 51, 889–895 (1995) 12. Leung, A.Y.T., Su, R.K.L: Fractal two-level finite element method for cracked kirchhoff’s plates using DKT elements. Eng. Fracture Mech. 54, 703–711 (1996) 13. Liu, D.S., Lin, Y.H.: Vibration analysis of the multiple-hole membrane by using the coupled DIEM-FE scheme. J. Mech. 32, 163–173 (2016) 14. Craig, J.R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6, 1313–1319 (1968) 15. Jain, J., Li, H., Cauley, S., Koh, C.K., Balakrishnan, V.: Numerically Stable Algorithms for Inversion of Block Tridiagonal and Banded Matrices, p. 357. Purdue University TR ECE (2007) 16. Li, R., Tian, Y., Wang, P.C., Shi, Y.F., Wang, B.: New analytic free vibration solutions of rectangular thin plates resting on multiple point supports. Int. J. Mech. Sci. 110, 53–61 (2016)

Part IX

Design, Analysis and Manufacturing Through Simulation

Chapter 26

Development of Smart-Technology for Forecasting Technical State of Equipment Based on Modified Particle Swarm Algorithms and Immune-Network Modeling Galina Samigulina and Zhazira Massimkanova Abstract The article is devoted to the development of Smart-technology for forecasting technical state of industrial equipment based on artificial intelligence methods. One of the most important tasks in forecasting is the creation an optimal set of descriptors that most fully characterize industrial equipment’s work. Preliminary data processing and the selection of informative descriptors based on modified particle swarm algorithms have been performed. The application of modified particle swarm algorithms allows to investigate a search space in more detail and to avoid premature convergence. The forecasting technical state of equipment and image recognition have been carried out based on immune-network modeling. The developed Smarttechnology is used to forecast the technical state of equipment based on real-life production data of TengizShevroil oil and gas company. The modeling results have been obtained on the basis of daily measurements from industrial Installation 300. Keywords Smart-technology · Forecasting technical state of industrial equipment · Modified particle swarm algorithms · Immune-network modeling

26.1 Introduction With rapid development of digital technologies, the effective and targeted implementation of innovative Smart-technologies in oil and gas industry is an actual task. There is especially important to apply the latest achievements of swarm intelligence (SI) for control objects of oil production, diagnostics the technical state of industrial equipment, optimization production, placement wells, forecasting exploration and G. Samigulina Institute of Information and Computational Technologies, Almaty, Kazakhstan e-mail: [email protected] Z. Massimkanova (B) Al-Farabi Kazakh National University, Almaty, Kazakhstan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_26

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transportation oil and gas. In recent years, various intelligent technologies are actively developed to create “smart” wells and to control oil production processes in real time in order to reduce production losses and extend the oilfield lifetime [1]. Intelligence technologies are used widely using neural networks, evolutionary algorithms, artificial immune systems, swarm intelligence algorithms, etc. The application of SI algorithms at implementation of Smart-technologies in oil and gas industry allows to decrease the amount of tests conducted to determine the characteristics of oilfields and reduce oil production cost. Hybrid systems based on modified SI algorithms are caused a special interest for analysis and processing of multidimensional data, optimization and solution the problem of prediction. Particle swarm optimization (PSO) algorithms are widely used to solve these problems. In work [2] the combination of PS algorithm and least squares support vector machine (LS-SVM) method is presented to forecast oil and gas production. The experimental results show that the proposed model has good convergence, training speed and high prediction accuracy. In paper [3] the application of PSO algorithm and support vector regression (SVR) is proposed for formation optimal set of parameters using real-life industrial datasets obtained during petroleum exploration from four oil wells. The comparison of proposed model to SVR models with random search (RAND-SVR) and trial and error approach (TE-SVR) shows the effectiveness of PSO-SVR model for formation of optimum combination of parameters. In research [4] the joint use of artificial neural network (ANN) and PSO algorithm is presented to forecast the productivity of horizontal wells under pseudosteady-state conditions in oil production. The proposed method is used to analyze parametric sensitivity in reservoir simulation. The paper [5] presents a hybrid system based on PSO and pattern search algorithms for determination of optimal power flow. The purpose of the research is to minimize the total generation cost, decrease real losses and improve the voltage stability index. Modeling outputs show the efficiency and potential of the proposed method. The work [6] describes the combination of PSO algorithm and support vector machine (SVM) for determination of oil recovery factor in the low-permeability reservoir. The accuracy and efficiency of proposed model are evaluated using 34 data sets and the model gives the best results with average absolute relative deviation of 3.79%. In article [7] PSO algorithm is used to predict the lifetime of a turbine and create an optimal schedule for maintenance and repair processes. The main disadvantages of classical PSO algorithm are premature convergence, local optimum and low convergence speed, therefore various modified PSO algorithms are actively developed, such as Inertia weight PSO (IWPSO), Cooperative PSO (CPSO), Modified PSO (MPSO), Local PSO (LPSO), Fully informed PSO (FIPSO), comprehensive learning PSO (CLPSO) and others. In work [8] the influence of inertia weight to the efficiency of IWPSO algorithm is analyzed. As a result of the study, it is noted the use of PSO algorithm with the proposed inertia weight allows to process a large amount of data quickly and provides the best solution. The paper [9] presents IWPSO algorithm, in which the inertia weight decreases with increasing number of iterations. In comparison with classical PSO algorithm, the proposed method improves the search of optimal values by solving the problem of

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premature convergence. In work [10] CCPSO-ISM (Competitive and cooperative PSO with information sharing mechanism) algorithm is proposed to prevent premature convergence at solution global optimization problem. In comparison with LPSO, FIPSO and CLPSO algorithms, the proposed algorithm has good ability of global search, because ISM (information sharing mechanism) approach allows mix up information of whole swarm when explore search space. In paper [11] CPSO algorithm based on k-means is applied to improve local search ability around the best position. One of the relevant task is the diagnosis of industrial equipment in oil and gas industry. There are published many articles about transformer fault diagnosis. The work [12] presents hybrid model based on modified evolutionary particle swarm optimisation-time varying acceleration coefficient (MEPSO-TVAC) and artificial neural network (ANN) for power transformer fault diagnosis. The MEPSO-TVAC model is used to optimize the performance of ANN. In paper [13] the combination of PSO algorithm and neural network is proposed to solve this problem. The modeling results show that the diagnosis accuracy is reached above 90%. In research [14] PSO algorithm is used to improve the fault diagnosis accuracy. The proposed method has good learning speed and more stable. The application of artificial immune systems (AIS) is especially relevant to solve various applied problems in oil and gas industry. Artificial immune systems represent a complex adaptive structure that imitates the behavior of human’s immune system in the process of protection the body from external factors and have such advantages as memory, learning, distribution, self-organization and a high degree of parallelism. The paper [15] describes the application of genetic algorithms, PSO algorithms and artificial immune systems for optimization of offshore oil production risers. All of the above confirms the relevance of application artificial intelligence approaches to solve the problem of diagnosis and forecasting the technical state of industrial equipment based on modified particle swarm algorithms and AIS approach. The article has the following structure: Sect. 26.2 presents problem formulation. Section 26.3 is devoted to the development of Smart-technology for forecasting the technical state of industrial equipment. Section 26.4 presents the modeling results of IWPSO and CPSO algorithms and comparative analysis. Conclusion and references are given at the end of article.

26.2 Problem Formulation The problem statement is formulated as following: it is necessary to develop Smarttechnology for forecasting the technical state of industrial equipment in oil and gas industry based on modified particle swarm algorithms and immune-network modeling. The research has been conducted using a real-life production datasets of the TengizShevroil oil and gas company. As an example the realization of Smart-technology for diagnostics of industrial equipment Installation 300 is considered, which designed

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Table 26.1 The fragment of daily measurements from industrial Installation 300 FIC31011 LIC31012 LT301031 PDT31001 PDT31008 ... 420.9 417.8 423.4 415.2 418.8 420.1 410.1 ... 413.3

79.8 79.9 79.3 80.5 80 80 80.1 ... 80.3

44.324 47.465 50.434 33.813 40.041 41.387 50.247 ... 50.942

5.556 6.093 5.206 6.14 5.649 6.023 5.603 ... 4.039

232.445 233.306 247.162 215.779 228.184 228.406 309.436 ... 234.538

... ... ... ... ... ... ... ... ...

TE301020 108.655 107.803 109.279 106.187 107.433 107.698 107.931 ... 104.254

for purification petroleum gases from acidic components and consists of several units (high-pressure absorber, low-pressure absorber, amine regenerator, amine cooler, etc.). There is used a data from 19 sensors of Installation 300: FIC31011—difference converter, LT301031—a buoy level gauge, TT31020—temperature converter, etc. The fragment of daily measurements from industrial Installation 300 is presented in Table 26.1. Classification the solutions for normal, border and emergency operation mode of Installation 300 is performed by experts.

26.3 Development of Smart-Technology for Forecasting the Technical State of Industrial Equipment The structural schema of Smart-technology for forecasting the technical state of industrial equipment based on modified PSO algorithms and AIS approach has been created (Fig. 26.1). Smart technology consists of several basic steps: preliminary data processing based on PSO algorithms [16], pattern recognition using immune-network modeling and forecasting the technical state of industrial equipment. According to the concept of multi-algorithmic approach [17], the modified swarm intelligence algorithm (IWPSO or CPSO) has been chosen, in which the minimum generalization error of AIS based on homologous proteins is given. The advantage of proposed immune-network modeling technology is the ability to recognize patterns on the boundaries of nonlinear classes.

26.3.1 Modified IWPSO-AIS Algorithm Modified algorithm (IWPSO-AIS) based on inertia weight particle swarm optimization (IWPSO) algorithm [18] and artificial immune systems (AIS) approach has been developed to solve the formulated problem.

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Fig. 26.1 The structural schema of Smart-technology for forecasting the technical state of industrial equipment

IWPSO-AIS algorithm: Step 1: Solution the problem of feature selection based on IWPSO algorithm. Step 1.1: Initialization of search space. Generation of agents’ population in Ddimensional space. Random generation of initial positions (xi ) and velocities (vi ) of agents: xi = (xi1 , xi2 , . . . , xi D ) and vi = (vi1 , vi2 , . . . , vi D ). Step 1.2: Calculation of fitness-function. Determination the start of inertia weight (wmax ) and the end of inertia weight (wmin ) of particles. Inertia weight (w) of particle is determined by the following Eq. (26.1): w = wmax −

wmax − wmin ∗ k, itermax

(26.1)

in which itermax —the maximum number of iterations, k—the current iteration. Step 1.3: Comparison of fitness-function values [19]: k+1 k+1 ) ≤ f ( pid ), then update pid = xid , • if f (xid • if f ( pid ) ≤ f ( pgd ), then update pgd = pid .

Step 1.4: Migration of agents. Agent velocities (26.2) and positions (26.3) are updated by the following equations: k+1 k k k = wvid + c1r1 ( pid − xid ) + c2 r2 ( pgd − xid ), vid

(26.2)

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(26.3)

where vid —particle’s velocity, k—the number of iterations, w—inertia weight, c1 , c2 —acceleration coefficients, r1 , r2 —random variables distributed on [0,1], which are used to save the population difference, pid —the local best value, pgd —the global best value, N —population size. Step 1.5: Checking of stop condition. The global best values of particles ( pgd ) are saved. Step 1.6: The reduction of low informative descriptors. The database of informative set of descriptors is formulated. Step 2: Solution the problem of pattern recognition on the basis of immune-network algorithm. Step 2.1: Creation an optimal immune-network model based on a dedicated set of descriptors. The optimal structure of immune network is a network created on the basis of the weight coefficients of selected informative descriptors that most fully characterize the system. The criterion is the maximum saving information at the minimum number of descriptors. Step 2.2: Creation the matrixes of standards and the matrixes of patterns, which formed from time series of descriptors. Step 2.3: Learning of AIS with teacher. Step 2.4: Singular matrix decomposition (SVD). The bond energies are determined. Step 2.5: Solution the problem of pattern recognition based on determining the minimum value of bond energy, energy error estimation on homologous proteins [20] and forecasting the technical state of industrial equipment and operation mode (normal, boundary, emergency mode of Installation 300).

26.3.2 Modified CPSO-AIS Algorithm Modified algorithm (CPSO-AIS) based on cooperative particle swarm (CPSO) algorithm [21] and artificial immune systems (AIS) approach has been developed for more efficient and faster research of multidimensional space and to reduce a forecasting time. CPSO-AIS algorithm: Step 1: Solution the problem of feature selection based on CPSO algorithm. Step 1.1: Initialization of search space. Population of master swarm and slave swarms are created. Random generation of initial positions and velocities of all agents is performed. Step 1.2: Parallel calculation of fitness-function every slave swarms is performed. The local best values of slave swarms ( pgs ) are determined and compared.

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Step 1.3: The best positions are transferred to the master swarm. The best velocities (viM ) (26.4) and positions (xiM ) (26.5)of master swarms are updated by the following equations [22]: viM = wviM (t) + r1 c1 ( piM − xiM (t)) + ψr2 c2 ( pgM − xiM (t)) + (1 − ψ)r3 c3 ( pgs − xiM (t)),

(26.4)

xiM (t + 1) = xiM (t) + viM (t), i = 1, 2, . . . n,

(26.5)

where t—iteration, w—inertia weight, M—master swarm, S—slave swarms, r1 , r2 ,r3 ,—random variables distributed on [0,1], c1 , c2 , c3 —acceleration coefficients, pgM —the global best position of master swarm’s particles, pgs —the global best position of slave swarms’ particles. Step 1.4: Migration of all agents are performed by formula (26.6): ⎧ ⎨

Gbest s < Gbest M , ψ = 0.5 Gbest s = Gbest M , ⎩ 1 Gbest s > Gbest M .

(26.6)

where ψ—migration factor, Gbest s —the fitness-function value determined for pgs , Gbest M —the fitness-function value determined for pgM . Step 1.5: The positions and velocities of slave swarms’ agents are updated. The stop condition is checked. The global best position of master swarm’s agents ( pgM ) is saved. Step 1.6: The reduction of low informative descriptors is performed. The database of informative set of descriptors is formulated. Step 2: Pattern recognition problem based on immune network algorithm is solved similarly to the IWPSO-AIS algorithm.

26.4 Modeling Results The construction of informative data set of daily measurements of Installation 300 sensors (Table 26.1) has been performed on the basis of IWPSO algorithm using WEKA software. As a modeling output of IWPSO algorithm, from 19 descriptors 6 informative descriptors are selected (Fig. 26.2). The software for creation informative set of descriptors using CPSO algorithm has been developed in Python 3.6 programming language. A visualization of selected informative descriptors based on CPSO algorithm has been illustrated in Fig. 26.3. As a modeling output of CPSO algorithm 5 informative descriptors are chosen from 19 descriptors.

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Fig. 26.2 Visualization of selected informative descriptors based on IWPSO algorithm

Fig. 26.3 Visualization of selected informative descriptors based on CPSO algorithm

26 Development of Smart-Technology for Forecasting Technical State … Table 26.2 Modeling results Algorithms Parameters IWPSO algorithm c1 = 1 c2 = 2 wmax = 0.9 wmin = 0.1 CPSO algorithm

c1 = 0.1 c2 = 0.1 w = 0.5

291

Iterations

Fitness cost

Descriptors

10 20 30 40 50 10 20 30 40 50

0.44 0.45 0.53 0.54 0.55 0.41 0.61 0.76 0.83 0.84

8 8 7 7 6 7 7 7 6 5

According to the described above algorithms (IWPSO-AIS, PSO-AIS) after selection of informative descriptors the forecasting of technical state of industrial equipment is performed. Comparison of modeling results of IWPSO and SO algorithms has been carried out with a different number of iterations for evaluation their effectiveness (Table 26.2). After comparison of IWPSO and CPSO algorithms the following findings are received: - in IWPSO algorithm the fitness-function value improves with the increasing number of iterations. The best fitness-function value is achieved at iteration 50 and 6 informative descriptors are selected. - in CPSO algorithm, agents explore the search space in detail. Similarly the fitnessfunction value increases with the increasing number of iterations. Due to the parallelization of computations, processing time for allocation an informative set of descriptors is reduced. The optimal solution is achieved with 50 iterations and 5 informative descriptors are selected. The efficiency of modified particle swarm algorithms depends on the type of analyzed data and the correct choice of fitness-function and parameters.

26.5 Conclusion Thus, the proposed Smart-technology is designed to predict the technical state of industrial equipment, which allows to determine emergency situations, adequately and quickly react to changes in operating conditions of installations and extend equipment lifetime, which significantly reduce financial costs and increase production efficiency. Notes and Comments. This work is carried out on the grant No. AP05130018 “Cognitive Smart-technology design for intelligent complex objects control system based

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on artificial intelligence approaches” (2018–2020) at the Institute of Information and Computational Technologies, the Committee of science of the Ministry of Education and Science of the Republic of Kazakhstan.

References 1. Korovin, I.S., Tkachenko, M.G.: Intelligent oilfield model. Procedia Comput. Sci. 101, 300– 303 (2016) 2. Qiao, Y., Peng, J., Ge, L., Wang, H.: Application of PSO LS-SVM forecasting model in oil and gas production forecast. J. Cognitive Inform. Cognitive Comput. (2017). https://doi.org/ 10.1109/ICCI-CC.2017.8109791 3. Akande, K.O., Owolabi, T.O., Olatunji, S.O., Abdul Raheem, A.: A hybrid particle swarm optimization and support vector regression model for modelling permeability prediction of hydrocarbon reservoir. J. Pet. Sci. Eng. 150, 43–53 (2017) 4. Ahmadi, M.A., Soleimani, R., Lee, M., Kashiwao, T., Bahadori, A.: Determination of oil well production performance using artificial neural network (ANN) linked to the particle swarm optimization (PSO) tool. J. Pet. 1(2), 118–132 (2015) 5. Berrouk, F., Bounaya, K.: Optimal power flow for multi-FACTS power system using hybrid PSO-PS algorithms. J. Control, Autom. Electr. Syst. 29(2), 177–191 (2018) 6. Han, B., Bian, X.: A hybrid PSO-SVM based model for determination of oil recovery factor in the low-permeability reservoir. J. Pet. 4, 43–49 (2018) 7. Osadciw, L.A., Yan, Y., Ye, X., Benson, G.: wind turbine diagnostics based on power curve using particle swarm optimization. In: Wind Power Systems: Applications of Computational Intelligence, pp. 151–165 (2010) 8. Umapathy, P., Venkataseshaiah, C., Arumugam, M.S.: Particle swarm optimization with various inertia weight variants for optimal power flow solution (2010). https://doi.org/10.1155/2010/ 462145 9. Bin, J., Zhigang, L., Xingsheng, G.: A dynamic inertia weight particle swarm optimization algorithm. Chaos, Soluti. Fractal 37, 698–705 (2008) 10. Li, Y., Zhan, Z., Lin, S., Zhang, J., Luo, X.: Competitive and cooperative particle swarm optimization with information sharing mechanism for global optimization problems. Inform. Sci. 293, 370–382 (2015) 11. Neshat, M., Yazdi, S.F., Daneyal, Y., Sargolzaei, M.: A new cooperative algorithm based on PSO and K-Means for data clustering. J. Comput. Sci. 8(2), 188–194 (2012) 12. Illias, H.A., Chai, X.R., Bakar, A.B.: Hybrid modified evolutionary particle swarm optimisation-time varying acceleration coefficient-artificial neural network for power transformer fault diagnosis. Measurement (2016). https://doi.org/10.1016/j.measurement.2016.04. 052 13. Li, H., Wang, F., Wang, R.: Transformer internal insulation fault diagnosis based on RBF neural network evolved by immune particle swarm optimization. Lect. Notes Electr. Eng. 1, 89–100 (2016) 14. Zhang, L., Yuan, J.: Fault diagnosis of power transformers using kernel based extreme learning machine with particle swarm optimization. Appl. Math. Inform. Sci. 9(2), 1003–1010 (2015) 15. Vieira, I.N., Jacob, B.P., de Lima, B.S.L.P.: Bio-inspired algorithms for the optimization of offshore oil production systems. Int. J. Numer. Methods Eng. 91(10), 1023–1044 (2012) 16. Samigulina, G.., Massimkanova, Zh.A.: Multi-agent system for forecasting based on modified algorithms of swarm intelligence and immune network modeling. In: Proceedings of the 12th International Conference Agents and Multi-agent Systems: Technologies and Applications (KES-AMSTA-18), pp. 199–208 (2018) 17. Samigulina, G.A., Samigulina, Z.I.: Modified immune network algorithm based on the random forest approach for the complex objects control. J. Artif. Intell. Rev. 1–17 (2018)

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18. Mu, A., Cao, D., Wang, X.: A modified particle swarm optimization algorithm. J. Nat. Sci. 1(2), 151–155 (2009) 19. Ahmad, I.: Feature selection using particle swarm optimization in intrusion detection. Int. J. Distributed Sensor Netw. 1 (2015). https://doi.org/10.1155/2015/806954 20. Samigulina, G.A.: Development of the decision support systems on the basis of the intellectual technology of the artificial immune systems. Autom. Remote Control 74(2), 397–403 (2012) 21. Hayashida, T., Nishizaki, I., Sekizaki, S., Koto, S.: Cooperative particle swarm optimization in distance-based clustered groups. J. Softw. Eng. Appl. 10, 143–158 (2017) 22. Niu, B., Zhu, Y., He X.: A multi-population cooperative particle swarm optimizer for neural network training. In: Advances in Neural Networks. Lecture Notes in Computer Science, vol. 3971, pp. 570–576 (2006)

Chapter 27

Method of Computer Simulation of Thermal Processes to Ensure the Laser Gyros Stable Operation Evgenii Kuznetsov, Yuri Kolbas, Yury Kofanov, Nikita Kuznetsov and Tatiana Soloveva

Modern and advanced electronic equipment is developing along the way of miniaturization of electronic devices (EDs) and increasing their package density, accompanied by growth of heat emission. At the same time, EDs are continuously required to become more reliable and trouble-free. In this paper we use the generally accepted definition of EDs as devices which operate utilizing electrical, thermal, optical or acoustic phenomena. One of such devices is a laser gyro (LG), composed of angular rotation sensors on ring lasers, and electronics to provide their functioning and signal processing. In recent years, thermal simulation is being used in the designing of EDs in order to create competitive equipment in the shortest possible time and at the lowest possible cost. It allows to check thermal calculations and to optimize the design at the early stages of projecting, to reduce the time of testing of developed devices and to exclude redesigning caused by pre-calculation errors. Analysis of thermal processes in complex multi-boards assemblies of stack and cassette types with large number of electronic components (ECs) is particularly challenging. Especial problems appear when Russian engineers design a system of complex interconnected EDs for space rocket which launch several geophysical and communication satellites onto the Earth orbit. Such EDs include up to 1 million ECs with a dense package on printed circuit boards (PCBs) operating in a wide temperature range. There are many foreign software systems (SSs) for analysis of thermal processes in mechanical engineering. Much fewer SSs are able to simulate E. Kuznetsov · Y. Kolbas · N. Kuznetsov · T. Soloveva (B) Research and Development Institute “Polyus” named after M. F. Stelmakh, 3 Vvedensky str., Moscow 117342, Russian Federation e-mail: [email protected] Y. Kofanov Moscow Institute of Electronics and Mathematics, National Research University Higher School of Economics, 20 Myasnitskaya Ulitsa, Moscow 101000, Russian Federation © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_27

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thermal processes in electronics. But none of them can be applied in Russia for the several reasons. First, descriptions of these SSs have no information on the analysis of thermal processes in the complex EDs featuring PCBs assemblies of stack and cassette types. Second, the use of foreign SSs will require their adaptation to the component database and typical designs used in Russia, as well as to Russian design standards. Consequently, high price of foreign SSs and time required for their adaptation will significantly increase cost and time of development. It is noteworthy that foreign literature does not provide any information on simulation of the distribution of thermal fields in 3-axis LG with electronics. Thermal fields, as it is well-known, strongly affect consistence of EDs’ performance, which makes thermal behavior at the component, board and system level increasingly important in EDs designing. In this study, the well-known Russian automated SS for ensuring the reliability and quality of apparatus (“ASONIKA”) is used to simulate thermal processes in the volume of LG. “ASONIKA” is widely used in Russian enterprises, which develop and produce complex EDs. Applying “ASONIKA” we made the first ever presentation of the thermal and vibration behavior of 3-axis LG with electronics. We describe a new method of joint thermal-mechanical modeling. This method allows to obtain data on both thermal fields and resonant mechanical deformations, the levels of which depend on temperature. One of the key characteristics of ED is the capability to withstand both external thermal loads and internal ones, arising from the self-heat emission. The actual distribution of the thermal field in ED is determined by combination of these factors. The most critical problem herewith is the local overheating of EC placed on PCB, since it may cause changes in the parameters and/or failure of the EC in operating ED. Overheating of up to 10 °C is known to reduce EC lifetime by 50% when working at high temperatures. Simultaneously, because of heating of PCB materials, the elastic modulus and other thermally depended mechanical parameters of PCB are changing. It is followed by appearance of additional resonant frequencies, which cause significant deformations of PCB. As a result, mechanical overloads exceeding the permissible acceleration for EC may occur that engineers could not pre-calculate during designing without computer simulation. This study presents a new method of simulation the thermal fields in complex EDs with SS “ASONIKA”. One of the distinctive features of this SS is a complete detailed analysis of thermal and mechanical fields in structurally complex EDs. Analysis includes determination of temperature of all ECs and identification of dangerously overheated and mechanically overloaded places of the device in various operation modes. On the basis of these data, recommendations for elimination of inadmissible overheating and resonance overloads of EC, parts of the device structure and the device as a whole are formulated. Recommendations can stipulate both replacement of EC by more resistant to thermal and mechanical influences and application of methods of

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additional heat removal through thermal tires (which are also additional stiffeners), as well as through radiators, heat pipes, etc. Subsystems of thermal simulation “ASONIKA-T” for 3D electronic blocks and “ASONIKA-TM” for printed circuit units (PCB with EC) use equations of the classical theory of heat transfer, interpreted in the finite difference form in application to ED. The “ASONIKA-TM” subsystem also determines the resonant frequencies by the finite difference presentation of the bi-harmonic equation of vibrations of PCB as a plate with additional masses of ECs, located on this plate. This allows to simulate the action of vibrations processes with due consideration of the thermal factor impact on the printed circuit unit. Engineers create thermal and mechanical models with “ASONIKA-T” as undirected graphs in the grid form of a set of vertices and edges. “ASONIKA-TM” produces such graphs automatically. The novelty of the proposed method of computer simulation of thermal processes consists in the expansion of the subsystem “ASONIKA-TM” to mechanical modeling of printed circuit assemblies with reduced elastic modulus (effect of material softening due to temperature influence) and thermal changes of other parameters of PCB materials. Consideration of thermal changes in the parameters permits to detect the shift of the resonances in the lower frequency area and the appearance of new resonance frequencies. As it is known, in real operating conditions, the temperature growth leads to a gain in accelerations on the EC case under vibrations, an increase in the deformation of the boards and mechanical stresses in their materials. The final purpose of the simulation is to compare the obtained increased values of physical parameters under consideration with the maximum permissible values specified in the reference documentation, and to prevent overloads. An important advantage of the subsystems “ASONIKA-T” and “ASONIKA-TM”, in addition to fitting to Russian standards, is much lower cost compared to foreign SSs. The considered subsystem is equipped with specialized graphical interfaces I/O information for complicated ED designs of cassette and stack types. It is important that the above subsystems are created on specialized algorithms for automatic synthesis of complex ED thermal models. Besides, they have databases on ECs and structural materials used in Russia. For “ASONIKA” system’s application in other countries, it is easy to fill the database with EC data and other reference data used in these countries. High level of automation and intelligence makes “ASONIKA” user-friendly for a wide community of engineers, not just specialists in thermal simulation. The advantages of “ASONIKA” have been numerously confirmed while it was being used in the design process of the most complicated multicomponent ED complexes in the aviation and space industries. Full paper presents the results of advanced “ASONIKA-T” and “ASONIKA-TM” subsystems application for analyzing and predicting the modern LG behavior under thermal effects. In LG, in addition to the three laser sensors of the angular rotation of the gyros, rigidly mounted in the aircraft, there is a set of electronic boards supplying the sensors operation and forming output signals for 3D channels X, Y, Z.

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The authors created a method to establish the algorithm for computer thermal simulation of LG. First, LG general macro-modeling is performed with “ASONIKAT” to obtain the temperatures of the LG supporting structures and the air surrounding the PCBs. Then the consequent full simulation of PCBs one by one is carried out with “ASONIKA-TM” with determination temperature and acceleration of each EC at the resonant frequency. For thermal simulation, a pre-calculation of the heat emission powers in each EC is necessary by analysis of the LG schematic diagram. At the same time, it should be taken into account that the heating of the EC cases often leads to a significant change in the electrical parameters of the EC, which, in turn, causes a change in heat emission levels. Thus, there is a circular interconnection of the electrical power emitted by the EC, with the temperatures of their cases. Usually, engineers apply the iterative modeling to exit this circular interconnection of powers and temperatures, working alternately in the electrical and thermal programs of “ASONIKA”. In this study, we proposed to combine algorithmically two programs (electrical and thermal), using mathematical analogies between mathematical descriptions of electrical and thermal fields. The analogy gives the possibility to organize the simultaneous operation of electrical and thermal modeling programs, to reduce the total simulation time and besides to improve the accuracy of the results. This leads to a decrease in the cost of designing LG. In the full paper, we demonstrate the simulation results in graphic view, which got by “ASONIKA” creating a virtual image of the ED in whole and in parts with superimposed continuous images of color physical fields of temperature, deformation and mechanical stresses, primarily at resonant frequencies. The adjacent color bars of matching colors and temperature values permit to visually qualitatively determine the most loaded places in the supporting structures and to quantify the thermal loads coefficients of each EC. When using the “ASONIKA” system for analyzing the behavior of LG under thermal effects, real levels of thermal loads on ECs and structural elements were obtained. As a result, “weak spots” were recognized, for example, ECs, the levels of actual thermal effects on which exceed the maximum allowable values for them (taking into account the established margins for reliability). After making changes in the design (replacing the ECs with more heat-resistant ones, installing radiators and heat tires), repeated computer simulations showed the effectiveness of these changes. Thus, our investigation proved “ASONIKA” to give a significant reduction in the devices development time, since even before the real research tests the actual temperature regimes of the ECs and the resonances of PCBs are available to be discovered, so that the necessary modifications of the design for eliminating the “weak spots” will be carried out. The resulting thermal fields clearly illustrate where the temperature sensors should be installed to check ECs temperature while LG operation for possible algorithmic thermal correction. Performed studies have shown that the modeling error is no more than 5–8%, being easy to take into account when assigning margin for reliability. The key action to further improve the accuracy of thermal modeling is to set an increased

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number of grid steps on the board drawing at the beginning of the simulation. However, this will either lead to a rising operating time of the program, or will require more powerful computers, so a compromise solution is being usually recommended to take optimal decision for a concrete project. Thus, the proposed method of modeling with “ASONIKA” significantly reduces the cost of designing and manufacturing of LG, permitting to decrease the amount of research tests, and also helps to improve the quality and reliability of the developed high-tech devices, reduce the cost of their maintenance, considerably diminish the risk of failure, ensuring the stability of the LG operation. In conclusion, as a result, we summarize the following most interesting new approaches of our research. 1. The calculation of heat power dissipation in each EC by analysis of schematic diagrams, which had earlier preceded thermal simulation, we first proposed to combine algorithmically with thermal modeling. We connected an electric model to a thermal model using mathematical analogies between mathematical descriptions of thermal and electric fields. This algorithm permitted to organize the simultaneous operation of the combined simulation program, thus to reduce the total simulation time and also to improve the accuracy of the results obtained. 2. We developed a new thermal-mechanical method of computer modeling under thermal effects, consisting in the up-growth of “ASONIKA-TM” ability for mechanical modeling. A new thermal-mechanical subsystem “ASONIKA-TM” simulates contemporaneously the thermal and vibration processes in the printed circuit assemblies with considering the elastic modulus decrease and changes in other parameters of the PCB materials due to temperature effects. 3. We created a method to establish the algorithm for a complete sequence of computerized electrical, thermal, and vibration modeling of LG. First, joint electrical and thermal macro-modeling of LG as a whole is carried out to obtain the total power of heat emission in each printed circuit unit, the temperatures of the supporting structures of LG and the air surrounding each printed circuit unit. Then full successive thermal and vibration modeling of the LG printed circuit assemblies in “ASONIKA-TM” is carried out with determination of the temperature and acceleration of each EC at the resonant frequency for the given parameters of influences. 4. We got the results of modeling by “ASONIKA” the thermal and vibration behavior of 3-axis LG with electronics as complex cyber-physical device. The developed method has good versatility and can be applied to virtual computer modeling in the design of a wide range of EDs. Thermal modeling with “ASONIKA” permits to analyze the thermal loads during the research of ED operation, simulating various flight test and real exploitation conditions.

Chapter 28

Numerical Approach of Viscous Flow Containing Short Fiber by SPH Method Nobuki Yamagata and Masakazu Ichimiya

Abstract The simulation methodology for the injection molding is proposed using SPH method in this paper. First, the specific algorism based on the SPH method is designed to simulate the solidification of the molding resin. The algorism is employed to simulate a plate of typical resin by two dimensional SPH method, focusing on the distribution of resin velocity and temperature in thickness direction. And then the viscous flows containing short fibers are simulated using the algorism. The results showed that the SPH method using the algorism can clearly simulate the formation of the solid layer along the mold surface and has a prediction potential of the resin solidification behavior. Keywords SPH · SMAC · Injectin molding · Resin flow velocity · Short fiber

28.1 Introduction In recent years, the injection molding method capable of producing a complex shape product in large quantities in a short period of time is widely used at the site of resin molding, and numerical simulation is carried out for studying the resin molding conditions and improving the quality [2, 4]. In particular, in order to clarify the phenomenon caused by the molding process in injection molding and to take measures against defects and improve quality, it is required to be able to sufficiently express the behavior peculiar to the resin in the mold. The SPH method (Smoothed Particle Hydrodynamics), which is one of the typical particle methods as a numerical simulation method, expresses a fluid or a solid as a group of a plurality of particles and calculates an interaction between the particles, thereby obtaining a fluid behavior and the interaction of both can be expressed N. Yamagata (B) Advanced Creative Technology Co., Ltd., Shibuya, Shibuya-Ku, Tokyo 150-0002, Japan e-mail: [email protected] M. Ichimiya The University of Tokyo, Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_28

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directly, and since it is a Lagrangian method which does not use a mesh, it has also a feature that it can easily analyze large deformation and movement of solid. For this reason, the SPH method has a potential to express the resin behavior in the molding process as a methodology, and it can be expected to demonstrate its effect in preventing defects and improving quality. In this paper, using the SPH particle method, we developed a methodology of the resin flow, the solidification simulation [2] and the resin flow containing short fibers, and confirmed that the methodology has the potential to express the behavior for the injection molding.

28.2 Numerical Approach by SPH Method In the simulation by SPH method, flow analysis and heat conduction analysis are weakly coupled.

28.2.1 Flow Analysis When the Navier-Stokes equation is expressed by the Lagrangian method, the following equation is obtained. Du/Dt = −1/ρ · grad P + ν∇2 u + f

(28.1)

where u is flow velocity, ρ is density, P is pressure, ν is kinematic viscosity coefficient, and f is volume force. Also, the pressure Poisson equation derived from the NavierStokes equations and continuous conditions is 2 P = divf

(28.2)

In this method, SMAC (Simplified Marker and Cell) algorithm is employed as a time integration method of the Navier-Stokes equation. Regarding the incompressibility condition, instead of divergence of the velocity field, the density constant condition is set.

28.2.2 Heat Transfer Analysis The SPH theory is applied to the heat conduction equation, considering heat transfer, heat transfer due to thermal conduction between resins, and internal heat generation of the material, the following equation is obtained. The final term represents the internal heat generation of the generalized Newtonian fluid.

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  ρc∂T/∂t = λ ∂ 2 T/∂x2 + ∂ 2 T/∂y2 + ∂ 2 T/∂z2 + η(∂γ/∂t)2

303

(28.3)

where ρ is density, c is specific heat, λ is thermal conductivity, η is viscosity, ∂γ/∂t is shear strain rate. The time increment t needs to be t < ξρch2 /λ from the stability condition, and η= particle distance, ξ = 0.15 here.

28.3 Numerical Analysis 28.3.1 Analysis Model The analysis target was a basic shape, and the cavity shape was a flat plate with a thickness of 3 mm so that injection molding of plastic could be simulated. We assumed that the width direction is uniform and decided to analyze it in two dimensions. The analysis model is shown in Fig. 28.1. Normally, the resin passes through the runner portion and flows into the cavity after passing through the narrow gate portion. In this model, the inflow velocity is uniform and the gate portion is deleted. At this time, it is assumed that the resin at 260 °C. from the left end is pushed out toward the cavity at 20 mm/s. Assuming that the flow velocity is uniform in the cavity, the resin flow velocity in the cavity is equivalent to 41.3 mm/s. The distance between particles was 0.05 mm.

Fig. 28.1 Analysis model

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Fig. 28.2 Analysis model (with fibers)

28.3.2 Material Propertiy and Boundary Conditions Assuming a plastic resin as the object material, material constants and thermophysical properties are as follows. Density: 1.0 × 10−6 kg/mm3 Specific heat: 1.856 × 103 J/kg/K Thermal conductivity: 0.183 W/m/K Viscosity depends sensitively on temperature and shear strain rate, but here it is set at a constant value of 0.1 Pa-s. The melting point or freezing point was set to 100 °C as the glass transition temperature. Fluidity disappears below melting point. The material property values of steel were used for the mold. It was assumed that the outer surface of the mold was specified at room temperature (Fig. 28.2).

28.3.3 Analysis Results (1) Flow analysis of resin without fibers The flow velocity, shear strain rate, temperature distribution and solidification growth of resin for about 0.1 s were investigated. The distribution of resin flow velocity in the cavity after 0.05 s is shown in Fig. 28.3a. Similarily the shear strain rate of the resin in the cavity after 0.05 s is shown in Fig. 28.3b. After 0.05 s, the resin in contact with the mold has already solidified because its temperature dropped blow freezing point. This is marked as “Solidified Layer” in the figure. The shear strain rate shows a large value at the interface between the solidified layer and the resin and is zero at the center of thickness. The temperature of the resin is destributed by the inflow heat from the upstream, the heat conduction between the resins, the heat conduction between the resin and the mold, and the internal heat generation of the resin. The internal heat generation is

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Fig. 28.3 a Flow velocity of resin. b Shear strain rate of resin

Fig. 28.4 a (left) Temperature distribution at 0.025 s. b (right) Temperature distribution at 0.05 s

influenced by the shear strain rate and the viscosity as shown in Eq. (28.3). Therefore, if the viscosity is constant in the resin, the amount of hear, which is not negligible low, is produced around the interface between the solidified layer and the resin. Figures 28.3 and 28.4 show the temperature distribution of the resin in cavity at time 0.025 and 0.05 s respectively. The resin portion in contact with the mold was solidified at about 0.03 s, and a solidified layer having a thickness of about 0.05 mm was formed. On the other hand, in the injection molding experiment [3] using polystyrene with a cavity thickness of 3 mm, a resin injection temperature of 260 °C and an injection rate of 6.3 cm3 /s (equivalent to a uniform injection rate of about 42 mm/s), the solidified layer thickness was about 0.2 mm (at the stage of 30 °C of mold initial temperature), which was different from the analysis result. This is because the viscosity defined in the analysis was a constant value of 0.1 Pa-s, the viscosity of polystyrene used in the experiment was about 1000 Pa-s (the shear strain rate in the vicinity of the solidified layer 100/s).

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(2) Flow analysis of resin containing fibers Figure 28.2 shows a resin flow analysis model in a cabity containing fibers. The shape and analysis conditions were the same as the analysis model shown in the previous Sect. 3.2. Here, the fiber material was modeled by connecting one particle and one particle by a spring. And a spring stiffness was set to so that the distance between two particles was constant. In order to verify the flow of the fiber material in the resin, as shown in Fig. 28.5, three fiber materials were located perpendicular to the flow direction and the resin flow analysis was performed. It is consistent with Jeffery’s theory that fiber orientation is aligned in the shear direction in shear flow [5]. Next, as shown in Fig. 28.6, the fibers were randomly located irrespective of the flow direction, and resin flow analysis was performed. The orientation of the fibers was also consistent with the Jeffery theory as in the previous analysis. It was found that the moving direction of fibers was becoming in parallel with the flow direction of resin due to the velocity gradient near the wall boundaries.

Fig. 28.5 Transition of fiber orientation (regular distribution)

Fig. 28.6 Transition of fibers orientation (randam distribution)

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28.4 Conlusions and Further Works (1) The methodology of the solidification simulation of resin by SPH method was proposed. (2) By applying this methodology to a typical thin plate of plastic resin, prediction of resin flow velocity and solidified layer progression were carrued out. As a result, despite simplification such as keeping the viscosity constant irrespective of the shear strain rate, the initial solidified layer evolution behavior close to the past experimental results could be reproduced. In order to further increase the degree of approximation, it is necessary to study the viscosity [1].

References 1. Ichimiya, M., Yamagata, N.: High viscous flow analysis in the 3D printer by SPH, ICCES 2019.3 (to be published) 2. Ichimiya, M., Shirazaki, M., Sakai, Y.: Develompent of solidification simulation in viscosity by SPH. JSCES, Vil.19, 2014.9 3. Seto, M., Wada, T., Sato, K., Okada, Y., Yamabe, M.: The influence of molding conditions on solidification growth during injection molding 19(9), 589–594 (2007) 4. Yashiro, S., Okabe, T., Matsushima, K.: A numerical approach for injection modeling. Adv. Compos. Mater 20(6), 503–517 (2011) 5. Jeffery, G.B.: The motion of ellipsoidal particles immersed in a viscous fluid. In: Proceedings of the Royal Society of London, Mathematical, Physical and Engineering Sciences, pp. 161–179 (1922)

Chapter 29

High Viscous Flow Analysis in the 3D Printer by SPH Masakazu Ichimiya and Nobuki Yamagata

Abstract The methodology of solidification simulation of high viscous resin using Smoothed Particle Hydrodynamics (SPH) was developed. An implicit scheme was adopted to calculate the viscous term of a high viscous non-Newtonian fluid. The methodology was applied to thermal melting, lamination and solidification of high viscous resin ABS injected from the 3-D printer nozzle. As a result, it is confirmed that this methodology has a potential to express lamination and solidification of high viscous resin. Keywords SPH · High viscous resin · Non-newtonian fluid · 3-D printer

29.1 Introduction In recent years, numerical simulation has been conducted to investigate resin molding conditions and improve quality [1]. In the simulation, it is required to be able to sufficiently express the behavior peculiar to the resin in the mold. In the Smoothed Particle Hydrodynamics (hereafter SPH) method, fluids or solids are represented as a group of particles, and calculation of the interaction between the particles enables direct representation of the behavior of the fluid, the solid and the interaction between the two. The SPH is a Lagrangian method without using a mesh, and it can be expected that there is a potential to express the resin behavior in its molding process because it can easily analyze various shape and large movement. Thus, a methodology of solidification simulation of high viscous non-Newtonian fluid using the SPH method was developed and applied to the analysis of lamination, and solidification process of 3-D printer. M. Ichimiya (B) The University of Tokyo, Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan e-mail: [email protected] N. Yamagata Advanced Creative Technology Co., Ltd., Shibuya, Shibuya-Ku, Tokyo 150-0002, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_29

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29.2 Numerical Approach by SPH Method In the simulation by the SPH method, flow analysis and heat conduction analysis are weakly coupled.

29.2.1 Flow Analysis The Navier-Stokes equation is expressed as follows by the Lagrangian method Du/Dt = −1/ρ · grad P + ν∇ 2 u + f

(29.1)

where, u is flow velocity, ρ is density, P is pressure, ν is kinematic viscosity coefficient, and f is body force. In this method, it is formulated by using SMAC (Simplified Marker and Cell) algorithm where the incompressibility is given by pressure Poisson equation derived from Navier-Stokes equations and continuous condition. uk+1 = u∗ +u

(29.2)

u∗ = uk + t [ν∇ 2 u + f]k

(29.3)

∇ 2 Pk+1 = ρ div u /t

(29.4)

In the pressure Poisson equation as shown in Eq. (29.4), the constant density approach was used instead of the velocity divergence free approach. In the case of a high viscous fluid, it is necessary to set the time step extremely small due to the diffusion number limit by the explicit scheme shown in Eq. (29.3). In order to avoid this cumbersome computation, an implicit calculation scheme [2] was adopted.

29.2.2 Heat Transfer Analysis Considering heat transfer by thermal conduction and internal heat generation of the material, the heat conduction equation is expressed as follows. The last term represents the internal heat generation of the generalized Newtonian fluid.   ρc∂T/∂t = λ ∂ 2 T/∂x2 + ∂ 2 T/∂y2 + ∂ 2 T/∂z2 + ηγ˙ 2

(29.5)

Here, ρ is density, c is specific heat, λ is thermal conductivity, η is viscosity 1/2    γ˙ ij γ˙ ji γ˙ = 1/2 i

j

(29.6)

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29.3 Simulation 29.3.1 Verification Analysis Since the implicit scheme of the viscous term is newly used in the flow analysis, program verification was carried out by analyzing the Poiseuille flow between two parallel stationary infinite plates. The high viscous fluid in the parallel plates of flow path width H = 1 mm, ρ = 10− 6 kg/mm3 is driven by a body force f = 2 × 107 mm/s2 and this was solved under periodic boundary conditions. First, Newtonian fluid with viscosity η = 9928 Pas was analyzed and it was confirmed that it agrees with the theoretical solution [2] as shown in Fig. 29.1. In this case, the body force and the pressure loss become balanced gradually, and arrive at a steady flow at about 10− 7 s. Next, non-Newtonian fluid was analyzed. It is assumed that the shear strain rate dependence of viscosity can be represented by the power law (Eq. (29.7)), n = 0.38 and η 0 = 9928 Pas with reference to the viscosity of the ABS resin [3]. η = η0 γ˙ n−1

(29.7)

In the present analysis, when the flow velocity increased with the body force, the shear strain rate increased and the viscosity decreased. The flow velocity itself continued to increase and did not arrive at a steady state. However, the non-dimensional flow velocity divided by average value V ave converged to the steady theoretical solution (29.8) [4] at each time except the initial state (Fig. 29.2).   V/Vave = (2n + 1)/(n + 1) · 1 − |1 − z/(H/2)|(n+1)/n

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29.3.2 Solidification Analysis of Resin Analysis of lamination and solidification of resin discharged from a hot melt lamination type 3-D printer nozzle model was carried out using the algorithm described in Sect. 29.2. In the hot melt lamination method, a thermoplastic resin melts easily by applying heat, and it laminated on an output stage to create a three-dimensional shape. ABS resin (acrylonitrile-butadiene-styrene), PLA (polylactic acid) etc. are used as the thermoplastic resin. In this analysis, amorphous ABS was chosen. The glass transition point Tg of ABS was set at 373 K. As shown in Fig. 29.3, the analysis Fig. 29.3 Configuration of 3-D printer model

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model has a nozzle diameter of 0.4 mm, and a distance between the output stage and the nozzle tip is 0.2 mm. The viscosity η is formulated as combination of a power law for the shear strain rate dependence and the WLF model [5, 6] for the temperature dependence. ˙ = η0 (T)γ˙ n−1 η(T, γ)

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  η0 (T) = D1 exp −A1 (T − T∗ )/(A2 + T − T∗ )

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where n = 0.38, D1 = 2.3 × 1011 (Pas), A1 = 23.87, A2 = 51.6 (K), T * = Tg = 373.0 (K). For the temperature boundary condition, the temperatures of the nozzle and the output stage were set to constant values of 503 and 353 K, respectively. The initial temperature of the resin is set at 503 K. The resin discharged from the nozzle is cooled by heat conduction with the output stage and natural convection heat transfer (heat transfer coefficient: 7.92 W/m2 K) with air (temperature: 300 K). Material properties are assumed as; c = 1500 J/kg/K, λ = 0.25 W/m/K, thermal expansion coefficient = 2.7 × 10 −4 /K. Fixing the output stage, first adhere the resin on the stage while moving the nozzle and container to the right (Figs. 29.4➀–➁), and lifted up the nozzle by 0.25 mm at the stage end (➁–➂). Then, laminate the resin on solidified resin (➂–➃) while inverting and moving to the left. The moving speed was set as shown in Fig. 29.4. The resin in the container was assumed to move downward and injected from the nozzle by the pressure caused by the upper push plate moving down at 15 mm/ sec. Analysis results for first and second layers are shown in Figs. 29.5, 29.6, 29.7, 29.8 and 29.9, respectively. They illustrate the states immediately after the turnarounding point and two layers stacking completion (Figs. 29.4➁, ➃). The resin velocity excluding the nozzle moving is the vertical direction. The resin adheres to the stage immediately after leaving the nozzle (Fig. 29.5). At the time of ➁, only a small amount of injected resin was solidified in the vicinity of the initial position (Fig. 29.4➀), however the unsolidified resin on the stage was cooled down while lifting up the nozzle at an extremely low speed, and all solidified except the vicinity of the nozzle at the point of ➂. Because, as the resin passes through the nozzle, it experiences a large shear strain rate at the interface with the nozzle (Fig. 29.6), its temperature rises slightly due to viscous heat (Fig. 29.7). The flow condition after returning (Fig. 29.8) is basically Fig. 29.4 Nozzle movement

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Fig. 29.5 Flow distribution prior to turnarounding

same as that just prior to the return point. Concerning the first layer resin at the time of completing the two-layer lamination, the portion sticking to the stage maintains solidified state, however the portion at the upper surface is redissolved due to the heat from the second layer resin (Fig. 29.9).

29.4 Conclusions (1) The methodology of solidification phenomenon of resin was developed using SPH. An implicit scheme was adopted in order to be able to deal with high viscous non-Newtonian fluid, and the validity of program was confirmed by comparing with theoretical solution of the Poiseuille flow. (2) The above methodology was applied to the analysis of the resin injected from a 3-D printer nozzle. As a result, it was found that the developed method had a potential for representation of lamination and solidification of high viscous resin.

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Fig. 29.6 Shear strain rate distribution prior to turnarounding

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Fig. 29.8 Flow distribution prior to completion of two layer lamination Fig. 29.9 Temperature distribution prior to completion of two layer lamination

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References 1. Yamagata, N., Ichimiya, M.: Numerical Approach of Viscous Flow Containing Short Fiber by SPH Method. ICCES, 2019.3 (to be publised) 2. Morris, J.P., Fox, P.J., Zhu, Y.: Modeling low reynolds number incompressible flows using SPH. J. Comput. Phys. 136, 214–226 (1997) 3. Kuroda, H., Shimohira, K.: Quantification of Weld-line and Quantitative Evaluation of its Molding Factors in Injection Molding, vol.2, pp. 159–165. Seikeikakou (1990) 4. Bharti, R.P., Chhabra, R.P.: Two-dimensional steady poiseuille flow of power-law across a circular cylinder in a plane confined channel: wall effects and drag coefficients. Ind. Eng. Chem. Res. 46, 3820–3840 (2007) 5. Williams, M.L., Landel, R.F., Ferry, J.D.: The temperature dependence mechanism in amorphous polymers and other glass-forming liquids. J. Am. Chem. Soc. 77, 3701–3707 (1955) 6. Boronat, T., Segui, V.J., Peydro, M.A., Reig, M.J.: Influence of temperature and shearrate on the rheology and processability of reprocessed ABS in injection molding process. J. Mater. Process. Technol. 209, 2735–2745(2009)

Chapter 30

Content Structure for Driving Object Parameters in Contextual Model of Engineering Structure László Horváth

Abstract Advanced industrial product is operated by cooperating systems and can be considered as cyber physical system (CPS). To cope with engineering of this product, system level model should be considered to replace the conventional physical level model. System level modeling was enforced by high level products among others at leading aircraft and car industries. This makes the conventional dialoguebased model definition challenging. Moreover, recent efforts for integration of formerly separated models of engineering product made physical level product model system very complex. As contribution to solving the problem generated by the above new situation, this paper introduces a new modeling method and model structure which assists engineers at the definition of system level product model. The proposed solution uses thematically structured driving intellectual content model which can be integrated in the structure and methodology of system enabled engineering modeling platform. In the organization of this paper first new concept of extended CPS model is introduced. This is followed by explanation of structure and operation of the proposed driving content model. In the rest of paper issues of integration of the proposed contribution in leading industrial engineering modeling platform and implementation issues are discussed. Recent idea of harmonizing well grounded theory and proven experience in industrial product model for innovation cycle and lifecycle of product is considered. Keywords System based engineering model · Extended cyber physical system concept · Thematically structured driving intellectual content model · Knowledge driven object model · Integration engineering areas in project

L. Horváth (B) Óbuda University, Budapest, Hungary e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_30

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30.1 Introduction Several achievements in computational intelligence and computer representation changed the world of engineering during the past two decades. Using these achievements, powerful computer model-based solutions were developed as answers to new challenges by smart industrial products such as consideration of all contexts, engineering with systems, design and application of advanced shapes including organic ones, and smart control of cyber physical system (CPS) [1]. Comprehensive modeling capabilities were organized in platform such as introduced in [2] replacing conventional software systems which served computer aided engineering, design, simulation, and manufacturing planning by contextually integrated system and intellectual property (IP) grounded solution. Smart products are operated by less or more systems real time cooperation of which is inevitable at every day usage of these products. There is no other way than starting product development projects with system level engineering. Recognizing this tendency systems related research started in five level abstraction [3], the concept of virtual engineering space (VES) [4], and organized representation of knowledge in the background of model generation for smart products and production [5] at the Laboratory of Intelligent Engineering Systems. This laboratory was founded by the Óbuda University for research and master and PhD level higher education programs in high abstraction modeling of industrial products in the year 2005. One of the main challenges in system-based product modeling is shifting of abstraction level. In [3], five levels of abstraction were proposed including intent of collaborating humans, meaning of concepts, engineering objectives, contexts, and decisions. It was concluded that high abstraction of model system was demanded by engineering for industrial products. Model of product represents and describes engineering structure (ES) and is developed in the course of multidisciplinary cooperation of participants from system engineering (SE), information engineering, mathematics, physics, and areas in engineering expertise. Other critical issue is integration. The story of integration the formerly separated engineering solutions was started by the integrated product information model (IPIM), ISO10303) [6] during the 90 s. IPIM applied the product model concept and included several new concepts such as application shape feature and its boundary representation, object-oriented model with engineering related object classes, resource-based model construction, and application protocol (AP). Product model concept was generalized as ES in [7] where ES may be industrial product, experimental structure for research or development, or virtual prototype. The new demand for contextual system organized integration was studied and the VES concept was developed [4]. VES was defined as a full contextual virtual version of a planned or existing engineering space and considered as utmost purpose of engineering model development. Development and application of VES is proceeded during the whole innovation and life cycle of an industrial product. Industrial modeling practice implemented the four levelled requirements, functional, logical, and physical (RFLP) structure from systems engineering (SE) as

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organization of system-based model of ES [6]. Related research at the Laboratory of Intelligent Engineering Systems focused on content structure which is suitable for driving object parameters in RFLP structured model of ES [5]. This way of research was found as very important because despite advanced modeling capabilities of industrial engineering systems the conventional dialogue-based definition of product model objects was still challenging to apply at definition of system-based model. The problem is that too much and always changing contexts must be considered at the definition of each functional and logical level component of ES (11). Four levelled driving content structure was developed and published in [7] with levels for initiative, behavior, context, and action structures on its levels. Based on the above discussed early modeling concepts and methods, this paper introduces recent results in the Laboratory of Intelligent Engineering Systems in extending of system-based engineering model to CPS, model of driving contexts for generation of RFLP structured model objects, integration of driving context model with industrial engineering modeling platform, and implementation of the proposed modeling in real world industrial, research, and higher education environments.

30.2 Concept of Extended CPS Model of ES Information technology support of ES has been organized into CPS. This means that ES consists of physical units to execute functions and cyber units which real time control parameters of physical units [8]. Real time control is supported by actual sensed parameters in physical units. Cyber units provide coordinated control for physical unit functions. Because ES model is developed and applied during life cycle of ES, it seems obvious that object and knowledge [9] representations in ES model can be utilized by cyber units of field operated CPS ES. At the same time, cyber units can provide information about real physical processes as experience. This would provide ES as lifecycle experimental environment to improve ES model and finally ES. Development of production systems rushes towards CPS. These smart production environments utilize new paradigm of Industry 4.x. Recent ES models include manufacturing model which represents activities, system, and resources. Consequently, three CPSs namely virtual ES, field operating ES, and ES manufacturing system need integration in the future. This section of paper is about a proposal for model to support this integration. Before introduction of the proposed integrated model, it must be stated, that one of the main critera for its implementation is availability of suitably advanced and comprehensive industrial engineering modeling platform in which role assignable modeling capabilities can be accessed at anywhere and anytime from professional dedicated cloud system as service. Engineering modeling system is under development at the Laboratory of Intelligent Engineering Systems in cooperation with the Doctoral School of Applied Informatics and Applied Mathematics which will fulfill the requirements of this purpose. This system is based on the 3DEXPERIENCE

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platform by the Dassult Systémes [10] and will be appropriate for research, solution development, and product (ES) development using advanced modeling capabilities. The new system will serve as research environment for master and doctoral level students. It is impossible to track changes of contexts acting inside ES and acting from outside of ES using conventional engineering methods. Recent model and realistic simulation [11] structures make it possible to develop model system which has the capability of self-modification reacting driving context changes [12]. For this reason, the proposed content model includes full context system to support this integrating feature of ES model. The CPS ES model which is extended by field operated CPS and CPS for the manufacturing of CPS ES is shown in Fig. 30.1. Extended CPS model consists of model sectors. Sectors are levels of system-based ES model (RFLP structure), levels of ES manufacturing model, levels of driving content system (DCS) model, cyber units of field CPS ES, and cyber units of manufacturing system. Figure 30.1. does not separate these sectors because content elements, ES components, and cyber entities are individually connected by contexts. New or modified context can not break any already decided (AD) context. Principle of full contextual modeling is applied which one of the main recent achievements in engineering [8]. In Fig. 30.1, groups of contextual connections are shown in shaded boxes. Within a group, individual contexts are defined between relevant parameters of connected objects. Driving object sends driving content to the driven object. Context may be value of the driven parameter, method to generate the driven parameter, or method placed in the driven parameter. In this way, any value, value range, discrete value, mathematical function, algorithm, etc. can be placed in context. The model in Fig. 30.1 constitutes contextually changing and developing structure. Changes in model system of ES are generated by affects from outside context sources. These sources are participants with roles, organized and approved IP, and higher priority affects. Role of participant authorizes for contribution to ES model using given set of modeling capabilities. Multiple role can be assigned to participant. IP is organized at the environment where ES is modeled, at other institutions, at companies and at independent experts. In case of proper authorization, context directly connects driving IP entity with driven ES model entity. Higher priority affects come from different sources, decided by the world outside of ES model and mandatory to apply at definition of the affected model entities. At the beginning of its definition an ES model is empty. ES model is generic so that it can be applied at generation of high number of ES model instances using purposeful set of active contexts. In this way, old method of retrieved and actualized typical solutions is avoided. ES model consists of three multilevel structures for DCS model, RFLP structured model, and manufacturing model of ES. Sequence of levels within each structure also means sequence of AD contexts. Connection between objects in different levels is free when AD contexts are not broken. This is required to maintain integrity of ES model. Sources of affects from outside of ES are contextual with DCS structure elements (E-DCS) or directly with RFLP structure components (E-RFLP). It is obvious that E-DCS and E-RFLP contexts are very different. E-RFLP context drives requirements

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Fig. 30.1 Extended CPS model

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(R), functional (F), logical (L), and physical (P) levels of RFLP structured model of CPS ES. When outside source affect context is placed on DSC structure element (E-DCS), relevant initiatives, system, behavior, and activity level elements of DCS model drive R, F, L, and P level components of RFLP structure by contexts DI-R, DS-F, DB-F, DB-L, and DP-P. DCS structure levels will be introduced and discussed in the next section of this paper. In the extended CPS model, model of ES manufacturing is contextual with physical level of RFLP structure (M-A) and activity level of DCS structure (D-A). Contextual connection of ES manufacturing objects from abstraction levels higher than these basic abstraction levels is planned to research in the future. This depends on development for level of abstraction in manufacturing models in the future. It is also assumed that manufacturing model receives all RFLP and DCS contexts by objects on its activity level. Manufacturing system and resource model levels are assumed to define in the context of activity level. Manufacturing model for ES is not an issue in this paper (Fig. 30.4). Model in Fig. 30.1 assumes that cyber units of ES are in contextual connection with objects in physical level of RFLP (M-C) and activity level of DSC (D-C) structures. M-C context communicates physical level object information with CPS ES and operating parameter related experience with P level entities in RFLP structure. This is not issue of this paper. In the present stage of research, it is supposed that the M-C contexts communicate ES physical component information while D-C contexts communicate the content behind component information. This paper introduces one of the possible sets of D-C contexts in Sect. 5 (Fig. 30.4). Cyber units of manufacturing CPS (10) called as cyber control in Fig. 30.1 is in contextual connection with manufacturing system model (M-P) and cyber units of ES (C-P) to exchange of manufacturing system and practical experience information, respectively. At the development of extended CPS model (Fig. 30.1) availability and application of the following modeling capabilities are presumed. Object modeling is required which allows for system level definition of ES in RFLP structure. All inside and outside contextual connections should be represented using conventional and smart knowledge representations to allow them to include in ES model. Functional and logical level models of ES must be virtually executable [13]. Means must be available to propagate any change of inside or outside context information or representation in the entire model system. Finally, capabilities must be available for the definition of contextual connections between component models within ES model system for the entire innovation and life cycles of ES.

30.3 Structure of DCS Model Driving content entities are organized in the DCS which is proposed as contextually integrated new sector of the extended CPS model. The CPS concept is defined among others as means for utilization organized contexts with ES model object parameters.

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Because DCS and its operation in the extended CPS model is one of the main contributions in this paper, it is obvious that the question why CPS is a separated sector must be answered here. The alternative would be to represent driving content within the RFLP model structure because RFLP structure is capable of representation both knowledge and driving contexts. The main argument for DCS as a separated sector is that purpose of DCS is much more than representation of knowledge and information in contextual chains for RFLP structured model of ES. While context structure in RFLP represents ES, context structure of DCS represents collected and specifically structured intellectual content. Although DSC structure provides driving for RFLP structure component parameters, its context structure is very different from context structure in RFLP. Difference can be seen from discussions below in this paper. A parameter of an RFLP structure element is often driven by two or more contexts. These driving contexts must not be contradictory. Contradictions must be dissolved within the DCS. Contexts are changed in their quantity, content and activation and are always tied to their source during the whole innovation and life cycle of ES. Recent variant of DCS structure (Fig. 30.2) was configured to support the extended CPS model (Fig. 30.1). This variant is introduced below. One of two former variants of this driving content structure supported only conventional physical level, while the other could drive RFLP structured system level [6] model of ES. Four contextual levels of DCS structure serve organized and continuously developed representation of content for outside context driven initiatives, structure of cooperating systems in ES, behaviors of ES for driving functional and logical components in RFLP structure, and physical actions for driving physical level of RFLP structured ES model and cyber units in field operating ES. Each level consists of contextual sequence of sublevels. Sublevel organizes content for a well-defined aspect of driving. Operation of this DCS structure is discussed in the next section of this paper. On the level of initiatives, aspects of driving are specification for ES, patterns those applied in ES, and configuration of units and main parameters of ES. Additional aspects are methods, processes, and procedures which are applied at ES entity definition. Element in these substructures is personally or institutionally dedicated in accordance with its source. Multilevel content structures were proposed for the representation of IP and higher priority affecting content in [14] to assist connection of affecting sources with elements in DCS structure sublevels. Affecting content from participants is communicated on user surface of the RFLP structure enabled modeling system. On the level of systems, driving is somewhat problematic because system definition in RFLP structure is inherently implicit [6]. Systems aspect was introduced to establish explicit structure of systems in ES. One of the purposes of this aspect is to avoid omitted systems as it can be seen in some recent system-based ES. This level structure and its contextual connections need high amount research in the future. The two other aspects on this level are decomposition of functions and organization of system level contexts. The next level of DCS was designed to drive RFLP structure using behavior structure. Structured behavior is inherently not included in RFLP structure despite

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Fig. 30.2 Levels and sublevels of DCS model

behavior representations play key role at concept level virtual execution of the ES model [6]. Behavior is defined for situation. Content for situations is organized in own aspect where situation is defined by its contexts within DCS. Novel attempt in this structure is organizing content for simulations. Actuality of this issue is given by recent developments in ES wide simulation structures and simulation-based definition of components and their structures in engineering modeling systems [15]. On the level of physical actions on ES model and field ES aspects are divided into two groups. One of these aspect groups is for the driving of generation physical level features using contexts with P level components in the RFLP structure. P level component structure is produced by execution of L level component structure in the RFLP structured model of ES. Execution of logical component structure is assisted by discipline specific ES features, as well as related feature generation processes and procedures. Therefore, four aspects here are disciplines, features, processes, and

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procedures. The other group of aspects serves two-way contextual driving between DCS and cyber units of CPS ES. These aspects are introduced in sector 5 of this paper (Fig. 30.4). Although aspect of controllers is in the second group, structured representation of equipment and device controllers is also applied at generation of P level components in the RFLP structure in case of ES which includes controller such as a robot. Moreover, controller model is necessary to generate control programs for the CPS which is included in manufacturing of ES.

30.4 Operation of DCS Model As it was introduced and discussed above in this paper, DCS model organizes always actual information and knowledge as content and applies this content as driving context on RFLP structure and CPS cyber unit object parameters. Driving contexts are always active and their changes are real time propagated in the ES model and in cyber units of connected CPS. Context system in DCS acts as extension to consistent context systems represented in RFLP structure and CPS units. In this way, the reported research is contribution to model integration in engineering which is a leading trend at application of advanced system-based informatics in engineering. RFLP structure level consists of structure of blocks. Each block is ready to define ports on it. Port can serve establish connection between blocks, transfer of content between blocks, and control of the connected block. This is a well proven methodology and the needed modeling capabilities are available in RFLP structure enabled industrial modeling systems. Because DCS structure is devoted as extension to RFLP structured ES model, modeling capabilities for RFLP seems practical to apply at creation and handling elements and structures in DCS. The required knowledge and other representations are also available at RFLP structure modeling. This solution supports contextual driving between DCS and RFLP structures. Driving contextual connections at the operation of DCS model are summarized and explained in Fig. 30.3 for blocks DIJK and DIJL on the DIJ sublevel of DI level of a DCS structure. Driving contexts are transferred between ports on blocks in case of RFLP and DCS structures, and ports on blocks of these structures and connection points on sources of affecting contexts and cyber units of CPS. Figure 30.3 includes port RIJKL on RIJK block in RIJ sublevel of RI level of an RFLP structure.

30.5 Contexts Between Model Sectors As it was discussed above, despite the already decided (AD) context restricted free definition of contexts, the extended CPS model (Fig. 30.1) is formally divided into model sectors. Consistent group of contexts between any pair of these sectors is developed in sector independent generic way. Groups of contextual connections between pairs of structural units of the extended CPS model sectors are shown in

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Fig. 30.3 Operation of DCS model

shaded boxes in Fig. 30.1. Connections between RFLP and DCS level elements and components and between sources of affects from outside of ES model system and DCS initiatives level were published in [16, 14], respectively. This section of paper concentrates on contextual connections between DCS model and cyber units of field operated CPS ES (D-C in Fig. 30.1). Inter-sector sets of driving contexts in the D-C relation are introduced in Fig. 30.4. D-C driving contexts are organized in three inter-sector sets for representation, operation status, and operation parameters of physical objects in field operating CPS ES. Driving context is defined between block in relevant DCS sublevel and relevant entity in CPS cyber unit. Content for sophisticated object representations can be utilized by

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cyber units. Beyond this transfer, mutual object definition assistance and evaluation of object representations can be applied as contribution to harmonize theory and experience. Status is a complex parameter to evaluate operation of physical units in CPS. Recognition and analysis of status can be applied at both sides using appropriate algorithms. Communication of status comes from CPS cyber units as information about actual real condition of physical processes in CPS. Essential means of control and evaluation of physical processes uses operating parameters in CPS. Definition and evaluation of parameters can be done on both of the sides. Driving contexts from DCS provide parameter values for CPS cyber units considering all of the available information including directly accessed sensor information from CPS. The schema in Fig. 30.4 is one of the most important area of future research. This area requires analysis of connection with really operating CPS model of which is available. This can be realized using physical laboratory CPS or field operating CPS. Laboratory is easier to access for research while field system provides real experience. Mixed solution seems to be the optimal. Anyway, following the structure in extended CPS model in Fig. 30.1 at practice is one of the possibilities to integrated utilization of theory, engineering model and simulation, experience proved experiment, as well as and field operation, diagnostic, improving, and troubleshooting of CPS. Important aspect of integration is that model system can serve engineering activities during the whole innovation and life cycle of ES. Integration of sources for theories, engineering models, experiments, and CPS operation activities needs project organization which is suitable for model centered cooperation of companies, institutions, and experts. Next section of this paper is about a proposal for these projects.

30.6 Organizing Multisector Projects for Innovation and Life Cycles of ES ES innovation cycle includes research, solution development, product development, virtual prototyping, and physical prototyping. The result is an ES lifecycle of which includes marketing, production, installation, customer services, and recycling. The above lists are case dependent. In the classical definition of product lifecycle management (PLM), the product lifecycle is often cited as integration of the two above cycles [17]. Conceptual schema of establishing project which can integrate work of participants from all engineering related activity sectors is shown in Fig. 30.5. Activity sectors were defined to include industry, institutional research, higher education, and individual experts. This concept allows flexible composing of tasks and participant list. Scheduled tasks determine the needed participant activity sectors. Recent higher education course, for example, often includes industrial or/and research tasks and participants beyond the classical academic. The project in which participants are invited from different activity sectors needs access from everywhere and at any time. Cloud environment is required where all

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Fig. 30.5 Cooperation of participants in activity sector organized engineering project

resources are given as service from the cloud system provider. Main stages of making connection of participants with a common active engineering model system are outlined in Fig. 30.5. Cloud connection needs cloud access administration and checking for minimum workstation configuration. Project administration makes and applies project definitions. Project definition includes activities, participants, and structure of the commonly managed engineering model. Project administration assigns roles for participants. Participants are authorized to access role dependent groups of modeling capabilities. Contribution by participant is restricted to modeling capabilities assigned for actual role. Applications which are available for modeling capabilities ensure completing, modification, and application of active engineering model system.

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30.7 Conclusions This paper includes recent research results in integrated smart engineering. Main purpose of the reported research is joining efforts to answer the question that how engineering changes in the era of smart model representation of ES, CPS based industrial products, and smart manufacturing systems. Main contributions in this paper are extended model of CPS, organized structure of driving content for better development of ES model, and ES model assisted operation of CPS. It is obvious that consistent, always actual and smart driving contexts are required for the whole scenario of CPS environment as it is shown in Fig. 30.1. Completing all components of this environment to make them capable of correct active reaction to any new or modified inside or outside context requires so much research in the future. The Laboratory of Intelligent Engineering Systems which was founded by the Óbuda University for activities in high abstraction modeling of industrial products works on several selected issues for future smart engineering using past published own results. One of main characteristics of this work is that any local solution must be developed considering contexts with the entire structure of the proposed extended CPS model.

References 1. Leitaoa, P., Colomboc, A.W., Karnouskose, S.: Industrial automation based on cyber-physical systems technologies: Prototype implementations and challenges. Comput. Ind. 81, 11–25 (2016) 2. Sharma, S., Segonds, F., Maranzana, N., Chasset, D., Frerebeau, V.: Towards cloud based collaborative design—analysis in digital PLM environment. In: Product Lifecycle Management to Support Industry 4.0. PLM 2018. IFIP Advances in Information and Communication Technology, vol. 540, pp. 261–270. Springer, Berlin (2018) 3. Horváth, L., Rudas, I.J.: Human intent representation in knowledge intensive product model. J. Comput. 4(10), 954–961 (2009) 4. Horváth, L., Rudas, I.J.: Virtual intelligent space for engineers. In: Proceedings of the 31st Annual Conference of IEEE Industrial Electronics Society, pp. 400–405, Raleigh, USA (2005) 5. Horváth, L.: Supporting lifecycle management of product data by organized descriptions and behavior definitions of engineering objects. J. Adv. Comput. Intell. Intell. Inform. 11(9), 1107–1113 (2007) 6. Kleiner, S., Kramer, C.: Model based design with systems engineering based on RFLP using V6. In: Smart Product Engineering, pp. 93–102. Springer, Berlin (2013) 7. Horváth, L., Rudas, I.J.: New approach to multidisciplinary content driving of engineering model system component generation. In: Book New Trends in Software Methodologies, Tools and Techniques, pp. 38–49. IOS Press, The Netherlands (2016) 8. Canedo, A., Schwarzenbach, E., Al Faruque, E.M.A.: Context-sensitive synthesis of executable functional models of cyber-physical systems. In: Proceedings of the 2013 ACM/IEEE International Conference on Cyber-Physical Systems (ICCPS), pp. 99–108. Philadelphia, PA, USA (2013) 9. Maksimovic, M., Al-Ashaab, A., Shehab, E., Flores, M., Ewers, P., Haque, B., Furian, R., von Lacroix, F., Sulowski, R.: Industrial challenges in managing product development knowledge. Knowl.-Based Syst. 71, 101–113 (2014)

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10. Gomez, E., Adli, H., Fernandes, C., Hagege, M.: System engineering workbench for multiviews systems methodology with 3DEXPERIENCE platform. the aircraft radar use case. In: Complex Systems Design and Management Asia, pp. 269–270 (2016) 11. Lefèvre, J., Charles, S., Bosch-Mauchand, M., Eynard, B., Padiolleau, É.: Multidisciplinary modelling and simulation for mechatronic design. J. Des. Res. 12(1–2), 127–144 (2014) 12. Horváth, L., Rudas, I.J.: Information content driven model for virtual engineering space. Acta Polytech. Hung. 15(2), 7–32 (2018) 13. Baughey, K.: Functional and logical structures: a systems engineering approach. In: SAE 2011 World Congress, SAE Technical Paper 2011-01-0517 (2011) 14. Horváth, L.: Contextual knowledge content driving for model of cyber physical system. In: Proceedings of the 15th International Conference on Control, Automation, Robotics and Vision, pp. 1845–1850. Marina Bay Sands Expo and Convention Centre, Singapore (2018) 15. Blades, E.L., Miskovish, R.S., Luke, E.A., Collins, E.M., Kurkchubashe, A.G.: A multiphysics simulation capability using the SIMULIA co-simulation engine. In: Proceedings of the 20th AIAA Computational Fluid Dynamics Conference, pp. 1–16. Honolulu, Hawaii, (2011) 16. Horváth, L.: New method for definition of organized driving chains in industrial product model. In: Proceedings of the 2017 IEEE International Conference on Industrial Technology (ICIT 2017), pp. 1183–1188. Toronto, Canada (2017) 17. Ambroisine, T.: Mastering increasing product complexity with collaborative systems engineering and PLM. In: Proceedings of the Embedded World Conference, pp. 1–8. Nürnberg, Germany (2013)

Chapter 31

Quasi Two Dimensional FEM Model for Form Rolling Analysis and Its Application with LS-DYNA Tomohiko Ariyoshi and Ken-ichi Kawai

Abstract A form rolling analysis model generally needs massive elements solving large plastic deformation of three dimensional problems with sufficient accuracy. We have developed a unique and compact analysis model. A small angle cut model can be a small size model but it will not be enough to analyze precise phenomena near the contact surface between the roller and the blank. We have developed a compact model composed of small angle one-layer solid elements given a certain angular velocity field with oblique roll contact. The computation time of the model indicated only about 1/2000 compared with that of full size model. This model will shorten both the model generation period and the span of the design change refining the performances of the product with form rolling process. We tried this method on an example to understand the precise material deformation history under the large deformation occurred according as the rolling deformation on a certain aluminum circular plate. In this trial we utilized the function “Component Analysis” of LS-DYNA(Crash and Structure Analysis software). This function is used for precise investigations to material deformations in some narrow compartments of the components. We aimed to survey the material structure precise changes under the working tool. Keywords Rotary forming · Form rolling · Numerical analysis · CAE simulation

31.1 Introduction Form rolling FEM models for precise understanding of large plastic deformations generally lead to fully circular precisely meshed models. Hence axi-symmetric models are often used, but this method likely lacks precise conditions around the deformation tools for instance deformation localities neighboring forming tools. T. Ariyoshi (B) ATORI CAE Co.Ltd., 3-18-19, Futaba, Shinagawa, Tokyo 142-0043, Japan e-mail: [email protected] K. Kawai Yokohama National University, 79-5, Tokiwadai, Hodogaya, Yokohama 240-8501, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_31

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We have already developed a compact and precise form rolling analysis model composed of axi-symmetric small angle one-layer solid elements given a certain angular velocity field representing oblique and localized roller contact [3, 6, 8]. This model can shorten both the model generation period and the span of the design change refining the performances of the product with form rolling process. In this paper we show a very compact axi-symmetric element model for analyzing the boss forming process of the aluminum circular blank including the calculation condition of angular velocity field set around the blank axis [1, 2, 4, 5, 7], and also introduce the LS-DYNA function named the component analysis to look into the precise deformation performances of the blank under the rolling tool.

31.2 Reference Experiments for the Analysis We show samples of a certain boss forming experiment with the aluminum circular plate. Figure 31.1 shows the experimental layout. The experiment was composed of an aluminum circular blank and a roller. The roller is set on the blank at an inclination angle of 45°, and rolled by the contacting force with the blank and gradually drawn near to the blank center. The experimental conditions are listed in Table 31.1. Mandrel Blank

ωb

Roller

ωr

v x

t0

dt

R S

Blank section (Test B)

Fig. 31.1 Boss forming experiment and blank deformation

Table 31.1 Experimental conditions

Item

Test A

Test B

Round-off radius of roller R [mm]

3

1

Indentation depth of roller dt [mm]

2.25

3.5

Blank rotational speed ωb [rpm]

190

190

Feed rate of roller v [mm/rev]

0.2

0.2

Blank thickness t0 [mm]

5

15

Stroke of roller S [mm]

35

35

31 Quasi Two Dimensional FEM Model for Form Rolling Analysis … Blank rotation :

ωb

337

Roller rotation : ω r

rb Roller velocity: v x

x

Δθ

z Fig. 31.2 Blank deformation by roller contact y

Roller surface

R Blank profile

Blank U-zone

Roller edge

x

x rb

Blank original profile

L-zone

rr

z (a) Boss forming side view

(b) U-zone upper view

Fig. 31.3 Zone classification of the contact area

31.3 Blank Deformation Process in the Boss Forming The blank deformation process on this experiment is shown in Fig. 31.2. This figure shows the plan-view of the contacting process between the blank and the roller. The symbol Δθ in Fig. 31.2 indicates the angle formed by roller contact with the blank in the midst of this boss forming process. The side view is shown in Fig. 31.3a. The blank upward zone is subjected to main radial compressive force by the roller and downward zone is subjected to shearing force by the roller feed to the blank center. With the indicated deformation profiles of the blank and roller position, we can specify relevant forces on the elements. The blank upward zone is mainly subjected to the load of form rolling force and the lower zone is subjected to the shearing load by the roller. From this point of view we determined the each load condition of this boss forming process.

31.4 Construction Plan of the Analysis Model For construcing the axi-symmetric analysis model, we prepared the block structure for the model representing the load conditions of this boss forming process. We constructed the deformation model for the upward zone of the blank of Fig. 31.3. First, we cut off one small angle slice part with unit thickness from the

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whole circular blank in the boss forming process, and drew out the local loads acting on this slice. Figure 31.4 shows the loads acting on this slice. This deformation phenomenon occurs cyclically in every slice of the whole blank as the roller revolution. F(x) and S(x) are radial forces acting on the slice and S(x) is the outer force through the element both edges of neighboring slices those are not directly contacted the roller. Then the equation of forces on this slice is as follows: F(x) = S(x)

(31.1)

At this point we intend to transform S(x) from the shear force on the slice-edges to the body forces inside of the fragmented slice part. And then we can change S(x) to Si(x) as shown in Fig. 31.5. The process means changing shear force S(x) to the local outer forces Si(x) composed of radially divided into many fragments. Next we change Si(x) to body force mi αi (α; a certain acceleration). This is done when we prepare pertinent αi, and we are able to obtain proper αi by introducing a certain angular velocity field along blank radius from the blank center. The angular velocity field giving the yielded state to the outer end of the slice is calculated by Eq. (31.2). f (r1 ) = r1 · θ · t · σ y = r1 · θ · σ y (t = 1)

(31.2)

According to Eq. (31.2), we obtain angular velocity field Ω y by Eq. (31.3). Ω y = (1/r1 ) · SQRT((3 · σ y )/m d ) Fig. 31.4 Concept of the blank model

Δθ

(31.3)

0.5 S(x)

Roller F(x)

Solid element 0.5 S(x)

Fig. 31.5 Shear force conversion to element-inertia forces

0.5S(3) 0.5S(2) 1 Solid element 0.5S(3) z

0.5S(1) F(x)

0.5S(2) 0.5S(1)

F(x) z

m3

α3

x

m2

α2

m1

α1

x

31 Quasi Two Dimensional FEM Model for Form Rolling Analysis … y

Angular velocity field set

339 Roller axis

Angular velocity field not set Adaptive meshes Blank center

Roller profile

Non-adaptive meshes x

Fig. 31.6 Angular velocity field

When setting angular velocity field Ω y we formed setting zone as shown in the Fig. 31.6. The reason is that we must set the field inversely proportional to the blank radius. In this time the analysis model was constructed most simply with axisymmetric shells.

31.5 Calculation Conditions of the Model Aiming reduction of calculation time of the model, the radial velocity of the roller was increased by about 50 times within the range not being harmful to arising too much inertia force of the blank material. At this time the roller speed was slowed down at both first and last stages of the contact with the blank for stability. Friction coefficients between the roller and the blank etc. were all set at 0.1.

31.6 Verification of the Calculation Results with the Experiment Figures 31.7 and 31.8 show the blank deformation results of the experiment and the calculation. Both results well correspond to the deformation mode. A small quantity of contractions occurred in the blank as shown in Fig. 31.8. Both results clearly appeared this phenomenon too. The analysis results indicated enough coincidence with the experimental results.

section of the deformed blank (analysis)

Fig. 31.7 Boss forming blank deformation results of experiment and analysis

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Fig. 31.8 Boss forming experimental (upper) and analysis (lower) results

Axial coordinate y / mm

340

a. Experiment Roll displacement 28mm 21mm Contraction

b. Analysis Roller displacement 14 mm Blank center Contraction

Roller displacement 21 mm

14mm

Radial coordinate r /mm 14 mm

21 mm

Roller displacement 28 mm 28 mm

31.7 The Component Analysis For a trial application we executed the component analysis of LS-DYNA. The function assists the precise understanding of the selected zone in the whole blank. We looked into the precise deformations under the roller path. Figure 31.9 shows the (color ; σy,

400MPa)

The component analysis zone Roller at 0.8s

a. The component analysis zone - original stress

b. The component analysis zone – small-pitch meshed stress

Fig. 31.9 The component analysis results

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analysis results of the area of the component analysis. The small-pitch meshed part indicates precise stress distributions of this zone.

31.8 Conclusions A compact boss forming deformation analysis model was developed with a certain angular velocity field. The angular velocity field simulates resistant forces of rotating blank surface formed by the roller. The results of the model were compared to the experimental results with practically enough correspondences in deformation mode. Besides the precise deformations of the selected zone under the roller were surveyed with the assist of LS-DYNA function named “Component Analysis”. This model is composed of compact axi-symmetric elements and consequently the calculation time was only about 2 h on the personal computer. This model will contribute much to the reduction of model developing time and model analysis time in planning or testing of the flat circular metal products.

References 1. Ariyoshi, T., Kawai, K.: Development of quasi two dimensional fem model for fast and effective boss-forming analysis. In: Proceedings Springer Conference Technology Plast, pp. 243–244 (2017) 2. Ariyoshi, T., Kawai, K.: Development of quasi two-dimensional fem model for bossforming analysis. J. Jpn. Soc. Technol. Plast. 56, 45–50 (2015) 3. Ariyoshi, T., Kawai, K.: Development of quasi-two-dimensional FEM model for form rolling analysis. J. Jpn. Soc. Technol. Plast. 55, 45–49 (2014) 4. Kawai, K. et al.: Adv. Thechnol. Plast. 2008, 1317 (2008) 5. Kawai, K. et al.: Key Eng. Mater. 344, 947 (2007) 6. Kim, W. et al.: J. Mater. Process. Tech. 194, 46 (2007) 7. Kawai, K. et al.: The Amada Foundation Report of Grant-Supported Researches, vol. 19, p. 1 (2006) 8. Kawai, K. et al.: Adv. Thechnol. Plast. 1, 408 (1993)

Chapter 32

The Study of Limit Load and Plastic Collapse Load Under Combined Loads Ying Zhang, Bin Zheng, Liping Zhang, Zhenyu Liu and Juan Du

Abstract In the process of mechanical analysis for equipment and pipe of reactor coolant system (RCS), due to the severe load condition, the stress analysis cannot satisfy the evaluation criteria at times, the limit analysis and plastic analysis is necessary. This paper focus on the limit load and the plastic collapse load of the straight pipe and three-way pipe of reactor coolant system (such as the nozzle of main equipment and the pipe of main system) under combined loads, the analytical method and the finite element (FE) method are adopted in this paper. Get the limit load curve and the plastic collapse load curve of straight pipe and three-way pipe under combined loads, the effect of load history and structure size is analyzed. Tangent intersection criterion (TI) and Plastic work-tangent criterion (PWT) are adopted to study the plastic collapse load curve of straight pipe under combined load of pressure and bending moment. Get the theoretical equation of the limit load and plastic collapse curve of straight pipe under combined loads; get the theoretical equation of the limit load surface of straight pipe under combined load of pressure, bending moment and torque. The effect of many factors is analyzed on three-way pipe. Keywords Limit load · Plastic collapse load · Limit load curve · Plastic collapse load curve · Limit load surface

Nomenclature PL ML TL PP MP

Limit load for pure pressure Limit load for pure bending Limit load for pure torsion Plastic collapse load for pure pressure Plastic collapse load for pure bending

Y. Zhang (B) · B. Zheng · L. Zhang · Z. Liu · J. Du State Key Laboratory of Reactor System Design Technology, Nuclear Power Institute of China, Chengdu 610213, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_32

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Wp σy σe εp E ν rm V n t Z D d l λ

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Plastic work Yield strength Von Mises equivalent stress Plastic strain Elastic modulus Poisson ratio Mean pipe radius Single element volume of FE model Total element number of FE model Thickness of pipe Plastic section modulus for bending Outside diameter of pipe Inside diameter of pipe Length of pipe Radius-thickness ratio

32.1 Introduction In the process of mechanical analysis for equipment and pipe of reactor coolant system (RCS), due to the severe load condition, the stress analysis cannot satisfy the evaluation criteria at times, the limit analysis and plastic analysis is necessary. These load-based analysis method have higher limiting value, easier to satisfy the evaluation criteria. Moreover, for structure with a complex geometry and/or complex loading, the categorization of stresses requires significant knowledge and judgment. Application of the limit or plastic analysis is recommended for cases where the categorization produce ambiguous results. Analytical method and finite element (FE) method are adopted in this paper to analysis the straight pipe which is a common structure of RCS (such as the nozzles of main equipment, the pipe of main system and etc.), the limit load and the plastic collapse load under combined loads will be studied. According to the ASME Boiler and Pressure Vessel Code ASME [1]], the theoretical bases of limit analysis and plastic analysis as follows: when the gross plastic deformation happened, the structural failure was adjudged; the limit load and the plastic collapse load is the critical point when the gross plastic deformation happened. When the material model of the structure is elastic-perfectly plastic and the small deformation theory is adopted, limit load is the critical load to balance the internal force and the external force, Fig. 32.1a is the structural response. To evaluate the plastic collapse load base on more complex theory, including strain hardening and large deformation effects, Fig. 32.1b is the structural response, and twice elastic slope criterion is adopted to defined the plastic collapse load in ASME code. The way to define the plastic collapse load is as follow: draw a straight line which has twice elastic slope (TES), the original point is its end point, and the intersection point of

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Fig. 32.1 Structural response of limit analysis and plastic analysis

the straight line and the load-deformation curve is the plastic collapse load, as shown in Fig. 32.2. Several other papers have given some other criteria to define the plastic collapse load. Save [2] proposed the tangent intersection (TI) criterion which is a similar criterion to the TES criterion, Gerden [3] and Muscat [4] proposed the plastic work (PW) criterion, and the plastic work-tangent criterion in reference Ying [5] which is base on the PW criterion. Different criteria have their own advantages, and TI criterion and Plastic work-tangent criterion (PWT) are adopted in this paper.

Fig. 32.2 Twice elastic slope (TES) criterion

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32.2 Study Object As pipes and nozzles are subjected to both internal pressure and mechanical load in the analysis process and often di not meet the requirements of the specification, two typical structures, straight pipe and three-way structure are selected for analysis in this paper.Commercial software ANSYS 17.2 [6] is used to establish the FE models for the limit analysis and the plastic analysis. 3D 20-node structural solid element (SOLID186) is adopted in the FE models. Five FE models (as shown in Figs. 32.3 and 32.4) are as follow:

Fig. 32.3 Straight pipe FE models

Fig. 32.4 Three-way structure FE models

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Model 1 : D = 108 mm, d = 100 mm, l = 500 mm, λ = 13 Model 2 : D = 108 mm, d = 96 mm, l = 500 mm, λ = 8.5 Model 3 : D1 = 50 mm, d1 = 40 mm, D2 = 50 mm, d2 = 40 mm Model 4 : D1 = 95 mm, d1 = 85 mm, D2 = 50 mm, d2 = 40 mm Model 5 : D1 = 140 mm, d1 = 130 mm, D2 = 50 mm, d2 = 40 mm where, radius-thickness ratio λ = rtm , D1 , d1 are the outer and inner radius of tube 1, D2 、 d2 are the outer and inner radius of tube 1. The material properties of models are as follows, elastic modulus E = 180.7G Pa, poisson ratio ν = 0.3, yield stress σ y = 205M Pa.

32.3 Normalized Load Parameters Before studying the limit load and the plastic collapse load under combined loads, it is necessary to calculate the limit load and the plastic collapse load under single load and take this as the normalized load parameters. This chapter mainly introduces the calculation methods and results of the limit load and the plastic collapse load of straight pipe under single load, compares and verifies the results of the FE method by means of analytical method. The calculation method of three-way structure is the same as that of straight pipe.

32.3.1 Limit Load and Plastic Collapse Load Under Single Load This chapter will calculate the limit load and the plastic collapse load of the straight pipe under unique load, due to this is the basis of studying the limit load and the plastic collapse load under combined loads. The calculation contents are as follows: • The calculation model is model 1 (the constitutive relationship is elastic-perfectly plastic material). Calculate the limit load for pure internal pressure PL , the limit load for pure bending moment M L , the limit load for pure torsion TL , the results are shown in Table 32.1; • The calculation model is model 2 (the constitutive relationship is elastic-perfectly plastic material). Calculate the limit load for pure internal pressure PL , the limit Table 32.1 The results of limit load under single load

rm /t

PL /MPa

13

18.212

M L /Nm 8874.8

TL /Nm 8089.1

8.5

27.860

12819.3

11669.9

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load for pure bending moment M L , the limit load for pure torsion TL , the results are shown in Table 32.1; • The calculation model is model 1 (the bilinear constitutive relation and the strain hardening is adopted). Calculate the plastic collapse load for pure internal pressure PP and the plastic collapse load for pure bending moment M P under different strain hardening ratios (2% strain hardening and 5% strain hardening) and different criteria (TI criterion and PWT criterion). For combined loads, non-proportional loading was applied, where bending moment was applied under the fixed internal pressure or internal pressure was applied under fixed bending moment. In these cases, the elastic phase of the bending moment-strain relation curve is too small, the TES criterion of the ASME code is difficult to apply, and using the TI criterion and the PWT criterion can a avoid this problem. Therefore, the TI criterion and the PWT criterion are applied in this paper to calculate the plastic collapse load as the base for studying the plastic collapse load under combined loads. The way of the TI criterion to define the plastic collapse load is as follow: two straight lines are drawn in the characteristic load-deformation curve, one tangent to the initial elastic response and one tangent to the plastic deformation region of the curve, the load corresponding to the intersection is the plastic collapse load, as shown in Fig. 32.5; The way of the PWT criterion to define the plastic collapse load is as follow: when the tangent variation of the plastic work-load parameter curve is tend to 0, the gross plastic deformation happened, and the corresponding load step is the plastic collapse load, as shown in Fig. 32.6. The results of different criteria and different strain hardening ratio are shown in Table 32.2. In the load-deformation parameter curves of Figs. 32.5 and 32.6, the load is internal pressure and the bilinear constitutive relation is adopted, the strain hardening ratio

Fig. 32.5 Tangent intersection criterion (TI), 5% strain hardening

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Fig. 32.6 Plastic work-tangent criterion (PWT), 5% strain hardening

Table 32.2 The results of plastic collapse load under single load rm /t = 13

TI PP /MPa

M P /N m

PP /MPa

M P /N m

2% strain hardening

18.218

8855

18.3

8880

5% strain hardening

18.274

8890.6

18.3

9120

PWT

is 5%. The deformation parameter of Fig. 32.5 is Von Mises equivalent strain, and the equation applied in FE analysis to get the plastic work of Fig. 32.6 is as follow: Wp =

n   1 1

2

  ε p σe + σ y · V

 (32.1)

32.3.2 Validation of the Finite Element Solution To validate the accuracy of the finite element solution, comparing the analytical solution with finite element solution is necessary. Chang and Yun [7] gives the theoretical formula of limit load under unique internal pressure for straight pipe, as follow: 2 t PL = √ σ y · rm 3

(32.2)

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Table 32.3 Comparison of limit loads computed by FE method and analytical method rm /t

13

Method

FE

Internal pressure/MPa

8.5 18.212

Analytical 18.209

Ratio (%)

FE

100.02

Analytical 27.860

27.849

Ratio (%) 100.03

Bending moment/Nm

8874.8

8873.5

100.01

12819.3

12811.7

100.05

Torsion/Nm

8089.1

8043.4

100.56

11669.9

11605.5

100.55

Note: ratio = FE result/analytical result

Lingyah, Taha and Dae [8] gives the theoretical formulas of limit loads under unique bending moment and unique torsion for straight pipe, as follows: ML = σy · Z where, plastic section modulus for bending Z =

(32.3) D 3 −d 3 . 6

σy TL = 2π √ · rm2 · t 3

(32.4)

The theoretical formulas above are based on a elastic-perfectly plastic material model and Von Mises yield criterion. The comparison of the results computed by the analytical method and the FE method are as shown in Table 32.3. The results in Table 32.3 show a very good agreement between analytical solution and finite element solution, the maximum difference between them is less than 0.56%. This validates reasonability of the finite element discretization and the accuracy of the finite element solution.

32.4 Limit Load and Plastic Collapse Load Under Combined Loads The main purpose of this paper, studying the limit load and the plastic collapse load under combined loads which includes the following aspects: 1. The effect of different load histories on the limit load and the plastic collapse load. 2. The effect of different shape and size on the limit load and the plastic collapse load. 3. The effect of different properties (strain hardening ratios) on the plastic collapse load. 4. The effect of different criteria on the plastic collapse load.

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Denominate PPL normalized limit pressure, MML normalized limit bending moment, T normalized limit torsion. Denominate the relation curve or the relation surface TL between the normalized limit loads limit load curve or limit load surface. Denominate PPP normalized plastic collapse pressure, MMP normalized plastic collapse bending moment; TTP normalized plastic collapse torsion. Denominate the relation curve between the normalized plastic collapse loads plastic collapse load curve.

32.4.1 Limit Load Under Combined Loads for Straight Pipe 32.4.1.1

Effect of Different Load Histories

This chapter will study the effect of different load histories through three different loading process. Model 1 is the calculation model, constitutive relationship is elasticperfectly plastic material. 1. First, apply fixed internal pressure on the structure, then apply gradually increase bending moment until the gross plastic deformation happened, we can get the limit bending moment under fixed internal pressure. After multiple calculations in this way (different fixed internal pressure), the results can be fitted with a curve, this curve is named limit load curve. 2. First, apply fixed bending moment on the structure, then apply gradually increase internal pressure until the gross plastic deformation happened, we can get the limit internal pressure under fixed bending moment. After multiple calculations in this way (different fixed bending moment), the results can be fitted with limit load curve. 3. Apply increasing internal pressure with non-proportion increasing bending moment until the gross plastic deformation happened, we can get the corresponding pair of limit internal pressure and limit bending moment. After multiple calculations in this way, the results can be fitted with limit load curve. The FE results are as shown in Fig. 32.7, fit the three groups of results respectively, we can get three curves, represented by Eq. (32.5). (P/PL )2 + (M/M L )2 = 1

(32.5)

Use the same method, we can get the curves under pressure-torsion combination and bending moment-torsion combination, as shown in Figs. 32.8 and 32.9, represented by Eqs. (32.6)–(32.7). (P/PL )2 + (T /TL )2 = 1

(32.6)

(M/M L )2 + (T /TL )2 = 1

(32.7)

352 Fig. 32.7 Limit load curves for different load histories (pressure-bending moment combination)

Fig. 32.8 Limit load curves for different load histories (pressure-torsion combination)

Fig. 32.9 Limit load curves for different load histories (bending moment-torsion combination)

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As you can see in Figs. 32.7, 32.8 and 32.9, we get the same limit load curve under different load histories. It proves that the load history has no effect on the limit load curve of the straight pipe.

32.4.1.2

Effect of Different Radius-Thickness Ratios

This chapter will study the effect of different radius-thickness ratios λ on the limit load. Model 1 and model 2 are the calculation models, constitutive relationship is elastic-perfectly plastic material. The FE results are as shown in Fig. 32.10, there are two groups of results for each load combination, fit them respectively, we can get a same curve, represented by Eqs. (32.5)–(32.7). As you can see in Fig. 32.10, we get the same limit load curve under different radius-thickness ratios. It proves that the radius-thickness ratio has no effect on the limit load curve of the straight pipe.

Fig. 32.10 Limit load curves for different radius-thickness ratios

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Fig. 32.11 Limit load surface for pressure-bending-torsion combination

32.4.1.3

Limit Load for Pressure-Bending-Torsion Combination

According to the previous analysis, the load history and the radius-thickness ratio has no effect on the limit load curve. This chapter will study the limit load for internal pressure-bending moment-torsion combination, model 1 is the calculation model, constitutive relationship is elastic-perfectly plastic material. The FE results are as shown in Fig. 32.11, fit the results in Fig. 32.11, we can get a surface, represented by Eq. (32.8). 

P PL

2

 +

M ML

2

 +

T TL

2 =1

(32.8)

As you can see in Fig. 32.11, limit load surface for internal pressure-bending moment-torsion combination is a eighth sphere.

32.4.1.4

Analytical Solution for Limit Load Under Combined Loads

Combine the analytical solution for limit load under unique load with the limit load relation equation which is obtained by fitting the FE results, we can get the analytical solution for limit load under combined loads. Limit load equation for pressure-bending combination: Plug Eqs. (32.2) and (32.3) into Eq. (32.5), the equation of the limit load for pressure-bending moment combination is as follow:

32 The Study of Limit Load and Plastic Collapse …

3λ2 2 9 P + M2 = 1 2 2 6 2 4σ y2 12λ + 1 t σ y

355

(32.9)

Limit load equation for pressure-torsion combination: Plug Eqs. (32.2) and (32.4) into Eq. (32.6), the equation of the limit load for pressure-torsion combination is as follow: 3λ2 2 3 P + T2 = 1 4σ y2 4π 2 λ4 t 6 σ y2

(32.10)

Limit load equation for bending-torsion combination: Plug Eqs. (32.3) and (32.4) into Eq. (32.7), the equation of the limit load for bending moment-torsion combination is as follow: 9 3 M2 + T2 = 1  2 2 4t 6σ 2 2 6 2 4π λ 12λ + 1 t σ y y

(32.11)

Limit load equation for pressure-bending-torsion combination: Plug Eqs. (32.2), (32.3) and (32.4) into Eq. (32.8), the equation of the limit load for pressure-bending moment-torsion combination is as follow: 3λ2 2 9 3 P + M2 + T2 = 1 2 2 2 4σ y 4π λ4 t 6 σ y2 12λ2 + 1 t 6 σ y2

(32.12)

32.4.2 Plastic Collapse Load Under Combined Loads for Straight Pipe 32.4.2.1

Effect of Different Load Histories

This chapter will study the effect of different load histories on the plastic collapse load, and the analysis method is the same as the limit analysis. Model 1 is the calculation model, the bilinear constitutive relation and large deformation theory are adopted. The FE results are as shown in Fig. 32.12, fit the results respectively, we can get three curves, represented by Eqs. (32.13)–(32.15). (P/PP )2 + (M/M P )2 = 1

(32.13)

(P/PP )2 + (T /TP )2 = 1

(32.14)

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Fig. 32.12 Plastic collapse load curves for different load histories

(M/M P )2 + (T /TP )2 = 1

(32.15)

As you can see in Fig. 32.12, the load history has no effect on the plastic collapse load curve of the straight pipe.

32.4.2.2

Effect of Different Radius-Thickness Ratios

This chapter will study the effect of different radius-thickness ratios λ on the plastic collapse load. Model 1 and model 2 are the calculation models, the bilinear constitutive relation and large deformation theory are adopted. The FE results are as shown in Fig. 32.13, fit the results respectively, we can get three curves, represented by Eqs. (32.13)–(32.15). As you can see in Fig. 32.13, the radius-thickness has no effect on the plastic collapse load curve of the straight pipe.

32 The Study of Limit Load and Plastic Collapse …

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Fig. 32.13 Plastic collapse load curves for different radius-thickness ratios

32.4.2.3

Effect of Different Strain Hardening Ratios and Criteria

This chapter will study the effects of different strain hardening ratios (elasticperfectly plastic, 2% strain hardening and 5% strain hardening) and criteria (TI criterion and PWT criterion) on the plastic collapse load for internal pressure-bending moment combination. Model 1 is the calculation model, and the bilinear constitutive relation is adopted. First, apply fixed internal pressure on the structure, then apply gradually increase bending moment until the gross plastic deformation happened, getting the plastic collapse bending moment under fixed internal pressure. Due to the two criteria need the elastic phase of load-deformation parameter curve, thus, when the plastic strain occurred only under fixed internal pressure, the loading process should be changed as follow: first, apply fixed bending moment on the structure, then apply gradually increase internal pressure until the gross plastic deformation happened, getting the plastic collapse internal pressure under fixed bending moment. The FE results of TI criterion are as shown in Fig. 32.14, and the FE results of PWT criterion are as shown in Fig. 32.15, fit the results in Fig. 32.14 and Fig. 32.15 respectively, we can get the curves, represented by Eq. (32.13).

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Fig. 32.14 Plastic collapse load curves for different criteria (TI)

Fig. 32.15 Plastic collapse load curves for different criteria (PWT)

As you can see in Figs. 32.14 and 32.15, we get the same plastic collapse load curve under different strain hardening ratios and different criteria. It proves that the strain hardening ratio and the criterion have no effect on the plastic collapse load curve of the straight pipe.

32.4.3 Limit Load and Plastic Collapse Load Under Combined Loads for Three-Way Structure 32.4.3.1

Effect of Different Load Histories

This chapter will study the effect of different load histories on the limit load and plastic collapse load for three-way structure, and the analysis method is the same as the limit analysis for straight pipe. Model 5 is the calculation model.

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When the constitutive relationship is elastic-perfectly plastic material. and small deformation theory is adopted, the results are as shown in Fig. 32.16. There are three groups of results, fit the groups respectively, we can get a same curve, represented by Eq. (32.5). When the bilinear constitutive relation and large deformation theory are adopted, the results are as shown in Fig. 32.17, three plastic collapse curve in all. As you can see in Fig. 32.16, we get the same limit load curve under different load histories. It proves that the load history has no effect on the limit load curve of the three-way structure. As you can see in Fig. 32.17, the load history has effect on the plastic collapse load curve of the three-way structure. Fig. 32.16 Limit load curves for different load histories

Fig. 32.17 Plastic collapse load curves for different load histories

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Effect of Different Size

This chapter will study the effect of different size (model 3–5) on the limit load and plastic collapse load for three-way structure. When the constitutive relationship is elastic-perfectly plastic material. and small deformation theory is adopted, the results are as shown in Fig. 32.18. There are three groups of results, fit the groups respectively, we can get a same curve, represented by Eq. (32.5). When the bilinear constitutive relation and large deformation theory are adopted, the results are as shown in Fig. 32.19, three plastic collapse curve in all. As you can see in Fig. 32.18, we get the same limit load curve under different size. It proves that the size has no effect on the limit load curve of the three-way structure. As you can see in Fig. 32.19, the size has effect on the plastic collapse load curve of the three-way structure. Fig. 32.18 Limit load curves for different size

Fig. 32.19 Plastic collapse load curves for different size

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32.5 Conclusions This paper focus on the limit load and the plastic collapse load of the straight pipe and the three-way structure under combined loads, the analytical method and the finite element (FE) method are adopted in this paper. The limit load curves for internal pressure-bending moment combination, internal pressure-torsion combination and bending moment-torsion combination is obtained, the limit load surface for internal pressure-bending moment-torsion combination is obtained too. The curves and the surface are represented by equations. Analyzed the effects of different load histories and radius-thickness ratios to the limit load curve; Analyzed the effects of different strain hardening ratios and criteria to the plastic collapse load curve. The conclusions by analysis are as follows: (1) For the straight pipe, different load histories, radius-thickness ratios, strain hardening ratios and criteria have no effect on the limit load curve and the plastic collapse load curve, it is a circular curve in the first quadrant. (2) For the three-way structure, different load histories and size have effect on the limit load curve and the plastic collapse load curve.

References 1. ASME: ASME Boiler and Pressure Vessel Code. New York: The American Society of Mechanical Engineers (2004) 2. Save, M.: Experimental verification of plastic limit analysis of torispherical and toriconical heads. Press. Vessel. Pip. Des. Anal. ASME, New York 1, 382–416 (1972) 3. Gerdeen, J.C.: A critical evaluation of plastic behaviour data and a united definition of plastic loads for pressure components, p. 254. Weld. Res. Counc. Bull, No (1979) 4. Muscat, M., Mackenzie, D., Hamilton, R.: A work criterion for plastic collapse. Int. J. Press. Vessel. Pip. 80, 49–58 (2003) 5. Zhang, Y.: a criterion for evaluating plastic collapse load: plastic work-tangent criterion. In: International Conference Nuclear Engineering Proeedingsc ICONE, vol. 1, No. ICONE25-66116 (2017) 6. ANSYS 17.2.: User’s Manual for Revision 17.2 (2016) 7. Oh, C.S., Kim, Y.J.: Closed-form plastic collapse load of pipe bends under combined pressure and in-plane bending (2006). PVP2006-ICPVT-11-93770 8. Yen, L., Al-Shawaf, T., Kim, D.J.: Effect of shear and torsion on the plastic collapse load of a pipe section with circumferential flaw (2010). PVP2010-25762

Chapter 33

Experimental Research on Uniaxial Tensile Behavior and Cyclic Deformation Behavior of TA17 Juan Du, Xuejiao Shao, Linyuan Kuang, Yuechuan Lu, Minda Yu and Jiang Lu Abstract Titanium alloys are widely used in various fields for its high strength, good corrosion resistance and high heat resistance. Many countries in the world have recognized the importance of titanium alloy materials, and have developed and applied them in practice. In order to accurately describe uniaxial tensile behavior and ratcheting behavior of material, a series of experiments of TA17 were conducted at room temperature and 350 °C, including the uniaxial tensile experiments of TA17 with different processing techniques, the strain controlled cyclic experiments and the stress controlled cyclic experiments. The uniaxial tensile curves of materials with different processing techniques are discussed. The effects of different strain amplitudes of cyclic straining were analyzed. The effects of different combinations of mean stress and stress amplitude were analyzed. Finally, the uniaxial stress and strain curves with different processing techniques, cyclic stress and strain curves under different strain amplitudes and stress amplitude were obtained. The results show that the stress and strain curves of longitudinal plates and forgings are the lowest, while the stress and strain curves of transverse plates and bars are the highest. TA17 exhibited significant cyclic stability under different strain amplitudes and has obvious ratcheting behavior under asymmetric stress control. The obtained conclusions are helpful to improve the mechanical properties of TA17 and establish the cyclic constitutive model in the following work. Keywords TA17 · Uniaxial tensile behavior · Ratcheting behavior · Experimental research

J. Du (B) · X. Shao · L. Kuang · Y. Lu · M. Yu · J. Lu State Key Laboratory of Reactor System Design Technology, Nuclear Power Institute of China, Chengdu 610213, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_33

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33.1 Introduction TA17 alloy, Ti-4Al-2 V, the international brand of the same type of alloy is IIT-3B, was developed by the Central Institute of Structural Materials Science of Prometheus, former Soviet Union, and successfully developed in China in response to the needs of engineering. TA17 alloy has excellent comprehensive properties, medium strength, good weldability and water corrosion resistance, and high hydrostatic and cyclic strength. It can be widely used in shipbuilding, chemical industry, aviation and atomic energy. As a structural material, the mechanical behavior of TA17 alloy should be studied in depth. Z. G. Wang et al. studied the microstructure and composition of TA17 alloy oxide layer at 300 °C and high temperature steam [1], B. S. Rodchenkov et al. studied the effect of radiation on the mechanical properties of TA17 alloy [2], A. R. Ebrahimi et al. studied the effect of thermal oxidation on the fatigue properties of TA17 alloy [3]. The application of TA17 in China is mainly focused on the research of thermophysical properties and fatigue properties, and the study of cyclic deformation behavior of TA17 is less. It is necessary to further carry out the investigation about the mechanics properties of TA17 titanium alloy. In order to accurately describe the ratcheting behavior of material at high temperature, a series of experiments of TA17 were conducted at 30 and 350 °C, including the uniaxial tensile experiments of TA17 with different processing techniques, the strain controlled cyclic experiments and the stress controlled cyclic experiments.

33.2 Experimental The basic mechanical properties and cyclic deformation characteristics of the material were obtained by uniaxial tensile test and cyclic loading test. The cyclic loading test material is TA17 rod, and the specimen size of TA17 rod is shown in Fig. 33.1. The samples were polished and polished after processing.

Fig. 33.1 Dimension of TA17 specimen (unit mm)

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The test equipment is MTS809 material testing machine. Tensile test at room temperature adopts MTS632.11c-21 (standard distance: 25 mm), high temperature tensile test adopts MTS632.68-08F (standard distance: 25 mm), high temperature test adopts MTS Lepel of heat generator and SHIMADEN SR53 of temperature controller. The tensile test system is shown in Fig. 33.2. The precision of high temperature extensometer and load sensor is 5‰. Lepel uses two thermocouples to measure the temperature. The temperature control accuracy is ±1 °C, and the temperature gradient in the range of sample standard distance is less than ±5 °C. The test is divided into two types: monotonic tension and cyclic loading. The monotonic tension test is controlled by displacement and the loading rate is 0.02 mm/s. The cyclic test is carried out under two control modes: strain control and stress control. The loading strain rate of the strain cycle is 0.2%/s. The loading rate of the stress cycle is 100 MPa/s, and the test waveform is shown in Fig. 33.3. The purpose of strain cycling test is to reveal the cyclic characteristics of materials, such as cyclic hardening, stabilization or softening, and stress cycling test is to reveal the ratcheting deformation behavior of materials. The test temperature is room temperature (about 30 °C) and 350 °C. During the high temperature test, the specimen is heated for 10 min before loading and then the strain extensometer signal is initially zeroed to eliminate the strain caused by the increase of temperature in the measurement strain.

Fig. 33.2 a MTS lepel thermal generator and temperature control device, b Sample, heating ring and extensometer

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ε σ t Fig. 33.3 a Strain cycle, b Asymmetric stress cycle

33.3 Results and Discussion 33.3.1 Uniaxial Tensile Test Figure 33.4 shows the uniaxial tension stress-strain curves of TA17 at T = 30 and T = 350 °C. Figures 33.5 and 33.6 show the engineering stress-engineering strain curves of TA17 longitudinal sheet at T = 30 and T = 350 °C under different processing conditions (the engineering stress-engineering strain curves of TA17 longitudinal sheet at T = 350 °C should be between T = 300 and 400 °C). It can be seen from the drawings that the engineering stress-engineering strain curves of longitudinal plates and forgings are the lowest, while those of transverse plates and bars are the highest. 1000

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33 Experimental Research on Uniaxial …

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Fig. 33.6 The monotonic tensile engineering stress-engineering strain curves of TA17 under different processing conditions (T = 350 °C)

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33.3.2 Strain-Controlled Cyclic Deformation Test Figure 33.7 shows the cyclic stress-strain curve of TA17 when the strain amplitude is 1.0%. Figure 33.8 shows the peak-valley stress curve of TA17 with cyclic cycles when the strain amplitude is 1.0%. It can be seen from the figure that the peak-valley stress hardly changes with the cycle number during the strain cycle, TA17 also exhibits obvious cyclic stability at 30 °C. Figure 33.9 shows the cyclic stress-strain curves of TA17 at different strain amplitudes at 30 °C. Figure 33.10 shows the stable cyclic stress-strain curve of TA17 at 30 °C. Compared with the uniaxial tension stress-strain

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Fig. 33.7 Cyclic stress-strain curve of TA17 with strain amplitude of 1.0% (T = 30 °C)

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Fig. 33.8 Peak-valley stress and cyclic cycle curve of TA17 with strain amplitude of 1.0% (T = 30 °C)

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curve, the uniaxial tension curve is slightly higher than the stable cyclic stress-strain curve. Figure 33.11 shows the cyclic stress-strain curves of TA17 at 350 °C with strain amplitude of 0.9%. Figure 33.12 shows Peak-valley stress curves of TA17 with cyclic cycles when strain amplitude is 0.9% at 350 °C. As can be seen from Figs. 33.11 and 33.12, TA17 exhibits obvious cyclic stability at high temperatures, similar to T = 30 °C, i.e. the peak-valley stress hardly changes with cyclic cycles. Figure 33.13 shows the cyclic stress-strain curves of TA17 at different strain amplitudes at 350 °C. Figure 33.14 shows the stable cyclic stress-strain curve of TA17 at 350 °C. Compared with the uniaxial tension stress-strain curve, the uniaxial tension curve is slightly higher than the stable cyclic stress-strain curve. 800

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Fig. 33.11 Cyclic stress-strain curves of TA17 with strain amplitude of 0.9% (T = 350 °C)

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33.3.3 Stress-Controlled Cyclic Deformation Test Figures 33.15 and 33.16 show the cyclic stress-strain curves and peak-valley strain curves of TA17 under stress control (50 ± 650 MPa) at 30 °C, respectively. It can be seen from the figure that TA17 has obvious ratcheting behavior at 30 °C: the peak-valley strain increases with the increase of cyclic cycles, and the increase rate is approximately constant due to the cyclic stability of the material. Figure 33.17 and Fig. 33.18 show the cyclic stress-strain curves and peak-valley strain curves of TA17 under stress control at 350 °C (30 ± 410 MPa) with cyclic cycles, respectively. It can be seen from the figure that TA17 has obvious ratcheting behavior at 350 °C: peak-valley strain increases with the increase of cycle number, peak-valley strain increases rapidly in the initial weeks, and then reaches a stable strain increase rate after about 2–3 cycles. With the further increase of cycle number, the strain increase rate increases slightly, but overall, the strain increase rate increases with the cycle number. The increase of cycle number is approximately linear. 1000

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Fig. 33.15 Cyclic stress-strain curves of TA17 under stress control of 50 ± 650 MPa (T = 30 °C)

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33.4 Conclusion At T = 30 and T = 350 °C, the uniaxial tension, strain and stress cycling tests of TA17 were carried out, and the following conclusions were drawn: (1) TA17 exhibits obvious cyclic stability under given test conditions. (2) TA17 exhibits ratcheting behavior under asymmetric stress cycles. (3) The engineering stress-engineering strain curves of longitudinal plates and forgings are the lowest, while those of transverse plates and bars are the highest.

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References 1. Wang, Z.G., Zu, X.T.: Characterization of oxide layers on Ti-2Al-2.5Zr and Ti-4Al-2 V alloys oxidized at 300 °C in a neutral water steam. J. Alloy. Compd. 384(1-2), 93–97 (2004) 2. Rodchenkov, B.S., Kozlov, A.V.: Irradiation behavior of Title of Ti-4Al-2 V(
T-3B) alloy for ITER blanket modules flexible attachment. J. Nucl. Mater. 367–370(2), 1312–1315 (2007) 3. Ebrahimi, A.R., Zarei, F.: Effect of thermal oxidation process on fatigue behavior of Ti-4Al-2 V alloy. Surf. Coat. Technol. 203(3–4), 199–203 (2008)

Chapter 34

Study on Uniaxial Tension and Cyclic Deformation Behavior of TA16 at Room Temperature and High Temperature Xuejiao Shao, Du Juan, Linyuan kuang, Yuechuan Lu, Jiang Lu and Mingda Yu Abstract TA16 is a new type of titanium alloy material, which is widely used in pipeline systems of aerospace, ship and nuclear reactor with high temperature and high pressure. In this paper, the basic mechanical properties and cyclic deformation characteristics of TA16 were obtained by uniaxial tensile test and cyclic loading test at room temperature and high temperature. The Chaboche constitutive model can reasonably describe the cyclic plastic cumulative effect of materials and is provided in ANSYS which is a commercial finite element analysis software. The constitutive model of Chaboche presents a very important evolution criterion of follow-up hardening for ratchet simulation. Based on steady cyclic stress-strain curves which is got by strain control at different temperatures, the material parameters of Chaboche model are obtained by least multiplication fitting for TA16 at room and high temperatures. The constitutive model can used in elastic and elastic finite element analysis of TA16 materials. The experiment results show that TA16 exhibits slight cyclic softening under symmetrical strain cycles, and obvious ratcheting behavior under asymmetric stress cycles. When ratcheting strain of the structure reaches a certain saturation state, plastic accumulation will lead to accidental damage of the structure and lead to serious consequences. The stable cyclic stress-strain curves of TA16 at room temperature and high temperature are simulated by Chaboche model. The simulation results are in good agreement with the experimental results, which provides the basic data for the finite element simulation of cyclic deformation behavior of TA16 titanium alloy structure. Keywords Uniaxial tensile test · Cyclic deformation · Chaboche constitutive model · Ratcheting behavior

X. Shao (B) · D. Juan · L. kuang · Y. Lu · J. Lu · M. Yu Science and Technology on Reactor System Design Technology Laboratory, Nuclear Power Institute of China, Chengdu 610213, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_34

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34.1 Introduction Titanium alloy is an ideal marine material because of its low density, high strength, non-magnetic, sound transmission and good shock and vibration resistance [1, 2]. The application of titanium alloys in warships extends the service life of warships, enlarges the carrying capacity of warships and reduces the damage accidents of seawater systems, improves tactical performance and meets the requirements of low maintenance of warships. TA16 titanium alloy, with nominal composition Ti-2Al2.5Zr, is a type-alpha titanium alloy developed in the former Soviet Union and the mid-1960s. It has the characteristics of high plasticity, moderate strength, good process performance, weldability and corrosion resistance. It is used in steam generator tubes of pressurized water nuclear power plant. Because of the special use of TA16 and the high cost of materials, there are very few published literatures on TA16 titanium alloy. At present, only the thermophysical properties of TA16 titanium alloy have been studied [3, 4]. Therefore, it is necessary to study the mechanical properties of TA16 titanium alloy and obtain the basic parameters of mechanical properties. Because of the particularity of TA16 titanium alloy application environment, such as heat exchange pipeline in nuclear reactor, asymmetric cyclic stress will lead to cyclic accumulation of plastic deformation, i.e. ratcheting effect [5–8]. Ratcheting effect is very important for the safety, reliability and life evaluation of structures. When ratcheting strain reaches a certain saturation state, plastic accumulation will lead to accidental damage of structures and have serious consequences. Therefore, it is necessary to carry out cyclic tests of strain control and stress control for TA16 titanium alloy. The cyclic deformation behavior of TA16 titanium alloy structure is further studied by numerical simulation of TA16 titanium alloy structure using Chaboche model in ANSYS. In this paper, uniaxial tension test, cyclic deformation test with different strain amplitudes and asymmetric stress control for TA16 titanium alloy have been carried out. The parameters of Chaboche model of TA16 titanium alloy were determined by analyzing the experimental data and studying the parameters of Chaboche model [9]. By comparing the experimental data with the results of finite element simulation, it is proved that the parameters determined can reasonably predict ratcheting behavior of TA16 titanium alloy.

34.2 Experimental Method 34.2.1 Experimental Equipment The TA16 specimen is a circular tube with 42Crmo material at both ends. Its size is shown in Fig. 34.1. The samples were polished after processing. The test equipment is MTS809 material testing machine. The normal temperature tension test adopts MTS632.11c-21 (standard distance: 25 mm), the high temperature

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Fig. 34.1 Dimension diagram of TA16 specimen (unit mm)

tension and torsion extensometer MTS632.68-08F (standard distance: 25 mm), and the precision of high temperature extensometer and load sensor is 5‰. MTS Lepel and SHIMADEN SR53 are used in the high temperature test, and the Lepel uses two thermocouples to measure temperature. The temperature control accuracy is ±1 °C, and the temperature gradient in the range of sample standard distance is less than ±5 °C.

34.2.2 Test Scheme The test is divided into two types: monotonic tension and cyclic loading. The monotonic tension test is controlled by displacement and the loading rate is 0.02 mm/s. The cyclic test is carried out under two control modes: strain control and stress control, and the test waveform is shown in Fig. 34.2. The loading strain rate of the strain cycle is 0.2%/s, and the loading rate of the stress cycle is 100 MPa/s. The purpose of strain cycling test is to reveal the cyclic characteristics of materials, such as cyclic hardening, stabilization or softening, and stress cycling test is to reveal the ratcheting deformation behavior of materials.

(a)

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The test temperature include the room temperature (about 30 °C) and the high temperature (about 350 °C). During the high temperature test, the specimen is heated for 10 min before loading and then the strain extensometer signal is initially zeroed to eliminate the strain caused by the increase of temperature in the measurement strain.

34.3 Test Results 34.3.1 Uniaxial Tensile Test of TA16 Figure 34.3 shows the uniaxial tension stress-strain curves of TA16 at 30 and 350 °C. The yield strength is 401 MPa at 30 °C, 210 MPa at 350 °C. The modulus of elasticity is about 100GPa at 30 °C, 89GPa at 350 °C. The yield strength at 30 °C is obviously higher than that at 350 °C.

34.3.2 Cyclic Test of TA16 Figure 34.4 shows the peak-valley stress curve of TA16 with cyclic cycles when the strain amplitude is 0.9%. It can be seen from Fig. 34.5 that the peak-valley stress increases slowly and decreases slightly with the cycle number, which indicates that TA16 exhibits slight cyclic softening at 30 °C. Figure 34.6 shows the cyclic stress-strain curves of TA16 under different strain amplitudes at 30 °C. The material exhibits slight cyclic softening under different strain amplitudes and has the same evolution law. Figure 34.7 shows the stable cyclic stress-strain curve of TA16 at 30 °C. Compared with uniaxial tension stress-strain curves, uniaxial tension curves and stable cyclic stress-strain curves almost coincide. Figure 34.8 show the cyclic stress-strain curves and peak-valley strain curves of TA16 under stress control (50 ± 450 MPa) at 30 °C, respectively. It can be seen from the 700

Fig. 34.3 Uniaxial tensile stress-strain curves of TA16 at different temperatures Stress σ,MPa

600

TA16, T=30 0C TA16, T=3500C

500 400 300 200 100 0

0

1

2

3

Strain ε,%

4

5

6

34 Study on Uniaxial Tension and Cyclic Deformation … Fig. 34.4 Cyclic stress-strain curve of TA16 at 30 °C with strain amplitude of 0.9%

Fig. 34.5 Peak and valley stress curves of TA16 with cyclic cycles when strain amplitude is 0.9% at 30 °C

Fig. 34.6 Cyclic stress-strain curves of TA16 at different strain amplitudes at 30 °C

Fig. 34.7 Stable cyclic stress-strain curve of TA16 at 30 °C

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Fig. 34.8 Experimental results under stress control (50 ± 450 MPa) at 30 °C

Fig. 34.9 Experimental results with the strain amplitude is 0.7% at 350 °C

figure that TA16 has obvious ratcheting behavior at 30 °C: the peak-valley strain increases with the increase of cyclic cycles, and the increase rate is approximately constant due to the cyclic stability of the material. Figure 34.9 shows the cyclic stress-strain curve of TA16 at 350 °C when the strain amplitude is 0.7%. As can be seen from Fig. 34.9, TA16 exhibits obvious cyclic stability at high temperature similar to that at T = 30 °C, i.e., peak-valley stress hardly changes with cyclic cycles. Figure 34.10 shows the cyclic stress-strain curves and the stable cyclic stressstrain curve of TA16 at different strain amplitudes at 350 °C. Compared with the uniaxial tension stress-strain curve, the uniaxial tension curve is slightly higher than the stable cyclic stress-strain curve. Figure 34.11 show the cyclic stress-strain curves and peak-valley strain curves of TA16 under stress control at 350 °C(25 + 225 MPa) respectively. It can be seen from the figure that TA16 has obvious ratcheting behavior at 350 °C: the peak-valley strain increases with the increase of cycle number, the peak-valley strain increases rapidly in the initial weeks, and then reaches a stable strain increase rate after about five cycles.

34 Study on Uniaxial Tension and Cyclic Deformation …

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Fig. 34.10 Experimental results with the different strain at 350 °C

(a) Cyclic stress-strain curve

(b) Peak-valley strain curve with cycle number

Fig. 34.11 Experimental results under stress control (25 ± 225 MPa) at 350 °C

34.4 Constitutive Model The results of uniaxial tension, strain and stress cycling experiments at room temperature show that TA16 titanium alloy exhibits obvious ratcheting behavior, which requires a reasonable description of the material and ratcheting behavior. In this paper, two constitutive models including multi-linear elastic-plastic constitutive model and Chaboche model are used to simulate materials, and the results are compared with the experimental results.

34.4.1 Multilinear Elastoplastic Model Metal material obeys von Mises yield criterion, and its yield surface can be expressed as: F(σ, p) = σeq − σ y ( p) = 0

(34.1)

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√ where, σeq = 1.5s : s is von Mises Equivalent stress and s is deviatoric stress tensor; σ y is the uniaxial tensile plastic yield stress for metal materials; p is cumulative plastic strain. Plastic strain rate can be expressed as: ε˙ p = λ˙

∂ Fy 9 s s : s˙ = 2 ∂σ 4Hi σeq

(34.2)

where, ε˙ p is Plastic strain rate, λ is Plastic multiplier; Plastic hardening behavior   ∂ F (σ, p) can be described by plastic hardening modulus Hi Hi = − y∂ p , i > 1, i ∈ N , this model can be called a multilinear elastic-plastic constitutive model, as shown in Fig. 34.12. The uniaxial tension curve of TA16 is simulated by using the multi-linear elasticplastic constitutive model shown in Fig. 34.12. The simulated results are shown in Fig. 34.13 which can be seen that the simulation results are in good agreement with the experimental results. However, the follow-up hardening model cannot simulate ratcheting behavior well and cannot reflect the plastic strain accumulation phenomenon of materials under asymmetric stress cycles. Fig. 34.12 Multilinear elastoplastic constitutive model

Fig. 34.13 Uniaxial tensile curve of TA16 simulated by multilinear elastoplastic constitutive model under stable cycle

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34.4.2 Chaboche Constitutive Model When the stress state exceeds the elastic stress state limit of the material and continues to load, the material will begin to harden. There are two hardening modes: one is the follow-up hardening caused by the displacement of yield surface in deviating stress space, the other is the isotropic hardening caused by the expansion of yield surface without displacement. Based on the A-F model, Chaboche proposed a very important evolution criterion of follow-up hardening for ratcheting simulation, which can describe the cyclic deformation behavior of materials, as shown in Fig. 34.14. The model has been embedded in the material library of ANSYS, a large-scale finite element software. The back stress evolution equation of the model is as follows: X=

M 

Xi

(34.3)

i=1

˙ i = 2 Ci ε˙ p − γi Xi p˙ + 1 dCi T˙ Xi (i = 1, 2, 3, 4, 5) X 3 Ci dT

(34.4)

where, X is back stress tensor, Ci and γi are temperature-dependent material constants which is obtained by fitting the steady cyclic stress-strain curves at different temperatures under strain control by the least square method. T is the temperature; T˙ is temperature change rate. The follow-up hardening law of chaboche has a high reputation in the world, and its ability to predict ratcheting behavior has been accepted by engineering field. Therefore, based on the relationship between stress and plastic strain in the cyclic stress-strain curve, the parameters of Chaboche model for TA16 at two temperatures are determined by repeated optimization, as shown in Table 34.1. The stable cyclic stress-strain curves of TA16 at different temperatures are simulated by using the material parameters in Table 34.1. The results are shown in Fig. 34.15. It can be seen from the figure that the simulation results are in good agreement with the experimental results.

Fig. 34.14 Chaboche cyclic plasticity model

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Table 34.1 Material parameters of chaobche model at different temperatures of TA16 Material TA16

C1 /MPa

γ1

C2 /MPa

γ2

30

13000

10

2000

50

350

20000

5

1000

50

Temperature/°C

Fig. 34.15 Stable cyclic stress-strain curve simulated by chaboche model

Figure 34.16 show the simulation results of strain and stress cycles of TA16 at T = 30 °C, respectively. Figure 34.17 show the simulation results of strain and stress cycles at T = 350 °C, respectively. It can be seen from the graph that the hysteretic loop moves forward under cyclic loading, and the Chaboche model can reasonably predict the ratcheting behavior of materials.

(a) Cyclic stress-strain curve with 0.9% strain amplitude (50±450MPa)

Fig. 34.16 Simulated results at 30 °C

(b) Ratcheting behavior under stress control

34 Study on Uniaxial Tension and Cyclic Deformation …

(a) Cyclic stress-strain curve with the strain amplitude is 0.7% (25±225MPa)

385

(b) Ratcheting behavior under stress control

Fig. 34.17 Simulated results at 350 °C

34.5 Conclusion At T = 30 °C and T = 350 °C, the uniaxial tension, strain and stress cycling tests of TA16 were carried out, and the following conclusions were drawn: (1) TA16 titanium alloy exhibits slight cyclic softening under symmetrical strain cycles. At the same loading rate and constant strain amplitude, the response stress amplitude decreases slightly with the increase of cycle number. (2) TA16 exhibits ratcheting behavior under asymmetric stress cycles. (3) Based on the stable cyclic stress-strain data of TA16, the material parameters of Chaboche model are determined. From the above analysis, it can be seen that TA16 titanium alloy will bear cyclic thermal load in complex cyclic stress state, such as TA16 titanium alloy applied to DC steam generator, and ratcheting behavior will inevitably occur. Due to ratcheting, plastic strain will accumulate cyclically in the local high stress area, which will lead to material size exceeding the standard and affect the service life of the structure. Therefore, in the structural design of TA16 titanium alloy, the ratcheting behavior must be reasonably considered. Based on the material parameters of Chaboche model in this paper, ANSYS software can be used to simulate ratcheting effect of TA16 titanium alloy structure reasonably.

References 1. Jun, Chen, Tingxun, Wang, Wei, Zhou, Qian, Li, Peng, Ge, Shewei, Xin: Domestic and foreign marine titanium alloy and its application. Titanium 32(6), 8–12 (2015) 2. He-xi, J.I.N., Ke-xiang, W.E.I., Jian-ming, L.I., Jian-yu, Z.H.O.U., Wen-jing, P.E.N.G.: Research development of titanium alloy in aerospace industry. Chin. J. Nonferrous Metals. 25(2), 280–292 (2015)

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3. Qiang, L.U.O., Li, W.A.N.G., Zhi-ru, Z.H.O.U., Si-wei, L.I.U.: Relationship between temperature and thermal physical properties of TA16 and TA17 titanium alloys. Mater. Mech. Eng. 33(10), 30–33 (2009) 4. Zhang, Y.-F., Yu, Z.-T., Yu S., He X.-J., Wen, B.-B., Liu, H.-Y., He, W.-M., Zhao, L.-Y.: Tensile properties and microstructure of TA16 alloy pipes. Guangzhou Chem. Ind. 46(9), 30–32 (2018) 5. Ohno, N.: Recent progress in constitutive modeling for ratchetting [J]. Mater. Sci. Res. Int. 3, 1–9 (1997) 6. Chen, Xu, Rong, Jiao, Tao, Tian: Research advances of ratcheting effects and cyclic constitutive models [J]. Adv. Mech. 33, 461–470 (2003) 7. Kang, G.Z., Li, Y.G., Gao, Q.: Non-proportionally multiaxial ratcheting of cyclic hardening materials at elevated temperatures: experiments and simulations [J]. Mech. Mater. 37, 1101–1118 (2005) 8. Kang, G.Z., Kan, Q.H., Zhang, J., Sun, Y.F.: Time-dependent ratcheting experiments of SS304 stainless steel [J]. Int. J. Plast 22, 858–894 (2006) 9. Chaboche, J.L., Nouailhas, D.: Constitutive modeling of ratchetting effects, PartII: possibilities of some additional kinematic rules. ASME J. Eng. Mater. Tech. 3, 409–416 (1989)

Part X

High-Performance and Intelligent Computing for Real World’s Applications (Celebrating the 60th Birthday of Shinobu Yoshimura)

Chapter 35

Implementation of Combined Ohno-Wang Nonlinear Kinematic Hardening Model and Norton-Bailey Creep Model Using Partitioned Stress Integration Technique Tomoshi Miyamura, Yasunori Yusa, Jun Yin, Kuniaki Koike, Takashi Ikeda and Tomonori Yamada Abstract The formulation of stress integration and the consistent tangent matrix for the combined Ohno-Wang and Norton-Bailey constitutive model is derived and it is implemented using a partitioned stress integration technique. The constitutive equation is included in a finite element analysis code, ADVENTURE_Solid Ver. 2, and it is verified by solving a simple illustrative example with one hexahedral element. Keywords Elastic-Plastic · Viscoelastic · High-Temperature · FEM

35.1 Introduction Structures such as vessels in thermal power plants, rocket engine nozzles, and components of internal combustion engine are used in a high-temperature environment. Materials in high-temperature environment exhibit both elastic-plastic behaviors such as thermal fatigue and ratcheting and viscoelastic behaviors such as creep deformation and stress relaxation. Therefore, development of combined elastic-plastic and T. Miyamura (B) Nihon University, Koriyama, Fukushima 963-8642, Japan e-mail: [email protected] Y. Yusa Tokyo University of Science, Noda, Chiba 278-8510, Japan J. Yin · K. Koike · T. Ikeda Advanced Simulation Technology of Mechanics R&D, Co., Ltd, Bunkyo-ku, Tokyo 112-0002, Japan T. Yamada The University of Tokyo, Bunkyo-Ku, Tokyo 113-8656, Japan © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_35

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viscoelastic constitutive model is important to achieve high precision finite element analysis of these structures. Ohno and Wang [1] proposed an elastic-plastic constitutive model with a nonlinear kinematic hardening model in which strain hardening and dynamic recovery are considered. The model can reproduce ratcheting under a cyclic load with nonsymmetrical tension-compression and with non-zero mean stress. In the Ohno and Wang model, over-prediction of biaxial ratcheting is less than previous models such as the Armstrong and Frederick model [2]. In addition, the consistent tangent matrix for the stress integration of the Ohno-Wang model is symmetrical. Therefore, the model is suitable for large-scale finite element analyses in which the conjugate gradient method is used as a linear solver for the implicit time integration. Kobayashi and Ohno [3] proposed a time-dependent non-unified constitutive model in which the inelastic strain rate consists of transient and steady parts. The transient rate is influenced by kinematic strain hardening, while the steady rate depends only on Von Mises stress. In fact, the Kobayashi-Ohno model is equivalent to a combined the Ohno and Wang model and the Norton creep model. The purpose of the present study is to develop a simulator for the structures in the high-temperature environment. A partitioned stress integration technique is devised in order to develop a framework for the simulator to couple various kinds of elastic-plastic constitutive model and viscoelastic constative model. As an example of the combined constitutive model, combined the Ohno-Wang elastic-plastic model and the time-hardening Norton-Bailey creep model that can consider both primary and secondary creep is formulated and implemented using the proposed stress integration technique.

35.2 Combined Ohno-Wang Elastic-Plastic Model and Norton-Bailey Creep Model 35.2.1 Ohno-Wang Model and Norton-Bailey Creep Model  T strainThe ε = εx x , ε yy , εzz , γx y , γ yz , γzx , is additively decomposed into elastic, plastic and creep parts: ε = ε e + ε p + ε cr

(35.1)

The constitutive equation is as follows:   σ = D e ε e = D e ε − ε p − ε cr

(35.2)

 T where σ denotes stress vector σx x , σ yy , σzz , σx y , σ yz , σzx , , and D e is the isotropic elastic stiffness matrix. The yield function for time-independent plasticity is defined as follows:

35 Implementation of Combined Ohno-Wang Nonlinear Kinematic …

 g = σ − σY = k

3 (S − α) · Q(S − α) − σY 2

391

(35.3)

where S is the deviatoric part of stress σ , σ¯ k is the equivalent Von Mises stress with respect to S – α, Q is an auxiliary matrix with non-zero elements only on its diagonal diag[1, 1, 1, 2, 2, 2], and σY is the yield stress that is a function of accumulated effective plastic strain ε¯ p . The backstress, α is the center of yield surface, and can be decomposed into several parts: α=

M 

αi

(35.4)

i=1

where M is the number of the parts. For the evolution of each part of the back stress α i , strain hardening and the dynamic recovery are considered in the Ohno-Wang model:  αi 2 · ε˙ p (35.5) α˙ i = h i Q −1 ε˙ p − ξi α i H ( f i ) 3 ri



p ε˙¯ i

where h i and ξi are material parameters for strain hardening and dynamic recovery, respectively. H denotes Heaviside step function,   the Macauley bracket, and ri = h i /ξi . f i is defined as follows: fi =

3 α i · Qα i − ri2 2

(35.6)

p It is similar to the Von Mises type plastic yield function. ε˙¯ i in Eq. (35.5) represents the dynamic recovery of each α i . For the time-dependent creep part, the following time hardening model of the Norton power law is utilized:

ε¯˙ cr = Aσ¯ n t m

(35.7)

where ε˙¯ cr is the equivalent creep strain rate, and A, n, and m are material constants. Note that σ¯ is the equivalent Von Mises stress, which is different from σ¯ k in Eq. (35.3) for kinematic hardening.

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35.2.2 Partitioned Stress Integration and Consistent Tangent Matrix The deviatoric stress tensor, t+t S at t + t is calculated simultaneously with the unknown plastic and creep strain increments during the time interval which are also related t+t S and t+t α. The backward Euler method combined with the elastic predictor and radial return method is utilized for the stress integration to calculate stress and strain increments. The deviatoric stress tensor, t+t S is calculated as follows: t+t

S=

t+t

  S∗ − 2G Q −1 ε p + ε cr

(35.8)

where ε p and εcr are incremental plastic and creep strains, respectively. t+t S∗ is the elastic predictor of deviatoric stress that is calculated as follows: t+t

S∗ = t S + 2Gεd

(35.9)

where εd is the deviatoric strain increment. ε p and εcr have to satisfy the following scalar equations: ⎡ χ cr t+t σ¯ k + ⎣3G + χ cr

M 

⎤ t+t θ h ⎦¯ε p = i i

i=1

 t+t 1+χp σ¯ + 3G¯εcr

   M  3 t+t θ t α  t+t S∗ − χ cr i i i=1 2

   M   3  t+t ∗ p t+t t = S −χ θi α i    2

(35.10) (35.11)

i=1

where  χ =

¯ε 3G t+t σ¯ k  M t+t ¯ε p p

p

1+

t+t σ ¯k

i=1

 ¯εcr χ cr = 1 + 3G t+t σ¯ t+t

θi =



θi h i 

1   t+t 1 + ξi H ( f i ) ri α i · ε p

(35.12) (35.13) (35.14)

¯ε p and ¯εcr are the increments of the equivalent plastic strain and the equivalent creep strain obtained from ε p and εcr , respectively. The global stiffness matrix that is consistent with the stress integration scheme is derived. Incremental displacement is obtained by the global Newton-Raphson iteration using the consistent tangent matrix. Then, the increments ε p and εcr

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that satisfy Eqs. (35.10) and (35.11) are calculated by local iteration of the NewtonRaphson method for a set of these two equations. In the present study, however, a partitioned technique is adopted to conduct the stress integration, that is, the local Newton-Raphson iteration is conducted separately for Eq. (35.10) with fixed εcr and Eq. (35.11) with fixed ε p in a staggered manner, and outer (local) iteration is performed until both Eqs. (35.10) and (35.11) are satisfied. In the implementation using the partitioned technique, existing codes for the stress integration of elastic-plastic constitutive models and viscoelastic constitutive models can be reused with little modifications since they are used separately in the partitioned technique. In addition, an integrated code of material models can be developed easily. For instance, by using macros of preprocessor of C language, functions of the stress integration for a single elastic-plastic model, a single viscoelastic model, and a combined elastic-plastic and viscoelastic model can be generated from a single code.

35.3 Verification of the Combined Constitutive Model The combined Ohno-Wang and Norton-Bailey constitutive model is implemented using the proposed partitioned stress integration technique. It is implemented in a finite element analysis code, ADVENTURE_Solid Ver. 2 [4], and it is verified by solving a simple problem under cyclic loading. Figure 35.1 shows the analysis model with one hexahedral element. Figure 35.1 shows the schedule of loading. Tables 35.1 and 35.2 show the material parameters. Figure 35.2 shows the loading schedule. The time increment is 1.0 s. First, the analysis is conducted using the single Ohno-Wang model without a creep model. The stress-strain relationship is shown in Fig. 35.3. Then, the analysis with the combined Ohno-Wang and Norton-Bailey constitutive model is conducted. The stress-strain relationship is shown in Fig. 35.4. In the result by the combined model, a drift of the loops due to the creep deformation is observed. Fig. 35.1 Analysis model

Cyclic point loads are applied at nodes on the upper face

Nodes on the lower face are fixed

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Table 35.1 Material parameters for the Ohno-Wang model 1.65 × 105

Young’s modulus (MPa) Poisson’s ratio

0.33

Yield stress (MPa)

240

Hardening coefficient (MPa)

1.65 × 103

Parameters for the back stress i(M = 4)

1

ξi

2

3

4

1110

370

200

125

115

57.4

24.3

13.3

ri = h i /ξi

Table 35.2 Material parameters for the Norton-Bailey model

2.84 × 10−18

A n

5.30

m

0.0

Load increment [N]

30 20 10 0 -10 -20 -30 1

21

41 Load step

61

Fig. 35.2 Loading schedule

35.4 Concluding Remarks A partitioned technique is proposed to conduct the stress integration of combined elastic-plastic and viscoelastic constitutive model. The formulation of the stress integration and the consistent tangent matrix of the combined Ohno-Wang and NortonBailey constitutive model is derived and it is implemented using the proposed partitioned technique. Acknowledgements The present study was supported by MEXT Post-K Project Priority Issue 6: Accelerated Development of Innovative Clean Energy Systems.

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600 500

Stress (zz) (MPa)

400 300 200 100 0 -100 -200 0

0.002

0.004

0.006 Strain (zz)

0.008

0.01

0.012

Fig. 35.3 Stress-strain relationship obtained by the Ohno-Wang model (Cycles 1–4.5)

600 500

Stress (zz) (MPa)

400 300 200 100 0

-100 -200 -300 0

0.005

0.01

0.015

Strain (zz)

Fig. 35.4 Stress-strain relationship obtained by the combined Ohno-Wang and Norton-Bailey model (Cycles 1 to 4.5)

References 1. Ohno, N., Wang, J.D.: Kinematic hardening rules with critical state of dynamic recovery, Part I: formulation and basic features for ratchetting behavior. Int. J. Plast 9, 375–390 (1993) 2. Frederick, C.O., Armstrong, P.J.: A mathematical representation of the multiaxial Bauschinger effect, materials at high temperatures 24(1), 1–26 (2007) (republished version of 1966)

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3. Kobayashi, M., Mukai, M., Takahashi, H., Ohno, N., Kawakami, T., Ishikawa, T.: Implicit integration and consistent tangent modulus of a time-dependent non-unified constitutive model. Int. J. Numer. Meth. Eng. 58, 1523–1543 (2003) 4. ADVENTURE project, https://adventure.sys.t.u-tokyo.ac.jp/. Last accessed 23 Dec 2018

Chapter 36

Cross Domain Recommendations Based on the Application of Fuzzy AHP and Fuzzy Inference Method in Establishing Transdisciplinary Collaborations Maslina Binti Zolkepli and Teh Noranis Binti Mohd. Aris Abstract Cross domain recommendation method is proposed by integrating Fuzzy Analytic Hierarchy Process (AHP) and fuzzy inference method to be applied in Bibliographic Big Data. Existing cross-domain recommendation tackles the problem of sparsity, serendipity, and individual issues found in single-domain, therefore the combination of fuzzy AHP and fuzzy inference method may be able to provide recommendations with a degree of connectedness between domains to initiate transdisciplinary collaborations. The cross domain recommendation will set a stage for efficient preparation for researchers who are interested to venture into other domains and disciplines to increase their research competency. The proposed method is applied to the DBLP bibliographic citation dataset that consists of 10 domains in the computer science discipline. Results show that the combination of fuzzy AHP and FIS as the multi-criteria decision making method is able to provide helpful guide for individuals who are interested in transdisciplinary collaborations to find matching and highly related domains they can collaborate with. Representation of the highly related domains is created using fuzzy visualization technique to overcome uncertainties in the matching result. The target users for the application of this method are individuals educated and knowledgeable in different disciplines, such as computer scientists, biologists, natural disaster experts, urban planners and more. Keywords Fuzzy AHP · Fuzzy inference system · Cross domain recommendation · Transdisciplinary collaboration

M. B. Zolkepli (B) · T. N. B. Mohd. Aris University Putra Malaysia, Serdang, Selangor, Malaysia e-mail: [email protected] T. N. B. Mohd. Aris e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_36

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36.1 Introduction Cross domain recommendation has garnered much interest from researchers in the field of data mining and knowledge discovery in the recent years. While the research in this area is flourishing, there is still very limited success in terms of realization for general research use and collaborations are still initiated through word of mouth or funding requirements. In the world where transdisciplinary collaborations are indispensable, a tool that can recommend potential collaboration among diversified fields is very much beneficial. The idea of transdisciplinary collaborations is that people from different fields utilize the knowledge and skills associated with their respective fields and apply them in integrated settings. Many real-world problems have become too complex to solve for a single expert out of one discipline [1], therefore solutions that go beyond disciplinary perspectives is desirable. Researchers are keen to do cross domain collaboration as it allows the researchers to enter a new field of research, keep their research competence up to date and also gives the opportunity to learn other domain research growing speed and potentials. Given the lack of cross domain collaboration recommendation tool available out in the market, a cross domain recommendation framework based on multi-criteria decision making is proposed. The proposed framework is beneficial to improve the task of finding relations and similarities among different domains quickly that is also supplied with in-depth information. Existing cross domain recommendation methods require analyzing each topic in desired domain and collecting information from many sources such as social circles and related research lab websites. There are several cross domain collaboration recommendation approaches that have been proposed in recent years, such as the cross domain topic learning (CTL) [2] that handles sparse connection, complementary exercise and topic skewness. A framework has been developed for folksonomies cross domain recommendation [3] that creates tags of users’ profile based on the relationship of tags among multiple domains. Content-Based Cross-Domain Recommendations framework using segmented models [4] can be implemented with various classifiers and transfer common information among different domains. Cross domain recommendation using multidimensional tensor factorization [5] trades off influence among domains optimally. It compares the sparsity, scalability, cold-start and serendipity issues found in single-domain recommendation system. Another cross domain recommender system [6] transfers knowledge from the source rating matrix to help increase the prediction accuracy of the recommender system on the target rating matrix to ensure consistent information transfer between domains. While these existing approaches gives recommendation for cross domain collaboration, they do not offer details on how much each recommended domain relates to the other domain. In this proposal, we aimed to develop a cross domain collaboration recommendation framework to solve multi-criteria decision making (MCDM) problems based on fuzzy analytic hierarchy process (fuzzy AHP) [7] and fuzzy inference

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[8, 9] methods. The proposed hybrid method will be able to provide users with visualized results that contain the degree of connectedness between domains and their desired collaboration topics. The degree of connectedness can be realized with feedbacks from experts, which different experts having different views on what is recommended. In order to evaluate the imprecise judgements of the experts’ decision makers when deciding which domain to collaborate with, a fuzzy AHP and fuzzy inference method decision making framework is proposed as an evaluation tool to make the cross domain recommendations. The proposed method is represented with fuzzy visualization [10–14] that is able to convey degree of precision of each recommendation given. This will help shorten the time taken for users to understand the structure and characteristic of the recommendation result. The rest of the paper is organized as follows: Sect. 36.2 describes the fuzzy AHP, fuzzy inference method and the integration of fuzzy AHP and fuzzy inference method. In Sect. 36.3 the DBLP citation big data with ten domains in computer science is explained. Section 36.4 describes the fuzzy visualization to present the recommendation result. In Sect. 36.5 the experiment result is explained by applying the integration of fuzzy AHP and fuzzy inference method algorithms on the DBLP citation big data.

36.2 Integration of Fuzzy AHP and Fuzzy Inference Method in Cross Domain Collaboration Recommendation Method Many methods are suggested for cross-domain recommendation. In the proposed framework, fuzzy AHP and fuzzy inference methods are selected for cross domain recommendation.

36.2.1 Fuzzy AHP The AHP [15] is a quantitative technique that structures a multi-attribute, multiperson and multi-period problem hierarchically so that solutions are facilitated. The main advantage of AHP is that it is effective in handling multiple criteria in order to make decisions and it is able to cater to qualitative and quantitative data [16]. It accepts the pair-wise comparison of each alternative that belong to each criterion and offers a decision support tool for multi-criteria decision making problems. In classic AHP method, the objective is situated in the first level, the criteria is situated in the second level, and the available alternatives is situated in the third level [17]. Classic AHP does not take the vagueness of the criteria into consideration when making

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decision. Therefore, fuzzy AHP was proposed to equip classic AHP with the ability to accept uncertainty and ambiguity in human thinking style [18]. Fuzzy AHP is a technique where the pair wise comparisons of the criteria and the alternatives are achieved by using fuzzy triangular numbers to represent linguistic variables [19]. One of the earliest application of fuzzy AHP provides more realistic results than the original crisp method [20] where the choice is between a number of alternatives under overlapping judgements. Fuzzy numbers was used to represent expert opinion in ranking alternatives across a set of criteria [21]. The fuzzy numbers were used to describe experts’ preferences and fuzzy arithmetic was used to compute fuzzy ranking. There are many more fuzzy AHP related techniques that are available, but the Buckley method was selected for this study to determine the fuzzy weights of importance for the criteria and the alternatives.

36.2.2 Fuzzy Inference Method A fuzzy inference is a method that uses fuzzy set theory to map inputs to outputs. There exist two fuzzy inference methods that are widely used, Mamdani method [8] and Sugeno method [22]. The difference between Mamdani and Sugeno methods are the output consequence is not computed by clipping an output membership function at the rule strength [23]. Fuzzy inference method have been used extensively in soft computing researches such as in music retrieval and recommendation [24] where fuzzy inference is used to infer user’s real situation by fusing interacted contextual circumstances surrounding music listeners through fuzzy rules. Fuzzy inference is also used for the estimation of the importance of online customer reviews [25] where author reputation, number of tuples features and quality words are used to produce an output of importance degree of the comments. Another example of fuzzy inference method application is in the speech recognition where it is used to adapt web voice interfaces dynamically in a vehicle sensor tracking application definition [26]. Fuzzy inference method is chosen in the proposed method as it will be able to capture the uncertain values given by fuzzy AHP in order to produce recommendation result for cross domain collaboration.

36.2.3 Integration of Fuzzy AHP and Fuzzy Inference Method Step 1—Cross domain recommendation evaluation criteria. In cross domain recommendation, the objective is to find out the best domain that matched the keywords entered by users. Figure 36.1 depicts the decision hierarchy for cross domain recommendation. Figure 36.2 shows the cross domain recommendation framework with fuzzy AHP and fuzzy inference method.

Fig. 36.1 Decision hierarchy for cross domain recommendation in the field of computer science

36 Cross Domain Recommendations Based on the Application … 401

Fig. 36.2 Fuzzy AHP and fuzzy inference method framework for cross domain recommendation

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403

Step 2—Determining cross domain recommendation evaluation criteria weights. Firstly, pair wise comparisons are performed in linguistic terms. Then the required data for the analysis are entered and the fuzzy comparison matrices are calculated. Once the weights’ values have been obtained, it will be evaluated along with the inputs for the fuzzy inference method. Step 3—Producing membership degree as recommendation value. The fuzzy inference method will accept 3 input variables that are fuzzified according to the membership function, and then evaluated by 27 fuzzy inference rules. The evaluation result is defuzzified to get a membership degree output that will become the recommendation value for each record in the DBLP bibliographic big data. Step 4—Visualization of the cross domain recommendation ranking to the users. Through the visualization system, the result of cross domain recommendation with preference order will be shown to users, in different colors to set the difference.

36.3 DBLP Bibliographic Big Data The dataset used to test the method is the DBLP and ACM citation data that are categorized into 10 domains in computer science. The domains are ‘Artificial Intelligence’, ‘Computer Graphics and Multimedia’, ‘Computer Networks’, ‘Database, Data Mining and Information Retrieval’, ‘High Performance Computing’, ‘Human Computer Interaction and Ubiquitous Computing’, ‘Information Security’, ‘Interdisciplinary Studies’, ‘Software Engineering’, and ‘Theoretical Computer Science’. Recommendation is discovered through publications published in each domain. By evaluating the publication year, venue, and citations, fuzzy AHP and fuzzy inference method are used to select the best recommendation for the researcher. Each domain has four attributes that can be specified by the researcher, which are the keyword, year of publication, venue of publication and total citation of the publication. In order to evaluate the domains to be recommended, three criteria has been selected for the performance assessment. The criteria are shown in Table 36.1. Year of publication determines how recent the paper is published. The publication venue is evaluated by its SCImago Journal Rank value, and citation count is also taken into consideration as it is regularly used to assess research impact by determining how often subsequent publications cite a specific publication [27]. Table 36.1 Criteria that defines the recommendation for collaboration Criteria

Description

Value range

Year published (C1 )

The year the research paper is published

1994–2014

SJR (C2 )

SCImago journal rank value of the publication venue

0.001–4.000

Citation Count (C3 )

Total citation of the research paper

0–30

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36.4 Experiment on Computer Science Bibliographic Dataset 36.4.1 Determining Criteria Weight Using Fuzzy AHP Via pairwise comparison, the fuzzy evaluation matrix [28] is constructed based on the proposed objective, as shown in Table 36.2. The consistency ratio is less than 0.1. Next, based on triangular fuzzy number’s operational law [30], the specific weight for each criteria is obtained, as shown in Table 36.3.

36.4.2 Fuzzification of Citation, Year, and SJR Criteria as Inputs For ‘Citation’ criterion, the citation is considered as ‘Very Few’ if the citation count is 10 or below, ‘Few’ is the citation count is between 5 and 25, and ‘Many’ if the citation count is 20 or more. Figure 36.3a shows the membership function for ‘Citation’ criterion. For ‘Year’ criterion, the publication is considered ‘Old’ if it is published in the last 10 to 30 years, ‘Recent’ if it is published between 5 to 15 years ago, and ‘Most Recent’ if it is published within the last 5 years. Figure 36.3b shows the membership function for ‘Year’ criterion. For ‘Scientific Journal Ranking’ criterion, the SJR of the journal is considered ‘Low’ if the score is between 0.000 and 1.800, ‘Medium’ if the score is between 1.000 and 3.000, and ‘High’ if the score is higher than 2.300. Figure 36.3c shows the membership function for ‘SJR’ criterion. Table 36.2 Fuzzy evaluation matrix

Table 36.3 Criteria weight triangular fuzzy number’s operational law

Criteria

C1

C2

C3

C1 C2

1, 1, 1

2/4, 5/4, 7/4

2/6, 4/6, 5/6

1/4, 2/4, 4/4

1, 1, 1

4/2, 5/2, 6/2

C3

5/5, 6/5, 7/5,

1/6, 2/6, 3/6

1, 1, 1

Criteria

Weight (WC )

C1

0.3119

C2

0.4812

C3

0.2068

(d) Recommendation

(c) SJR

Fig. 36.3 Membership functions for, a ‘Citation’, b ‘Year’, c ‘SJR’ criteria and d ‘Recommendation’ output

(b) Year

(a) Citation

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36.4.3 Output Defuzzification of Cross Domain Recommendation Defuzzification process gives a single value output of the aggregate fuzzy set. In the proposed recommendation method, the centroid method [29] is used for defuzzification method as defined by (36.1) Y =

n 

yµ Bi (y)/

i=1

n 

µ Bi (y)

(36.1)

i=1

where the input for the defuzzification process is a fuzzy set µ Bi (y) (the aggregate output fuzzy set) and the output is a single number Y that will become the Recommendation value. The recommendation output is divided into 3 categories. The recommendation is considered good if the output is between 0.0 and 0.25. The recommendation is considered very good if the output is between 0.25 and 0.75. The recommendation is considered excellent if the output is between 0.75 and 1.0. Figure 36.3d shows the defuzzification phase for Recommendation output. Figure 36.4a displays the dependency of output on Year and SJR inputs while Fig. 36.4b displays the dependency of output on SJR and Citation inputs. As the values of Year, SJR, and Citation increase then the Recommendation values also increase. To help users distinguish items based on their importance, color coding is used in the nodes to show the membership degrees of each node that is displayed. In Fig. 36.5, 20 levels of color coding starts with color blue, followed by green, yellow and orange respectively and ends with color red. Color red is used to denote the items with the highest membership degree. The membership degree decreases as it reaches the color blue. As the membership degree ranges from 1 to 100, it is divided into 20 equal intervals and each interval is represented by one color in the schema. The highest membership degree which is the interval between 96 and 100 is represented with the color red, which is hottest color tone in the scale. As the membership degree decreases, corresponding color tones become closer to the color blue, which represent the least membership degree interval, which is between 1 and 5.

36.5 Results and Discussions In the experiment, several keywords have been used to find domains that are related to the keyword to be recommended. For the keyword ‘Cross Domain’, there were 3 papers found that matches the keyword. Paper #3480399 from Multimedia domain achieves the highest membership degrees, 50%. Paper #779453 and #778704 from

Fig. 36.4 Surface view of the criteria. a Year & SJR and b Citation & SJR

(a) Year & SJR

(b) Citation & SJR

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Fig. 36.5 Range of colors used to visualize recommendation result

Artificial Intelligence domain achieves the lowest score of membership degrees, 8.0%. Table 36.4 shows the membership degrees score for each paper that have been found based on the supplied keyword ‘Cross Domain’. For the keyword ‘Multicriteria Decision’, there were 6 papers found that matches the keyword. Paper #857751 from Artificial Intelligence domain achieves the highest membership degrees, 39.7%. Paper #837709 from Theoretical Computer Science domain achieves the lowest score of membership degrees, 8.0%. Table 36.5 shows the membership degrees score for each paper that have been found based on the supplied keyword ‘Multicriteria Decision’. For the keyword ‘System Development’, there were 19 papers found that matches the keyword. Paper #3161643 from Theoretical Computer Science domain and paper #997111 from Software Engineering domain achieves the highest score of membership degrees at 50%. Table 36.6 shows the full membership degrees score for each paper that have been found based on the supplied keyword ‘System Development’. Based on the membership degrees score, the result of the recommendation is shown in network visualization on the Cross Domain Recommendation Visualization System, as shown in Fig. 36.6. In the prototype, users will enter a keyword, select domain preference, if any, and click the Find button. Table 36.4 Recommendation result for the keyword ‘Cross Domain’ Domain

Paper number

SJR

Year

Citation

Membership degrees

Multimedia

3480399

0.968

2012

19

0.5000

Artificial intelligence

779453

0.744

2004

0

0.0800

Artificial intelligence

778704

0.744

2000

0

0.0800

Table 36.5 Recommendation result for the keyword ‘Multicriteria Decision’ Domain

SJR

Year

Citation

Membership degrees

Theoretical Comp. Sc.

Paper number 837709

0.823

2000

0

0.0800

Artificial intelligence

857751

1.806

2007

8

0.3970

Artificial intelligence

2919425

1.806

2010

0

0.0800

Artificial intelligence

880000

1.506

2003

0

0.0889

Artificial intelligence

857728

1.806

2007

0

0.0946

Database

884509

1.068

2006

0

0.0847

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Table 36.6 Recommendation result for the keyword ‘System Development’ Domain

Paper number

SJR

Year

Citation

Membership degrees

Theoretical Comp. Sc.

3161643

null

2010

17

0.5000

Software engineering

1075897

0.456

1999

0

0.0800

Software engineering

997582

0.64

1984

0

0.1123

Software engineering

997111

0.64

2004

25

0.5000

Software engineering

997048

0.64

1983

0

0.1173

Software engineering

997044

0.64

1992

0

0.0824

Software engineering

996559

0.64

1996

0

0.0800

Software engineering

995754

0.64

1996

0

0.0800

Interdisciplinary

783282

0.349

1999

0

0.0800

HCI

948028

0.334

2000

0

0.0800

HCI

831661

0.751

2005

3

0.0863

Multimedia

1143179

0.593

2008

4

0.0902

Artificial intelligence

3395084

0.271

2000

0

0.0800

Artificial intelligence

831661

0.751

2005

8

0.3900

Artificial intelligence

3321716

0.316

2007

0

0.0946

Database

2998626

1.628

1993

5

0.0946

Database

2998588

1.628

1993

0

0.0837

Database

2998492

1.628

1995

0

0.0837

Database

2998501

1.628

1995

0

0.0837

Fig. 36.6 ‘Cross domain recommendation visualization system’ prototype

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Each paper will be represented by a vertex that shows the paper number and domain. Each vertex is colored according to the paper’s membership degree score. Users can select any vertex and the paper’s description such as its authors, title, publication year, publication venue, and citation count will be shown. Based on this information, users can decide on their future actions.

36.6 Conclusion A hybrid multi-criteria decision making technique is proposed where fuzzy AHP is integrated with fuzzy inference method to evaluate cross domain collaboration alternatives in order to present the highly related domain with the input supplied by initial collaborator. When the criteria weights and performance ratings are uncertain, Fuzzy AHP and fuzzy inference method are able to handle the uncertainties of the inputs. The hybrid multi-criteria decision making technique is targeted to overcome the limitations of singular method that might overlook the accuracy of the decision making. The recommendation results are visualized to show the ranking of each related domains and researchers are able to find domains that are highly related to their interest to collaborate with. Based on the input from the researchers, the result is visualized in network form that shows each related domain. Each node represents a paper in the domain and colored according to its priority level. The visualization helps researchers to focus on the domain that are highly related to their desired criteria. When the node is selected, further information regarding the publication is displayed such as paper title, authors, publication year, and the name of the journal or conference proceedings. To further improve the proposed hybrid method, the time taken to produce the visualization result should be shorter. Time can be reduced if the analytics part of the bibliographic big data is improved. The amount of bibliographic data is increasing rapidly each day, hence the process to find and recommend potential cross domain collaboration gets more challenging by the day, and this make it an interesting research to focus on. Researchers are still in need for big data visualization that is quick to comprehend yet give a thorough vision of what they are looking for. Users who can benefit from the proposed method are individuals educated and knowledgeable in different disciplines, such as computer scientists, business analysts, biologists, natural disaster experts, urban planners and others who are interested to form collaborations with other domains for transdisciplinary collaborations. Acknowledgements This study was supported by Geran Putra (GP/2017/9569500) from Universiti Putra Malaysia.

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References 1. Domik, G., Fischer, G.: Transdisciplinary Collaboration and Lifelong Learning: Fostering and Supporting New Learning Opportunities, Rainbow of Computer Science, pp. 129–143. Springer, Berlin (2011) 2. Tang, J.: Cross-domain collaboration recommendation. In: Proceedings of the 18th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ‘12, pp. 1285–1293 (2012) 3. Guo, Y., Chen, X.: A framework for cross-domain recommendation in folksonomies. J. Autom. Control Eng. 1(4), 326–331 (2014) 4. Sahebi, S., Walker, T.: Content-based cross-domain recommendations using segmented models. In: Workshop on New Trends in Content-based Recommender Systems CBRecSys@RecSys. pp. 57–64 (2014) 5. Taneja, A., Arora, A.: Cross domain recommendation using multidimensional tensor factorization. Expert Syst. Appl. 92, 304–316 (2018) 6. Zhang, Q. et al.: A cross-domain recommender system with consistent information transfer. Decision Support Systems (2017) 7. Wang, Y.M., Chin, K.S.: Fuzzy analytic hierarchy process: a logarithmic fuzzy preference programming methodology. Int. J. Approx. Reason. 52(4), 541–553 (2012) 8. Amindoust, A., Ahmed, S., Saghafinia, A., Bahreininejad, A.: Sustainable supplier selection: a ranking model based on fuzzy inference system. Appl. Soft Comput. 12(6), 1668–1677 (2012) 9. Mamdani, E.H.: Application of fuzzy logic to approximate reasoning using linguistic synthesis. In: Proceedings of the Sixth International Symposium on Multiple-Valued Logic (MVL ‘76). IEEE Computer Society Press, Los Alamitos, CA, USA, pp. 196–202 (1976) 10. Pham, B., Streit, A., Brown, R.: Visualization of Information Uncertainty: Progress and Challenges. In: Trends in Interactive Visualization. Springer, London (2009) 11. Zolkepli, M., Dong, F., Hirota, K.: Visualizing fuzzy relationship in bibliographic big data using hybrid approach combining fuzzy c-means and Newman-Girvan algorithm. JACIII 18(6), 896–907 (2014) 12. Zolkepli, M., Dong, F., Hirota, K.: Automatic switching of clustering methods based on fuzzy inference in bibliographic big data retrieval system. Int. J. Fuzzy Log. Intell. Syst. 14(4), 256–267 (2014) 13. Zolkepli, M., Dong, F., Hirota, K.: Visualization of fuzzy relationship using clustering algorithms in bibliographic big data. In: 14th International Symposium on Advanced Intelligent Systems (2013) 14. Zolkepli, M., Dong, F., Hirota, K.: Application of fuzzy inference engine as an automatic switch between ensembles of clustering methods. In: Soft Computing and Intelligent Systems (SCIS) 2014 Joint 7th International Conference on and Advanced Intelligent Systems (ISIS), pp. 1164–1169 (2014) 15. Saaty, T.L.: Analytic Hierarchy Process. McGraw Hill, New York (1980) 16. Kahraman, C., Beskese, A., Ruan, D.: Measuring flexibility of computer integrated manufacturing systems using fuzzy cash flow analysis. Info. Sci. 168(1–4), 77–94 (2004) 17. Kilincci, O., Onal, S.A.: Fuzzy AHP approach for supplier selection in a washing machine company. Expert Syst. Appl. 38(8), 9656–9664 (2011) 18. Chamodrakas, I., Batis, D., Martakos, D.: Supplier selection in electronic marketplaces using satisficing and fuzzy AHP. Expert Syst. Appl. 37, 490–498 (2010) 19. Duran, O., Aguilo, J.: Computer-aided machine-tool selection based on a fuzzy-AHP approach. Expert Syst. Appl. 34(3), 1787–1794 (2008) 20. Van Laarhoven, P.J.M., Pedrycz, W.: A fuzzy extension of Saaty’s priority Theory. Fuzzy Sets Syst. 11(1–3), 199–227 (1983) 21. Buckley, J.J.: Fuzzy hierarchical analysis. Fuzzy Sets Syst. 17(3), 233–247 (1985) 22. Sugeno, M.: An introductory survey of fuzzy control. Info. Sci. 36, 59–83 (1985) 23. Fuzzy Inference Systems, http://www.cs.princeton.edu/courses/archive/fall07/cos436/ HIDDEN/Knapp/fuzzy004.htm, last accessed 2018/10/24

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24. Dridi, R., Zammali, S., Arour, K.: Fuzzy rule-based situational music retrieval and recommendation. In: 14th International Conference on Computer Systems and Applications (AICCSA), pp. 549–556 (2017) 25. Sousa, R.F.D, Rabêlo, R.A.L., Moura, R.S.: A fuzzy system-based approach to estimate the importance of online customer reviews. In: IEEE International Conference on Fuzzy Systems (FUZZ-IEEE), pp. 1–8 (2015) 26. Cueva-Fernandez, G., Espada, J.P., García-Díaz, V., Crespo, R.G., García-Fernández, N.: Fuzzy system to adapt web voice interfaces dynamically in a vehicle sensor tracking application definition. Soft. Comput. 20(8), 3321–3334 (2016) 27. Hwang, C.L., Yoon, K.: Multiple attributes decision making methods and applications. Springer, Berlin (1981) 28. Carpenter, C.R., Cone, D.C., Sarli, C.C.: Using publication metrics to highlight academic productivity and research impact. Acad. Emerg. Med. 21(10), 1160–1172 (2014) 29. Chang, D.Y.: Applications of the extent analysis method on fuzzy AHP. Eur. J. Oper. Res. 95, 649–655 (1996) 30. Lee, C.: Fuzzy logic in control systems: Fuzzy logic controller, Parts I and II. IEEE Trans. Syst. Man Cybern. 20, 404–435 (1990)

Chapter 37

Merging Behavior Simulation of Vehicular Platoon Eisuke Kita, Miichiro Yamada and Daisuke Ishizawa

Abstract In vehicle platoon, vehicles travel in a row at short inter-vehicle distance in order to increase the traffic capacity and enhance the traffic safety. The aim of this study is to control velocity of the platoon vehicles when the platoon of two vehicles merges into the other platoon. The velocity control model is defined by means of the vehicle following model. The effectiveness of the model is discussed in the experiments of four LEGO MINDSTORM. The experimental result is compared with the simulation result. The results show that the trajectories of the vehicles in the platoon are similar both in experiment and in simulation although the vehicle movement in the experiment is delayed more than that in the simulation. Keywords Traffic flow · Vehicle following model · Multi vehicle following model · Vehicle robot

37.1 Introduction In vehicle platoon, vehicles travel in a row at short inter-vehicle distance. It is very effective for increasing the traffic capacity of the road effectively and safely. The aim of previous studies is to realize the stable and successive vehicle platoon. Recently, the vehicle platoon in the several typical traffic situations becomes more important. This study focuses on the vehicle platoon in the case of vehicle merging, which is one of the typical traffic situations. The vehicle velocity control model is defined by means of the vehicle following model. The model controls the vehicle velocity according to the velocity difference and the inter-vehicle distance from the other vehiE. Kita (B) · D. Ishizawa Graduate School of Informatics, Nagoya University, Nagoya 464-8601, Japan e-mail: [email protected] M. Yamada Graduate School of Information Science, Nagoya University, Nagoya 464-8601, Japan © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_37

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cles in the platoon. The model parameters are determined from the stability analysis of the velocity control model. The effectiveness of the present model is confirmed in the computer simulation and the experiments of the LEGO MINDSTORMS [1].

37.2 Velocity Control Model 37.2.1 Traffic Situation Two lanes meet at the center of the road network and four vehicles travel on the network (Fig. 37.1). Firstly, four vehicles are separated into two platoon of two vehicles. The leading vehicle of the platoon on the main lane is referred to as a vehicle M1 and the following vehicle is referred to as a vehicle M2, The leading vehicle of the platoon on the merging lane is the vehicle S1 and the following vehicle is the vehicle S2. Two platoons are merged into one platoon of four vehicles at the merging point. Before merging, the vehicle M1 travel at a constant velocity. The vehicle M2 and S1 follow the vehicle M1 and the vehicle S2 follow the vehicle S1. The velocity control depends on Helly model.

37.2.2 Helly Model Helly model is given as follows [2] x¨n (t + Δt) = α1 (x˙n−1 (t) − x˙n (t)) + β1 (xn−1 (t) − xn (t) − Dn (t))

(37.1)

where xn (t) is the position at the time t of the vehicle n and Δt is the response delay time. The parameters α1 and β1 are the sensitivities with respect to the velocity and the position of the vehicle n, respectively. The vehicle n − 1 is the nearest preceding vehicle of the vehicle n. The notation (˙) and (¨) represent the first and the second derivatives with respect to time, respectively. The function Dn (t) is the ideal headway distance at the time t of the vehicle n, which is given by

Fig. 37.1 Traffic situation around merging point

37 Merging Behavior Simulation of Vehicular Platoon

Dn (t) = 0.029x˙n2 (t) + 0.3049x˙n (t)

415

(37.2)

where the parameters a, b, and c represent constants. The stability analysis of Helly Model is as follows. When Laplace transform is performed on the Eq. (37.1), the following equation is obtained se T s Vn (s) = α(Vn−1 (s) − Vn (s))     (37.3) 1 1 1 +β Vn−1 (s) − Vn (s) − γ0 + γ1 Vn (s) + γ2 sVn (s) s s s

Substituting γ0 = 0 and applying Pade approximation to e T s , the following transfer function is derived Vn (s) = G(s)Vn−1 (s)

(37.4)

where the transfer function G(s) is given as G(s) =

αs + β D(s)

(37.5)

Characteristic equation D(s) is as follows D(s) = a4 s 4 + a3 s 3 + a2 s 2 + a1 s + a0

(37.6)

where the coefficients a0 , a1 , a2 , a3 and a4 are as follows a0 a1 a2 a3 a4

= 12β = 12α − 6T α + 12βγ1 = 12βγ2 − 6T α + T 2 β − 6Tβγ1 + 12 = 6T + T 2 α − 6Tβγ2 + T 2 βγ1 = T 2 + T 2 βγ2

(37.7)

Stability range of the parameters α and β is shown in Fig. 37.2 when T = 0.1, γ1 = 2 and γ2 = 0.

37.2.3 Two Vehicle Following Helly Model Two vehicle following Helly model is the simple extension of Helly model so as to refer to the lead vehicle and the nearest frontal vehicle of the platoon. The model is given as follows

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Fig. 37.2 Stable region for Helly model

Fig. 37.3 Stable region for two vehicles following Helly model

x¨n (t + Δt) = α1 (x˙n−1 (t) − x˙n (t)) + β1 (xn−1 (t) − xn (t) − k1 Dn (t)) + α2 (x˙0 (t) − x˙n (t)) + β2 (x0 (t) − xn (t) − k2 Dn (t))

(37.8)

where the variables xa and xb denote the positions of the vehicle a and the vehicle b, respectively. The parameters α j and β j are the sensitivities for the velocity and the position of the vehicle. According to the similar formulation for Helly model, stability range for two vehicles following Helly model is shown in Fig. 37.3 when T = 0.1, α1 = α2 , β1 = β2 , γk1 = 2k and γk2 = 0.

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417

37.3 Experiment In Eq. (37.8), the velocity x˙0 (t) is obtained via Bluetooth communication from the lead vehicle. The term x˙n−1 (t) − x˙n (t) on the right side is approximated as follows x˙n  (t) − x˙n (t) ≡ Δx˙n  (t) 

Δxn  (t) − Δxn  (t − Δt) Δt

(37.9)

where the variable Δxn  (t) represents the distance between vehicles n  and n at time t as (37.10) Δxn  = xn  (t) − xn (t). The vehicle M1 travels at a constant velocity. The vehicle M2 and S1 follows the vehicle M1. Their velocity is controlled as follows: 1. 2. 3. 4. 5. 6. 7.

t = 0. Assign the initial headway distance. t = t + Δt. Obtain x M1 from the ultrasonic sensor. Calculate Δx˙ M1 (t) from the Eq. (37.9). Calculate x¨n (t + Δt) from the Eq. (37.1) and change the velocity. Return to step 3.

The velocity of the vehicle S2 is controlled according to the vehicle S1 and M2: 1. 2. 3. 4. 5. 6. 7.

t = 0. Assign the initial headway distance. t = t + Δt. Obtain x˙ M2 (t) from the vehicle M2 via Bluetooth. Calculate Δx˙ S1 (t) from data obtained through the ultrasonic sensor. Calculate x¨n (t + Δt) from Eq. (37.8) and change the velocity. Return to step 3.

Fig. 37.4 Vehicle positions in experiment of MINDSTORM

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Fig. 37.5 Vehicle positions in computer simulation

In the experiments of MINDSTORM, the velocity change of each vehicle is shown in Fig. 37.4. Computer simulation result is shown in Fig. 37.5. The trajectory of vehicle M1 is identical both in experiment and in simulation. The trajectories of other vehicles M2, S1 and S2 are similar both in experiment and in simulation although the vehicle movement in the experiment is slightly delayed.

37.4 Conclusion In vehicle platoon, vehicles travel in a row at short inter-vehicle distance in order to increase the traffic capacity and enhance the traffic safety. The aim of this study is to control the vehicle velocity in the platoon when two vehicles merge into the platoon. The velocity control model is defined by means of the vehicle following model. The effectiveness of the model is discussed in the experiments of four LEGO MINDSTORM. The experimental result is compared with the simulation result. The results show that the trajectories of the vehicles in the platoon are similar both in experiment and in simulation although the vehicle movement in the experiment is delayed more than that in the simulation. Acknowledgements Part of this research was supported by Japan Society for the Promotion of Science Scientific Research Fund (Basic Research (C) No.15K05760).

References 1. LEGO. Lego mindstorms education. http://www.legoeducation.jp/mindstorms/ 2. Helly, W.: Simulation of bottlenecks in single lane traffic flow. In: Theory of Traffic Flow Symposium, pp. 207–238 (1961)

Part XI

Theory and Application of Meshless Methods for the Numerical Solution of Engineering and Scientific Applications

Chapter 38

P-FEM Based on Meshless Trial and Test Functions: Part I-MLS Approximation Xiang Li, Wei Guo and Xiaoping Chen

Abstract In this paper the essential features of the P-FEM methods for solving linear elliptic equations using variational principles was addressed from the point of view of approximation space enrichment using meshless approximation. As meshless trial and test functions, MLS approximation was used as generalized p-version convergence. By using this generalized p-version convergence, along with the FEM paradigm, a new numerical approach is proposed to deal with differential equations. Through numerical examples, convergence tests are performed and numerical results are compared with MLPG and analytical solutions. The analysis has shown that the numerical solution obtained by using this method will converge as the order of MLS approximation increases. P-FEM can be directly used for higher order equations because there are no difficulties in construction shape function of any regularity. Adaptive procedures can be realized through the adaptive construction of meshless trial and test functions. The present method possesses a tremendous potential for convergent improved compared with traditional h- or p-version FEM. Keywords P-FEM · MLS · Galerkin approximation · Meshless trial function · Meshless test function

38.1 Introduction The development of approximate methods for the numerical solutions of partial differential equations has attracted the attention of engineers, physicists and mathematicians for a long time. There are two fundamentally different approaches to obtain approximation solutions [1]. The first strategy uses polynomial approximation spaces and relies on meshes and are refined near singularities and high gradients for non-smooth solutions. The second strategy is to “enrich” a polynomial approximation space such that it meshes independently. Obviously, the enriched method could construct shape functions meshless or mesh based. An efficient way to achieve X. Li (B) · W. Guo · X. Chen University of Electronic Science and Technology of China, Chengdu, Sichuan, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_38

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the first strategy is primal finite elements, which involve displacement-type nodal shape functions, are widely accepted and applied in computer modeling of physical problems. This is because of their simplicity, efficiency, stability and established convergence. However, the disadvantages of finite elements method are also wellknown, such as unsatisfactory performance in problems which involve constraints, low convergence rate for problems which are of singular nature, difficulty to satisfy higher-order continuity requirements sensitivity to mesh distortion, etc. [2]. The latest developments in this field indicate that its future lies in adaptive higher-order methods, which successfully respond to the increasing complexity of engineering simulations and satisfy the overall trend of simultaneous resolution of phenomena with multiple scales. Among various adaptive strategies for finite elements, the best results can be achieved using goal-oriented hp-adaptivity which is based on the combination of spatial refinements (h-adaptivity) with simultaneous variation of the polynomial order of approximation (p-adaptivity) [3]. In spite of the practical success of these hp-adaptivity algorithms, there are still lots topics of active research, such as posterior error estimators [4]. The h-adaptivity which based on an adaptation of the finite element mesh with the aim of improving the resolution of a specific quantity of interest concentrates primarily at the nodes. The h-adaptivity uses a low order interpolating polynomial, which usually is linear or quadratic. This caused a discontinuity in the stress field between elements, and will lead to inaccurate values for the maximum local and global stresses. The process of mesh refinement will lead to a fairly large set of differential equations, and then simultaneous linear algebraic equations, moreover, its inability to adapt to shape extremes in terms of skew and large size variation. The p-adaptivity is well recognized as a powerful way, which shows the merits of hierarchical nature, well-conditioned system of equations, fast convergence rate and simple modeling. While in the p-adaptivity, the same mesh is retained but the order of the interpolating polynomials is increased and the domain of interest is divided into convex subdomains and the polynomial approximants are piecewise smooth only over individual convex subdomains. This lead to higher rate of convergence of the p-adaptivity of the FEM over the h-adaptivity of the FEM [5]. However, there still are issues that need to be addressed before the hp-FEM can become standard in computational engineering software. One of the well-known drawbacks of the method is its algorithmic complexity and relatively high implementation cost [6]. Recently developed methods research various problems in hp-FEM [7–9], such as optimal higher-order shape functions for various types of continuous, automatic load balancing and problems with boundary layers, etc. To overcome complexity of higher-order method and make use of finite element method, a new method using meshless approximation to archive trail and test function was proposed in this paper in combination with meshed based integral. Meshless methods [10–12], have gained much attention in recent years, not only in the mathematics but also in the engineering community. Meshlsee approximation method is to enrich a polynomial approximation space such that the non-smooth solutions can be modeled independent of the mesh. However, the regularity of the shape functions is higher than low-order finite element ones. Moreover, the computational cost of

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these approaches is still higher than finite elements, and the parameters involved in the formulation are not always easy to select a priori. Recently developed mesh based enrichment method have proved to be an effective answer for alleviating meshing constraints and improving the accuracy of the solution by means of the physical knowledge of its behavior [13]. These methods include the generalized finite element method (GFEM) [14, 15] and the extended finite element method (XFEM) [16, 17]. Both methods use the partition of unity to as meshless approximation, and geometry is defined explicitly and implicitly respectively to uncoupling the geometry and the approximation. This meshless approximation, possess intrinsic nonlocal properties. The shape functions possess high order continuity will lead to large numbers of degrees of freedom including nodal displacements, nodal rotations (i.e. first order gradients of displacement), and even higher order derivatives [18]. For two-dimensional problems, such as involving plate and shell analysis, C -continuous methods are very complicated. In this paper, we try to keep C0 and C -continuity to alleviate high computational cost and a large number of degrees of freedom. However, non-local properties of meshless approximations confer an arbitrary degree of smoothness on solutions and have been applied to various problems. While an analysis of a class of meshless methods provided a possibility to cope with high order elliptic problem conveniently. This smooth interpolation procedure, by using Moving Least Square (MLS) method, one can easily achieve a global conforming Cm () interpolation field of desired order, which is very difficult to obtain by regular finite element method [19]. The objective of this contribution is to take advantage of mesh based enrichment approach, not only for the improved convergence properties but also to be able to solve PDE with complex geometries. In this paper, new P-FEM method with MLS approximation is illustrated through solving PDEs. Different from p-adaptivity FEM, this “P” means approximation order of MLS which controlled by the consistency of basis function, support size and smoothness of weight function. Increasing approximation order of MLS amount to increase the order of the polynomial interpolation of each node based FEM mesh. Essential boundary conditions are enforced while using the meshless approximations, approximately by using a penalty formulation as in [12], and non-element interpolation schemes, MLS was critically reviewed with an emphasis on the characteristics of the resulting nodal shape function. However, the nodal shape functions from meshless interpolations, such as MLS, Shepard, PU, or hp clouds, are highly complex in nature, which makes an accurate numerical integration of the weak form highly difficult. We present here integrating the local symmetric weak form of P-FEM over each element triangle in 2-D to compute the global stiffness matrix directly, and “assembly” as in the Galerkin finite element method. To perform the convergence for P-FEM with MLS, we show some numerical results for the rates of convergence in the problem of Laplace equation and potential flow. The numerical examples in Sect. 38.5 illustrate the performance of the P-FEM method, in two-dimensional potential problems. We show that the basic framework of the P-FEM method is very versatile indeed, and holds a great promise to extend finite element method.

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38.2 The P-FEM Scheme Although the present approach is fully general in solving general boundary value problems, only the 2-dimensional linear Poisson’s equation is used in the following, to demonstrate the formulation. The Poisson’s equation can be written as u,ii − f (x) = 0 in  i = 1, 2 (sun on i)

(38.1)

Or  ∇ 2 u − f (x) = 0; ∇ 2 =

∂2 ∂2 + ∂x12 ∂x22

 (38.2)

With the boundary conditions u(x) = u¯ (x) at u

(38.3)

∂u = u,i ni = ¯t (x) at t ∂n

(38.4)

The condition (38.3) is often referred to as the Dirichlet boundary condition and (38.4) as the Neumann boundary condition. In Eq. (38.1),  is the 2-D geometrical domain where the partial differential equation is defined, and in Eqs. (38.3) and (38.4), u and t are the segments of the boundary ∂ of , where u and ∂u/∂n respectively, are prescribed, ni are the components in the Cartesian coordinates (x1 and x2 ) of a unit outward normal n to the boundary ∂; (∂u/∂n) is the derivative of u along the oriented direction n. The “global-nodal-shape functions”, for the Galerkin Finite Element Method (FEM) are clearly functions such that they have a value of unity at the node in question, and go to zero at all the immediately surrounding nodes, as well as at the boundary of the star surrounding the node J in question, as shown in Fig. 38.1. And MLS approximation corresponding to the nodal point J is chosen such that it is nonzero over the support size. The support of the nodal point J is usually taken to be a circle of radius R. In P-FEM, let u be the trail function, and v be the test function. The weak form of the differential Eq. (38.1) may be written as:  

[u,ii − f (x)]vd  = 0

(38.5)

To balance the requirements on the trail and test functions, u, and v respectively, we integrate Eq. (38.5) by parts once and using divergence theorem, to obtain: 

 

[−u,i v,i − f (x)v]d  +

∂

(ni u,i v)d  = 0

(38.6)

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Fig. 38.1 FEM shape function corresponding to unit values and meshless trial and test function at node J

Substituting the boundary conditions (38.4) into the weak form (38.6) directly, we obtain the following combined symmetric weak form for the differential Eq. (38.1) and the boundary conditions (38.4): 

 

[u,i v,i + f v]d  −

t

¯t (x)vd  = 0

(38.7)

We note that in arriving at the combined weak form (38.7), we have been forced to select a trial function u that satisfies the boundary condition (38.3) at u identically, a prior; and, correspondingly, we have been forced to select a test function v that satisfies the condition, v = 0 at u , a priori. When we assume C 0 continuous u and v over triangular element, higher-order polynomial shape functions are created by increasing the number of nodes per element. Such higher-order C 0 Lagrange elements are a p-adaptivity method. Convergence with order p of the C 0 Lagrange element shows in Fig. 38.2b. We now consider the equivalent “global” trial and test functions uI∗ and vI∗ that can be built up from the “local” trial and test functions ui and vi , respectively, for node I

Fig. 38.2 Two different strategy of approximation spaces: a Mesh refined (h-adaptivity), b Mesh based enrichment (p-adaptivity), c Meshless enrichment (P-FEM)

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and J. The global trial function u and the global test function v may be represented as:   uI∗ qI∗ ; v = vJ∗ pJ∗ ; I , J = 1, 2, . . . , N (38.8) u= I

J

where N denotes the number of arbitrarily distributed nodes in . When we use Eqs. (38.7) in (38.8), we obtain:   N N   J =1









 

¯t (x) vJ∗ pJ∗ dS = 0 (38.9) uI∗,i qI∗ vJ∗,i pJ∗ + f vJ∗ pJ∗ d  − St

I =1

Equation (38.9) must hold true for all arbitrary values of pJ∗ [Note that the condition = 0 for nodes J lying on v is assumed to be satisfied, a priori since v must vanish a priori at u ]. Likewise, we assume, for now, that the condition qJ∗ = u¯ J for nodes J lying on u is assumed to be satisfied, a priori, since u = u¯ at u a priori. Thus, we obtain the following algebraic system of equations: pJ∗

N   I =1



   ¯t (x)vJ∗ d  − uI∗,i vJ ,i d  qI∗ = f vJ∗ d ; J = 1, 2, . . . , N (38.10) t



Or N 

KIJ∗ qI∗ = QJ∗ ; J = 1, 2, . . . , N

(38.11)

I =1

where KIJ∗ QJ∗

 =

t

 =



∗ ∗  uI ,i vJ ,i d 

¯t (x)vJ∗ d  −

 

f vJ∗ d 

(38.12) (38.13)

We define the “star”, sJ surrounding node J, as the piecewise linear-segmented boundary that connects the node J to its immediately neighboring nodes that were used to mesh the domain  with triangular element. Thus, KIJ∗ ≡ 0 unless node I is on sJ

(38.14)

When node I falls on the star sJ , i.e., the star surrounding the node J, Eq. (38.9) can be written as:

38 P-FEM Based on Meshless Trial and Test Functions …

 sI ∪sJ

 ∗ ∗ ∗ ∗  uI ,i vJ ,i qI pJ + f vJ∗ pJ∗ d  −

 t

427

 ¯t (x) vJ∗ pJ∗ d  = 0

(38.15)

For arbitrary pJ∗ , Eq. (38.15) leads to:  sI ∪sJ

 ∗ ∗ ∗  uI ,i vJ ,i qI + f vJ∗ d  −

 t

¯t (x)vJ∗ d  = 0

(38.16)

By integration by parts, and the use of the divergence theorem, we may write Eq. (38.16) as:  sI ∪sJ

 ∗ ∗  −uI ,ii qI + f vJ∗ d  +

 ∂(sI ∪sJ )

ni uI∗,i vJ∗ qI∗ d  −

 t

  ¯t (x) − ni uI∗,i vJ∗ d  = 0 (38.17)

If we choose a C 0 continuous linear trial function uJ in each element, and when f = 0. The error in Eq. (38.17) is identically zero. Thus, as the mesh is refined, the global trail function becomes C 1 continuous all over . This is the nature of convergence as the h-adaptivity method shows in Fig. 38.2a. P-FEM methods, which may also be required to preserve the desirable local character of the finite element, but use a local interpolation or approximation to represent the trail and test function, using the values (or perhaps the fictitious values) of the unknown variable at surrounding nodes in the local vicinity, we can use mesh to evaluate the integrals. These interpolation schemes possess all good property of meshless method. Moreover, the continuity and consistency conditions are related to the convergence of the interpolant-based weak-form method, shows in Fig. 38.2c. On the other hand, increasing the local meshless-based interpolation or approximation order, such as numbers of nodes in each interpolation domain, approximation order of local interpolation schemes (i.e. MLS), can achieve much better convergence than either the h-adaptivity or p-adaptivity FEM based on mesh based approximations.

38.3 Local Symmetric Weak Form for PFEM In the Galerkin finite element, and element free Galerkin methods, which are based on the global Galerkin formulation, one uses the global weak form over the entire domain  to solve the problem numerically. 

 

(∇ 2 u − p)vd  − α

u

(u − u¯ )vd  = 0

(38.18)

In Eq. (38.18), a penalty parameter α ≥ 1 is used to impose the essential boundary conditions, as the MLS approximation will be used to approximate the trial function, and it is not easy to directly impose the essential boundary conditions, a priori, in

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MLS approximation. Using ∇ 2 u v = u,ii v = u,i v,i − u,i v,i , and the divergence theorem, yields the following expression: 

 ∂

ni u,i vd  −





(u,i v,i + pv)d  − α

u

(u − u¯ )vd  = 0

(38.19)

in which ∂ is the boundary of the domain  and n is the outward unit normal to the boundary ∂. Making use of Galerkin method in which trial and test function are chosen from the same space. Trial function is written as: uh (x) = T (x) · uˆ =

n 

φj (x)ˆuj

(38.20)

φj (x)ˆvj

(38.21)

i=1

And test function: vh (x) = T (x) · vˆ =

n  i=1

where φi (x) is usually called the shape function of the MLS approximation corresponding to the nodal point. And uˆ i , vˆ i are called fictitious nodal values. Imposing the natural boundary condition, q = q¯ , and noticing that u,i ni = ∂u/∂n ≡ q in Eq. (38.19), we obtain: 

 u

qvd  +

q

 q¯ vd  −



 u,i v,i + pv d  − α

 u

(u − u¯ )vd  = 0

(38.22)

Rearranging as: 

 

u,i v,i d  + α



u

uvd  −

 u

qvd 

 q

q¯ vd  + α

 u

u¯ vd  −



pvd  (38.23)

Let’s substitute the trial and test functions which approximated by MLS into the Eq. (38.23) for all nodes leads to the following discretized system of linear equations: K · uˆ = f

(38.24)

where, the stiffness matrix K and the load vector f are defined by 

 Kij=



φj,k (x)φi,k (x)d  −  f i=



 u

nk φj,k (x)φi (x)d  + α

pφi (x)d  + α



 u

u¯ φi (x)d  +

q

u

φj (x)φi (x)d  (38.25)

q¯ φi (x)d 

(38.26)

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It is noted that it is easy to integrate using mesh, as FEM.

38.4 Numerical Tests By using the presently developed P-FEM approximation procedure, several numerical examples are worked out to investigate the numerical characteristics of the proposed method. The calculated results are compared with analytical solutions. An adaptive Gaussian quadrature rule is used in each element and boundary; Gaussian quadrature points will increase with approximation order to evaluate the entry of the stiffness matrix accurately. To observe the convergence, three relative error norms are measured. They are defined as Sobolev norms k as:  u0 =

 21 u2 d 

(38.27)

 21

2 u2 + u d 

(38.28)



And  u1 =



The relative errors are defined as  num  u − uexact k rk = , k = 0, 1 uexact k

(38.29)

r0 and r1 also defined as L2 and H 1 error norm respectively in some literature.

38.4.1 Patch Test Consider the standard patch test in [12]. The value of u at any interior node is given by the same linear function; and that the derivatives of the computed solution are constant in the patch. The nodal arrangements of regular and irregular are tested which mesh was generated by Delaunay triangulation. Present P-FEM method based on MLS passes all the patch test.

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38.4.2 Laplace Equation The second example solved here is the Laplace equation in the 2 × 2 domain, with the exact solution, a cubic polynomial, as

2

2

3 3 u = − x1 − x2 + 3 x1 x2 + 3x1 x2

(38.30)

A Dirichlet problem is solved, for which the essential boundary condition is imposed on all sides, and a mixed problem, for which the essential boundary condition is imposed on top and bottom sides and the flux boundary condition is prescribed on the left and right sides of the domain. The MLS approximation with constant, linear and quadratic bases as well as Spline weight functions are employed in the computation. The size of support for both weight functions are taken to be 5h with h being the mesh size. In order to compare with MLPG method, regular meshes of 9(3 × 3), 36(6 × 6) and 64(8 × 8) nodes are used to study the convergence of the method. Also, 15 Gauss points are used on each section of , and 37 points are used in the local domain  for numerical quadrature. The size (radius) of the support size for each node is chosen as 5h in the computation. The h-convergence with mesh refinement of the present method is studied for this problem. The results of relative errors and convergence for norms ·0 and ·1 are shown in Fig. 38.3 for the Dirichlet problem and in Fig. 38.4 for the mixed problem, respectively. In this figure “CS”, “LS” and “QS” denote “Constant Spline”, “Linear Spline” and “Quadratic Spline” respectively. In order to demonstrate P-convergence based P-FEM, we use fixed mesh of 64(8 × 8), and 3 group figures are used. In Figs. 38.5, 38.6, 38.7 constant, linear and quadratic polynomial pm bases, the value of 5 times nodal distance (5h) for support size, are used. The results show that the convergence increasing according to the higher continuity of Spline weight functions. Although m0 = 0, m1 = 0 (C 0 ), m0 = 0,m1 = 1 (C 0 ) and m0 = 0, m1 = 2 (C 0 ) have same continuity, convergence increasing with higher order Spline weight function.

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-1

-1.5

Log 10 (r0 )

-2

-2.5

-3

-3.5

P-FEM,CS P-FEM,LS P-FEM,QS MLPG,LS MLPG,QS

-4

-4.5 -0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Log 10 (h) -1

Log 10 (r1 )

-1.5

-2

-2.5

P-FEM,CS P-FEM,LS P-FEM,QS MLPG,LS MLPG,QS

-3

-3.5 -0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Log 10 (h)

Fig. 38.3 Comparison of relative errors for Dirichlet problem of Laplace equation with MLPG, a For norm ·0 , b For norm ·1

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Log 10 (r0 )

-2

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P-FEM,CS P-FEM,LS P-FEM,QS MLPG,LS MLPG,QS

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Log 10 (h)

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Log 10 (r1 )

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P-FEM,CS P-FEM,LS P-FEM,QS MLPG,LS MLPG,QS

-3

-3.5 -0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Log 10 (h)

Fig. 38.4 Comparison of relative errors for mixed problem of laplace equation with MLPG, a for norm ·0 , b for norm ·1

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Constant Bases Function -1.2

-1.4

Log 10 (r0 )

-1.5 -1.6

-2 -1.8

-2.5 0

-2

0

-2.2

1

m0

1

2

m1

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-0.8

Log 10 (r1 )

-1 -1

-1.5

-1.2

-2 0

-1.4

0

-1.6

1

m0

1

2

m1

Fig. 38.5 Convergence for Laplace equation, using constant bases function and with the value of 5h for the support size, and different m0 and m1 for different continuity of Spline weight functions

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-1.8

Log 10 (r0 )

-2

-2

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-2.2

-3

-2.4

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0 -3

1

m0

1

2

m1

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-1.2

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Log 10 (r1 )

-1.5

-1.6

-2

-1.8

-2.5

-2

-3 0

-2.2

0 1

m0

1

2

-2.4

m1

Fig. 38.6 Convergence for Laplace equation, using linear bases function and with the value of 5h for the support size, and different m0 and m1 for different continuity of Spline weight functions

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Quadratic Bases Function

Log 10 (r0 )

-2

-2

-2.2

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-3

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1

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1

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m1

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Log 10 (r1 )

-1.5

-1.8

-2

-2

-2.5

-2.2

-3 0

-2.4

0 1

m0

1

2

-2.6

m1

Fig. 38.7 Convergence for Laplace equation, using quadratic bases function and with the value of 5h for the support size, and different m0 and m1 for different continuity of Spline weight functions

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38.5 Concluding Remarks In this paper, the P-FEM method has been presented. For the meshless approximation scheme in P-FEM, the conventional FEM is generalized. Increase the order of approximation amount to increasing the order of the C 0 polynomial interpolation of each node. The analysis has shown that the numerical solution obtained by using this method will converge as the order of MLS approximation increases. P-FEM can be directly used for higher order equations because there are no difficulties in construction shape function of any regularity. Adaptive procedures can be realized through adaptive construction of meshless trial and test functions. The present method possesses a tremendous potential for convergent improved compared with traditional hor p-version FEM. From our perspective, up to this point, the P-FEM method is not yet a mature numerical tool. There is still much room for improvement so that the method can fit various computational tasks. Nevertheless, whatever the modification might be, the key issue is to increase the computation accuracy and efficiency. If this problem can be properly handled without losing its original technical merits, there is no doubt that the P-FEM method will open up a brand new line of research in computational mechanics. Acknowledgements This research is supported by National Natural Science Foundation of China (grant No. 51405066) and Natural Science Foundation of China (grant No. 51405063).

References 1. Fries, T.P., Belytschko, T.: The extended/generalized finite element method: an overview of the method and its applications. Int. J. Numer. Meth. Eng. 84(3), 253–304 (2010) 2. Dong, L., Atluri, S.N.: Development of T-Trefftz four-node quadrilateral and Voronoi cell finite elements for macro-& micromechanical modeling of solids. Comput. Model. Eng. Sci. (CMES). 81(1), 69–118 (2011) 3. Šolín, P., Segeth, K. et al.: Higher-Order Finite Element Methods, Taylor & Francis (2003) 4. Melenk, J.M., Wihler, T.P.: A Posteriori error analysis of $ hp $-FEM for singularly perturbed problems. arXiv:1408.6037 (2014) 5. Babuska, I., Szabo, B.A., et al.: The p-version of the finite element method. SIAM J. Numer. Anal. 18(3), 515–545 (1981) ˇ 6. Šolín, P., Cervený, J., et al.: Arbitrary-level hanging nodes and automatic adaptivity in the hp-FEM. Math. Comput. Simul. 77(1), 117–132 (2008) 7. Šolín, P., Vejchodský, T., et al.: Imposing orthogonality to hierarchic higher-order finite elements. Math. Comput. Simul. 76(1), 211–217 (2007) 8. Laszloffy, A., Long, J., et al.: Simple data management, scheduling and solution strategies for managing the irregularities in parallel adaptive hp finite element simulations. Parallel Comput. 26(13), 1765–1788 (2000) 9. Pardo, D., Demkowicz, L.: Integration of hp-adaptivity and a two-grid solver for elliptic problems. Comput. Methods Appl. Mech. Eng. 195(7), 674–710 (2006) 10. Atluri, S.N., Shen, S.: The meshless local Petrov-Galerkin (MLPG) method, Crest (2002)

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11. Shen, S.N.A., Shengping.: The meshless local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods. Comput. Model. Eng. Sci.3, 11–51 (2002) 12. Atluri, S.N., Zhu, T.: A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech. 22(2), 117–127 (1998) 13. Legrain, G., Chevaugeon, N. et al.: High order X-FEM and levelsets for complex microstructures: uncoupling geometry and approximation. Comput. Model. Eng. Sci. 241, 172–189 (2012) 14. Strouboulis, T., Babuška, I., et al.: The design and analysis of the generalized finite element method. Comput. Methods Appl. Mech. Eng. 181(1), 43–69 (2000) 15. Strouboulis, T., Copps, K., et al.: The generalized finite element method: an example of its implementation and illustration of its performance. Int. J. Numer. Meth. Eng. 47(8), 1401–1417 (2000) 16. Belytschko, T., Black, T.: Elastic crack growth in finite elements with minimal remeshing. Int. J. Numer. Meth. Eng. 45(5), 601–620 (1999) 17. Dolbow, J., Belytschko, T.: A finite element method for crack growth without remeshing. Int. J. Numer. Meth. Eng. 46(1), 131–150 (1999) 18. Tang, Z., Shen, S., et al.: Analysis of materials with strain-gradient effects: A meshless local Petrov-Galerkin (MLPG) approach, with nodal displacements only. Comput. Model. Eng. Sci. 4(1), 177–196 (2003) 19. Liu, W.-K., Li, S., et al.: Moving least-square reproducing kernel methods (I) Methodology and convergence. Comput. Methods Appl. Mech. Eng. 143(1–2), 113–154 (1997)

Chapter 39

Thermodynamic Performance Analysis on Various Configurations of Organic Rankine Cycle Systems Jun Fen Li, Hang Guo, Biao Lei, Yu Ting Wu, Fang Ye and Chong Fang Ma

Abstract The efficiencies of single screw expander and multi-stage centrifugal pump are obtained by fitting experimental data, and which substitute the constant efficiencies of the expander and the pump in most researches. Modelling analysis of four configurations of organic Rankine cycle system (conventional organic Rankine cycle, organic Rankine cycle with a regenerator, extraction organic Rankine cycle and extraction organic Rankine cycle with a regenerator) is conducted, the effects of evaporation pressure and condensation temperature on the thermal efficiency of different cycle configurations are investigated and compared. Extraction pressure and extraction ratio are introduced to analyze the thermal efficiency of the latter two cycle configurations. Result shows that the extraction organic Rankine cycle with a regenerator has the highest thermal efficiency at the same operation condition; evaporation pressure has a positive effect on the thermal efficiency, while condensation temperature has a negative effect under a certain range. This study can provide a reference for the selection of the cycle configuration and design of operation parameters for a given system. Keywords Organic rankine cycle · Evaporation pressure · Condensation temperature · Extraction · Regenerator · Thermal efficiency

39.1 Introduction Organic Rankine cycle (ORC) as a promising technology to recovery low-grade energy gets more attention, the study of ORC concludes two aspects, experimental and numerical research. Numerical method as an efficient way can obtain qualitative rules at a lower cost compared with experimental method. In recent year, many numerical studies on ORC are conducted, including working fluid, operation parameters, J. F. Li · H. Guo (B) · B. Lei · Y. T. Wu · F. Ye · C. F. Ma MOE Key Laboratory of Enhanced Heat Transfer and Energy Conservation, and Beijing Key Laboratory of Heat Transfer and Energy Conversion, College of Environmental and Energy Engineering, Beijing University of Technology, Beijing 100124, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_39

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and cycle configurations and so on. For instance, according to the characteristic of engine exhausted heat, Shu et al. [1] proposed a dual-loop ORC system with a regenerator, the effects of condensation temperature and inlet temperature of the expander on net power output and exergy loss were investigated. Liu et al. [2] investigated the influence of working fluid on the thermal efficiency for waste heat recovery system, results showed that wet fluids were inappropriate for ORC system; the lower critical temperature, the lower thermal efficiency of ORC system; and assuming the waste heat as a constant heat source might result in considerable deviation from the reality. An ORC system with a regenerator was introduced to recovery the waste heat of PEM fuel cell, the effects of working fluid mass flow rate, expansion pressure ratio and working fluid on the system performance were investigated by Mohamad [3]. A two-stage ORC system was adopted to recover the geothermal heat of 90–120 °C. The relationships of evaporation temperature, mass flow rate, volumetric flow ratio and Jakob number were revealed [4]. Delgado-Torres et al. compared the system efficiency of conventional ORC system with ORC system with a regenerator for low temperature solar thermal power generation, and the system efficiency of ORC system with a regenerator was higher than the conventional ORC system [5]. In most researches, the expander efficiency and organic working fluid pump efficiency are regarded as constant [6–8], which will result in the derivation of the reality. Lei B investigated the single screw expander efficiency and the centrifugal pump efficiency at different operation conditions, results show that the shaft efficiency and isentropic efficiency of the single screw expander are influenced by the expansion ratio, and the efficiency of the centrifugal pump is influenced by the mass flow rate of the working fluid [9], therefore the shaft efficiency and isentropic efficiency of the expander and the organic working fluid pump efficiency should be considered as variables at various operation conditions. In this paper, the model of the ORC system generating 10 kW of electric energy is established. R123 is chosen as the circulating working fluid [10], and the single screw expander and multi-stage centrifugal pump are introduced. The effects of operation parameters on the thermal efficiencies of four configurations of ORC system are investigated.

39.2 Different Configurations of ORC System Four kinds of organic Rankine cycle systems are investigated, the schematic diagrams of these four configurations are shown in Fig. 39.1. All four kinds of systems have been applied in exhausted gas heat recovery, solar thermal power generation, biomass heat recovery and geothermal heat utilization. The working process of ORC systems has been introduced in other researches; I will not reiterate it here. These four configurations of ORC systems are numerically explored at the same model hypotheses. The model parameters are given in Table 39.1. Furthermore, some assumptions of the mathematic model are presented as following: (1) All components operate in stable state condition; (2) The heat loss

39 Thermodynamic Performance Analysis …

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Expander Expander

Evaporator

Evaporator

Regenerator Pump Pump

Condenser

Condenser

(a) Conventional ORC system

(b) ORC system with a regenerator Expander

Expander

Evaporator

Evaporator

Regenerator Condenser

Pump 2 Feed-water heater Pump 2

Feed-water heater

(c) Extraction ORC system

Pump 1 Condenser

Pump 1

(d) Extraction ORC system with a regenerator

Fig. 39.1 Schematic diagrams of four configurations of ORC systems

Table 39.1 Model parameters settings for four ORC systems

Evaporation pressure range/kPa

800–1250

Superheating degree/°C

5

Condensation temperature range/°C

30–45

Subcooling degree/°C

3

Generator efficiency

0.8

Isentropic efficiency of multi-stage centrifugal pump

0.85

Electric output/kW

10

and pressure drop of each heat exchanger are neglected; (3) Mathematic models of the expander and centrifugal pumps are derived from the reference [9]. (4) The entropy difference in shell side is equal to that in tube side of the regenerator [10]; (5) The outlet working fluid of feed-water heater is saturated liquid. (6) The evaporation pressure range and condensation temperature range are 800–1250 kPa and 30–45 °C, respectively. In the reference [9], the shaft efficiency and isentropic efficiency of the expander at different expansion ratios and the efficiency of the centrifugal pump at different mass flow rates were calculated by the experiment. We obtained the formulas of the

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expander efficiency and pump efficiency by the data fitting method; the results are given as follows: The isentropic efficiency of the single screw expander when the expansion ratio n is less than 9 ηise = −0.001181 × n 4 + 0.03578 × n 3 − 0.4086 × n 2 + 2.081 × n − 3.245 (39.1) The shaft efficiency of the single screw expander when the expansion ratio n is less than 9 ηsh = −0.001412 × n 5 + 0.04702 × n 4 − 0.6145 × n 3 + 3.9411 × n 2 − 12.33 × n + 15.48

(39.2) The efficiency of the centrifugal pump when the mass flow rate m is less than 2800 kg/h ηpum = −4 × 10−8 × m 2 + 0.0002 × m + 0.0459

(39.3)

The relationship between extraction pressure and extraction ratio for the extraction ORC system can be obtained, as shown in Fig. 39.2. As the extraction pressure increases, the extraction ratio increases at the same evaporation pressure and condensation temperature to ensure that the outlet working fluid of the feed-water heater is saturated liquid. The smaller the evaporation pressure and condensation temperature, the higher the extraction ratio. For the extraction ORC system with a regenerator, the variation tendency of extraction pressure versus extraction ratio is shown in Fig. 39.3,

0.32 0.28

Extraction ratio

0.24 0.20 0.16 0.12 evaporation pressure 1250 kPa, condensation temperature 45 °C evaporation pressure 1250 kPa, condensation temperature 40 °C evaporation pressure 1050 kPa, condensation temperature 45 °C evaporation pressure 1050 kPa, condensation temperature 40 °C

0.08 0.04 0.00

200

300

400

500

600

700

800

Extraction pressure (kPa)

Fig. 39.2 Relationship between extraction pressure and extraction ratio for extraction ORC at different evaporation pressures and condensation temperatures

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0.32 0.28

Extraction ratio

0.24 0.20

evaporation pressure 1250 kPa, condensation temperature 45 °C evaporation pressure 1250 kPa, condensation temperature 40 °C evaporation pressure 1050 kPa, condensation temperature 45 °C evaporation pressure 1050 kPa, condensation temperature 40 °C

0.16 0.12 0.08 0.04 0.00 200

300

400

500

600

700

800

Extraction pressure (kPa)

Fig. 39.3 Relationship between extraction pressure and extraction ratio for extraction ORC with a regenerator at different evaporation pressures and condensation temperatures

and which is similar with the extraction ORC system. When the extraction pressure is smaller, the extraction ratio is not presented in this graph, which means that there is no need to adopt the extraction process, the outlet working fluid of the condenser can reach the saturation state.

39.3 Effects of Evaporation Pressure and Condensation Temperature on the Thermal Efficiency The condensation temperature and evaporation pressure are two vital parameters for the thermal performance of ORC system; therefore, the effects of evaporation pressure and condensation temperature on the thermal efficiency are investigated. Figure 39.4a shows the variation tendency of thermal efficiency at different evaporation pressures and condensation temperatures for the conventional ORC system. With the increase of evaporation pressure, the thermal efficiency increases in the experimental range; when the condensation temperature increases, the thermal efficiency decreases. When the evaporation pressure and the condensation temperature are 1000 kPa and 31 °C, respectively, the thermal efficiency reaches the highest, 0.1054. In Fig. 39.4a, the portion of missing data is caused by the application scope of the expander efficiency and pump efficiency. In reference [9], only the isentropic efficiency and shaft efficiency on the expander ratio less than 9 and the efficiency of the centrifugal pump on the mass flow rate less than 2800 kg/h are given, so when the expander ratio is more than 9 and mass flow rate through the centrifugal pump is over than 2800 kg/h, the thermal efficiency is not calculated in the modeling process. The effects of evaporation pressure and condensation temperature on the thermal

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(a) Conventional ORC system

(b) ORC system with a regenerator

Fig. 39.4 Effects of evaporation pressure and condensation temperature on thermal efficiency

efficiency for ORC system with a regenerator are shown in Fig. 39.4b. The variation tendency on the thermal efficiency of the ORC system with a regenerator is similar with that of the conventional ORC system. For both two configurations of ORC systems, the thermal efficiency increases with the decrease of the condensation temperature; and as the evaporation pressure increases, the thermal efficiency increases. When the evaporation pressure is 1250 kPa, and the condensation temperature is 37 °C, ORC system with a regenerator has the highest thermal efficiency, which reaches 0.1232. It is obvious that the thermal efficiency of the ORC system with a regenerator is obviously higher than that of the conventional ORC system at the same operation parameters. The addition of the regenerator decreases the heat absorption of the working fluid in the evaporator at the same condensation temperature, which boosts the thermal efficiency compared with the conventional ORC system. For the extraction ORC system, the extraction pressure and ratio should be introduced. In the mathematical model, the extraction pressure is directly given, and the extraction ratio can be calculated according to the relationship between the extraction pressure and extraction ratio in the feed-water heater. The highest thermal efficiency for this two configurations can be obtained according to the model simulation, and extraction ORC system has the highest thermal efficiency when the condensation temperature is 35 °C; for the extraction ORC system, the highest thermal efficiency occurs when the condensation temperature is 37 °C. Figure 39.5a presents the effect of extraction pressure on thermal efficiency at different evaporation pressures for the extraction ORC system. When the condensation temperature is 35 °C, the thermal efficiency increases firstly and then decreases with the increase of extraction pressure. A major contributor to this result is that with the extraction pressure increases, the decrease of shaft power generation is less than the decrease of the heat absorption in the evaporator when the extraction pressure is small, however with extraction pressure increases, the extraction ratio will increases, with the interaction of the extraction pressure and extraction ratio, the decrease of the heat absorption in the evaporator will not offset the power generation decrease, so the thermal efficiency decreases. When the evaporation pressure is 1150 kPa, and the extraction pressure is 360 kPa, the thermal efficiency reaches the highest, 0.1145. In the extraction ORC system

39 Thermodynamic Performance Analysis …

(a) Extraction ORC system Condensation temp rature is 35 °C

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Extraction ORC system with a regenerator Condensation temperature is 37 °C

Fig. 39.5 Effects of extraction pressure and evaporation pressure on thermal efficiency

with a regenerator, the same variation tendency can be observed. The impact of the evaporation pressure and extraction pressure when the condensation temperature is 37 °C is shown in the Fig. 39.5b. Compared with the extraction ORC system, the thermal efficiency of extraction ORC system with a regenerator is slightly improved. The highest thermal efficiency reaches 0.1269, when the evaporation pressure is 1250 kPa, and extraction pressure is 630 kPa. It is clear that not all the simulation dada is shown in the Fig. 39.5a, b, which is caused by the application scope of the expander efficiency and pump efficiency. In all configurations of ORC system, the extraction ORC system with a regenerator has the highest thermal efficiency at the same evaporation pressure and condensation temperature. However, the thermal efficiency is not the only evaluation indicator to assess the performance of ORC system, more evaluation indicators, e.g. exergy efficiency, economic performance and environmental effect should be considered further.

39.4 Conclusion The mathematic model on shaft efficiency and isentropic efficiency of the single screw expander and the efficiency of the multi-stage centrifugal pump are introduced. ORC system models with different configurations, which is closer to the reality, are established. Four configurations of ORC system at different operation parameters are investigated. 1. The computational formulas on shaft efficiency and isentropic efficiency of the single screw expander and efficiency of the centrifugal pump are obtained by fitting the experimental data. The introduce of real isentropic and shaft efficiency of the expander and the efficiency of the centrifugal pump has an important influence on the thermal efficiency of different configurations of ORC system, and the efficiencies of the expander and pump regarded as a constant is inappropriate.

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2. Compare the thermal efficiency of conventional ORC system with the ORC system with a regenerator, the addition of a regenerator reduces the rejection temperature of the condenser and decreases the heat absorption of the evaporator, thus the thermal efficiency of the ORC system with a regenerator is improved. 3. In the application scope of the expander and centrifugal pump, the thermal efficiencies of extraction ORC system and extraction ORC system with a regenerator both increase firstly and then decrease with the increase of the extraction ratio. However, the thermal efficiency increases with the evaporation pressure throughout the process for extraction ORC system and extraction ORC system with a regenerator. Acknowledgements This work was supported by the National Key R&D Program of China [Grant number 2016YFE0124900]

References 1. Shu, G., Liu, L., Tian, H., et al.: Analysis of regenerative dual-loop organic Rankine cycles (DORCs) used in engine waste heat recovery. Energy Convers. Manag. 76, 234–243 (2013) 2. Liu, B.T., Chien, K.H., Wang, C.C.: Effect of working fluids on organic Rankine cycle for waste heat recovery. Energy 29(8), 1207–1217 (2004) 3. Mohamad, A.S., Soheil, M.A.S., Aghajani, D.M.: Waste heat recovery from a 1180 kW proton exchange membrane fuel cell (PEMFC) system by Recuperative organic Rankine cycle (RORC). Energy 157, 353–366 (2018) 4. Li, T., Yuan, Z., Li, W., et al.: Strengthening mechanisms of two-stage evaporation strategy on system performance for organic Rankine cycle. Energy 101, 532–540 (2016) 5. Delgado-Torres, A.M., García-Rodríguez, L.: Analysis and optimization of the low-temperature solar organic Rankine cycle (ORC). Energy Convers. Manag. 51(12), 2846–2856 (2010) 6. Le, V.L., Kheiri, A., Feidt, M., et al.: Thermodynamic and economic optimizations of a waste heat to power plant driven by a subcritical\ORC (Organic Rankine Cycle) using pure or zeotropic working fluid. Energy 78, 622–638 (2014) 7. Oyewunmi, O.A., Ferré-Serres, S., Lecompte, S. et al.: An assessment of subcritical and transcritical organic Rankine cycles for waste-heat recovery. Energy Procedia 105, 1870–1876 (2017) 8. Mago, P.J., Chamra, L.M., Srinivasan, K., et al.: An examination of regenerative organic Rankine cycles using dry fluids. Appl. Therm. Eng. 28(8), 998–1007 (2008) 9. Lei, B.: Theoretical and experimental study on the performance improvement of organic Rankine cycle and single screw expander. Beijing University of Technology (2016) 10. Roy, J.P.: Parametric optimization and performance analysis of a regenerative organic Rankine cycle using R-123 for waste heat recovery. Energy 2012(39), 227–235 (2012)

Part XII

Data-Driven Estimation and Control of Flow Fields in Engineering Applications

Chapter 40

Prediction of Rubber Friction on Wet and Dry Rough Surfaces Using Flow Structure Coupling Simulation Takayoshi Kubota, Yusuke Mizuno, Shun Takahashi , Ryota Asa, Reina Sagara, Yuji Kodama and Shigeru Obayashi Abstract In the design and development of tires, the robust grip performance is a key role in different road conditions such as dry or wet load. The objective of this study is accurate prediction of friction coefficient on the wet road by using numerical analysis. Therefore, we developed a finite element analysis (FEA) solver for hyperelastic materials to apply flow–structure coupling simulations. In this paper, we investigated the validity and applicability of the structure and flow solver. As a result, present numerical method we developed showed good agreements with theoretical and experimental values. Keywords Friction · Rubber · Viscoelasticity

40.1 Introduction In the design and development of tires, the robust grip performance is a key role in different road conditions such as dry or wet load. The hydroplaning is widely known as a critical situation that can lose frictional force due to the effect of the liquid film on the load. The phenomenon has been investigated by using coupled simulation based on numerical analyses [1]. The objective of this study is to predict the rubber friction on wet and dry rough surfaces using flow–structure coupling simulation. Though the detailed mechanism of the friction is based on tribology, it was approximated in this study. We focus on the micro scale behavior and try to predict the frictional coefficient by using finite element method for the deformation. Furthermore, the friction force especially for the wet road is estimated by using the coupling simulation with flow simulation T. Kubota (B) · Y. Mizuno · S. Takahashi · R. Asa · R. Sagara Tokai University, 4-1-1Kitakaname, Hiratsuka, Kanagawa 259-1292, Japan e-mail: [email protected] Y. Kodama Yokohama Rubber CO., LTD, 2-1 Oiwake, Hiratsuka, Kanagawa, Japan S. Obayashi Tohoku University Institute of Fluid Science, 2-1-1, Katahira, Aoba, Sendai, Miyagi, Japan © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_40

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based on finite difference method. In this paper, we discuss validation results of the developed coupling solver of fluid-structure interaction (FSI) as a preliminary study for the pre-diction of the rubber friction.

40.2 Numerical Method 40.2.1 Constitutive Law The developed FEA solver employs three–dimensional computational elements represented by 20-node isoparametric brick elements. The dynamic explicit analysis is conducted based on the stiffness equations including the inertial force described by   [M ]{¨u} = F˙ − [K]{u} − [C]{˙u}

(40.1)

where {M}, {C} and {K} are the lamped mass matrix, the lumped dumping matrix and the global stiffness matrix, and {u} is the nodal displacement velocity vector. The time integration scheme in present solver employs central differential method from the point of the code simplicity. The hyperelastic material model is defined by the compressible Neo–Hookean model [2] given by W =

μ λ (I1 − 3 − 2 ln J ) + (J − 1)2 2 2

(40.2)

where W, μ and λ are the energy density and the Lame’s parameters given by μ=G λ=

2Gν 1 − 2ν

(40.3) (40.4)

where G and ν are the shear modulus and Poisson’s ratio. I 1 is the invariant given by I1 = λ21 + λ22 + λ23

(40.5)

where λ1 , λ2 and λ3 are the stretching ratio of principal axis. J is the volume change ratio given by J = λ1 λ2 λ3

(40.6)

Therefore, the material moduli tensor of the neo–Hookean model with respect to the reference configuration D are given as

40 Prediction of Rubber Friction on Wet and Dry Rough Surfaces …

∂S ∂2W ∂S =2 =4 ∂E ∂C ∂C∂C = λJ (2J − 1)C−1 ⊗ C−1 + 2[μ − λJ (J − 1)]C∗−1

451

D=

(40.7)

In Eq. (40.7), each component is given as −1 , Dijkl = λJ (2J − 1)Cij−1 Ckl−1 + 2[μ − λJ (J − 1)]Cijkl

(40.8)

where S, E, C−1 and C*−1 are second Piola–Kirchhoff tensor, Green–Lagrange strain tensor, inverse matrix of right Cauchy–Green deformation tensor and the special tensor given by −1 = Cijkj

 1  −1 Cil + Cjk−1 2

(40.9)

40.2.2 Viscoelasticity Model We use Generalized Maxwell model as viscoelasticity model in this study. In each Maxwell elements, stressing rate of second Piola–Kirchhoff stress is given by ˙ − Q, S˙ vis = Q τ

(40.10)

where, τ and Q are relaxation time and internal stress of the Maxwell element. If the viscoelasticity has only affected to isovolumetric stress, Eq. (40.10) is convert to ˆ Q ˙ ˙ˆ Sˆ vis = Q − , τ

(40.11)

where S and Q are isovolumetric components of viscoelastic stress tensor and internal stress tensor, respectively. If energy density function of a Maxwell model has same tendency of the isovolumetric component of Eqs. (40.2), (40.11) is convert to   Q ˆ ˙ ˆ˙ C ˆ − , Sˆ vis = Q τ

(40.12)

ˆ is isovolumetric component of Green–Lagrange strain tensor. Therefore, where, C the isovolumetric component of Eq. (40.7) is given by  1 ∗ 1 1 −1 1 −1 −1 −1 ˆ D = 2μvis I trC − I ⊗ C − C ⊗ I − C ⊗ C trC det C (40.13) 3 3 3 9

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In Eq. (40.13), each component is given as  1 ∂Cij 1 1 −1 1 −1 −1 −1 ˆ trC − δij Ckl − Cij δkl − Cij Ckl trC det C (40.14) Dijkl = 2μvis 3 ∂Ckl 3 3 9 In the developed solver, variable μvis in Eq. (40.14) and τ in Eq. (40.11) are given from Prony series, shown in Eq. (40.15) G(t) = G∞ +

n

i=1

 t Gi exp − τi

(40.15)

40.2.3 Contact Model In the developed solver, contact between rubber and road is treated with simplified rigid contact model, shown in Fig. 40.1. This contact model gives nodal position to each node instead of nodal force. In each step, the distance between each nodes and center of the sphere is given as R∗ =

xo∗2 + yo∗2 + zo∗2

(40.16)

where, x *o , y*o and z*o are coordinates of the node which from the center of the sphere. If the distance is shorter than sphere radius, z-axis position of the node is corrected by Eqs. (40.17) and (40.18) as follows; Fig. 40.1 Rigid contact model

40 Prediction of Rubber Friction on Wet and Dry Rough Surfaces …

R= zo =

453

xo∗2 + yo∗2 + zo2

(40.17)

R2 − R∗2 + zo∗

(40.18)

If colliding velocity of the sphere is much smaller than elastic wave velocity, displacement in a step is much smaller than minimum nodal distance. Therefore, if radius of the sphere is greater than minimum nodal distance, change of the normal direction of the node on the sphere surface in a step is very small. Therefore In this correction, horizontal deformation and balance of force in tangential direction are neglected.

40.2.4 Flow Solver The governing equations in flow solver are three dimensional continuous Eq. (40.19), Navier-Stokes Eq. (40.20), advection equation of conservative level set function (CLSF) (40.21) compression equation of CLSF (40.22). ∇ ·v =0

(40.19)

Dv 1 = f − ∇p + v∇ 2 v Dt ρ

(40.20)

1   1 + exp ϕε

(40.21)

t + u · ∇ = 0

(40.22)

=

Convection term is discretized by second–order skew symmetric scheme with small numerical dissipation for stabilization. Pressure and diffusive terms are discretized by second–order central difference scheme. Poisson equation of the pressure term is solved by Successive Over Relaxation (SOR) method. Computational grid is defined by equally-spaced Cartesian grid. Liquid surface is defined by CLSF. Object boundary is defined by level set and ghost cell method [3, 4].

40.3 Validation of Constitutive Law 40.3.1 Conditions Three types of tensile tests were carried out to validate the developed FEM solver from the points of the tensile, compression and pure shear behaviors by the uniaxial

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Model

Young’s modulus (MPs)

1.075

Poison’s ratio (–)

0.475

Density

(kg/m3 )

910

tensile test, equibiaxial tensile test and pure shear test. The size of the uniaxial tensile test model and the pure shear test model are fixed at 40 mm × 20 mm × 2 mm. The size of the equibiaxial tensile test model is set to 40 mm × 40 mm × 2 mm. The tensile velocity of these tests is set to be 500 mm/min. Figure 40.1 shows geometry of test models. Table 40.1 shows mechanical properties of test models (Fig. 40.2).

40.3.2 Results and Discussion Figure 40.3 shows the stress–strain curves. The nominal stress σ of the incompressible Neo–Hookean model under the uniaxial extension, the equibiaxial extension and pure shear are expressed by Eqs. (40.23)–(40.25) [5], respectively  1 σuniaxial = G λ − 2 λ  1 σequibiaxial = G λ − 5 λ  1 σshear = G λ − 3 λ

(40.23) (40.24) (40.25)

The results obtained from the uniaxial and pure shear extension cases show the same with theoretical curves. The maximum error of the stress are 0.072, 0.031 and 0.10% in uniaxial tensile, equibiaxial tensile and pure shear tests at nominal strain ε = 1, hence the developed solver can predict the hyperelastic deformation with high accuracy.

40.4 Validation of Viscoelasticity Model 40.4.1 Conditions Two types of the tests were carried out to validate the developed viscoelasticity model from the points of the hysteresis loop and frequency spectrum by the cyclic loading tests.

40 Prediction of Rubber Friction on Wet and Dry Rough Surfaces …

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(a) Uniaxial tensile test.

(b) Equibiaxial tensile test.

(c) Pure shear test. Fig. 40.2 Geometry of tensile tests

First, we validate finite strain viscoelasticity model by comparing with previous numerical results [6] using cyclic loading test, as shown in Fig. 40.4. Conditions and mechanical properties are summarized in Tables 40.2 and 40.3. In this test, sinusoidal displacement is given to the top of the analysis model. The period of the cycles are set to be four second. At each cycle the amplitude is kept constant. The amplitude for the first cycle is given as 0.01 m. The second, third and fourth amplitudes are set to 0.02, 0.03 and 0.04 m. After the four cycles, displacement at top of the model is fixed.

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Fig. 40.3 Stress–strain curves of the present model

Fig. 40.4 Cyclic loading test

Table 40.2 Conditions

Properties

Model

x–axis element length, x (m)

0.1

y–axis element length, y (m)

0.1

z–axis element length, z (m)

0.1

Amplitude A (m)

0.01, 0.02, 0.03, 0.04

Period T (s)

4.0

40 Prediction of Rubber Friction on Wet and Dry Rough Surfaces … Table 40.3 Conditions

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Properties

Model

Young’s modulus E ∞ (MPa)

1.259

Young’s modulus E 1 (MPa)

1.259

Young’s modulus E 2 (MPa)

1.259

Relaxation time, τ 1 (s)

0.4

Relaxation time, τ 2 (s)

0.2

Poisson ratio (–)

0.495

Density (kg/m3 )

1000

Fig. 40.5 Maxwell model of the analyze model

The analysis model is defined by a five element Maxwell model as shown in Fig. 40.5. Relaxation time of the element 1 and 2 are set to be 0.4 and 0.2 s. Material moduli tensor of the Maxwell element 1 and 2 are same as that of the static isovolumetric components.

40.4.2 Results and Discussion Figure 40.6 shows cyclic strain history of the developed viscoelasticity model. Developed model overestimated peak of the Cauchy stress than that of the previous numerical result due to the difference of the hyperelastic model. However, the final stress at the end of the loop was the same with the previous result, that is, the current numerical method could predict the viscoelasticity with sufficient accuracy. Figure 40.7 shows time history of the stress in the viscoelastic validation. The stress relaxation after the last cycle and phase lag were successfully captured. The Cauchy stress at end of the cycles, 0.6459 MPa, was in good agreement with the previous result, 0.6013 MPa. Next, the finite strain viscoelasticity model was compared with experimental results using cyclic phase angle spectrum. Figure 40.8 shows frequency spectrum of

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Fig. 40.6 Cyclic strain history of the viscoelastic model

Fig. 40.7 Stress time history of the viscoelastic model

the stress–strain phase–lag. The phase–lag of the developed model is in good agreement with experimental results. Therefore, the developed solver has a capable of the accurate prediction of the viscoelastic behavior.

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Fig. 40.8 Phase angle spectrum of the model

40.5 Validation of Contact Model 40.5.1 Conditions Contact problem between the rigid sphere and elastic wall were carried out to validate the developed contact model from contact stress and contact radius. The numerical conditions are summarized in Table 40.4. Radius of the rigid sphere is set to be 200 mm and the sphere moves opposite to z–axis direction at 0.25 m/s. Initial position of center of the sphere is set to x = 0 mm, y = 0 mm and z = 224 mm. The analysis models are as defined cubic shapes which side length is set to 24 mm. Three cases with difference sizes were carried out to investigate the influence of the element size to the stress distribution. Coarse, medium and fine elements are set to be 2, 1 and 0.5 mm. Mechanical properties of the model are same as tensile tests. Contact stress distribution between rigid sphere and the elastic materials can predict theoretically [7]. Contact load P and Contact radius c are given as function of indentation depth δ, shown in Eqs. (40.22) and (40.23). Table 40.4 Conditions

Properties

Coarse

Medium

Fine

x–axis model length (mm)

24

24

24

y–axis model length (mm)

24

24

24

z–axis model length (mm)

24

24

24

Element size (m)

2.0

1.0

0.5

Sphere radius (m)

200

200

200

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 δ=

16 P 2 (1 − ν 2 )2 9 E2R √ c = δR

(40.22) (40.23)

where, E, ν and R are Young’s modulus, Poisson’s ratio and radius of the rigid sphere, respectively. Contact stress distribution p(y) is given by  p(y) = pmax 1 −

y2 , c2

(40.24)

where, pmax maximum contact stress given by pmax =

3P 2π c2

(40.25)

40.5.2 Results and Discussion Figure 40.9 shows stress distribution of the model. Contact stress distribution of medium and fine conditions are in good agreement with theoretical solutions. However, con-tact stress distribution of coarse condition underestimated the theoretical Fig. 40.9 Geometry of the collision model

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Fig. 40.10 Contact stress distribution of the present model

solution. Therefore, it is confirmed that developed contact model can estimate the contact stress and the contact area with medium element sufficiently (Fig. 40.10).

40.6 Validation of Flow Solver 40.6.1 Conditions We conduct numerical validation for the flow solver by using sinking sphere problem. In this validation, we compare maximum sinking depth of sphere into the water with experimental results. The analysis model is defined as cubic shapes which side length is set to 200 mm. The water depth is set to 100 mm. The sphere diameter D is set to 30 mm and three different initial positions are set to 165, 185 and 205 mm. Three different mesh sizes are prepared to investigate the mesh dependency. Coarse (CM), medium (MM) and fine mesh (FM) conditions are set to dx = 3 mm, 1.5 mm and 0.75 mm, respectively.

40.6.2 Results and Discussion Figures 40.11 and 40.12 show flow field at impacting and maximum sinking of the simulation and experiment. In the result of simulation, Fig. 40.11 shows the similar water inter-face at the impact to the water surface. However, shape of the wave surface from the simulation is different from one of the experiment. One of the cause of the difference can be come from the contact angle. In this simulation, the contact angle is kept constant at 90°.

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Fig. 40.11 Interface and sphere at impacting. a MM; b FM; c Exp

Fig. 40.12 Interface and sphere at maximum sinking. a MM; b FM; c Exp

Figure 40.13 shows the maximum sinking depth H. The vertical and horizontal axes are maximum sinking depth and falling height to the water surface, respectively. Experimental result showed that the falling height is proportional to the maximum sinking depth. It can be considered from the kinetic energy of the sphere at the colliding. Numerical results except coarse mesh condition show similar trends with experiment, however, the maximum sinking depth are different from the experimental results. The discrepancies becomes smaller as mesh resolution finer, that is, the mesh resolution is insufficient to capture the phenomenon.

40.7 Concluding Remarks We developed the hyperelastic finite element analysis (FEA) solver for prediction of the rubber friction on rough dry and wet surfaces. From the validations, we confirmed the FEA solver can accurately predict the stress, viscoelasticity and contact problem. Now we develop the adhesion model, the hysteresis friction model, and the flow–structure coupling solver.

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Fig. 40.13 Position of sphere at maximum sinking

Acknowledgements Part of the present simulations were implemented by the High Performance Computer Infrastructure (HPCI) hp150130, hp160150, hp170111 and jh180051-NAJ. This work was supported by JSPS KAKENHI Grant Number 18K03937. Part of the work was carried out under the Collaborative Research Project J15052 “Study for accurate prediction of unsteady aerodynamic characteristics around moving objects” with the Institute of Fluid Science, Tohoku University. This study was partially supported by the Research Project of Tokai University “Development of measurement system for unsteady flows around moving objects by numerical simulations and experiments”.

References 1. Wagner, P., Wriggers, P., Veltmaat, L., Clasen, H., Prange, C., Wies, B.: Tribol. Int. 111, 243–253 (2017) 2. O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method Volume 2: Solid Mechanics Fifth edition, (2000), 341 3. Takahashi, S., Nonomura, T., Fukuda, K.: J. Appl. Math. 2014, 21 (2014) 4. Mizuno, Y., Takahashi, S., Nonomura, T., Nagata, T., Fukuda, K.: Math. Probl. Eng. 2015, 21 (2015) 5. Treoloar, L.R.G.: Trans. Faraday Soc. 40, 59–70 (1944) 6. Holzapfel, G.A.: Trans. Int. J. Numer. Meth. Eng. 39, 3903–3926 (1996) 7. Timoshenko, S., Goorier, J.: Numerical Theory of Elasticity, 3rd edn, McGRAW–HILL, New York, pp. 417–434 (1970)

Chapter 41

Zonal Reduced-Order Modeling of Unsteady Flow Field Takashi Misaka

Abstract The utilization of real-world data in cyberspace is becoming attractive in various fields due to the massive growth of sensing and networking technologies. It is expected to utilize such a data-rich environment to improve engineering simulations in computer-aided engineering (CAE). Data assimilation is one of methodologies to statistically integrate a numerical model and measurement data, and it is expected to be a key technology to take advantage of measured data in CAE. However, the additional cost of data assimilation is not always affordable in CAE simulations. In this study, we consider the cost reduction of numerical flow simulation with the help of a reduced-order model, which encodes a flow field into a low-dimensional representation. Since the prediction accuracy of existing ROMs are limited in complex flow fields, we investigate here a zonal hybrid approach of a full-order model and a reduced-order model. Keywords Reduced-Order model · Proper orthogonal decomposition · CFD

41.1 Introduction Computer-aided engineering (CAE) plays an indispensable role in modern research and development processes. CAE realizes the reduction of design and development lead-time and enables design optimization aiming for better performance of products. Increase of computing power further improves the accuracy and ability of CAE, however, it also exposes the importance of the treatment of uncertainties contained in a CAE model, computational conditions and measurement data. On the other hand, downsizing and commoditization of sensors are now making easy to gather measured data to a computing server. The movement of the internet of things (IoT) would make the collection of measurement data easier. One of the approaches to utilize measurement data for the improvement CAE simulation is the use of a data assimilation T. Misaka (B) National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8564, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_41

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method. Modern data assimilation methods such as ensemble Kalman filter, particle filter and four-dimensional variational method demand several to several tens more computational resources than the original CAE simulation, therefore, there exists a wall that prevents the implementation of statistical methods such as data assimilation in the real-world applications. To alleviate the above-mentioned difficulty, two approaches can be considered, i.e., the development of efficient data assimilation techniques or the cost reduction of the original simulation model. We consider here the latter approach, which is so-called reduced-order model (ROM) [1–3]. In fluid problems, proper orthogonal decomposition (POD)-based ROM is often used, where a flow field is reconstructed by using the POD spatial bases. Fast and accurate prediction is possible especially in the case of periodic flows if the pre-constructed spatial bases are appropriate. However, it is difficult to represent a complex flow field without having appropriate bases. To overcome this, a hybrid approach, where a full-order model (FOM) and a ROM are switched depending on the accuracy of ROM prediction, was proposed in a relatively simple problem [4]. In their research, FOM and ROM are switched in time for a whole domain. In the case of a complex flow field, the modelling fidelity can be switch in space using the framework of multi-block computational infrastructure [5]. In this study, we consider a simple flow around a circular cylinder with low Reynolds number, which is discretized by a multi-block Cartesian mesh solver. Block-wise switching of FOM and ROM is conducted to realize FOM/ROM hybrid prediction of the flow field.

41.2 Numerical Methods 41.2.1 Building-Cube Method Solver as a Full-Order Model Cartesian mesh-based CFD simulation is recently gaining attentions due to its capabilities of fast mesh generation around complex geometries and efficient solution procedures. The building cube method (BCM) is one of Cartesian mesh-based approaches that employs a simple block structured mesh called ‘cube’ [5]. A flow field is divided by an assemblage of multi-level cuboids, where those cubes join with the size ratio of 1/2, 1 or 2. Each cube has the same number of mesh points so that the local resolution is determined by the cube size. For flow simulation, we employ the incompressible Navier-Stokes equations: ∂u j ∂ S jk ∂u j 1 ∂ p + uk =− + 2ν , ∂t ∂ xk ρ ∂x j ∂ xk

(41.1)

∂u j = 0, ∂x j

(41.2)

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where u j and p  represent the velocity components in three spatial directions ( j, k =  1, 2, or 3), the  pressure deviation from the reference state p = p0 + p , respectively. S jk = ∂u j /∂ xk + ∂u k /∂ x j /2 denotes a strain rate tensor. The summation convention is used for the velocity components u j . ρ and ν denote density and kinematic viscosity, respectively. Equation (41.1) can be normalized by introducing the Reynolds number: Re = U L/ν, which represents the ratio of inertial and viscous effects in a flow field and governs the flow structure including the onset of flow transition. The above equations are discretized by the fully-conservative fourth-order central difference scheme [6], where the staggered arrangement of velocity components is employed. Time integration is performed by the third-order low storage RungeKutta scheme [7]. For comparison, we also employ third-order upwind scheme of Kawamura-Kuwahara [8] as described in the later section. The velocity-pressure iterative solver of Hirt et al. [9] is used for obtaining a pressure field. In this study, we employ an immersed boundary method (IBM) that use image points [10]. The kinematic viscosity in Eq. (41.1) is defined by the sum of molecular viscosity and eddy viscosity obtained by a subgrid-scale model [11]. The code has been validated in previous study [12].

41.2.2 POD-Based Reduced-Order Model Proper Orthogonal Decomposition (POD). The proper orthogonal decomposition (POD) is a methodology similar to the principal component analysis [1]. The POD decomposes a flow field based on the variance of deviation, that is,   arg minϕ  X − ϕϕ T X ,

(41.3)

where X contains a certain data set and ϕ is a POD basis vector. In the case of a flow simulation data set, the matrix X tends to be very large, therefore, a snapshot-based approach is employed. During an unsteady flow simulation, a flow field is extracted every several time steps as a snapshot. The matrix X is composed as, ⎡

u 11 · · · ⎢ .. X =⎣ . u 1n

···

⎤ um 1 .. ⎥, . ⎦

(41.4)

um n

where a superscript m indicates the number of snapshots and n is the number of variables (number of grid points multiplied by the number of variables) in the case of flow simulation. The minimization in Eq. (41.3) results in an eigenvalue problem as follows, X T X ϕ i = λi ϕ i ,

(41.5)

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Using m snapshots of a flow field, we have m eigenvalues λi and eigenvectors ϕ i . The POD basis vector which has the same size as the original field variable is defined as,  (41.6) Ψ i = X ϕ i / λi . This POD basis vector can be used to construct the reduced-order model as described in the following section. Using the basis vectors, the original velocity field can be reproduced by, ¯ u(x, t) = u(x) +

r

ai (t)Ψ i (x).

(41.7)

i=1

¯ where u(x) is the average field subtracted from the matrix X before composing the matrix X T X and r denotes the number of POD modes used for reconstructing the flow field. The temporal coefficients ai (t) can be obtained from the original snapshots by the inner-product of the velocity vector and the POD basis vector as, ¯ · Ψ i (x). ai (t) = (u(x, t) − u(x))

(41.8)

The temporal coefficients obtained from the original snapshots are used to construct a surrogate-based reduced-order model. Reduced-Order Model. A well-known approach to construct a reduced-order model from an unsteady flow field is the Galerkin projection method. The inner product of the POD bases and governing equations is calculated to generate a set of ordinary equations, which can be in the form of, ai (t + 1) = f i [a1 (t), a2 (t), . . . , ar (t)].

(41.9)

Due to the orthogonality of POD bases, the derived ordinary equations are easy to solve and the number of equations corresponds to that of POD bases considered. Using the obtained time-varying coefficients ai , a velocity field is reproduced by Eq. (41.7). The Galerkin projection approach is attractive since the mathematical background is well established, however, there exist several drawbacks such as a computational cost for calculating inner products, a stability problem for high Reynolds number flows. One alternative would be the use of surrogate models for the time development of the time coefficients. A radial basis function (RBF) is one of such surrogate models to develop the temporal coefficients in time [13], i.e., the temporal coefficients of POD mode i can be represented as a linear combination of r radial basis functions φ, ai (t + 1) =

r j=1

  wi, j · φ r j .

(41.10)

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   j where wi, j are weight coefficients and r j is a distance r j = ai (t) − aˆ i  from a set of j

data points aˆ i calculated from the snapshots by Eq. (41.8). The temporal coefficient In of mode i at time level n − 1, ain−1 is advanced to that at time n with Eq. (41.10).   2 this study, we adopted a multi-quadratic function is employed: φ r j = r j + r02 , where r0 is a parameter to be determined.

41.2.3 Hybrid ROM/FOM Prediction Since the BCM framework allow us to switch numerical scheme between cubes as in the previous attempt [12], we consider here a switching of FOM and ROM [14]. The switching is done by simple rules shown in Fig. 41.1. Initially, all the cubes are with FOM, i.e., Navier-Stokes equations discretized by 4th-order central scheme as described above. During FOM simulation, the online POD is conducted monitoring whether the flow state of a cube can be accurately represented by a certain number of POD modes. Once the error of POD reconstruction becomes smaller than a threshold, POD-based ROM is constructed and the error between FOM and ROM is monitored. The above procedure is conducted for each cube and if a cube is surrounded by the FOM/ROM cube, the cube is switched to ROM only cube, where FOM is not used anymore. The ROM cube is switched back to FOM/ROM cube if more than one of surrounding cube is FOM cube. And FOM/ROM cube is changed to FOM cube if ROM prediction error is larger than a threshold. (1) FOM Cube

(Online POD)

(2) Both FOM/ROM (Error evaluation)

(3) ROM cube

(ROM prediction only)

The flow field is represented by POD bases

Own cube is surrounded by FOM/ROM cubes

FOM/ROM error is larger then a threshold

One of surrounding cubes contains a FOM cube

Fig. 41.1 Procedures to switch cube status, (1) FOM cube where a Navier-Stokes solver is activated along with online POD analysis, (2) both FOM and ROM are solved to evaluate ROM prediction error, and (3) ROM cube where only ROM prediction is conducted

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41.3 Results 41.3.1 Computational Setting A computational domain is composed of a rectangular domain with a circular cylinder of unit diameter as in Fig. 41.2. A uniform flow is specified in the inlet boundary (the left boundary in Fig. 41.2) and the Neumann boundary condition is used on the outlet boundary (the right boundary). The Neumann boundary condition is applied to the top and bottom boundaries. The nonslip boundary condition is applied for the wall of the circular cylinder.

41.3.2 FOM/ROM Hybrid Prediction Figure 41.2 shows instantaneous streamwise velocity distribution obtained by FOM/ROM hybrid approach in Fig. 41.2a, where black solid lines show cube boundary. As shown in Fig. 41.2a, the velocity contour lines are smoothly connected across cube boundaries. Figure 41.2b shows the velocity distribution only of ROM cubes, i.e., other part of cubes is handled by FOM. Figure 41.2c shows an example of cube status, where light blue cubes are FOM cubes, green cubes are FOM/ROM cubes, and yellow cubes corresponds to ROM cubes. Figure 41.3 shows the number of ROM cubes during FOM/ROM hybrid simulation, where ROM cube is initially zero before construction of POD bases and ROM. The ROM cubes then increase after constructing ROM in some part of the computational domain. The root-mean-square error of velocity components evaluated in FOM/ROM cubes. The instantaneous streamwise velocity distribution obtained by FOM with 4thorder central scheme in Fig. 41.4a and that with 3rd-order upwind scheme in Fig. 41.4b, which can be compared with FOM/ROM hybrid prediction in Fig. 41.2a. The flow field of FOM/ROM hybrid prediction appear quite similar to that of FOM with 4th-order central scheme. Figure 41.5 shows the root-mean-square errors evaluated by velocity components of a whole computational domain, which compare FOM/ROM hybrid prediction versus 4th-order central scheme and that versus 3rdorder upwind scheme.

Fig. 41.2 Instantaneous streamwise velocity distribution by a FOM/ROM hybrid approach, b cubes predicted by ROM, and c an example of cube status

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160

# of ROM cubes

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ROM error

1.5E-03

120 100 1.0E-03 80 60

ROM error

Number of ROM cubes

180

5.0E-04

40 20 0 0

1000

2000 3000 Time step

4000

0.0E+00 5000

Fig. 41.3 The number of ROM cubes and the error of ROM prediction during FOM/ROM hybrid simulation

41.4 Conclusions In this study, we considered a simple flow field around a circular cylinder, which was discretized by a multi-block Cartesian mesh solver. Block-wise switching of FOM and ROM was conducted to realize FOM/ROM hybrid prediction of the flow field. The dynamic switching of FOM and ROM was demonstrated and the accuracy of the FOM/ROM hybrid prediction was evaluated by comparing with FOM with the same flow conditions. It is expected with the present approach that data-driven method such as the POD-based ROM can be utilized to accelerate time consuming unsteady flow simulation. Acknowledgements This work was partially supported by a Grant-in-Aid for Scientific Research on Innovative Areas (No. 16H015290) from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

Fig. 41.4 Instantaneous streamwise velocity distribution obtained from a 4th-order central scheme and b 3rd-order upwind scheme, which can be compared with FOM/ROM hybrid prediction in Fig. 41.2a

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4thCEN vs FOM/ROM hybrid 4thCEN vs 3rdKK

RMSE

1.E+00

1.E-01

1.E-02

1.E-03 0

5000

10000

15000

20000

25000

30000

35000

Time step

Fig. 41.5 Root-mean-square errors evaluated by velocity components of a whole computational domain, which compare FOM/ROM hybrid prediction versus 4th-order central scheme and that versus 3rd-order upwind scheme

References 1. Sirovich, L.: Turbulence and the dynamics of coherent structures. I - Coherent Struct. Q. Appl. Math. 45(3), 561–571 (1987) 2. Couplet, M., Basdevant, C., Sagaut, P.: Calibrated reduced-order POD-galerkin system for fluid flow modelling. J. Comput. Phys. 207(1), 192–220 (2005) 3. Kikuchi, R., Misaka, T., Obayashi, S.: Real-time prediction of unsteady flow based on POD reduced-order model and particle filter. Int. J. Comput. Fluid Dyn. 30(4), 285–306 (2016) 4. Williams, M., Schmid, P., Kutz, J.: Hybrid reduced-order integration with proper orthogonal decomposition and dynamic mode decomposition. SIAM Multiscale Model. Simul. 11(2), 522–544 (2013) 5. Nakahashi, K.: High-density mesh flow computations with Pre-/Post-data compressions, AIAA Paper 2005–4876 (2005) 6. Morinishi, Y., Lund, T.S., Vasilyev, O.V., Moin, P.: Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys. 143, 90–124 (1998) 7. Williamson, J.H.: Low-storage Runge-Kutta schemes. J. Comput. Phys. 35, 48–56 (1980) 8. Kawamura, T., Takami, H., Kuwahara, K.: Computation of high Reynolds number flow around a circular cylinder with surface roughness. Fluid Dyn. Res. 1, 145–162 (1986) 9. Hirt, C.W., Cook, J.L.: Calculating three-dimensional flows around structures and over rough terrain. J. Comput. Phys. 10, 324–340 (1972) 10. Mittal, R., Iaccarino, G.: Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261 (2005) 11. Kobayashi, H.: The subgrid-scale models based on coherent structures for rotating homogeneous turbulence and turbulent channel flow. Phys. Fluids 17, 045104–1–045104–12 (2005) 12. Misaka, T., Sasaki, D., Obayashi, S.: Adaptive mesh refinement and load balancing based on multi-level block-structured cartesian mesh. Int. J. Comput. Fluid Dyn. 31(10,) 476–487 (2017) 13. Xiao, D., Fang, F., Pain, C., Hu, G.: Non-intrusive reduced-order modelling of the NavierStokes equations based on RBF interpolation. Int. J. Comput. Fluid Dyn. 79, 580–595 (2015) 14. Misaka, T., Obayashi, S.: Zonal reduced-order modelling toward prediction of transitional flow fields. J. Phys.: Conf. Ser. 1036(1), 012012–1–9 (2018)

Chapter 42

Real-Time Prediction of Wind and Atmospheric Turbulence Using Aircraft Flight Data Ryota Kikuchi, Takashi Misaka and Shigeru Obayashi

Abstract A new technique that integrates low dimensional model (LDM) based on proper orthogonal decomposition (POD) and the flight data of a commercial aircraft is proposed to realize real-time prediction of wind and atmospheric turbulence for aviation safety and efficiency. The proposed technique sequentially assimilates flight data into LDM and predicts the wind and atmospheric turbulence at lower computational cost than the general numerical weather prediction (NWP). Actual experiments were conducted for two cases: first, weather conditions of an extratropical cyclone approaching Japan, and second, stationary front in the sea near Japan. The actual experiments consisted of two cases: under the condition of an extra-tropical cyclone approaching Japan (Case 1) and a stationary front at Pacific Ocean near Japan (Case 2). In Case 1, the proposed method was able to produce matches between the areas predicted for turbulence and the locations where turbulence was actually encountered. The proposed method is able to correct these spatiotemporal uncertainties by using the flight data. In Case 2, NWP predicted weaker wind than the flight data, and the difference between the wind rates of the NWP and the flight data was about 10 ms−1 at 55 min after the take-off, which is the time of maximum wind magnitude by the flight data. The proposed method was able to correct this difference, and predict the maximum wind magnitude accurately. Keywords Atmospheric turbulence · Aircraft flight data · Data assimilation

R. Kikuchi (B) Doer Research, Chiba 289-1106, Japan e-mail: [email protected] T. Misaka National Institute of Advanced Industrial Science and Technology, Ibaraki 305-8064, Japan S. Obayashi Institute of Fluid Science, Tohoku University, Miyagi 980-8577, Japan © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_42

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42.1 Introduction Atmospheric turbulence poses a potential risk to aircraft operation, and the accident rate related to turbulence has gradually increased [1]; in Japan, accidents caused by turbulence accounted for 48% of aircraft accidents involving commercial airplanes from 2003 to 2012 [2]. Thus, turbulence prediction systems are essential in terms of aviation safety to minimize the risks of turbulence-related accidents. Numerical weather prediction (NWP), which is an essential tool for aircraft operation, can forecast the weather condition for days and even weeks ahead and output broader area weather in formation compared to radar and lidar. The prediction uncertainty of turbulence using NWP is considerably large because the turbulence is too small to resolve explicitly using NWP [3, 4]. It is necessary to quantify the uncertainty to ensure efficient operation [5]. The weather information with the uncertainty is provided for the aviation community at present, however this uncertainty may lead to bad effects for aviation safety and efficiency. Thus, there remains a need to correct an efficient method that can reduce the prediction uncertainty of the weather information at before or after providing for the aviation community. Data assimilation (DA) is vital in the field of meteorological and oceanic research because differences in the models initial and boundary conditions can lead to exponential divergence in the predicted results over time for chaotic systems [6]. DA is employed to reduce the uncertainty of a numerical model by assimilating the observations into the simulation [7]. DA can reduce the uncertainty of NWP directly before providing the weather information for the aviation community. However, the DA update interval of operational mesoscale NWP is generally a few hours. Therefore, the prediction results are updated every few hours [8, 9]. The results predicted by these NWP models are updated a few hours later after obtaining the observation data. This paper focuses on the real-time prediction of wind field and atmospheric turbulence, i.e., frequently updating weather prediction, by using a low dimensional model (LDM) based on proper orthogonal decomposition (POD) [10–13] and the flight data of a commercial aircraft in flight. This method is able to correct the uncertainties of the weather information after providing the information for the aviation community, which differ from the general data assimilation. This method is the postprocessing for reducing the uncertainties of the weather information. POD is known as a statistical recognition tool [11] or post processing tool that is used in experiments such as particle image velocimetry [12] or dominant components analyses in meteorology, called the empirical orthogonal function (EOF) [14]. POD is a technique for extracting coherent structures from a weather field and expressing them as POD basis vectors. In this research, LDM is defined as the sum of POD basis vectors and the weighting factors. The POD basis vectors are obtained from NWP results before flight, and the weighting factors are optimized using the flight data in flight. LDM can predict the weather field for a much lower computational cost than NWP models. We test the proposed method and investigate its effectiveness as a real-time prediction of wind and atmospheric turbulence fields. We conduct the case studies of actual experiments that use actual flight data from commercial aircrafts that have

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encountered turbulence for two cases. The actual experiments use the actual flight data for the prediction of wind and turbulence by using the proposed method, and the predicted results are compared with the flight data.

42.2 Numerical Methodology for Real-Time Prediction 42.2.1 Numerical Weather Prediction In this study, the Japan meteorological Agency non-hydrostatic model (JMA-NHM) is employed as NWP [8, 15]. JMA-NHM uses fully compressible Navier-Stokes equations on an Arakawa C grid and a hybrid terrain-following vertical coordinate. The horizontal grid resolution is 5.0 km, the horizontal mesh is 400 × 400, and the model top is 40 hPa with 60 vertical layers. The improved Mellor-Yamada level 3 scheme is used for the turbulence closure. On the surface layer, turbulent fluxes are computed from the bulk method formulated by Beljaars and Holtslag. Further, the Kain-Fritsch convective parameterization is used and the 3-ice bulk micro-physics scheme is employed. Topography data is the global 30 arc-second elevation dataset (GTOPO30), and land use data is the global land cover characterization (GLCC). The daily sea-surface temperature (SST) for the bottom boundary is merging satellite infrared and microwave SSTs. JMA-NHM was run from the initial and boundary condition of the 3-h mesoscale objective analysis data, which are produced with a mesoscale four dimensional variational (4D-VAR) DA system by JMA. The computational domain of JMA-NHM are shown in Fig. 42.1. Figure 42.1 includes the Fig. 42.1 Computational domain of JMA-NHM. The trajectories of the actual experiments; white circles and squares indicate locations of airport departures and arrivals, and white triangles indicate locations where the aircraft encountered turbulence

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aircraft trajectories of the actual experiments that are thick lines. The detailed information of these trajectories is described below.

42.2.2 Low Dimensional Model Based on Proper Orthogonal Decomposition POD can be used as a technique for extracting coherent structures from the weather fields, and for creating a set of orthogonal spatial basis functions (POD basis vector) and a set of eigenvalues as reference weighting factors. POD basis vectors and reference weighting factors are calculated from the weather fields computed by JMANHM. In this study, snapshot POD [10] is employed, which requires lower computational cost to determine the POD basis vectors compared to ordinary POD. The snapshot POD can construct POD basis vectors using considerably smaller matrices of K × K (K is the total number of snapshots). The model variables U t are the velocity components of a wind velocity field at every step in time t = 1, …, t K , where K is the number of wind fields used for POD. In snapshot POD, the POD basis vectors are obtained by solving the K x K eigenvalue problem. The K x K eigenvalue problem is established as  E = U t1 , U t2  ≡

(U t1 , U t2 )d V ≡

K  K 

U t1 U t2 xt1 yt1 z t1

(42.1)

t1 =1 t2 =1

V

Ex i = λi x i

(42.2)

where V is the area of the JMA-NHM computational domain, dV is the infinitesimal area, and xt1 yt1 z t1 is the grid spacing of JMA-NHM for correcting the difference of cell volume. In addition, xi is the eigenvector corresponding to the eigenvalue li of the covariance matrix E, and λi is always positive because E is a symmetric matrix. The POD basis vectors i are then computed as M = (U t1 , . . . , U t K )

(42.3)

M xi i = √ λi

(42.4)

where the i-th eigenvalue is a measure of the kinetic energy contained within the i-th mode. The reference weighting factors are f i (t) are then computed as ref

ai (t) = U t , i  ref

(42.5)

where ai (t) is calculated by the inner product of a wind field U t and POD basis ref vectors i . The wind field U(x, t) can be constructed by i (x) and are ai (t) as

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U(x, t) =

r 

ref

ai (t)i (x)

479

(42.6)

i=1

where r (r ≤ K) is the number of POD basis vectors used for the prediction of a weather field. LDM of the wind field is expressed as this equation, and the wind field is ref reconstructed by replacing the reference weighting factors ai (t) with the optimized opt opt weighting factors ai (t). The optimized weighting factors ai (t) are estimated by minimizing the difference of the horizontal wind velocity components between flight data and the prediction result of LDM. The prediction using the LDM requires the calculation of weighting factors for a number of POD basis vectors. Therefore, LDM can predict weather at a much lower computational cost than general NWP, and it can update the entire wind field in real-time in the flight.

42.2.3 Definition of Cost Function A cost function is required to optimize the weighting factors of the LDM. In this study, the difference of the horizontal wind velocity components between flight data and the prediction result of the LDM along a flight trajectory is considered for the cost function. The cost function J is defined as J=

T    t 2  t 2  t u obs (x, y, z) − u tsim (x, y, z) + vobs (x, y, z) − vsim (x, y, z) t=1

(42.7) where t is the flight time from take-off, T is the total time required for the optimization, and (x, y, z) denote the three-dimensional position of the aircraft (longitude, t latitude, and height, respectively). u tobs (x, y, z) and vobs (x, y, z) are the horizontal wind velocity components (east-west and north-south winds) of flight data that are the t observation data of aircraft in flight. u tsim (x, y, z) and vsim (x, y, z) are the horizontal wind velocity components (east-west and north-south winds) of the LDM prediction result. In this study, the wind data of the LDM along the aircraft trajectory cannot be obtained directly because the horizontal grid resolution of the LDM is much coarser than the observation interval of the aircraft. The horizontal grid resolution of the LDM is 5.0 km, and the aircraft observes wind data every 1 s, i.e., about 250 m t for the speed of a cruising aircraft. Therefore, u tsim (x, y, z) and vsim (x, y, z) on the aircraft position are estimated using trilinear interpolation. The weighting factors of LDM are optimized by particle swarm optimization (PSO) and the cost function [16, 17]. The PSO algorithm is modeled on the behavior of birds and fish, and it can find a global optimal solution rapidly.

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42.3 Actual Experiments 42.3.1 Turbulence Indicators The actual experiments used the actual flight data of commercial aircrafts that encountered turbulence in order to evaluate the effectiveness of the proposed method. The trajectories of the actual experiments are shown in Fig. 42.1. The white circles and squares are the locations for airport departures and arrivals, the white triangles represent the locations where the aircraft encountered turbulence. For predicting turbulence, the turbulence indicator, vertical wind shear (VWS), was used in this study. The expressions for calculating VWS is given as  VWS =

∂u ∂z m

2

+

∂v ∂z m

2 (42.8)

where u and v are the horizontal wind velocity components (east-west wind and north-south wind), z m represents the vertical positions of the computational grid with the distances being denoted in meters. The turbulence kinetic energy equation includes the production term related to the wind shear, VWS. Therefore, the generation of turbulence kinetic energy cause by wind shear. In addition, the domestic Significant Weather prognostic charts (FBJP) provided by JMA, which show the forecast information regarding turbulence, lightning, and other significant weather events, were used. For predicting turbulence through the FBJP, the threshold value of VWS is 16 kt/1000 ft (2.7 ms−1 /100 m). The FBJP show the areas with values larger than this threshold value as areas with turbulence. The threshold value of VWS is used for estimating the turbulence prediction area in this study.

42.3.2 Case1: An Extratropical Cyclone Over a Pacific Ocean Near Japan The first case involved an approaching extratropical cyclone, with its fronts located over the Pacific Ocean near Japan; this condition was used in the numerical experiments as well. In this case, the airports for the departure and arrival were Saga airport (HSG) and Tokyo international airport (HND). The time series data for the vertical acceleration and flight altitude of the aircraft are shown in Fig. 42.2. The flight time is about 83 min between 1037 UTC 26 December 2006 and 1157 UTC 26 December 2006; Fig. 42.2 shows the 60-min flight data between 10 min and 70 min after the take-off. The aircraft encountered the turbulence around 30 min and 45 min after the take-off. First turbulence is right turbulence, and second turbulence is larger fluctuation than first one. When the aircraft encountered the second turbulence 45 min after the take-off, the aircraft descended by 800 m. In the proposed method, 16 POD

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Fig. 42.2 Time series data for the vertical acceleration and flight altitude of the aircraft

basis vectors were used throughout. In addition, the wind information was updated by executing the proposed method every 3 min during the flight. JMA-NHM that was compared with the proposed method was updated at 0600UTC 26 December 2006 by 4D-VAR of JMA system in Case 1. The weather field for case 1 was predicted by using the proposed method and the results were compared with those obtained by JMA-NHM. Figure 42.3 shows the horizontal distributions of wind and VWS fields at an altitude of 12 km; these are the results at 27 min after the take-off, which is 18 min before encountering the second turbulence. In the wind fields shown in Fig. 42.3a, c, there are some differences between the values obtained through JMA-NHM and the ones estimated using the proposed method. The proposed method predicted a weaker wind field than JMANHM. Similarly, the distributions of VWS changed significantly. In Figs. 42.3b, d, the black counter areas indicate the predicted areas of turbulence, with threshold values larger than VWS > (2.7 ms−1 /100 m). JMA-NHM predicted turbulence to be distributed over thin and long stretches of areas toward north, while turbulence was not predicted along the aircraft trajectory at this altitude. On the other hand, the area predicted by the proposed method was distributed more toward south. In addition, turbulence was predicted along the trajectory at this altitude; thus it was concluded that the proposed method produced close matches between the predicted locations and the actual locations where the second turbulence was encountered. Figure 42.4 shows the vertical distributions of VWS field along the aircraft trajectory. Figure 42.4 a, b provide the results obtained through JMA-NHM and the proposed method at 27 min after the take-off, and Figs. 42.4c, d show the results obtained through JMANHM at 60 and 120 min before the take-off. Figure 42.4a shows some differences between the actual locations with both first and second turbulence and the areas predicted for turbulence. The area predicted by JMA-NHM is located at a lower altitude than the location where second turbulence was actually encountered. On the other hand, the area that was predicted by the proposed method is close to the

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Fig. 42.3 The horizontal distribution of wind and VWS field a and b JMA-NHM, c and d proposed method, a and c wind magnitude and direction, b and d VWS field, white and red circles: locations of airport departure and arrival, white triangles: locations where turbulence was encountered, purple line: aircraft trajectory

20 location where both first and second turbulence was encountered. The proposed method was able to predict the turbulence area more accurately than JMA-NHM at 27 min after the take-off, which was 18 min before encountering the second turbulence. Next, the areas predicted in Fig. 42.4c, d are closer to the ones predicted in Fig. 42.4b than in Fig. 42.4a. In addition, the shape of the predicted area in Fig. 42.4d is similar to the one in Fig. 42.4b. The above findings suggest that the results of JMA-NHM had some spatiotemporal uncertainties in terms of the predicted area of turbulence, and thus the results were not accurate. This also suggests that the proposed method corrects the spatiotemporal errors such as phase error in JMA-NHM by using the flight data.

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Fig. 42.4 The vertical distribution of the VWS field along the trajectory, a the result of JMA-NHM at 27 min after the take-off, b the results by the proposed method at 27 min after the take-off, c and d the results of JMA-NHM at 60 min and 120 min before the take-off, vector point: the actual location of turbulence, dotted line: aircraft trajectory

42.3.3 Case2: A Stationary Front at Pacific Ocean Near Japan The weather condition in the second case was assumed to be a stationary front at the Pacific ocean near Japan. In this case, the airports for departure and arrival were the new Chitose airport (CTS) and Chubu Centrair international airport (NGO). The aircraft encountered turbulence 70 min after the takeoff, when it started descending for landing. The time series data of the vertical acceleration and flight altitude of the aircraft are shown in Fig. 17. The flight time is about 88 min between 0605 UTC 6 June 2006 and 0733 UTC 6 June 2006, and Fig. 42.5 shows the data during the 70 min of flight between 10 min and 80 min after the take-off. In the proposed method, sixteen POD basis vectors were used throughout. In addition, the wind information was updated by executing the proposed method every 3 min during the flight. JMANHM that was compared with the proposed method was updated at 0300 UTC 6 June 2006 by 4D-VAR of JMA system in Case 2. In other word, the result of JMA-NHM was updated 3 h and 5 min before takeoff. The weather field for case 2 was predicted using the proposed method and the results were compared with those of JMA-NHM. Figure 42.6 shows the horizontal distributions of wind at altitudes of 5.5 km and the vertical cross-sections at longitude of 136.8 E that the aircraft encountered the turbulence, and the values correspond

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Fig. 42.5 The time series data for the vertical acceleration and flight altitude of the aircraft

Fig. 42.6 The horizontal distribution and vertical cross-section of the wind field, a and b as predicted by JMA-NHM, c and d as predicted by the proposed method, a and c the horizontal distribution at altitude of 12 km, b and d the vertical cross-section at longitude of 136.8 E, white and red circles: airport locations of departure and arrival, white triangles: locations where the aircraft encountered turbulence, purple line: aircraft trajectory

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Fig. 42.7 Time series data of wind magnitude by the proposed method, JMA-NHM, and the flight data along the trajectory at 40 min after the take-off

to 40 min after the take-off, which is 30 min before encountering the turbulence. On the whole, the proposed method predicted a stronger wind field over the Pacific ocean near Japan than the one predicted by JMA-NHM. This area with a strong wind filed was affected by a subtropical jet stream. The vertical cross-sections show that the jet stream is located to altitude of 12 km, and the proposed method updated the magnitude of the jet stream by using the flight data. Figure 42.7 shows the time series data of the wind magnitude given by the proposed method, JMA-NHM, and the flight data along the trajectory at 40 min after the take-off. JMA-NHM predicted weaker wind than the flight data, and the difference between the wind magnitutde predicted by the JMA-NHM and the flight data is about 10 ms−1 at 55 min after the take-off, which is the time of maximum wind magnitude according to the flight data. On the other hand, the proposed method was able to correct this difference, and predict the maximum wind magnitude accurately. Therefore, it can be said that the wind magnitude predicted by the proposed method matched that of the jet stream.

42.4 Conclusions A method capable of predicting the wind field and atmospheric turbulence in realtime was proposed. The proposed method that integrates a low dimensional model based on proper orthogonal decomposition and flight data was investigated through actual experiments. The actual experiments consisted of two cases: under the condition of an extratropical cyclone approaching Japan (Case 1) and a stationary front at the Pacific ocean

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near Japan (Case 2). In Case 1, the proposed method was able to produce matches between the areas predicted for turbulence and the locations where turbulence was actually encountered; however, JMA-NHM was not able to predict the locations of turbulence accurately. This was because JMA-NHM suffered from spatiotemporal uncertainties in terms of the predicted areas of turbulence. The proposed method is able to correct these spatiotemporal uncertainties by using the flight data. In Case 2, JMA-NHM predicted weaker wind than the flight data, and the difference between the wind rates of the JMA-NHM and the flight data was about 10 ms−1 at 55 min after the take-off, which is the time of maximum wind magnitude by the flight data. The proposed method was able to correct this difference, and predict the maximum wind magnitude accurately. In terms of turbulence prediction, while JMA-NHM did not predict turbulence around 70 min after the take-off, which was the time when turbulence was actually encountered, the proposed method predicted the same. The low dimensional model is based on the proper orthogonal decomposition by using NWP results that are updated before flight. The proposed method leverages NWP results effectively for real-time prediction such as nowcasting. It is possible to compensate for the problem of NWP update interval.

References 1. Federal Aviation Administration, Preventing Injuries Caused by Turbulence (2006). https:// www.faa.gov/regulationspolicies/advisorycirculars/index.cfm/go/document.information/ documentID/99831/ 2. Japan Transport Safety Board, 2003–2012: Aircraft Accident Reports. http://www.mlit.go.jp/ jtsb/.] 3. Sharman, R., Tebaldi, C., Wiener, G., Wolff, J.: An integrated approach to mid-and upper level turbulence forecasting. Weather Forecasting 21, 268–287 (2006) 4. Kim, J.H., Chun, H.Y., Sharman, R.D., Keller, T.L.: Evaluations of upper-level turbulence diagnostics performance using the graphical turbulence guidance (GTG) system and pilot reports (PIREPs) over East Asia. J. Appl. Meteor. Climatol. 50, 1936–1951 (2011) 5. Steiner, M., Bateman, R., Megenhardt, D., Liu, Y., Pocernic, M., Krozel, J.: Translation of ensemble weather forecasts into probabilistic air traffic capacity impact. Air Traffic Control Quart. 18, 229–254 (2010) 6. Lorenz, E.: Section of planetary sciences: The predictability of hydrodynamic flow. Trans. NY Acad. Sci. 25, 409–432 (1963) 7. Kalnay, E., 2003: Atmospheric modeling, data assimilation and predictability. Cambridge, 638 pp 8. Saito, K., Ishida, J., Aranami, K., Hara, T., Segawa, T., Narita, M., Honda, Y.: Nonhydro static atmospheric models and operational development at JMA. J. Meteor. Soc. Japan 85B, 271–304 (2007) 9. Gebhardt, C., Theis, S., Paulat, M., Ben Bouallegue, Z.: Uncertainties in COSMO-DE precipitation forecasts introduced by model perturbations and variation of lateral boundaries. Atmos. Res. 100, 168–177 (2011) 10. Sirovich, L.: Turbulence and the dynamics of coherent structures. I-Coherent Struct. Q. Appl. Math. 45, 561–571 (1987) 11. Mohammed, A., Minhas, R., Jonathan Wu, Q., Sid-Ahmed, M.: Human face recognition based on multidimensional PCA and extreme learning machine. Pattern Recogn. 44, 2588–2597 (2011)

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12. Murray, N.: An application of gappy POD. Exp. Fluids 42, 79–91 (2007) 13. Kikuchi, R., Misaka, T., Obayashi, S.: Assessment of probability density function based on POD reduced-order model for ensemble-based data assimilation. Fluid Dyn. Res. 47(051), 403 (2015) 14. Wu, B., Zhou, T., Li, T.: Seasonally evolving dominant interannual variability modes of East Asian climate. J. Climate 22, 2992–3005 (2007) 15. Kikuchi, R., T. Misaka, Obayashi, S.: Real-time flow prediction of low-level atmospheric turbulence. In: 33rd Wind Energy Symposium, AIAA SciTech, AIAA, pp. 2015–1469 (2015) 16. Kennedy, J., Eberhart, R.: Particle swarm optimization. In: The 1995 IEEE International Conference on Neural Networks. IEEE, pp. 1942–1948 (1995) 17. Rini, D., Shamsuddin, S., Yuhaniz, S.: Particle swarm optimization: technique, system and challenges. Int. J. Comput. Appl. 14, 19–26 (2011)

Part XIII

Structural Damage Identification, Maintenance and Life-Cycle Integrity Management

Chapter 43

Stress-Strain-Based Approach to Catastrophic Failure of Steel Structures at Low Temperatures V. M. Kornev

Abstract The paper concerns propagation of cracks in structures. The coupled fracture criterion is used when plotting diagrams of quasi-brittle failure of structures made of elastic-plastic material having ultimate strain. Diagrams of quasi-brittle fracture for commonly occurring simplest components of structures just as homogeneous, so with welded joins are given. Analysis of parameters entering into the proposed model has been performed. Diagrams of quasi-brittle fracture were used in analyzing catastrophic failures of steel structure components operating at temperatures lower the cold-brittleness threshold. Parameters of the model are selected from two laboratory experiments (critical stress intensity factor and classical stress-strain diagram) performed at appropriate temperatures. It has been established that weld structures with cracks in the vicinity of a welded joint possess decreased crack toughness. The break-down effect of construction under monotonic loading is clearly expressed: critical loadings significantly decrease with increasing crack lengths. The attention is given to the parameter characterizing plastic material deformation and exhaustion of plasticity resource under preliminary plastic material deformation. After the plasticity resource is exhausted, the temperature of brittleness threshold approaches a room temperature. Keywords Quasi-brittle failure · Elastic-plastic materials · Parameters of cold-brittleness · Embrittlement · Ultimate strain

43.1 Introduction Accidents and failures of steel structures at low temperatures are well known. Some of accidents have been described in publications [2, 5, 25, 29, 30, 33]. Catastrophic failures of bridges, large tanks, gas pipes, and ships are mentioned among the accidents, the steel structures being, as a rule, fabricated with using welding. V. M. Kornev (B) Lavrentyev Institute of Hydrodynamics, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_43

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The analysis of catastrophic failures of the constructions mentioned above is given in the second chapter of monograph [33]. The typical centers of stress concentrations and incipient cracks induced by those are shown schematically in Fig. 43.6 borrowed from [Uzhik 33]. These cracks may provoke complete break-down of a structure at reducing temperature. Crack initiation (focuses of failures) may be induced: “…(1) welded joints or sites adjacent to welding points; (2) different areas of stress concentrations caused by the shape of structure components; (3) various local damages (scratches, hollows, etc.) arising during manufacturing some components of structures or due to false machining” [33, p. 20]. Catastrophic failures described in works [2, 5, 25, 29, 30, 33] have specific signs of brittle failure even when temperatures under service conditions of structures were essentially higher than the cold-brittleness threshold of original steel. Using the linear elastic fracture mechanicals (LEFM) does not explain catastrophic brittle failures of steel structures at low temperatures [3]. When catastrophic failures are studied, it is reasonable to pay attention to loading: it is, as a rule, stress-control one. Under action of loadings, the break-down of structure (up to the final failure) occurs due to finite sizes of structure components with a crack. Except for indicated foci of break-down having the mechanical nature of origin, degradation of properties of structural steels attributed to structural transformations during sustained operation should be mentioned. Studies on metal degradation of pips from gas transmittal pipeline are described in [Syromyatnikova 31]. Panin et al. [24] discuss the scientific frameworks of cold-brittleness of structural steels and the methods allowing the influence of ductile-to-brittle transition on temperature at failure. Here the references describing also degradation of properties of structural steels are given. It is apparent from the short overview of the results obtained on catastrophic failure that any methods that allow for influence of cold-brittleness on mechanical parameters in the right direction are very valuable. Panin et al. [23] assessed the effect of the material structure degradation at low temperatures on the decrease in material lifetime during fatigue tests. The effect of linear and nonlinear deformation of materials is discussed in work [6]. Attention should be given to the studies on highly entropy alloys CrMnFeCoNi at low temperatures. The review [18] and the paper [32] typically represent this line of inquiry, which includes the full reference list. The point of the conflict between strength and fracture toughness, including that for steels, is discussed in detail by Ritchie [26]. In this work, paths of crack toughness increase have been outlined.

43.2 Diagrams of Quasi-brittle Fracture of Compact Specimens Judging from catastrophic break-downs, all structures have, in one form or another, nucleus defects such as cracks. Crack lengths didn’t reach critical lengths at room

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temperatures, therefore structures continued to be used. Mechanical characteristics of the material (steel) change with decreasing the temperature. The non-classic scheme of material fracture is applied in the model proposed in Kornev [7, 11]. Suppose that when a macro-specimen was tested in laboratory experiment, a stress–strain diagram of quasi-brittle material was obtained for various temperatures T, i.e., σ = σ (T ), ε = ε(T ). Take the simplest approximation of a real σ − ε diagram of a tested material when this diagram is approximated by two-link polygonal line. During approximation, the original material is changed by elastic ideally-plastic material, which has an ultimate strain. Parameters of this approximation E, σY , ε0 , ε1 , μ are as follows: E = const is modulus of elasticity, σY (T ) is yield tensile strength and constant stresses acting in modified fashion of the Leonov-Panasyuk-Dugdale model [17, 4], ε0 (T ) is the maximum elastic material strain (σY = Eε0 ), ε1 (T ) is the maximum material strain, μ = const is the Poisson coefficient. The σ − ε approximation of the diagram within the range ε0 < ε < ε1 may be interpreted as a perfect plasticity. It has been accepted in the study proposed below that the modulus of elasticity and Poisson coefficient are independent of temperature, that approximately corresponds to behavior of steels at low temperatures. Let r be the grain diameter, more exactly, effective diameter of fracture structure for material with a regular structure [7, 11]. The Neuber-Novozhilov approach permits solutions having a singular component with integrable singularity to be used for structural media [21, 22]. An opening mode crack is under consideration. Let this plane opening mode crack be propagating rectilinearly. Apart from a length l0 of a real crack-cut, introduce into consideration a length of a model crack-cut. The lengths of model cracks are l = l0 + , the pre-fracture zone with the length  being located on the continuation of a real crack. The fracture problem has two linear scales: if the grain diameter r is defined by a material structure, then the second linear scale is developed by the system itself. It is the pre-fracture zone length that is the second linear scale , which varies in accordance with the fact how the following parameters are changed (i) a real crack length, and (ii) load intensity. It should be emphasized that the critical pre-fracture zone length ∗ is a well-defined parameter (l ∗ = l0 + ∗ is the critical macro-crack length) under single loading. Short, long, and medium length macro-cracks are considered. Sufficient fracture criteria [7, 11] are used, when plotting diagrams of quasi-brittle fracture. The sufficient (coupled [16, 34]) criterion may be represented in the form of two relations for short macro-cracks. 1 r

r σ y (x, 0)d x = σY ,

(43.1)

0

2ν(−∗ , 0) = δ ∗ .

(43.2)

Here σ y (x, 0) are normal stresses on the crack continuation; O x y is the rectangular coordinate system, the coordinate origin being coincident with a model crack tip in the modified Leonov-Panasyuk-Dugdale model, where the x-axis is directed along

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the crack plane and the y-axis is directed along the normal to the crack plan; 2ν = 2ν(x, 0) is the model macro-crack opening (x < 0); δ ∗ is the critical opening of this crack (CTOD); ∗ is the critical length of a pre-fracture zone (critical values derived via sufficient and necessary fracture criteria are labeled by superscripts ∗ and i 0, respectively). Attention is given to the fact that the proposed criterion (43.1) and (43.2) is two-parametric criterion. A field of normal stresses σ y (x, 0) on the continuation of model cracks x > 0 can be represented as a sum of two summands σ y (x, 0) ∼ = K I /(2π x)1/2 + σnom , K I = K I∞ + K I > 0, K I∞ > 0,

(43.3)

where σnom are nominal stresses, in other words, estimates of regular members of solutions in the vicinity of model crack tips, and these members have no singularities; K I∞ is the stress intensity factor (SIF) generated by specified test conditions; K I is SIF generated by constant stresses −σY acting in a pre-fracture zone. The first and second summands in relation (43.3) are singular and regular parts of the solution, respectively. The total SIF K I at the model crack tip is positive inasmuch as the low-scale yielding is considered. The specimen with a crack is shown in Fig. 43.1 (P are forces applied during tests, l is the model crack length, w and t are the width and thickness of the specimen, respectively). The proposed approximation of normal stresses at tip of model crack for compact specimens with a sharp crack has following form −1  σnom = σ1nom + σ2nom , σ1nom = P[t(w − l)]−1 , σ2nom = 3P(w + l) t(w − l)2 . (43.4) Fig. 43.1 The compact specimen

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Here σ1nom and σ2nom are nominal stresses under tension and bending, respectively. The singularity of nominal stresses σnom for l → w reflects the break-down process of a structure at catastrophic failure of steel structures under conditions at low temperatures. Under conditions of low-scale yielding for compact specimens with sharp crakes, we have for the total SIF [3, 20]. K I = K I∞ (P/tw, l, α) + K I (l, , σY ) > 0,

(43.5)

 √ K I∞ = (P/tw)Y (α) πl, K I ≈ −2σY 2/π ,   2+α 0.886 + 4.64α − 13.32α 2 + 14.72α 3 − 5.6α 4 , 3/2 √ πα (1 − α) 0.2 ≤ α = l/w ≤ 1. Y (α) =

Opening of a model crack 2v in a compact specimen with a sharp crack is represented in the form  η+1 −x 3−μ 2v(−x, 0) ≈ K I (l, , α) , K I > 0, ηd = 3 − 4μ, ηs = . G 2π 1+μ (43.6) Coefficients ηd and ηs are given for the plane strain and plane stress states, G = E/2(1 + μ) is shear modulus and μ is the Poisson ratio. The critical opening of model cracks δ ∗ in relation (43.2) is calculated as [8, 9],     δ ∗ = (ε1 − ε0 )a, a = [K I∞0 ]2 / 2π (σY )2 3/2 + (1 − 2μ)2 , aw = a/2. (43.7) Identify the pre-fracture zone width in relation (43.7) with the width of plasticity zones at the real crack tip for compact specimens with sharp cracks made of a homogeneous material a and material with weld joint aw (the joint plane is located in the tip plane [13]). A pre-fracture zone can only be located in the weakest material. Furthermore, such cases are considered when the yield strength of original material is less than that of a welding joint material. The plasticity zone width aw in the material with weld joint is half as large than the plasticity zone width in the homogeneous material, i.e., aw = a/2. We take into account relations (43.3)–(43.7) when small-scale yielding ∗ terms with the l0 is implemented. Under transformation of a system of equations, √ ∗ /l ∗ 1 multiplayer are omitted, while terms with the ∗ /l ∗ multiplayer remain. If the dimensionless representation is applied, the approximate solution of this system for a homogeneous specimen has the following form

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     2     −1 1 − α ∗ + 3 1 + α ∗ / 1 − α ∗ + 1 − Mε p Y α ∗ 2l¯∗ .

(43.8)

   2 ∗ ¯ ∗ ≈ π 2 /8 M¯ε p σ¯ ∞  Y α ∗ l¯∗ ,

(43.9)

M¯ε p < 1, ∗ σ¯ ∞ =

∗ σ∞ l∗ ∗ ∗ ∗ l ∗ ε1 − ε0 3 + 2(1 − 2μ)2 ¯ =  , ε¯ p =  , l¯∗ = ,  . ,α = ,M = 2 σY r r w ε0 8π 1 − μ (43.10)

The last inequality (43.10) is restriction at which quasi-brittle fracture occurs under low-scale yielding conditions of homogeneous material in a pre-fracture zone. For a specimen with welding joint, in the case, when the crack is located along bi-material boundary, relations (43.8)–(43.10) may be used. Let yield strengths of original material and material with welding joint are different, then the parameter will be aw = a/2. Therefore, critical loads δ ∗ of a specimen with welding seam is ∗ ∗ ∗ of a homogeneous specimen, i.e., σ¯ ∞w < σ¯ ∞ . essentially less than critical loads σ¯ ∞ 0 0 Under conditions of critical stresses σ¯ ∞ = σ∞ /σY corresponding to the necessary fracture criterion, the following relation can be written

0 σ¯ ∞



≈ (1 − α0 ) + 3(1 + α0 )/(1 − α0 ) + Y (α0 ) 2l¯0 2

−1

, l¯0 = l0 /r, α0 = l0 /w. (43.11)

0 for homogeneous structures and structures with welded joint Critical stresses σ¯ ∞ are coincident. If in two laboratory experiments, the critical SIF K Ic and classical σ −ε diagram (more precise, its approximation), then, by three parameters r, σY , ε¯ p , with allowance made for the Poisson ratio μ, it is possible to construct two critical curves ∗ 0 , σ¯ ∞ in the plane “crack length – stress” within the wide range of variation of crack σ¯ ∞ lengths. For effective diameter of fracture structure for brittle r0 and quasi-brittle r materials, such representations are valid if critical SIFs K Ic and approximation of σ − ε diagram are known, and restrictions (43.10) are met

2 2  2   r0 = (2/π ) K Ic /σY , r = (2/π ) K Ic /σY 1 − M¯ε p .   0 ¯ l, 0, α Given in Fig. 43.2a, b are conditional, dimensionless, critical stresses σ¯ ∞   ∗ ¯ l, ε¯ p , α (curves 2, 3, 4, and 5) for compact specimens with sharp (curves 1), σ¯ ∞ cracks; solid and dotted lines reflect behavior of specimens made from homogeneous material and from that with a welding joint, respectively. It is assumed that yield strength of material with welding joint exceeds the yield strength of original material. Under concrete implementation of calculations, the parameters w/r = 500 for curves in Fig. 43.2a and those w/r = 2500 for curves in Fig. 43.2b (α = l/w = (l/r )/(w/r )) are chosen. Straight lines 6 in Fig. 43.2a, b in log-log

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0 ,σ ∗ ,σ ∗ , and σ LMF Fig. 43.2 Critical stresses σ¯ ∞ ¯∞ ¯ ∞w ¯∞

  LFM ¯ l, ε¯ p constructed in the context of linear coordinates are critical stresses σ¯ ∞ elastic fracture mechanics (LEFM) for an edge crack in a plate of infinite width (in calculation, it is assumed that μ = 0 and ε¯ p = 3),      −1 LFM ¯ σ¯ ∞ l, ε¯ p = 1.12 1 − 5¯ε p /8π l¯∗ . Pairs of curves 1–4 and 1–5 in Fig. 43.2 are diagrams of quasi-brittle fracture for the type of a specimen being considered when the specimen is made from homogeneous material with the plasticity factors ε¯ p = 3 and ε¯ p = 4.5, respectively (μ = 0). Pairs of curves 1–2 and 1–3 present the essence of quasi-brittle fracture diagrams for the type of a specimen under consideration when the specimen is made from material with welding joint (the welding seam being located in a crack plane and original material possess plasticity factors ε¯ p = 3 and ε¯ p = 4.5, respectively). The plotted curves corresponding to relations (43.8) and (43.11) constructed on the plane

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Fig. 43.3 Ratios of critical stresses L M F /σ ∗ ,σ L M F /σ ∗ , σ¯ ∞ ¯∞ ¯∞ ¯ ∞w L M F /σ 0 ¯∞ and σ¯ ∞

  ¯ σ¯ ∞ depend on the α ratio. Let the load intensity σ¯ ∞ be given. Then the diagram 2l, of quasi-brittle fracture permits one to estimate the system state. Two curves divide   0 ¯ σ¯ ∞ into three subareas: (i) the area σ¯ ∞ < σ¯ ∞ , where there is no the plane 2l, 0 ∗ failure, (ii) the area σ¯ ∞ < σ¯ ∞ < σ¯ ∞ , where damage accumulation in a pre-fracture ∗ , where a specimen fails under zone material takes place, and (iii) the area σ¯ ∞ > σ¯ ∞ monotonic loading. Figure 43.3 demonstrates the curves characterizing the ratio of critical stresses in LMF ∗ to critical stresses σ¯ ∞ for homogeneous structures (curve 1), terms of LEFM σ¯ ∞ ∗ to critical stresses σ¯ ∞w for structures with welding joints (curve 2), and to the lower 0 (curve 3). The calculations have been performed for parameters stress threshold σ¯ ∞ w/r = 2500, ε¯ p = 3, and μ = 0. The analysis of relations (43.8), (43.11), and curves in Figs. 43.2 and 43.3 shows that LEFM poorly describes the states of homogeneous structures, which correspond to CT specimens with extensive cracks. Of particular concern are welded structures. Curve 3 characterizes damage accumulation in the pre-fracture zone material. The classical LEFM does not allow one to obtain a lower threshold above which initiation of plastic deformations occurs near a crack tip.

43.3 Diagrams of Quasi-brittle Fracture of Most Often Used Specimens Such as Counterparts of Structures One of typical loading schemes of structures has been given in the previous section in which construction of quasi-brittle fracture diagrams of compact tension (CT) specimen is discussed in detail. Except the loading scheme suitable for CT specimen, other loading schemes should be also considered. Given below are schemes of loading application to specimens, expressions for critical stresses, representations of fracture diagrams for homogeneous structures, and those with welded joints (some designations coincide with those in the previous section). All cracks are located in

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the planes of a welded joint for welded structures. For homogeneous structures in Figs. 43.5, 43.6, 43.7, 43.8, 43.10 and 43.11 solid curves are used, whereas dashed curves are used for structures with welded joints.

43.3.1 Center-Cracked Panel (CCP) The scheme of load application to tensile specimen of the finite width with an internal crack is shown in Fig. 43.4. ∗ 0 , σ¯ ∞ of a tensile specimen of the width w with internal crack Critical stresses σ¯ ∞ of length 2l has the form [7, 11]. ∗ = σ¯ ∞

 −1     −1 1 − λ∗ + 1 − M¯ε p Y λ∗ 2l¯∗ ,

0 σ¯ ∞

 −1 −1 = (1 − λ0 ) + Y (λ0 ) 2l¯0 ,

(43.12) (43.13)

 √ K I∞ = σ∞ Y λ∗ πl ∗ , λ∗ = 2l ∗ /w, λ0 = 2l0 /w, Y (λ) = cos−1/2 λπ/2. Here w is the specimen width, expression for Y (λ) is borrowed from handbook [20]. In the limit as ε¯ p → 0 (ε1 → ε0 ), the expression corresponding to the necessary 0 0 is obtained from the relation (43.12). Critical stresses σ¯ ∞ stand fracture criterion σ¯ ∞ for the lower threshold stress above which damage accumulation occurs in the pre0 for homogeneous structures and structures fracture zone material. Critical stresses σ¯ ∞ with welded joint are coincident. Fig. 43.4 Scheme of load application to tensile specimen of finite width with internal crack

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The plots, i.e., diagram of quasi-brittle fracture for the specimen types under  ¯ σ¯ ∞ depend on the λ ratio. Let the consideration, the plotted curves on the plane 2l, fracture permits one to load intensity σ¯ ∞ be given. Then the diagram of quasi-brittle   ¯ σ¯ ∞ into three subareas: estimate the system state. Two curves divide the plane 2l, 0 0 ∗ , where there is no failure, (ii) the area σ¯ ∞ < σ¯ ∞ < σ¯ ∞ , where (i) the area σ¯ ∞ < σ¯ ∞ damage accumulation in a pre-fracture zone material takes place, and (iii) the area ∗ , where a specimen fails under monotonic loading. σ¯ ∞ > σ¯ ∞ ∗ of welded structures are significantly less than critical ones Critical loads σ¯ ∞w ∗ ∗ ∗ < σ¯ ∞ ). σ¯ ∞ of homogeneous structures (σ¯ ∞w 0 (curves 1) and critical Figure 43.5 demonstrates the lower threshold of stresses σ¯ ∞ ∗ ∗ stresses for homogeneous σ¯ ∞ (curves 3) and welded σ¯ ∞w (curves 2) structures. LFM (curve 4) are constructed in terms of the classical LEFM for Critical stresses σ¯ ∞ plates of the infinite width with internal cracks. Curves 1–3 are constructed applying the proposed model (43.12), (43.13) for plates of the finite width w/r = 100, plastic properties of the quasi-brittle material are estimated by the parameter ε¯ p = 3 with 0 , above which permanent μ = 0. Curve 1 is the lower threshold of stresses σ¯ ∞ residual deformations begin to occur in pre-fracture zone material in the vicinity of a real crack. LFM for critical stresses in terms of LEFM. Then, if the Introduce the notation σ¯ ∞ results given in Fig. 43.5 are used, curves 1–3 in Fig. 43.6 can be obtained. Curves LFM ∗ LFM ∗ /σ¯ ∞ and welded σ¯ ∞ /σ¯ ∞w 1 and 2 conform to the ratios for homogeneous σ¯ ∞ LFM 0 structures, whereas curve 3 is the σ¯ ∞ /σ∞ ratio. The analysis of curves 1 and 2 shows that the classical LEFM significantly underestimates critical stresses of the quasi-brittle fracture in the case when ultimate material strains are not taken into account. Curve 3 characterizes damage accumulation in the pre-fracture zone material. The classical LEFM does not allow obtaining a lower threshold above which initiation of plastic deformations occurs near a crack tip. Ratios of critical stresses obtained in terms of the LEFM to those obtained via the proposed model are within

∗ ,σ ∗ ,σ 0 , and σ LFM Fig. 43.5 Critical stresses σ¯ ∞ ¯ ∞w ¯∞ ¯∞

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L F M /σ ∗ ,σ L F M /σ ∗ ,σ L F M /σ 0 Fig. 43.6 Ratios of critical stresses σ¯ ∞ ¯∞ ¯∞ ¯ ∞w ¯∞ ∞

the range 1.8 ÷ 3.0, whereas ratios of critical stresses obtained in terms of the LEFM to the lower threshold of stresses obtained with respect to the proposed model are essentially larger.

43.3.2 Panel with Two Edge Cracks Tensile specimen of finite width with two edge cracks is described in [12]. ∗ ∗ 0 , σ¯ ∞w , σ¯ ∞ of a tensile band of the width w with two edge Critical stresses σ¯ ∞ cracks of the length l¯ can be written as relations (43.12), (43.13), if Y (λ) = 1, 122 − 0154λ+0, 807λ2 −1, 894λ3 +2, 494λ4 . Here the expression for Y (λ) was borrowed 0 from handbook by [20]. Figure 43.7 represents the lower threshold of stresses σ¯ ∞ ∗ ∗ (curves 1) and critical stresses for homogeneous σ¯ ∞ (curves 3) and welded σ¯ ∞w Fig. 43.7 Critical stresses ∗ ,σ ∗ ,σ 0 , and σ LFM σ¯ ∞ ¯ ∞w ¯∞ ¯∞

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L F M /σ ∗ ,σ L F M /σ ∗ , and σ L F M /σ 0 Fig. 43.8 Ratios of critical stresses σ¯ ∞ ¯∞ ¯∞ ¯ ∞w ¯∞ ∞

(curves 2) structures. Curve 4 is drawn in terms of the classical LEFM for plates of the infinite width with edge cracks. Curves 1–3 are drawn with using the proposed fracture model for plates of the finite width w/r = 100 with two edge cracks. Plastic properties of a quasi-brittle plate material are estimated by the parameter ε¯ p = 3 with μ = 0. ∗ If using the results demonstrated in Fig. 43.7, it is possible to obtain σ L∗ F M /σ∞ ∗ 0 and σ L F M /σ∞ ratios for a band with two edge cracks. These ratios are given as plots LFM ∗ /σ¯ ∞ in Fig. 43.8: curves 1 and 2 correspond to the ratios for homogeneous σ¯ ∞ LFM ∗ LFM 0 and welded σ¯ ∞ /σ¯ ∞w structures, and curve 3 is the σ¯ ∞ /σ∞ ratio. The curves in Figs. 43.4 and 43.7 are scarcely different.

43.3.3 Three Point Bending (TPB) Specimen In Fig. 43.9, the scheme corresponding to loading of bending specimens having edge cracks is shown. Under symmetrical three-point bending, put the case that the bending moment M is in the form M = P L/4 for L = 4b, where P are applied loads, L is the span length of a bending beam, and b is the height of a bending specimen. Fig. 43.9 Scheme of loading application for bending specimen

43 Stress-Strain-Based Approach to Catastrophic Failure …

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Dimensionless critical moments M¯ ∗ and M¯ 0 of specimens with edge crack l have the form (t is the beam thickness) [10]. 

−1   ∗  2 1 − M¯ε p Y λ λ∗ b¯ , + π (43.14)  2     −1 0   1 1 2 6M 0 1− 1 − M¯ε p Y (λ0 ) λ0 b¯ M¯ = 2 + = , ¯ − λ0 ) tb σY 1 − λ0 π b(1 (43.15)   1, 99 − λ(1 − λ) 2, 15 − 3, 93λ + 2, 7λ2 , λ = l/b, b¯ = b/r. Y (λ) = (1 + 2λ)(1 − λ)3/2

6M ∗ = M¯ ∗ = 2 tb σY

1 1 − λ∗

2  1−

1 ¯b(1 − λ∗ )





The expression for the coefficient Y (λ) is borrowed from the handbook [28]. Plots in Fig. 43.10 have been constructed using relations (43.14), (43.15), where: curve 1 is the lower threshold of critical moments M¯ 0 , curves 2 and 3 are critical moments for welded M¯ w∗ and homogeneous M¯ ∗ structures, respectively, and curve 4 for homogeneous M¯ L F M structures is plotted in terms of the classical LEFM for a semi-infinite homogeneous plate with the edge crack. Curves 1–3 are plotted using the proposed fracture model for a beam of the finite height b/r = 100 with edge cracks, plastic properties of a quasi-brittle material are estimated by the parameter ε¯ p = 3 with μ = 0. Curve 1 is the lower threshold of critical moments above which plastic deformations of pre-fracture zone material occur. Using the results given in Fig. 43.10, curves 1–3 shown in Fig. 43.11 are plotted. Curves 1 and 2 correspond to relations for homogeneous M¯ L F M / M¯ ∗ and welded M¯ L F M / M¯ w∗ structures, and curve 3 is the M¯ L F M / M¯ 0 ratio. The analysis of curves 1 and 2 in Fig. 43.9 shows that the classical LEFM significantly underestimates critical stresses of quasi-brittle fracture, when ultimate material strains are not taken into consideration. Curve 3 characterizes damage accumulation in the pre-fracture zone material. The classical LEFM does not allow obtaining a lower threshold above Fig. 43.10 Critical moments M¯ ∗ , M¯ w∗ , M¯ 0 and M¯ L F M

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Fig. 43.11 Ratios of critical moments M¯ L F M / M¯ ∗ , M¯ L F M / M¯ w∗ , and M¯ L F M / M¯ 0

which initiation of plastic deformations occurs near a crack tip. The ratios of critical stresses obtained in terms of the LEFM and the proposed model are within the range 1, 5 ÷ 2, 5, whereas the ratios of critical stress obtained in terms of the LEFM and the lower threshold of stresses obtained using the proposed model are significantly larger. The break-down of construction is most pronounced in a beam under bending. Minimum ratio magnitudes in response to error terms of critical stresses and critical moments of specimens are listed in Table 43.1. These magnitudes depend on the specimen loading schemes under examination. These errors are large enough for compact specimens, so extended cracks are considered for this type of specimens. The calculations have been performed for parameters w/r = 2500 (compact specimen), w/r = 100 (tensile specimen with internal crack), b/r = 100 (three-point bending specimen), ε¯ p = 3, μ = 0. Thus, based on the analysis results given in Figs. 43.3, 43.5, 43.7, 43.10 and Table 43.1, it can be argued that: (i) the classical LEFM essentially underestimates critical states of structures under quasi-brittle fracture if both the fracture toughness and strength characteristics of material are taken into account; (ii) nothing has been said in the classical LEFM as to damage accumulation of material in the vicinity of Table 43.1 Ratios of critical stresses and moments of specimens Compact specimen

Tensile specimen of finite width with internal crack

Three-point bending specimen

L F M /σ ∗ minσ¯ ∞ ¯∞

7.3

L F M /σ ∗ minσ¯ ∞ ¯ ∞w

12.3

L F M /σ 0 minσ¯ ∞ ∞

17.3

L F M /σ ∗ minσ¯ ∞ ¯∞ L F M ∗ minσ¯ ∞ /σ¯ ∞w L F M /σ 0 minσ¯ ∞ ∞ ¯ min M L F M / M¯ ∗

min M¯ L F M / M¯ w∗ min M¯ L F M / M¯ 0

1.8 2.65 3.3 1.4 1.7 2.1

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a crack tip; (iii) critical loads of welded structures are essentially lesser than those of homogeneous structures; (iv) critical stresses and critical moments vanish when a crack length tends to the specimen width, i.e., break-down of the structure occurs under loading (break-down of structure is especially strongly pronounced under beam bending).

43.4 Application of Quasi-brittle Fracture Diagrams When Analyzing Catastrophic Failures of Simplest Structures ∗ Critical stresses σ¯ ∞ in relations (43.8), (43.12) and critical moments M¯ ∗ in relation (43.14) depend on three governing parameters r, σY , and ε¯ p of a quasi-brittle material and one accessory parameter μ. The third parameter from the governing ones has been derived on the basis of two initial parameters ε1 and ε0 . The parameters r = r (T ), σY = σY (T ), ε1 = ε1 (T ), ε0 = ε0 (T ) depend on temperature T at which the structure has been in operation. These parameters can be obtained using the results of laboratory tests on determination of critical SIF K Ic (T ) and the classical σ (T )−ε(T ) diagram (more exactly, its approximation). Effective diameters r = r (T ) of fracture structures of quasi-brittle materials are calculated via relation

2  2  r (T ) = (2/π ) K Ic (T )/σY (T ) 1 − M¯ε p (T ) ,

(43.16)

having regard to three parameters σY , ε1 , ε0 . Refer to some experimental results on behavior of steels at low temperatures [1, 15, 19, 27]. It can be considered that Young’s modulus E is independent of temperature. Some increase of material yield stress σY = σY (T ) takes place when the temperature is reduced and, therefore, the parameter ε0 = ε0 (T ) increases as temperature is reduced. The parameter ε1 = ε1 (T ) decreases due to embrittlement at the reduced temperature. Consequently, the dimensionless parameter ε¯ p (T ) = [ε1 (T ) − ε0 (T )]/ε0 (T ) characterizing plastic material properties significantly decreases with reducing temperature. If all the parameters r, σY , ε¯ p , and μ have been obtained, then diagrams of quasibrittle fracture of simplest structural components can be constructed using various relations (43.8), (43.12), (43.14). The schemes of load application for these diagrams are given in Figs. 43.1, 43.3 and 43.9. But in order to take advantage of these schemes when estimating a possibility of fracture structural components, it is necessary to have reliable lengths l0 of initial cracks taking into consideration the operating life of structural components at temperatures T0 above the cold-brittleness threshold. Attention should be given to the fact that the parameter ε¯ p plays the crucial role in generation of the system response to the applied loading. The parameter ε¯ p characterizes plastic properties not only of the original material, but the material after preliminary plastic deformation in the pre-fracture zone under low-cycle loading [7,

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11]. The effect of preliminary plastic deformation of low-carbon steels on fracture load is mentioned in [Moskvichev et al. 19]: in some cases, the cold-brittleness threshold T0 shifted to the region of room temperatures. When manufacturing cold-formed profiles, a strain-hardening range occurs [19]. The margin of material plasticity is already partly exhausted at points of profile bending. Using such profiles provokes crack initiation when operating under conditions of low temperatures. Sometimes, due to strain hardening, the-brittleness threshold occurs at temperatures close to room ones (about ≈ 20 ◦ C). The situation becomes worse if bending places are adjacent to a welded joint. When calculating, it is desirable to select the parameter ε¯ p , for which the minε1 is used with account for plastic deformation of low-carbon steels. The values of effective diameters r of fracture structures for quasi-brittle materials (steels at low temperatures) are listed in Table 43.2. The limited set of experimental results is available in handbook [14], inasmuch as data on both the critical SIF K Ic and σ − ε diagram at appropriate temperatures are needed for processing. The proposed model of quasi-brittle fracture may be used only if limitation (43.10) is satisfied. For calculations, it is conventional to take E = 200000M Pa and μ = 0.3. In the Table there are no rows for steel 50X at the temperatures 293 K and 223 K, as well for steel 50XN at the temperature 203 K because limitation (43.10) is violated. In the next-to-last column of the Table 43.2, there are given effective diameters r of steel fracture structures. These diameters were obtained via the relation (43.16) for μ = 0 (the plane stress state) and μ = 0.3 (the plane strain state). The proposed model (43.8), (43.9) of quasi-brittle material fracture includes one ¯ ∗ , characterizing the critical pre-fracture zone length. This more critical parameter  ∗ ¯ critical parameter  completes diagrams of the quasi-brittle fracture. If the component of a structure contains a crack of the length l¯0 , then, in order to obtain a conservative estimate of critical loading, it is necessary to add the pre-fracture zone ¯ ∗ to this crack length inasmuch, the margin of plasticity of an original matelength  rial may be exhausted almost completely in the pre-fracture zone. Properly speaking, as state below, the last loading circle is considered, when operation conditions of the structure are changed due to abrupt change of the temperature regime. ¯ ∗ /l¯∗ for a specimen with Let us derive crude estimates of the critical parameter  the crack using relation (43.9). Given in Fig. 43.12 are curves 1–6 for specimens made of homogeneous material: curves 1, 2, 3 and 4, 5, 6 have been constructed for ε¯ p = 3, ε¯ p = 4, ε¯ p = 4, 5, respectively; solid lines (curves 1–3) and dash-and-dot lines (curves 4–6) correspond to specimens made of structured materials w¯ = 100 Table 43.2 Steels Material steel 50X

Temperature (K)

√ KIc M Pa m

σY M Pa

ε0 (%)

ε1 (%)

ε¯ p

r (mm)

[14]

183

27

1920

0.96

4.3

3.48

0.012–0.024

pp. 94–95

77

25

2030

1.02

3.2

2.14

0.032–0.041

pp. 94–95

50XN

77

22

2020

1.01

4.2

3.16

0.01–0.022

pp. 96–97

30XGSN2A

77

16.4

840

0.42

1.6

2.81

0.047–0.071

pp. 100–101

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Fig. 43.12 Relative pre-fracture zone lengths of specimens w

and w¯ = 1000. Based on the calculation results shown in Fig. 43.11, the relative pre¯ ∗ /l¯∗ may be about one-third or one-fourth of the crack length fracture zone length  for short cracks, and therefore, when calculations are performed, an initial short crack of the length l¯0 must be increased by one-third. Loading regimes present the greatest danger when damage accumulation takes place under low-cycle loading at normal temperatures, since the yield stress σY of material at the normal temperature is less than yield stress σY of this material under conditions of low temperatures. In what follows, we retain the notation l¯0 as the length of a calculated crack under conditions of low temperatures. We now proceed to estimate operating capability of steel structure components (homogeneous and welded) under conditions of low temperatures using diagrams of quasi-brittle fracture for structural components. When obtaining values of critical ∗ ∗ and σ¯ ∞w , as well as moments M¯ ∗ and M¯ w∗ , the following relations are used stresses σ¯ ∞ (43.8), (43.12), and (43.14), into which the calculated crack length l¯0 is introduced. From comparison of theoretical parameters with known ones, the estimate of working capacity of a structure under conditions of low temperature is obtained. The comparison of results on critical states of different structures presented in Table 43.1 are unilateral, since there are critical SIFs K Ic (T ) for Russian steels and there are no data on critical CTOD δ ∗ at lower-temperatures. To fill this gap, Fig. 43.13 is represented. Plots given in this Figure characterize the ∗ ∗ 0 . Recall that σ¯ ∞ → σ¯ ∞ relative effect of parameters K Ic and ε¯ p on critical stresses σ¯ ∞ is for ε¯ p → 0, that is, in the limit, an elastic ideally-plastic material, having ultimate strain ε1 (T ), turns to ideally-elastic material with the ultimate strain ε0 (T ). Curves 1, 2, 3, 4, and 5 in Fig. 43.13 have been plotted via relation (43.12) for tensile specimen of finite width with internal crack, the curves represent parameters ε¯ p = 0, 1, 2, 3,, ∗ essentially depend on ε¯ p , but only for short and 4, w/r = 100. Critical stresses σ¯ ∞ cracks.

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∗ versus the plasticity factors ε¯ Fig. 43.13 Critical stresses σ¯ ∞ p

43.5 Conclusion The comprehensive approach (stress-strain-based approach) is proposed to be used for study of mechanical behavior of materials under the effect of deformation and fracture. Diagrams of quasi-brittle fracture of simplest structures have been constructed. The coupled fracture criterion was used for plotting these diagrams that allowed conflicting requirements placed on fracture toughness and strength of structure to be partly reconciled. The conservative estimates for critical stresses and critical moments of homogeneous structure and those with welded joints have been obtained. When analyzing parameters including into the proposed model, the data on deformation of steels at temperatures near and below the cold-brittleness threshold are used. Diagrams of quasi-brittle fractures are proposed to be used when analyzing catastrophic fractures of components of steel structures operating at temperatures below the cold-brittleness threshold. Quantitative estimates of decreasing critical stresses have been indicated for simplest structures with welded joints as compared to those for homogeneous structures. The break-down effect of a structure under monotonic loading is strongly pronounced inasmuch as critical loads are significantly decreased with increasing a crack length. The attention is given to the preliminary steel deformation that takes place in technological processes or at the initial stage of operation of structures with cracks under low-cycle loading. Only the limited data set on characteristics of steels at low temperatures is available. Additional laboratory examinations are needed for the cases when fracture toughness is measured simultaneously with determination of strength characteristics of steels at low temperatures. Acknowledgements This work was supported by the Russian Science Foundation (Grant No. 19-19-00126).

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References 1. Barsom, J.M.: Relationship between plain-strain ductility and K Ic for various steels. J. Eng. Ind. 1209–1215 (1971) 2. Brown, A.L., Smith, J.B.: Failure of spherical hydrogen storage tanks. Mech. Eng. 66(6), 392–397 (1944) 3. Dowling, N.E.: Mechanical Behavior of Materials, Engineering Methods for Defomation, Fracture, and Fatigue. International edition (2013) 4. Dugdale, D.S.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960) 5. Graf, O.: The strength of welded joints at low temperature and the selection and treatment of steels suitable for welded structures. Welding J. (Supplement), 508-s–517-s (1947) 6. Karihaloo, B.L., Xiao, Q.Z.: Asymptotic crack tip fields in linear and nonlinear materials and their role in crack propagation. Phys. Mesomech. 22(1), 18–31 (2019) 7. Kornev, V.M.: Quasi-brittle fracture diagrams under low-cycle fatigue (variable amplitude loadings). Engine. failure analysis 35, 533–544 (2013) 8. Kornev, V.M.: Delamination of bimaterial and critical curves of quasi-brittle fracture in the presence of edge cracks. Adv. Mater. Sci. Applic. 3(4), 164–176 (2014) 9. Kornev, V.M.: Damage accumulation and fracture of welded joints under low-cyclic loading conditions. Appl. Mech. Mater. 784, 179–189 (2014) 10. Kornev, V.M.: Quasi-brittle fracture diagrams for bending elements of constructions in the presence of cracks. J. Mach. Manuf. Reliab. 44(2), 140–146 (2015) 11. Kornev, V.M.: The coupled criterion for description of fatigue fracture. Material embrittlement in pre-fracture zone. In: MATEC, Web of Conferences, vol. 165, p. 13008 (2018). https://doi. org/10.1051/matecconf/201816513008 12. Kornev, V.M., Demeshkin, A.G.: Quasi-brittle fracture diagram of structured bodies in the presence of edge cracks. J. Appl. Mechan. Tech. Phys. 52(6), 152–164 (2011) 13. Kornev, V.M., Kurguzov, V.D., Astapov, N.S.: Fracture model of bimaterial under delamination of elasto-plastic structured media. Appl. Compos. Mater. 20(2), 129–143 (2013) 14. Kovchik, S.E., Morozov, E.M.: Characteristics of Short-Term Fracture Toughness of Materials and Methods of Their Identification, vol. 3, Mekhanika razrusheniya i prochnost’ materialov. Naukova Dumka (in Russian) (1988) 15. Krasovsky, A.Y.: Brittleness of Metals at Low Temperatures. Naukova Dumka (in Russian) (1980) 16. Leguillon, D.: Strength or toughness? a criterion for crack onset at a notch. Eur. J. Mech. A. Solids 21(1), 61–72 (2002) 17. Leonov, M.Y., Panasyuk, V.V.: The smallest cracks growth in solid. Prikl. Mekh. 5(4), 391–401 (1959). (in Russian) 18. Miracle, D.B., Senkov, O.N.: A critical review of high entropy alloys and related concepts. Acta Mater. 122, 448–511 (2017) 19. Moskvichev, V.V., Makhutov, N.A., Chernyaev, A.P.: Fracture Toughness & Mechanical Properties of Engineering Materials of Technical Systems. Nauka (in Russian) (2002) 20. Murakami, Y.: Stress Intensity Factors Handbook (in 2 Volumes). Pergamon Press (1987) 21. Neuber, G.: Kerbspannunglehre: grunglagen fur genaue spannungsrechnung. Springer (1937) 22. Novozhilov, V.V.: On necessary and sufficient criterion of fracture strength. Prikl. Mat. Mekh. 33(2), 212–222 (1969). (in Russian) 23. Panin, S.V., Marushchak, P.O., Vlasov, I.V., Eremin, A.V., Byakov, A.V., Syromyatnikova, A.S., Stankevich, R.: Structure and fatigue durability of 09Mn2Si pipe steel after long-term operation in far north conditions. Diagn. Resour. Mech. Mater. Struct. (4), 81–85 (2017) 24. Panin, V.E., Derevyagina, L.S., Lebedev, M.P., Syromyatnikova, A.S., Surikova, N.S., Pochivalov, Y.I., Ovechkin, B.B.: Scientific basis for cold brittleness of structural BCC steels and their structural degradation at below zero temperatures. Phys. Mesomech. 20(2), 125–133 (2017)

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25. Raevsky, G.V.: On brittle fracture of welded tanks and other structures. Avtom. Svarka 6, 3–18 (1955). (in Russian) 26. Ritchie, R.O.: The conflicts between strength and toughness. Nat. Mater. 11, 817–822 (2011) 27. Rolfe, S.T., Barsom, J.M.: Fracture and Fatigue Control in Structures Application of Fracture Mechanics, p. 520. ASTM STP, Philadelphia (1977) 28. Romaniv, O.N., Yarema, S.Y., Nikiforchin, G.N., Makhutov, N.A., Stadnik, M.M.: Fatigue and fatigue crack growth of engineering materials, vol. 4. Mekhanika razrusheniya i prochnost’ materialov: in 4 volumes. Kiev: Naukova Dumka (in Russian) (1990) 29. Shabalin, V.I.: Some cases of destruction of welded tanks at low temperature. Avtogen. Delo. 6, 29–30 (1948). (in Russian) 30. Shank, M.E.: A critical survey of the brittle failure in carbon plate steel structures other than ships. In: Symposium on Effect of Temperature on the Brittle Behavior of Metals With Particular Reference to Low Temperature, ASTM, vol. 158, pp. 45–110 (1953) 31. Syromyatnikova, A.S.: Degradation of physical and mechanical condition of the main gas pipeline metal at long operation in the conditions of the cryolitzone. Phys. Mesomech. 17(2), 85–91 (2014) 32. Thurston, K.V.S., Gludovatz, B., Hohenwarter, A., Laplanche, G., Georgef, E.P., Ritchie, R.O.: Effect of temperature on the fatigue-crack growth behavior of the highentropy alloy CrMnFeCoNi. Intermetallics 88, 65–72 (2017) 33. Uzhik, G.V.: Durability and Plasticity of Metals at Low Temperatures. Publisher AS USSR, Moscow (1957). (in Russian) 34. Weißgraeber, P., Leguillon, D., Becker, W.: A review of finite fracture mechanics: crack initiation at singular and non-singular stress raisers. Arch. Appl. Mech. 86(1–2), 375–401 (2016)

Chapter 44

Cement Failure Caused by Thermal Stresses with Casing Eccentricity During CO2 Injection Xuelin Dong, Deli Gao and Zhiyin Duan

Abstract Carbon capture and sequestration (CCS) is one of the most promising technologies to mitigate greenhouse gas levels. To ensure an effective underground storage, well integrity is critical to isolating the injected fluid between different zones or back to the surface. Among the wellbore components, the cement sheath is the most important sealing element for zonal isolation. However, cement is vulnerable and prone to cracking that may provide leakage pathways for CO2 . Both laboratory study and field test show that thermal stresses caused by the temperature variation in the wellbore are a major factor for the mechanical integrity loss of cement. This work focuses on the mechanical response of the casing-cement-formation section above the injection zone. We firstly propose a wellbore flow model to predict the temperature distribution along the well depth. Then we calculate the induced stress in cement during injection by a finite element simulation. To identify the cement failure mode, we introduce failure factors by the Mogi-Coulomb criterion, tensile strength and interfacial strength corresponding to shear compressive failure, radial cracking and debonding at the casing/cement or cement/formation interfaces, respectively. A parametric study is conducted to investigate the influence of the injection temperature and rate as well as casing eccentricity on failure factors. The results show that radical cracking and debonding at the cement/formation interface are the main failure modes during CO2 injection. Both the two failure factors would increase linearly as the injection temperature decreases while they grow non-linearly with the injection rate. In addition, the casing eccentricity exacerbates the risk of cement integrity loss by increasing failure factors. This study provides a failure assessment of CO2 geological sequestration and guidelines for injection operations. Keywords Well integrity · Thermal stress · Failure mode X. Dong (B) · D. Gao Key Laboratory of Petroleum Engineering, China University of Petroleum, Beijing 102249, China e-mail: [email protected] Z. Duan Beijing Key Lab of Heating, Gas Supply, Ventilating and Air Conditioning Engineering, Beijing University of Civil Engineering and Architecture, Beijing 100044, China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_44

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44.1 Introduction Carbon capture and sequestration (CCS) has been recognized as an attractive technology to mitigate greenhouse gas levels in the world [1, 2]. Injecting large tonnages of CO2 into saline aquifers or depleted oil and gas reservoirs through wells onshore or offshore is an effective method of long-term storage [3]. Although several research, pilot or even commercial CO2 storage projects have been successfully implemented, there are still concerns on their safety among scientists, engineers and publics [4]. Leakage of the stored CO2 or re-emerging to the surface is one of the most concerned issues. Therefore, maintaining the well integrity to prevent leakages is crucial to a long-term storage. CO2 injection and storage pose various challenges to well integrity. The injected cold fluid would induce complex geochemical and geomechanical interactions between barrier materials, reservoir formation and underground fluids [5, 6]. Damages to barrier materials, especially to the cement sheath would cause embedded cracks or interfacial debonding, which provide potential pathways for CO2 leakage. Both laboratory experiments and field tests have demonstrated that the temperature fluctuation caused by the temperature difference between the cold injected fluid and hot formation is a primary factor to induce large thermal stresses in cement [7–9]. The induced stress once surpasses the material’s strength, cement failure will occur. Hence, a proper estimation of thermal stress in cement is valuable to evaluate the cement integrity. Many researchers have studied thermal stresses in injection wells using analytical or numerical methods. Thiercelin et al. [10] proposed an analytical mechanical model based on linear elastic theory to investigate the role of thermal perturbations on the mechanical response of the cement. They showed that cement failure could be avoided by selecting proper thermo-elastic properties of wellbore materials. Yu et al. [11] carried out coupled thermo-poromechanical multi-phase simulations to study the effect of thermal stresses on the caprock integrity. They concluded that injecting CO2 at the temperature close to the aquifer significantly reduces the risk of caprock fracturing. Nygaard et al. [12] evaluated the integrity of CO2 injection well by a 3D finite-element model, in which the cement and formation are treated as poroelastoplastic materials. They suggested that lower Young’s modulus and Poisson’s ratio of the cement would reduce the risk of debonding and tensile failure. Aursand et al. [13] proposed a coupled flow and heat conduction model to determine injection parameters’ effect on temperature variations in wells. Particularly, they showed that longer pauses between injections will induce a larger thermal stress which is enough to cause debonding at the casing/cement interface. Roy et al. [14] considered the initial damage in the cement and studied the impact of thermal stresses on cracking through stress intensity factors. They found that the existence of in-situ horizontal stresses has a positive effect on preventing damage evolution in cement. This paper firstly presents a wellbore flow model that is used to provide the temperature profile along the well depth. Then we construct a mechanical model

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based on thermoelastic theory to estimate the stress components in a casing-cementformation section above the injection zone. Failure factors are defined to identify the failure modes according to material strength criteria. Finally, we estimate failure factors under different injection operations and investigate the effect of the casing eccentricity.

44.2 Wellbore Flow Model Studies for well integrity of CO2 storage have indicated that the temperature difference between the cold injected CO2 and the hot surrounding rock will cause large thermal stresses in barrier materials. In particular, once the induced stress in cement exceeds its strength, damages or interfacial debonding would occur to undermine the well integrity. To estimate the stress in cement during CO2 injection, it is of great importance to obtain the temperature profile along the well. This can be achieved by wellbore flow models. Numerous models have been proposed to investigate the flowing temperature and pressure during CO2 injection [15]. These models intend to describe CO2 wellbore flow and heat transfer in different working conditions including single-phase flow and two-phase flow [16]. In this study, we assume that the injected CO2 is in a liquid state. Industry practices suggest injecting liquid CO2 that is more efficient due to its higher density than its supercritical gaseous counterpart. In addition, we suppose the flow along the well depth is steady while the radial heat transfer is unsteady as illustrated in Fig. 44.1. Hence, the governing equations for wellbore flow are presented as [17]. Fig. 44.1 The configuration of the wellbore

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d(ρf vf ) =0 ds

  d ρf vf2 d pf − + ρf g cos θ − f w = 0 ds ds    d d 1 − ρf vf ef + vf2 − ( pf vf ) + ρf gvf cos θ − vf f w − q = 0 ds 2 ds −

(44.1a) (44.1b) (44.1c)

where pf and vf represent the pressure and velocity of the injected fluid respectively, ρ f and ef are the fluid’s density and specific internal energy respectively, g is the gravitational acceleration, s and θ are the well depth and deviation angle respectively. Equations (44.1a)–(44.1c) are the continuity, momentum and energy conservation equations for fluid flow respectively. f w in Eq. (44.1b) is the frictional force between the viscous fluid and the tubing wall which is calculated as fρ f v2 f /(4r ti ), where f is the friction coefficient and r ti is the inner radius of the tubing [18]. The remained term q denotes the radial heat transfer per unit control volume from the surrounding formation to the injected fluid and is estimated as q = 2U tot (T f −T ei )/r ti . T f and T ei are the fluid temperature and the initial temperature of the formation respectively, and U tot is the overall-heat-transfer coefficient based on the tubing inside area. Many wellbore flow models provide efficient methods to obtain U tot [18, 19]. It should be noted that none of Eqs. (44.1a)–(44.1c) contains the fluid temperature. We introduce it in the above governing equations through the fluid specific enthalpy hf = ef + pf /ρ f . The gradient of the fluid specific enthalpy is related to the fluid temperature and pressure as [20]: dTf d pf dh f = c pf − c pf CJf ds ds ds

(44.2)

where cpf and C Jf are the fluid specific heat capacity and Joule-Thomson coefficient, respectively. Thermal properties of CO2 have to be determined to solve Eqs. (44.1a)–(44.1c). This work adopts the Span-Wagner equation of state to calculate fluid properties [15]. Solutions to Eq. (44.1) provide the temperature profile and the concerned temperature difference between the injected fluid and the formation.

44.3 Stress Analysis for the Well Section Figure 44.2 presents a typical geometry of the well cross section composed of casingcement-formation. Since wells are often as long as several kilometers, the deformation in the horizontal plane as shown in Fig. 44.2 is much larger than the one along the axial direction. Therefore, we undertake a plane-strain approach to evaluate the stress state in the cement. Here, we focus on the cement behind the non-perforated casing above the injection zone, and consider good and poor cementing conditions

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Fig. 44.2 Casing-cementformation section with boundary conditions and casing eccentricity

corresponding to concentric cemented casing and eccentric cemented casing respectively. The casing eccentricity e is defined as a distance between the centers of the casing and wellbore and the angle ϕ from the horizontal line to the two centers’ connection. The symmetry of geometry and loads restricts that 0 ≤ e ≤ 1 and 0 ≤ ϕ ≤ 90°. The annular fluid exerts a pressure pi inside the casing while the well section is subjected to in-situ stresses with σ H > σ h . T si and T fo are the temperatures at the inner surface of casing and the formation near the wellbore respectively, which can be determined from the wellbore flow model given in Sect. 44.2. Under geomechanical loads and temperature fluctuation depicted in Fig. 44.2, stresses would be caused in cement. The stress field can be evaluated by the equilibrium equation and thermoelastic constitutive equation as follows. Neglect the body force, the equilibrium equation is given as [21]: 1 ∂τr θ σr − σθ ∂σr + + =0 ∂r r ∂θ r

(44.3a)

∂τr θ 1 ∂σθ 2τr θ + + =0 ∂r r ∂θ r

(44.3b)

where (r,θ ) represents the polar coordinates with the origin at the wellbore center as shown in Fig. 44.2, σ r , σ θ and τ rθ are the radial, hoop and shear stress components, respectively. It should be noted that the casing is eccentric that breaks the axisymmetric symmetry and introduces θ and τ rθ in Eq. (44.3). For simplicity, we assume the casing, cement and formation are all homogenous, isotropic linear elastic, and then the materials’ stress can be expressed in terms of the strain tensor as: σr =

E [(1 − ν)εr + νεθ − (1 + ν)α(Tc − Tc0 )] (1 + ν)(1 − 2ν)

(44.4a)

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σθ =

E [(1 − ν)εθ + νεr − (1 + ν)α(Tc − Tc0 )] (1 + ν)(1 − 2ν)

(44.4b)

σz = ν(σr + σθ ) − Eα(Tc − Tc0 )

(44.4c)

τr θ =

E εr θ 1+ν

(44.4d)

where σ z is the axial stress component, T c and T c0 are the temperature distribution in cement and its initial temperature, respectively, εr , εθ and εrθ are the radial, hoop and shear strain components respectively. E, ν and α are the materials’ Young’s modulus, Poisson’s ratio and thermal expansion coefficient, respectively. To solve Eq. (44.3), boundary conditions should be provided. From Fig. 44.2, the mechanical boundary conditions are the pressure inside the casing as well as in-situ stresses. Analytical solutions to Eq. (44.3) have been provided for the symmetry situation without a casing eccentricity. However, for asymmetric geometries, it is very difficult to give a solution in close-form. Finite element (FE) methods are convenient to simulate the mechanical response for complex conditions. In this work, the temperature is obtained from the flow model while the stress state is estimated through an FE simulation. Not simultaneously calculating the fluid-structure interaction would reduce computational cost.

44.4 Failure Factors As aforementioned, the induced stress in cement may cause integrity failure that poses a leakage risk for the stored CO2 . Many researchers and engineers have identified that the primary failure modes of cement include shear compressive failure, radial cracking and interfacial debonding as illustrated in Fig. 44.3. Shear failure occurs when the equivalent stress in the cement is greater than the material’s strength. There are many criteria that characterize this kind of failure for cement. Among them, the Mogi-Coulomb criterion is adopted in this work since it has been proven to be applicable to several types of rocks. It defines a shear failure envelope as [22]: Fig. 44.3 Failure modes of cement for shear failure, radial cracking and interfacial debonding

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τ8 ≤ τmax τ8 =

 1 (σ1 − σ2 )2 + (σ2 − σ3 )2 + (σ3 − σ1 )2 3

τmax =

σuc (1 − sin φ) sin φ(σ1 + σ3 ) + 2 cos φ 2

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(44.5a) (44.5b) (44.5c)

where τ 8 is the octahedral shear stress and τ max is the maximum material’s allowable shear stress, σ i (i = 1, 2, 3) are the principle stress components in cement, σ uc represents the unconfined compressive strength of cement, and φ is the material’s internal friction angle. According to Eq. (44.5), the failure factor for shear compressive strength can be defined as: ηs =

τ8,max τmax

(44.6)

where τ 8,max is the maximum octahedral shear stress calculated by the FE simulation. ηs ≥ 1 indicates a high risk of shear failure in cement. When the tensile hoop stress in cement surpasses its tensile strength σ t , radial cracks may generate. Similarly, the failure factor for radial cracking is: ηr =

σθ,max σt

(44.7)

where σ θ ,max is the maximum tensile hoop stress under certain conditions. The failure factor for interfacial debonding compares the tensile radial stress at the interfaces between the casing and cement or the cement and formation to the interface strength, which is given as: ηsc =

σr sc,max σr cf,max , ηcf = σsc σcf

(44.8)

where ηsc and ηcf denote the interfacial failure factors for the casing/cement and cement/formation interfaces respectively, σ rsc,max and σ rcf,max are the maximum tensile radial stresses at the casing/cement interface or the cement/formation interface respectively, and σ sc and σ cf are interfacial strengths of these two interfaces. Revealing evolutions of the above failure factors with injection operations such as injection temperature and rate could identify the main failure mode during CO2 injection and provide guidance for maintaining well integrity.

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44.5 Wellbore Parameters and Injection Operations Equation (44.1) provides the temperature of the well structure which could be used to estimate the induced stress in cement and failure factors from an FE simulation. Then the obtained failure factors quantify the impact of CO2 injection on the well integrity. To concrete our approach, we consider an injection well of CO2 with typical loading cases. The geometry of the well is listed in Table 44.1. The deviation angle of the well is set to be zero i.e. a vertical well. The surface temperature is 4.5 °C with a temperature gradient of 4.2 °C/100 m. From Eqs. (44.1) and (44.3), thermal and mechanical properties should be supplied to calculate the temperature and stress as presented in Table 44.2. The thermodynamic properties of the injected CO2 are determined from the Span-Wagner equation of state by iteration in each calculation step. Previous studies have indicated that the injection temperature and rate have important effects on the temperature profile [14]. Here, we investigate the influence of these two injection parameters on the temperature in the cement above the injection zone and their impact on failure factors. In practice, the CO2 temperature varies from 25 °C for onshore pipeline transport to −50 °C for offshore pipeline transport. Hence, we consider an injection temperature range of −20 to 20 °C. We fix the annual injection as 1t/a per well and change the injection rate from 1 kg/s to 20 kg/s, and the injection Table 44.1 Geometry of the well Parameter

Value

Well depth (m)

2000

Inner radius of tubing (mm)

31.0

Outer radius of tubing (mm)

36.5

Inner radius of casing (mm)

60.68

Outer radius of casing (mm)

69.85

Radius of wellbore (mm)

107.95

Table 44.2 Thermal and mechanical properties of materials Properties

Tubing/Casing

Cement

Formation

Annular fluid

Thermal conductivity (W/m K)

47

0.72

2

0.6

Specific heat (J/kg K)

/

/

900

4100

Density (kg/m3 )

/

/

2 500

1000

Viscosity (Pa s)

/

/

/

6 × 10−4

Young’s modulus (GPa)

200

1

1

/

Poisson’s ratio

0.3

0.23

0.2

/

Thermal expansion coefficient (10−6 /°C)

12

10

11

/

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time varies accordingly. During injection, the casing inside is subjected to a pressure caused by the annular fluid that can be estimated by the well depth and fluid density. It has been demonstrated that the existence of in-situ stresses would benefit interfacial strength. To study the worst situation, we assume σ H = σ h = 0 in our research.

44.6 Results and Discussions It can be inferred that a larger temperature difference between the barrier materials would cause a greater stress in cement. We firstly discuss the influence of injection operations on this temperature difference, and then we go to cement’s failure factors. Figure 44.4 presents the temperature difference between the tubing inside surface and the formation along the well depth with different injection rates (vinj ). The injection time (t inj ) is also illustrated. Figure 44.4 demonstrates that the maximum temperature difference occurs at the well section just above the injection zone (not including the well section below the packer). In addition, the temperature difference increases with the well depth. With slow injection rates, this increase exhibits a non-linear behavior, while it is nearly a straight line at faster injection rates (vinj > 5 kg/s). Faster injection rates would induce a larger temperature difference at deeper positions. It should be noted that as the injection rate increases further, the temperature differences corresponding to variant injection rates are close to each other. For example, T si −T e is −38.86 °C for vinj = 5 kg/s and it is −40.94 °C and −41.75 °C when vinj is 10 kg/s and 20 kg/s, respectively. To reveal the influence of injection rates on the temperature difference between the casing and formation clearly, Fig. 44.5 plots T si −T e at the bottom section as a function of the injection rate with different injection temperatures. It is obvious that the temperature difference will reach a steady value as the injection rate becomes faster and faster. At high injection rates, the heat transfer from the surrounding formation to the injected fluid goes quickly. It takes a short time to get the system Fig. 44.4 The temperature difference between T si and T e along the well depth for various injection rates

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Fig. 44.5 The temperature difference between T si and T e as a function of the injection rate

to thermal equilibrium. Figure 44.5 also tells that the injection temperature plays an important role in the temperature difference. Specifically, T si −T e is −21.55 °C for T inj = 20 °C while is −41.75 °C for T inj = −20 °C, it drops 93.7%. Our main purpose in this study is to illustrate the effects of injection operations on cement’s failure factors defined in Eqs. (44.6)–(44.8). At first, we consider a good cementing quality i.e. e = 0. Then we investigate the fluctuation of failure factors with poor cementing. Figure 44.6a and b provide evolutions of failure factors with the injection temperature and rate with a good cementing job, respectively. From Fig. 44.6a, it can be seen that the shear compressive failure factor ηs changes very slowly (from 0.16 to 0.19) as the injection temperature grows from −20 °C to 20 °C (vinj is fixed as 20 kg/s). However, ηr , ηsc and ηcf all decrease linearly as T inj increases. In particular, ηr changes from 0.934 to 0.797 which is close to 1. ηsc is no more than 1 as the injection temperature goes down. The most dangerous situation is that when T inj is below −10 °C, ηcf is larger than 1 that means debonding would occur at the cement/formation interface very likely. As the fluid is injected faster (1 kg/s ≤ vinj ≤ 20 kg/s and T inj is fixed as −20 °C), ηs decreases from 0.186 to 0.162 that is a tiny variation. Similar to Fig. 44.6a, ηr , ηsc and ηcf go higher and higher when injection rate increases. ηr still changes a little and is near the dangers value. ηsc is always smaller than 1, while as vinj goes beyond 3 kg/s ηcf becomes greater than 1. For both injection parameters, radial cracking and debonding at the cement/formation interface are more dangerous than the other two failure modes. It needs to be paid more attention to guarantee the bonding strength between the cement and formation when CO2 is injected at a low temperature and a fast rate. Previous studies have showed that a casing eccentricity will cause stress concentration in cement, which would rise failure risk for cement integrity. From Fig. 44.6 we can learn that when T inj = −20 °C and vinj = 3 kg/s, the failure factors for radial cracking and cement/formation interface debonding are 0.896 and 0.966, respectively. Both two failure modes are in a dangerous zone. We estimate failure factors under this injection operation with different casing eccentricities as shown in

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Fig. 44.6 Evolutions of cement’s failure factors with injection operations with a good cementing job: a injection temperature and b injection rate

Fig. 44.7. In general, there is no clear relation between failure factors and the casing eccentricity. For the given injection parameters, ηs is 0.169 with a good cementing quality. Figure 44.7a shows that ηs is safe since its maximum value is 0.186 when e = 0.99 and ϕ = 75°. For radial cracking, ηr is enlarged to 1.07 when e = 0.99 and ϕ = 45°. In the range of eccentricity, ηsc is negative as shown in Fig. 44.7c. Again, the cement/formation debonding has the highest risk with casing eccentricity that it will increase to 1.20 at e = 0.99 and ϕ = 60°. It is worth noting that a larger degree of eccentricity would induce a higher ηcf . All the above failure factor increases are obtained at e = 0.99, which is a very extreme condition. In fact, modern oil and gas industry has made a great progress in drilling and completion, and e is normally no more than 0.25. For this eccentricity degree, when ϕ varies in the range of [0, 90°], ηcf increases by 11.1% at most.

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Fig. 44.7 Effects of casing eccentricity on cement’s failure factors: a shear failure b radial cracking c casing/cement debonding d cement/formation debonding

44.7 Conclusions In this paper, we present a wellbore flow model to predict the temperature profile along the well depth during CO2 injection. An FE simulation is carried out to estimate the stress state in cement above the injection zone. To evaluate the cement integrity, we define failure factors for different failure modes including shear compressive failure, radial cracking and interfacial debonding. Then we present effects of the injection temperature, injection rate and casing eccentricity on the defined failure factors. The relevant results show that the temperature difference between the casing inside and formation is higher at deeper positions. Lower injection temperatures and faster injection rates would induce a larger temperature difference. For CO2 injection wells, radial cracking and cement/formation debonding are the potential failure modes during operation. The shear failure factor will increase with the injection temperature and decrease with the injection rate. The other failure factors would rise at a lower injection temperature and a faster injection rate. Severe casing eccentricities would enlarge failure factors in a large extent and the cement/formation debonding is the most dangerous failure mode due to a poor cementing job.

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Acknowledgements This work was supported by the National Key R&D Program of China (Grant No. 2018YFB0605502), the Natural Science Foundation of Beijing (Grant No. 2182062), and the National Natural Science Foundation of China (Grant No. 11872378).

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Chapter 45

Optimal Design for Toggle Brace Damper Systems Based on Virtual VD Modal Bingjie Du, Xin Zhao and Hao Li

Abstract Viscous damper (VD) is a kind of speed dependent damper without stiffness. It has the advantages of strong energy dissipation ability, no additional stiffness and multifunctional vibration reduction, which can absorb and dissipate seismic vibration and wind-induced vibration efficiently. In this paper, the geometric parameter optimization model of the reverse toggle brace viscous damper is established. A virtual VD model with additional damping ratio is put forward, based on which a VD priority placement analysis method is developed. And then the optimal design for toggle brace damper systems is proposed. The VD system of a super tall building structure is designed using the proposed method. It’s revealed that the geometric parameters are optimized by the proposed method, and under the premise of satisfying a series of design constraints, the deformation magnification coefficients of VD is close to the maximum value, which improves its working efficiency effectively; the VD priority placement analysis method proposed in this paper can obtain the priority arranging order of VD efficiently and accurately; the optimal design method for toggle brace damper systems designs VD in two directions at the same time, which considers the mutual action of VD in different directions, compared with traditional method, it’s more efficient and accurate. Keywords Super tall buildings · Toggle brace damper system · Virtual VD model · Optimal design

B. Du · X. Zhao (B) · H. Li Department of Structural Engineering, Tongji University, No. 1239 Siping Road, Shanghai 200092, China e-mail: [email protected] X. Zhao Tongji Architectural Design Group, No. 1230 Siping Road, Shanghai 200092, China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_45

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45.1 Introduction As the height of super-tall buildings continues to rise, the lateral stiffness of the structure decreases constantly, which makes the stiffness of the structure under earthquake action and wind load becomes a prominent issue. Viscous damper (VD) is a kind of speed dependent damper without stiffness. It has the advantages of strong energy dissipation ability, no additional stiffness and multi-performance vibration reduction. A large number of engineering applications proved that VD is an effective damping equipment to absorb and dissipate seismic vibration and wind-induced vibration. VD has been applied in many engineering projects, such as Boston 111, the four-season hotel in San Francisco and Boston millennium plaza project. Although VD can dissipate energy under small deformation, the efficiency is very low using the traditional diagonal connection and herringbone connection, so it usually requires motion amplification device. The toggle brace motion amplification device is one of which to enlarge the relative displacement and relative speed of VD under seismic action, improving the vibration reduction efficiency. So far, there have been many studies on VD location, quantity and damping coefficient optimization, but these studies mainly focus on the arranging optimization problem of linear viscous damper on plane frame structure, and rarely can solve the arranging optimization problem of nonlinear viscous damper on super high-rise structure. In this paper, the inverse toggle brace viscous damping system is taken as an object, and its geometric parameter mathematical optimization model is established. A virtual VD model with additional damping is proposed. Based on this model, a VD prioritization analysis method is developed. Then, an optimal design method of toggle brace viscous damping system based on virtual VD model with additional damping is proposed, which considers both directions of VD arranging at the same time. The VD system of a supper-high rise structure is designed by using the optimal design method of reverse toggle brace viscous damping system proposed in this paper, to verify its effectiveness and applicability.

45.2 Geometric Parameter Optimization 45.2.1 Toggle Type Viscous Damper As shown in Fig. 45.1, the toggle brace viscous damper is composed of one damping rod and two supporting rods, which can be divided into three types according to their different structures: the upper toggle brace type, the lower toggle brace type and the reverse toggle brace type. The deformation amplification coefficient of the toggle brace viscous damping system depends on geometric parameters, and usually can reach 2–4. Compared with the reverse toggle brace type, both the upper toggle

45 Optimal Design for Toggle Brace Damper Systems …

(a) Upper toggle brace type

(b) Lower toggle brace type

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(c) Reverse toggle brace type

Fig. 45.1 Geometric structures of toggle brace damper system

brace type and the lower toggle brace type have some disadvantages: in order to obtain a larger deformation amplification coefficient, one end of the damping rod is connected to the floor, which is adverse to the beam design. Occupy bigger space, affect the openings for door and window. Therefore, the reverse toggle brace viscous damper is selected as the damping device in this paper.

45.2.2 Geometric Parameter Optimization The main geometric parameters of the reverse toggle brace viscous damper are shown in Fig. 45.2, which mainly include the length of two supporting rods and VD bars L 1 , L 2 , L 3 and three angles θ1 , θ2 , θ3 . The deformation amplification coefficient of reverse toggle brace type viscous device is:   sin θ2 cos(θ3 − θ1 ) (45.1) + sin θ1 f =− cos(θ1 + θ2 ) In order to obtain the larger deformation coefficient, it is usually taken that θ1 = θ3 , which means VD is perpendicular to the supporting rod L 1 . At this point, once the parameters θ1 are determined, other geometric parameters of the inverted toggle Fig. 45.2 Reverse toggle brace viscous damper

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brace viscous damper can be determined according to the constraint conditions. In addition, when θ1 + θ2 = 90◦ , the displacement amplification coefficient reaches an infinite value, which means that the reverse toggle brace system becomes an instantaneously changeable system, and any small deformation of the damper will be amplified infinitely and the damper will be destroyed. In order to avoid excessive deformation amplification coefficient, this study suggests that when θ1 change ±0.3°, the variation of the displacement amplification coefficient should be less than 0.2, and the deformation amplification coefficient should not be more than 3. At the same time, the rationality of the actual value of the length of the three bars and the affection of the openings for doors and Windows should be considered. Therefore, the geometric parameter optimization mathematical model is obtained: max f (θ1 )

(45.2)

s.t. arctan(H/B) ≤ θ1 ≤ 80

(45.3)

L 1 = H sin θ1 ≥ 1.5

(45.4)

 L2 =

(L 3 cos θ1 )2 + (B − L 1 cos θ1 )2 ≥ 1.5

(45.5)

L 3 = H cos θ1 ≥ 1.5

(45.6)

| f (θ1 ) − f (θ1 − 0.3)| ≤ 0.2

(45.7)

f (θ1 ) ≤ 3

(45.8)

θ2 ≥ θ2,min = arctan[X 2/(H − Y 2)]

(45.9)

θ1 ≤ θ1,max = 90◦ − arctan(Y 1/ X 1)

(45.10)

When optimizing VD geometric parameters, firstly we determine the range of θ1 and set value at interval of 0.3°, then compute other parameters and verify constraints to find all the optional value, choose the value with maximum deformation amplification factor as the optimal.

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45.3 Priority Order of VD 45.3.1 Additional Damping Virtual VD Model As shown in Fig. 45.3, model 1 is the actual VD structure. Equaling the VD effect to the additional damping effect, we have model 2, which considered the natural damping of the structure and the additional damping of VD, then model 1 is equal to model 2. If virtual VDS are added to all the optional locations of model 2, additional damping virtual VD model (model 3) is formed. Virtual VD refers to the VD whose damping coefficient and damping index are equal to 1. Virtual VD is linear VD, no matter how many VD placed in the structure, the speed of structural analysis will not be affected. In addition, the damping coefficient of virtual VD is very small, which basically has no impact on the main structure, that is, virtual VD is “virtually set”, which only plays the role of data extraction, and can be regarded as “deformation measuring device”.

45.3.2 VD Priority Order Analysis In order to find the optimal VD arranging scheme, the most accurate method is to select each scheme in order of the number of VDS from less to more. However, this method is extremely inefficient, and when the number of VD optional position is large, the number of scheme combinations is huge, which is almost impossible to realize. Therefore, it is necessary to find an index to judge the priority of VD arranging order, to add up VD until the objective additional damping ratio is reached. In this

(a) Model 1

(b) Model 2

Fig. 45.3 Virtual VD model with additional damping ratio

(c) Model 3

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paper, a priority arranging order analysis method based on additional damping virtual VD model is proposed. Since both the damping coefficient and damping exponent of the virtual VD are 1, its output expression is: F1 = C1 |v1 |α1 =|v1 |

(45.11)

The output expression of the actual VD is: F2 = C2 |v2 |α2

(45.12)

Since model 1 and model 2 are approximately equivalent and model 3 and model 2 are equal, so model 3 and model 1 are approximately equivalent. Model 3 can be used to estimate the structural response of model 1, that is, the following relationship exists between virtual VD and actual VD: u1 = u2

(45.13)

v1 = v2

(45.14)

According to the virtual VD output force F1 the in model 3, the deformation speed |v1 | can be calculated. After knowing |v1 |, the actual VD output force F2 of can be calculated according to the damping coefficient C2 and the actual VD damping index α2 of. And the virtual VD deformation u 1 is equal to the actual VD deformation u 2 . Therefore, based on the calculation results of virtual VD, the energy consumption of VD after the actual VD is placed at the virtual VD can be calculated, and then the priority order of VD arrangement can be determined according to the relative energy consumption.

45.4 Optimized Design Process of VD System The optimal design process of the VD system proposed in this paper is shown in Fig. 45.4. When the additional damping ratio in a certain direction is less than 0.85 times the additional damping ratio objective, the VD in that direction is iterated quantitatively. When the additional damping ratio in both directions is greater than 0.85 times the additional damping ratio objective, the damping coefficient correction iteration is carried out for VD. When the error between the additional damping ratio in both directions and the objective additional damping ratio is within 5%, the iteration stops and the optimization design process ends.

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Fig. 45.4 Optimization process for toggle brace viscous system

This optimization design method designs VD in two directions in the same time, considering the influence of VD in one direction on the response of structure in the other direction. Compared with the traditional design method, the design efficiency of VD system can be significantly improved, and the coupling effect of VD in both directions is more consistent with the actual situation.

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45.5 Case Study Taking a 170 m super tall building as an engineering case, the optimal design method of toggle brace viscous damping system proposed in this paper is verified. The structure is a frame-core tube structure system, and the main lateral force resisting members include CFT giant column, steel supported core tube and outriggers, as shown in Fig. 45.5. The seismic fortification intensity is 8 degrees, the basic seismic acceleration is 0.30 g, the characteristic site period is 0.40 s, and the natural damping ratio of the structure under the action of small earthquakes is 3%. Under the action of small earthquake, the objective additional damping ratio in both X and Y directions is 2%, and 7 small earthquake time-history waves are used for analysis, including 2 artificial waves and 5 natural waves. In order to prevent excessive concentration of VD arrangement, VD is arranged at regular intervals, totaling 15 floors. Each layer has 8 optional positions, including 4 in X and 4 in Y directions, as shown in Fig. 45.5. The structure consists grids of two dimensions: 9.2 m wide and 5.4 m high, and 13.2 m wide and 5.4 m high. The optimization model proposed in this paper is used to design the geometric parameters of the grids. The results are shown in Table 45.1. The deformation amplification coefficients of both grids are close to 3.0. The virtual VD model with additional damping proposed in this paper is used to analyze the VD priority arranging order. The energy consumption of each group

VD5

VD6

VD7

VD8

VD1

VD2 Structure system Outer frame

Core

Outrigger

VD4

Fig. 45.5 Structural system

Table 45.1 Optimization results of geometric parameters Number

Width

Height

Deformation amplification factor

θ1 = θ3

θ2

1

9.2

5.4

2.967

38.51

66.26

2

13.2

5.4

2.919

33.65

72.27

45 Optimal Design for Toggle Brace Damper Systems … 45

45 EQ1X EQ2X EQ3X EQ4X EQ5X EQ6X EQ7X Mean value(Nm)

Energy dissipation (Nm)

35 30 25

EQ1X EQ2X EQ3X EQ4X EQ5X EQ6X EQ7X Mean value(Nm)

40 35

Energy dissipation (Nm)

40

20 15

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5

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0

0 0

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(a) X direction

(b) Y direction

Fig. 45.6 Dissipating energy of VD under earthquakes

is shown in Fig. 45.6, in which groups 1–15 are in X direction and groups 16–30 are in Y direction. The results show that the relative energy dissipation of VD in each group is different under different seismic waves, and that VD in one direction mainly reduces the vibration of this direction. This paper takes the average energy consumption of each group of VD under 7 seismic waves as the index to judge the priority arranging order, and it can be obtained that the optimal VD arranging position of X direction is 13F, 11F, 19F…. The optimal VD arranging position of Y is 29F, 27F, 31F… (Table 45.2). After 4 iterations, it can be obtained that: when 12 sets of VD are placed in the X direction on layer 13F, 11F and 19F, and 12 sets of VD are placed in the Y direction on layer 29F, 27F and 31F, the additional damping ratio in the X direction of the structure reaches 1.98% and 2.05% in the Y direction. and the VD arranging scheme of each iteration step is shown in Table 45.3. To verify the correctness of the results, three cases were set: time history analysis for the structure with additional damping (case 1), time history analysis for the Table 45.2 Priority for placing VD Number

1

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5

23

Table 45.3 VD placing schemes Number

Story X

Y

Damping coefficient

Additional damping ratio

X

X (%)

Y

Y (%)

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18F

27F

700

700

0.71

0.87

2

18F/13F

27F/29F

700

700

1.42

1.59

3

18F/13F/15F

27F/29F/31F

700

700

2.14

2.09

4

18F/13F/15F

27F/29F/31F

740

620

2.00

2.02

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structure with VDs (case 2), and response spectrum analysis for the structure with additional damping (case 3). The comparison of story drifts and story shear force of the structure was shown in Fig. 45.7. The results show that the structure response of the additional damping time history case is consistent with VD time history case, indicating that the VD system designed by this method has better accuracy. In addition, the story drifts and story shear force of the structure of the additional damping response spectrum case are greater than the results of time-history analysis, which indicates that the structure tends to be safe when the response spectrum analysis is used for structural optimization. 180

180

160

160

140

140

120

120

100

100

80

80

60

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40

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20

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0 0.5

1

1.5

2.5

2

0

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2.5

2

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1

-3

-3

x 10

x 10

(a) Story drifts of X direction

(b) Story drifts of Y direction

180

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10 7

(a) Story shear forces X direction Fig. 45.7 Structure analysis results

0

0

1

2

3

4 x

107

(b) Story shear forces Y direction

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45.6 Conclusion This paper established the geometric parameter optimization model of the reverse toggle brace viscous damper. A virtual VD model with additional damping ratio is put forward, based on which a VD priority placement analysis method is developed. And then the optimal design for toggle brace damper systems is proposed. The VD system of a super tall building structure is designed using the proposed method. The following conclusion can be obtained: (1) After the analysis of the geometric parameter optimization model proposed in this paper, the deformation amplification coefficient of the viscous damper is close to the maximum under a series of constraints, effectively improving its working efficiency; (2) This paper proposes a VD priority placement analysis method based on virtual VD model, which can obtain the priority arranging order of VD efficiently and accurately; (3) VD in one direction mainly reduces the vibration in that direction; (4) The optimization design method proposed in this paper designs VD in both directions simultaneously, considering the interaction effects between them. Compared with the traditional design method, it can significantly improve the design efficiency of VD system and is more consistent with the actual situation.

References 1. Huang, H.C.: Efficiency of the motion amplification device with viscous dampers and its application in high-rise building. Earthq. Eng. Eng. Vib. 8(4), 521–536 (2009) 2. Wang, Z.H., Chen, Z.Q.: New installations of viscous dampers in high rise buildings. World Earthq. Eng. 26(4), 135–140 (2010) 3. Hwang, J.S., Huang, Y.N., Hung, Y.H.: Analytical and experimental study of toggle-bracedamper systems. J. Struct. Eng. 131(7), 1035–1043 (2005) 4. Hwang, J.S., Huang, Y.N., et al.: Design formulation for supplemental viscous dampers to building structures. J. Struct. Eng. 134(1), 22–31 (2008) 5. CTBUH Seismic Working Group. Recommendations for the seismic design of high-rise buildings. Council on Tall buildings and Urban Habitat (2008) 6. Zhao, X., Zhang, H.W.: Optimal placement for buckling-restrained braces in super tall building structures based on grid shear deformation. Ind. Constr. 44(S1), 248–256 (2014) 7. Zhao, X, Shi, T.: Optimal Placement of Viscous Dampers in Super Tall Buildings Based on Grid Shear Velocity. In IABSE Symposium Report 2014 (Vol. 102, No. 42, pp. 193–198) 8. Constantinou, M.C., Tsopelas, P., Hammel, W., et al.: Toggle-brace-damper seismic energy dissipation systems. J. Struct. Eng. 127(2), 105–112 (2001)

Chapter 46

Viscous-Tuned Hybrid Structural Vibration Mitigation System: Wind-Induced Response Analyses Method and Case Study Yue Yang, Xin Zhao and Weixing Shi Abstract The wind-induced vibrations of super tall buildings become excessive due to strong wind loads, super building height and high flexibility. Tuned mass dampers (TMDs) and viscous dampers have been widely used to control vibrations for actual super tall buildings for decades. Sometimes people tend to use TMDs and viscous dampers simultaneously, for example the space is not sufficient for TMDs. An innovative supplemental damping system which is composed of both viscous damper and TMD, namely viscous-tuned damper system (VTD), is proposed to control the wind-induced vibrations of tall buildings. The governing equations are generated for the motion of both the primary structure and the VTD and solved to anticipate the dynamic response of the VTD-structure system. A two story frame is studied to verify the design method. The wind-induced vibration mitigation efficiency of super tall buildings equipped with both viscous dampers and tuned mass dampers (VTD) are investigated in this study. The results show that the VTD system has sound vibration mitigation capacity. Considering the practical engineering situation, the VTD system is a competitive option for wind-induced vibration control of super tall buildings. Keywords Hybrid vibration mitigation · Viscous damper · Tuned mass damper · Parameter optimization

46.1 Introduction The wind-induced vibrations of tall buildings become excessive due to strong wind loads, super building height and high flexibility. Tuned mass dampers (TMDs) and viscous dampers have been widely used to control vibrations for actual structures for decades. Sometimes people tend to use TMDs and viscous dampers simultaneously, Y. Yang · X. Zhao (B) · W. Shi Tongji University, No. 1239 Siping Road, Shanghai 200092, China e-mail: [email protected] Y. Yang · X. Zhao Tongji Architectural Design Group, No. 1230 Siping Road, Shanghai 200092, China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_46

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for example the space is not sufficient for single type devices. The need for VTD is discussed in Sect. 46.2.3 in detail. The study in hybrid vibration mitigation is rare. Xu proposed that with the combined application of TMD and TLCD, the structural vibration will cause TMD motion and TLCD motion. When the two movement is not synchronous, the sloshing of liquid column may inhibit damper system [1]. Zhao and Wang introduced collaborative loss factor to evaluate the damping performance of combination of TMD and TLCD [2, 3]. In application area, Chen analyzed the vibration control effect of TMD and viscous dampers in Tianjin R & F Guangdong Mansion [4]. The optimal parameters of VTD including frequency ratio of sub structure system, mass ratio of sub structure system, damping ratio of sub structure system and damping ratio of main structure are analyzed in this article. Different from the previous studies, which aim to minimize the vibration response under one or several given excitation, the control target of this study is to minimize the peak vibration response under arbitrary input, for the response is really sensitive to the input near the resonate zone. And in this paper the viscous damper is applied as additional damping coefficient of main structure, by which way the overall discipline that can guide the system design will be highlight. The governing equations are generated for the motion of both the primary structure and the VTD and solved to anticipate the dynamic response of the VTD-structure system. Optimal system parameters are also deducted in this article.

46.2 Hybrid Application of Tuned Dampers and Viscous Dampers 46.2.1 Viscous Dampers Viscous damper with small velocity index has been widely used in the vibration control of super high-rise building, dynamic energy can be consumed under relatively small wind load while the damping force under earthquake will not increase too fast to reduce the impact on the main structure. Also viscous damper do not introduce additional stiffness on the main structure, which means additional internal force input is avoided [5]. The viscous dampers can be applied both for the earthquake and wind load vibration mitigation.

46.2.2 Tuned Dampers Typical tuned damping devices including tuned mass damper (TMD), tuned liquid damper (TLD), tuned liquid column damper, and so on. This kind of devices is simply and easy to implement in structure. But it calls for high accuracy of frequency

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estimation and is fit to control the single frequency resonant response [6]. Tuned damping generally control the first mode of structure when served to decline the wind vibration, for the wind load can mainly excite the first mode of structure. While the pendulum type TMD and tuned liquid damper is usually only used for structural human comfort response control rather than control strong wind vibration for the dynamic characteristics deviation of the damper under the large deformation and the violent shaking.

46.2.3 Hybrid Application Requirement In engineering application, the need for mix use of tuned damping and energy dissipation system can be summarized as follows: Firstly, from the point of view of loading, pendulum TMD or TLD is not suitable for earthquake and strong wind control, so the viscous damper should be added when the earthquake and strong wind control is needed. Secondly, from the perspective of cost control, TLD is much cheaper. Thirdly, for the space limit of architecture, single control method may be not able to deduct the response to the limit, the combination of several control methods are more effective to satisfy different requirements simultaneously.

46.2.4 Derivation of Frequency Response Function As shown in the picture, the structure is assumed as a lumped mass with damping and stiffness, the TMD is assumed as an added mass with damping and stiffness, and the viscous damper is assumed as additional damping coefficient which is added to the main structure damping coefficient (Figs. 46.1, 46.2). Motion equation of two degree of freedom system: m u¨ + cu˙ + ku = F(t) + (cd u˙ d + kd u d )

(46.1)

m d u¨ d + cd u¨ d + kd u d = −m d u¨

(46.2)

Fig. 46.1 Hybrid damping system

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Fig. 46.2 Lumped mass model of the system

Table 46.1 The symbols and parameters in this system

Excitation

ρ = /ω ˆ 4 Fπ F¯ = 2 ω Dm

Main structure system

Sub structure system

Reaction

ω2 = k/m c ξ= 2ωm

m¯ d = m d /m  ωd2 = kd m d

uπ ˆ 4 D uˆ d π 4 u¯ d = D

f d = ωd /ω cd ξd = 2ωd m d

u¯ =

ˆ it . The response can be written as: Adopt spectral analysis method: F(t) = Fe u = ue ˆ it , u d = uˆ d eit , and u t = uˆ t eit . Introduce the following symbols shown in Table 46.1. The equation is written as follows:     1 − ρ 2 + 2iρξ u¯ − 2iρξd f d m¯ d + f d2 m¯ d u¯ d = F¯

(46.3)

  −ρ 2 m¯ d u¯ − ρ 2 m¯ d − 2iρξd f d m¯ d − f d2 m¯ d u¯ d = 0

(46.4)

and solved as follows: Frequency response function of structure

H TMD

 2 ρ 2 − f d2 + 4ρ 2 ξd2 f d2  = C12 + C22

(46.5)

Frequency response function of TMD ρ2 HdTMD =  C12 + C22

(46.6)

C1 = m¯ d ρ 2 f d2 − ρ 4 + ρ 2 + ρ 2 f d2 − f d2 + 4ρ 2 ξ ξd f d

(46.7)

where:

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(46.8)

46.3 Optimal Parameters Analysis The five independent parameters except ρ = /ω listed in the parameter table are analyzed respectively to get the optimal response of the main structure. For the excitation is broadband, the optimal response refers to the minimal peak response value under arbitrary excitation rather than minimal response value under one given excitation. It means that the frequency ratio of excitation ρ = /ω will take any value. As for the other four parameters we assign three parameters based on experience then study the other one.

46.3.1 Frequency Ratio of Sub Structure System and Optimal Response We assign m d = 0.04, ξ d = 0.1 and ξ = 0.01 · f d varies between 0.8 to 1.2. The response spectrum figure is shown as below. Comparison of Fig. 46.3a, b and c can be concluded as follows: (1) When the excitation frequency ratio is about 0.75–1.3, the structural response is significantly enlarged. (2) Structural response is affected by frequency ratio of substructure system f d . When ρ = 1 the response is significantly reduced by TMD, but when ρ deviate from 1, e.g. ρ = 0.95, f d = 0.8, the response is enlarged significantly. So there will be an optimal f d which can make the peak response minimal. It is the value who can make the left peak value equals to the right peak value. For example, in Fig. 46.3 the point (0.96, 1.075, 7.409) is the minimal peak response of the structure. And the f d = 0.96 Near the optimal response, the peak response curve is steep, which is very sensitive to f d . That means the frequency ratio of TMD has significant influence on the damping effect. Thus, there is one f d minimize the peak response.

46.3.2 Damping Ratio of Sub Structure System We assign m d = 0.04, f d = 0.96 and ξ = 0.01 · ξd varies between 0 to 1. The response spectrum figure is shown as below (Fig. 46.4).

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(a) Structure response without TMD (2d) (b) Structure response with TMD (3d)

(c) Structure response with TMD (2d)

(d) Structure response with TMD (2d)

Fig. 46.3 Response spectrum of frequency ratio of sub structure system

(a) Structure response with TMD (3d) (b) Structure response with TMD (2d)

(c) Structure response with TMD (2d view) Fig. 46.4 Response spectrum of damping ratio of sub structure system

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When f d = 0.96 (the optimal response), the peak structural response is closely related to the damping ratio of substructures. When the damping of substructure is small, the responses change along with damping ratio and frequency ratio sensitively. The robustness of structure control can be improved by increasing the damping ratio of substructures. When the damping ratio is greater than 0.15, the response of the structure increases along with damping ratio. When the damping ratio is infinity, the TMD acts as if it were fixed to the main structure. An optimal ξd can minimize the peak response.

46.3.3 Mass Ratio of Sub Structure System We assign ξd = 0.1, f d = 0.96 and ξ = 0.01 · m d varies between 0 to 0.15. The response spectrum figure is shown as below (Fig. 46.5). The optimal response is obtained at m d = 0.04. When the m d changes, the peak value is no longer equal, which means that the optimal frequency ratio changes.

(a) Structure response with TMD (3d)

(b) Structure response with TMD (2d)

(c) Structure response with TMD (2d view) Fig. 46.5 Response spectrum of mass ratio of sub structure system

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When the mass ratio is 0, it is equivalent to no TMD structure. When the mass is relatively small, the structural response varies significantly with the mass ratio. When the mass ratio increases to a certain extent, the structural responses no longer change obviously with the mass ratio. There is one optimal m d that can control the structure response without too much material consumed.

46.3.4 Damping Ratio of Main Structure System We assign ξd = 0.1, f d = 0.96 and m d = 0.04 · ξ varies between 0 to 0.1. The response spectrum figure is shown as below (Fig. 46.6). When the damping ratio of the main structure increases, the left and right peak values are no longer equal, and the optimal frequency ratio will change. One can see that the optimal frequency ratio is related to the damping ratio of the main structure. When the damping ratio of the main structure increases, the peak responses of the structure decrease monotonically. Thus, increase the viscous damping in the VTD system is always helpful.

(a) Structure response with TMD (3d)

(b) Structure response with TMD (2d)

(c) Structure response with TMD (2d) Fig. 46.6 Response spectrum of damping ratio of main structure system

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46.3.5 Solving Optimal Parameters Optimal parameters when main structure system has no damping ratio T MD Know ξ = 0, the peak response H peak can be got through ∂ H∂ρ = 0. From the left 1 peak equals to the right peak, f dopt = 1+m can be obtained. d   T MD ∂ H peak ( f dopt ) 3m d T MD = 0, ξ = and H = 1 + m2d can be obtained. From dopt opt ∂ξd 8(1+m d ) Thus, we can know that the optimal structural response of the undamped main structure depends on the mass ratio of the substructure system. T MD

Optimal parameters when main structure system has damping ratio If the main structure has damping ratio, getting the analytical solution of the above problem, we need to solve the nonlinear equation. Reference [7] gives empirical formula, when controlling the peak acceleration response, the optimal parameters are as follows: 1 + (0.096 + 0.88m d − 1.8m 2d )ξ 1 + md + (1.34 − 2.9m d + 3m 2d )ξ 2

f dopt = √

 ξdopt =

(46.9)

3m d (1 + 0.49m d − 0.2m 2d ) + (0.13 + 0.72m d + 0.2m 2d )ξ 8(1 + m d )

+ (0.19 + 1.6m d − 4m 2d )ξ 2

(46.10)

We can see that these are the optimal parameters for VTD system. The optimal response can no longer be written as analytical solution. But still, the response is determined by ξ and m d for the optimal parameter ξd and f d is determined by them too.

46.4 Conclusions A viscous-tuned hybrid structural vibration mitigation system is proposed in this paper. The governing equations are generated for the motion of both the primary structure and the VTD. Wind induced response analysis method is developed to solve the governing equations to anticipate the dynamic responses of the VTD-structure system. Case study was also investigated in this paper. The following conclusions are obtained: (1) The viscous damper is always helpful in reducing the peak response under arbitrary excitation, which means the VTD system is effective in controlling the vibration.

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(2) As for the optimal parameters, the optimal frequency ratio and damping ratio of TMD is relative to the TMD mass ratio and main structure damping coefficient. An optimal damping ratio can exist to minimize the peak response. There is one economic optimal mass ration that can control the structure response without too much material consumed. The final decision depends on the price of viscous dampers and TMD and the space for these two kinds of devices. (3) A two story frame is studied to verify the design method. The wind-induced vibration mitigation efficiency of super tall buildings equipped with both viscous dampers and tuned mass dampers (VTD) are investigated in this study. The results show that the VTD system has sound vibration mitigation capacity. (4) Considering the practical engineering situation, the VTD system is a competitive option for wind-induced vibration control of super tall buildings.

References 1. Xu, Y.L., Samali, B., Kwok, K.C.S.: Control of along-wind response of structures by mass and liquid dampers. J. Eng. Mech. (J.ASCE) 118(1) (1992) 2. Zhao, X., Wang, L., Zheng, Y.: Combined tuned damper-based wind-induced vibration control for super tall buildings. J. J. Tongji Univ. (Nat. Sci.) 44 (2016) 3. Wang, L.: A combined tuned damper and an optimal design method for wind-induced vibration control for super tall buildings. The Structural Design of Tall and Special Buildings (2016) 4. Cheng, P., Yongqi, C.: Discovery for a new system using fluid viscous damper and tuned mass damper (TMD) together in high-rise structure for wind-induced vibration reduction. J. Build. Struct. 43 (2013) 5. Da-yang, W., Yun, Z., Ye-hua, W., Kun, D.: State-of-the-Art of research and application on structures with viscous damper. J. Earthq. Resist. Eng. Retrofit. 28 (2006) 6. Kaynia, A.M., Veneziano, D., Biggs, J.M.: Seismic effectiveness of tuned mass dampers. J. Struct. Div. ASCE 107, 1465–1484 (1981) 7. Li, A.: Vibration Mitigation and Control of Engineering Structure [M], pp. 236–243. China Machine Press, Beijing (2007)

Chapter 47

Optimal Placement of Friction Dampers in High Rise Buildings Under Seismic Excitation Apetsi Ampiah and Xin Zhao

Abstract In the optimal design of damper placement in buildings, various methods exist. and of such methods exists different forms of algorithms for the optimal placement of damping devices. The use of such algorithms is sometimes complex and design engineers would need accurate and simple to use methods to determine the best placement of damping devices. The method proposed in this paper aims to address this. This is achieved by using the shear deformation of the building as the main criteria of damper placement since the shear deformation is easily computed. With the shear deformation known, the dampers can be sequentially placed throughout the building. In this paper, the design of a ten (10) storey building frame is investigated as a case study using optimally placed friction. The responses are obtained in the time domain using a nonlinear time history analysis of seven (7) random ground motions records. The analysis was run using 3D FEM software alongside a computational routine developed in the Python programming language. The optimal locations of the dampers are found using the sequential search algorithm using the grid shear deformation as the placement criterion. It is shown that the performance of the frame, before and after the use of the friction dampers is greatly improved and a reduction in the time required for the analysis. Furthermore, the optimal placement of dampers is sensitive to the type of seismic excitation and device parameters of friction dampers namely the slip load. Keywords Friction dampers · Optimization · Sequential search algorithm · Seismic excitation

A. Ampiah · X. Zhao (B) Tongji University, No. 1239 Siping Road, Shanghai 200092, China e-mail: [email protected] X. Zhao Tongji Architectural Design Group, No. 1239 Siping Road, Shanghai 200092, China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_47

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47.1 Introduction There has been a vast number of research works on reducing the seismic response of buildings using friction dampers or viscous dampers. This is because; both friction and viscous dampers are effective in dissipating large amounts of energy. Even though the principles of operation of both devices are different, their combined benefit in a building cannot be understated. Pall and Marsh introduced a new concept of seismic design of steel framed buildings by incorporating sliding friction devices in the braces of the framed building [8]. Lee et al. proposed a seismic design methodology of friction dampers based on the storey shear force distribution of an elastic building structure [4]. Fu and Cherry proposed a lateral force procedure for the design of friction damped structures [2]. Seong and Min developed an analytical approach for the design of structures equipped with friction dampers [9]. This was achieved by transforming the analysis model into an equivalent mass-spring-dashpot system by approximating nonlinear friction damping force with an equivalent viscous damping force. Mualla and Belev introduced a new type of rotational friction damper with a variable slip moment [7]. Miguel et al. implemented the backtracking search optimization algorithm for the simultaneous optimization of force and placement of friction dampers [6]. Whittle et al. compared the effectiveness of five viscous damper placement methods for improving seismic design [11]. McNamara and Taylor showed the application of fluid viscous dampers in high-rise buildings to reduce responses against wind-induced accelerations [5]. Works done by Aydin et al. illustrated the variations of optimal damper placement using different objective functions [1]. Halperin et al. showed the use of Newton’s optimisation method for enhanced optimal viscous damper design for seismically excited structures [3]. Experimental simulations on a scaled model were carried out by Symans and Constantinou to analyse the behaviour of passive fluid viscous dampers [10].

47.2 Optimal Design of High Rise Buildings with Friction Damper Inspired by the friction brakes used in automobiles, A. Pall invented the friction damper to be used for buildings. Friction dampers are the most common means used in dissipating the kinetic energy from structures. The design process of a high rise building with dampers can generally be divided into two main steps. In the first step, the structure is conventionally designed first based on best engineering judgement and is then assumed fixed while the damper is designed to satisfy a given performance requirement. In the second step, the damper and the structure are then redesigned to achieve a common goal as prescribed by the performance objective used in the first step.

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For the friction damper to be very effective, it should be placed in a location that results in the best response required by some given design conditions e.g. inter storey drift, storey acceleration or base shear. Various methods exist for the optimal placement of dampers and this decision falls on the design engineer as to which method would be favourable. In this paper, a detailed description of the damper placement method is described in Sect. 47.3. Estimating the damper number has a close correlation to the damper positions. Ideally, once the damper positions are determined it follows that the damper number is known unless there is a constraint on the total number of dampers that can be installed. If there is a constraint on the total number of dampers that can be installed, there exist a number of algorithms that can be applied to determine the best friction damper parameters for the structure under the given constraint. The friction damper works best when placed at the location which tends to cause it to dissipate the greatest amount of energy. The shear deformation of the grid or placement location provides a close enough approximation for this quantity and since it is easily computed it makes for a good analysis case.

47.3 Optimal Placement of Friction Damper The sequential search algorithm can be conceptually described as follows: at the onset, the seismic response of the bare structure (with no dampers added) is obtained by performing a non-linear time history analysis, using ground acceleration history compatible with the expected seismic events at the given site. The optimal location indices are given by: τi = [(u b )i − (vb − va )i .Hi /L i ]

(47.1)

where τ i is the optimal location index of the ith storey, (ub)i is the horizontal relative deformation of the joint b to the joint d, va and vb are the vertical relative deformation of the joint a and b to the joint d pertaining to the ith storey, H i is the storey height and L i is the span of the grid (Fig. 47.1). The greatest value of τ i indicates the optimal placement location of the first damper. The stiffness and/or damping properties of the first damper are included in the mathematical model of the structure and new optimal location indices are determined by performing non-linear time history analysis of the newly revised model. The optimal location for the second damper is evaluated again using Eq. (47.1) above. Fig. 47.1 Grid for damper placement

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The procedure is repeated until all the available damper locations have been filled. The theory behind this is that, the maximum shear deformation has a correlation with the maximum energy dissipated by the friction damper (Figs. 47.2 and 47.3).

Fig. 47.2 Design flow chart for optimal placement

Fig. 47.3 Variation of slip load and performance

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47.4 Case Study In this study, the building structure considered was a ten (10) storey steel building which can be considered as a medium rise building. The frame consists of a storey height of 3.5 m and bay width of 6 m. The structure is shown in Fig. 47.4 below with the available damper placement locations. Where the numbers in the grid represent the grid of the storey which is available for damper placement. Previous studies done by Lee et al. [4] showed that the optimal slip load of the damper has a correlation with the storey shear. In the works done by the aforementioned authors, it was proven that the response performance of the structure did not improve once the slip load exceeded thirty (30%) per cent of the storey shear. Seven (7) random ground motions are used to perform the numerical simulations in this study. A plot of these ground motions are shown below and will be subsequently referred to by the given names i.e. simply as TH1, TH2, TH2, TH3, TH4, TH5, TH6, TH7. It must be noted that the ground motions used are classified as frequent earthquakes by the Chinese Code for Seismic Design (GB50011-2010) (Fig. 47.5 and Table 47.1).

Fig. 47.4 Frame under investigation

Fig. 47.5 Ground motion records used for the analysis

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47 Optimal Placement of Friction Dampers … Table 47.1 Summary of damper properties

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Damper name

Slip load (N)

TH1

12812.90

TH2

15115.30

TH3

14253.94

TH4

14190.01

TH5

10927.42

TH6

10683.25

TH7

16684.33

The analysis of the structure was carried out using 3D FEM SOFTWARE. Also, since the design process is highly iterative, a program was written in the Python programming language to interface with the 3D FEM SOFTWARE program. This is possible by using the Application Programming Interface (API) of 3D FEM SOFTWARE (Figs. 47.6 and 47.7, Table 47.2).

Fig. 47.6 Variation of drifts with number of dampers

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Fig. 47.7 Storey drifts for optimal damper placement and number

Table 47.2 Percentage reduction in drift Number of dampers Ground motion

1

2

3

4

5

6

7

8

9

10

TH1

33.05

28.80

38.31

41.29

59.22

64.77

72.04

80.20

81.02

80.60

TH2

−1.85

14.41

21.84

31.69

33.61

54.40

68.65

74.65

75.98

75.46 70.68

TH3

24.02

−24.71

29.24

51.42

60.55

62.71

63.33

69.54

71.18

TH4

19.37

19.98

22.35

22.53

17.68

47.55

59.99

66.67

67.35

67.96

TH5

−9.17

21.53

31.79

27.11

20.08

40.91

57.09

55.79

59.61

59.53

TH6

−41.93

1.84

12.75

38.03

18.86

29.63

47.81

61.40

65.16

64.66

TH7

−7.72

5.33

25.88

30.78

28.16

50.54

55.50

66.66

68.24

67.54

The optimum number of dampers was chosen with the criteria of satisfying the drift limit requirement as specified in the Chinese Code for seismic design. According to the code, the drift limit should not exceed 1/550. This drift limit is applied to all the ground motion and various analysis carried out. A summary of the optimum number of dampers to meet drift requirement are shown below (Table 47.3).

47.5 Conclusion In this paper, an optimal design procedure for friction dampers is used to improve the performance of the structure during a seismic event. An optimal placement method is also developed to find the best places to install the friction dampers. The following conclusions can be obtained in this paper:

47 Optimal Placement of Friction Dampers … Table 47.3 Final damper positions and slip loads

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Ground motion

Number of dampers

Slip load

Locations

TH1

6

12812.9

TH2

6

15115.3

TH3

8

14253.94

2,3,4,5,6,1,7,8

TH4

6

14190.01

2,4,3,5,6,1

TH5

6

10927.42

2,4,3,5,6,1

TH6

7

10683.25

2,4,3,6,5,1,7

TH7

8

16684.33

2,4,3,5,6,1,7,8

2,4,3,5,1,6 3,2,4,5,6,1

(1) The damper location is highly dependent on the ground motion used for the analysis; (2) The best locations for the placement of the dampers usually start at the lower storeys when the shear deformation of the placement grid is used; (3) The design procedure using the shear deformation is very simple and can be easily applied to any engineering project.

References 1. Aydin, E., Boduroglu, M.H., Guney, D.: Optimal damper distribution for seismic rehabilitation of planar building structures. Eng. Struct. 29(2), 176–185 (2007). https://doi.org/10.1016/j. engstruct.2006.04.016 2. Fu, Y., Cherry, S.: Design of friction damped structures using lateral force procedure. Earthq. Eng. Struct. Dynam. (2000). https://doi.org/10.1002/1096-9845(200007)29:7%3c989:AIDEQE950%3e3.0.CO;2-7 3. Halperin, I., Ribakov, Y., Agranovich, G.: Optimal viscous dampers gains for structures subjected to earthquakes. Struct. Control. Health Monit. (2016). https://doi.org/10.1002/stc.1779 4. Lee, S.H., Park, J.H., Lee, S.K., Min, K.W.: Allocation and slip load of friction dampers for a seismically excited building structure based on storey shear force distribution. Eng. Struct. 30(4), 930–940 (2008). https://doi.org/10.1016/j.engstruct.2007.03.020 5. McNamara, R.J., Taylor, D.P.: Fluid viscous dampers for high-rise buildings. Struct. Des. Tall Spec. Build. 12(2), 145–154 (2003). https://doi.org/10.1002/tal.218 6. Miguel, L.F.F., Miguel, L.F.F., Lopez, R.H.: Simultaneous optimization of force and placement of friction dampers under seismic loading. Eng. Optim. (2016). https://doi.org/10.1080/ 0305215X.2015.1025774 7. Mualla, I.H., Belev, B.: Performance of steel frames with a new friction damper device under earthquake excitation. Eng. Struct. 24(3), 365–371 (2002). https://doi.org/10.1016/S01410296(01)00102-X 8. Pall, A.S., Marsh, C.: Response of friction damped braced frames. J. Struct. Div. ASCE (1982). Retrieved from http://www.palldynamics.com/fr/pdf/40Pall_doc1.pdf

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9. Seong, J.Y., Min, K.W.: An analytical approach for design of a structure equipped with friction dampers. Procedia Eng. 14(Supplement C), 1245–1251 (2011). https://doi.org/10.1016/j. proeng.2011.07.156 10. Symans, M.D., Constantinou, M.C.: Passive fluid viscous damping systems for seismic energy dissipation. J. Earthq. Technol. (1998) 11. Whittle, J.K., Williams, M.S., Karavasilis, T.L., Blakeborough, A.: A comparison of viscous damper placement methods for improving seismic building design. J. Earthq. Eng. (2012). https://doi.org/10.1080/13632469.2011.653864

Chapter 48

Prospect of Using Nano Particles in Compatible Water for EOR Application M. Al-Samhan, F. Jasim, F. Al-Attar and J. AL-Fadhli

Abstract When different waters are mixed, it is necessary to evaluate their compatibility prior to the injection into the reservoir. The individual waters may be quite stable under all system conditions and present no scale deposition problems. However, once they are mixed, the reaction between ions dissolved in the individual waters may form insoluble products that cause permeability damage in the near wellbore area. In addition, the properties and composition of contaminants in produced water vary considerably in different geological formations. Many studies are currently focusing on adding Nano Particles (NP’s) to the injected water to enhance its characteristics that consequently maximize oil recovery by increasing wettability or decreasing Interfacial Tension (IFT). The most commonly used nano particles for water flooding are Aluminium oxide (Al2 O3 ) and other metal oxides, the selection of the suitable NP depends on the water mixture and the reservoir rock properties. This paper will cover compatible water preparation and analysis, in addition description of NP’s and their expected effect on enhance oil recovery. Keywords Nano particles · Water injection · Water compatibility · Scale

48.1 Introduction Kuwait is one of the major oil-producing countries with an estimated oil production of around three million barrels/day. The increased oil production has resulted in the production of large amounts of produced water that is causing a major problem to Kuwait Oil Company. This huge amount of produced water deserves a special attention and a study of it characteristics to understand and determine how it can be treated and later used for suitable application such as water injection for Enhance Oil Recovery (EOR) or environmental concerns. Two of the more difficult problems in designing proper water for EOR operation are the predetermination of chemical M. Al-Samhan (B) · F. Jasim · F. Al-Attar · J. AL-Fadhli Petroleum Research Center, Kuwait Institute for Scientific Research, P.O. Box 24885, 13109 Safat, Kuwait e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_48

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incompatibilities of waters used in the flood and the forecast of these incompatibility effects on future field operations [1, 2]. When different waters are mixed, it is necessary to evaluate their compatibility prior to the injection into oil wells [3]. Jar test is conventionally performed in the laboratory to estimate formation damage by scale formation in the wellbore and facility [4, 5]. The individual waters may be quite stable under all system conditions and present no scale problems. However, once they are mixed, the reaction between ions dissolved in the individual waters may form insoluble products that cause permeability damage in the vicinity of the wellbore [6]. It is well documented that many factors influence the produced water properties (physical and chemical), such as the geographic location of the field, the geologic formation from which the water was produced, and the type of hydrocarbon product being produced [7]. In some hydrocarbon reservoirs where water injected into the hydrocarbon formation to increase hydrocarbon production, it was noticed that the properties of the produced water may change drastically as a result of chemical reaction between the injected waters. Having already known the main produced water constituents, the following characteristics should be analyzed and measured: – The salt content, oil and grease, inorganic and organic toxic compounds and naturally occurring radioactive material (NORM) Solving scaling formation problems when mixing water is not the ultimate goal, as increasing oil recovery is the main purpose for applying water injection technique. Many studies are currently focusing on adding Nano Particles (NP’s) to the injected water to enhance its characteristics. NP’s are defined as small objects that behave as a whole unit with respect to its transport and properties, and they are classified according to diameter. Therefore these nano particles, when mixed with water with an expectable level of suspension, it is anticipated that it will maximize oil recovery by increasing wettability or decreasing Interfacial Tension (IFT) [8–10]. Some studies have shown that using “nano-water” instead of water alone in the wetting cycle of the process enhances oil recovery. It has been found that in a six-monthlong cycle with five months of nano-water, oil recovery rose by 13% compared to conventional alternative water and gas enhancement [11]. Most of the NP’s study has related the increase of the oil recovery to the modification of interfacial tension and rearrangement of wettability [12, 13]. This paper systematically investigates water incompatibility by mixing sea water with effluent water indifferent ratios and predicts their performance in terms of scale formation. In addition the paper will reflect on the recent updates of using nano particles in water for enhancing oil recovery.

48.2 Experimental Procedure and Materials Sea and Effluent Water samples were collected from Kuwait Oilfield’s and analyzed in the Petroleum Research Centre Laboratories, KISR. The physical characteristics of the water samples were determined by ICP analysis. All selected water mixed

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ratios were analyzed before and after mixing to measure any changes in the major characteristics of the water mix. The detailed physical characterization of water from these oilfields is necessary in predicting the possible effects of water scaling and corrosivity. Experimental steps are described below: • Before pouring the sample in the cell, the system was cleaned and rinsed with DI water. Then the Autoclave cell was filled with 1.2 L of water sample. The sample was prepared and filtered before mixture, Sea water and Effluent water samples were collected from field and mixed with different Ratios of (50:50, 60:40), respectively. • A high pressure Nitrogen Cylinder was connected to the inlet valve, then the system was purged with Nitrogen to remove air bubbles through the outlet, and after 3 min the outlet valve was closed. Using the pressure regulator the Nitrogen pressure f was set to 1000 Psi. The temperature was set to 75 °C from the controller and the stirrer to 60 RPM and left for 3 h until Temperature stabilization. After 24 h the stirrer was switched off, temperature was decreased to 25 °C and the pressure was released carefully. After subjecting the water mix to downhole conditions, samples were collected and the filtered precipitates were characterized by Energy Dispersive Spectroscopy (EDS), X-ray diffraction (XRD) to get the information about crystal structure, where EDS gives the information about composition analyses. Scanning electron microscopy (SEM) was used for micrographs of the precipitates.

48.3 Results and Discussion Various parameters such as ionic composition of the water, chemical incompatibilities as in scales, precipitates and suspended solids content were described as the potential quality factors in the injection water process. The obtained results from this study indicated that the main constituents of the scale deposited at a typical water mixing ratio were strontium sulphate, calcium carbonate, and barium sulphate. The EDS results for different mixed ratio of waters showed that the composition of the 60:40 sample was the best mix in terms of Fe and Ca deposition Figs. 48.1 and 48.2. However, the other characteristics are almost similar composition, that could be attributed to the fact that they were obtained from the same oil field. Scales and precipitates can be formed in the produced waters from a number of root causes. Changes in the temperature and pressure of the produced waters, as it comes to surface, may initiate pH changes, which may begin the formation of scales. The scale filtered samples were examined by SEM and XRD to observe the particle size and morphology of the precipitates. The formations of CaSO4, SrSO4, and BaSO4 were detected by SEM micrographs Figs. 48.3 and 48.4. The possible formation of scale is clearly synergized with the results produced from EDS, especially the calcium sulphate. The XRD analysis of the membranes showed barium sulfate scale for 50/50 samples Fig. 48.5, while in other samples

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Fig. 48.1 EDS composition analysis of water mix (50:50)

Fig. 48.2 EDS composition analysis of water mix (60:40)

(60/40) was not detected because of an amorphous structure, since XRD detects crystalline structures. Overall, mixed ratio of waters introducing the typical scale-inducing anions like sulphate and carbonate can cause carbonate and sulphate type scales, especially in sea water, where sulphate concentrations are high. However, regarding the compatibility

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Fig. 48.3 Micrograph of detected sulphates in the water mix (50:50)

Fig. 48.4 Micrograph of detected sulphates in the water mix (60:40)

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Fig. 48.5 XRD results for (50:50) precipitate of calcium sulfate hydrate (blue), iron oxide hydroxide (grey), aragonite (red)

test procedure used in this study, it is important to mention that high pressure was a key variable added to learn about the actual behavior of water when subjected to downhole conditions.

48.4 Compatible Water with NP’s for Enhancing Oil Recovery Solving the scaling problem is not the only target when the water injection technique is discussed, it is also of high importance to increase oil recovery. Many studies are currently focusing on adding Nano Particles (NP’s) to the injected water to enhance its characteristics to maximize oil recovery by increasing wettability or decreasing Interfacial Tension (IFT). However, NP’s like any other enhancing additive come in many grades and types that vary in effect depending on rock formation and the recovered oil API. The most commonly used nano particles for water flooding are Aluminium oxide (Al2 O3 ), Silicon oxide (SiO2 ), and Nickel oxide, depending on the water mixture and the reservoir rock properties [12–15]. Wang et al. [16] observed that the addition of nanoparticles to the injected fluid can enhance oil recovery by altering the wettability pattern of the reservoir rock. The study also revealed that nanoparticles may also serve as inhibitors for asphaltene precipitation. The roles of interfacial forces were believed to be significantly altered in nanofluids rather than in pure liquids due to the spontaneous phenomenon of nanoparticle adsorption at bubble

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interfaces. Because of the nanoparticle layer, bubbles submerged in nanofluids would partially behave like a rigid sphere and develop a rotation movement [17]. The main mechanism for the spreading of nanofluids to produce oil, which was investigated by Wasan [18], Chengera et al. [19], and Wasan et al. [20], is called the structural disjoining pressure gradient (Aµ), which is directed towards the wedge from the bulk solution. Using artificial cores which simulated geologic conditions of a certain factory of a selected oilfield, decompression and augmented injection experiments of SiO2 nano-powder were performed after waterflooding, best injection volume of SiO2 nano-powder under the low-permeability condition was selected. It has shown that SiO2 nano-powder inverted the rock wettability from hydrophilicity to hydrophobicity [21]. The best injected concentration is respectively 0.5%, the corresponding value of EOR is 6.84% and decompression rate is 52.78%. Due to the combination of many factors such as type of water, rock formation, oil API, NP’s concentration, and the geographical concern further studies and continues investigation in this is required to contour the optimum selection for the water injection. Moreover, the deep understanding of the NP’s characteristics when mixed with the injection water and behavior in the rock formation will enable in the future the design of NP’S to fit exactly the needed task.

48.5 Conclusion One of the primary concerns in the implementation of an effective waterflood is the compatibility of the water to be injected. It is important to mention that in the compatibility test procedure, high pressure was a key variable added to better understand the extent of the formation of insoluble products during water injection in the rock formation. The results obtained for different mixed ratio of waters showed that the composition of 60:40 sample was the best mix in terms of Fe and Ca deposition (scale formation). The formations of CaSO4, SrSO4, and BaSO4 were detected by SEM micrographs. Many studies are currently focusing on adding Nano Particles (NP’s) to the injection compatible water to enhance its characteristics for maximum oil recovery. Enhancing recovery by adding common NP’s such as Aluminium oxide (Al2 O3 ), Silicon oxide (SiO2 ), and Nickel oxide is believed to happen through wettability alteration and IFT modification. The roles of interfacial forces were believed to be significantly altered in nanofluids rather than in pure liquids due to the spontaneous phenomenon of nanoparticle adsorption at bubble interfaces. The type of water, rock formation, oil API, NP’s concentration, and the geographical concern further studies and continue investigation in this direction is required to contour the optimum selection for the water injection.

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Chapter 49

Performance-Based Design Optimization of Steel Braced Frame Using an Efficient Discrete Algorithm X. Wang, Q. Zhang, X. Qin and Y. Sun

Abstract Performance-based design optimization (PBDO) of steel braced frames (SBF) is a computationally intensive task, especially when nonlinear time history analysis is applied. In this paper, an efficient discrete optimization algorithm is proposed for PBDO of SBF utilizing a deformation-based method. Two difficulties exist in PBDO of SBF, and the first one is that multiple performance constraints are imposed on the structures. To avoid tackling all constraints simultaneously, a strategy is proposed in which the deformation constraints of beams and columns are strictly followed throughout the optimization process and the brace deformation constraints are checked again at the end of the optimization. The second difficulty is that the search for an optimum design is conducted in a discrete design space since the structural elements are usually taken from standard sections. A common practice is to use regression models for the sections, at the expense of removing sections that cannot fit into the regression models from the design space. In this paper all standard sections are preserved by using the cross-sectional area (Area) and moment of inertia (Ix) as the design variables, thus any standard section can be uniquely defined by its Area and Ix. To investigate the effectiveness of the proposed algorithm, three numerical examples are presented. Compared to the results achieved by genetic algorithm (GA) and differential evolution (DE), the proposed algorithm can achieve better or comparable structural designs. Furthermore, the convergence rate of the proposed algorithm is much higher than GA and DE, proving that the proposed algorithm is an efficient optimization method for PBDO of SBF. Keywords Performance-based design optimization · Steel braced frame · Efficient discrete algorithm

X. Wang · Q. Zhang (B) · X. Qin · Y. Sun College of Mechanical Engineering, Tongji University, Shanghai, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_49

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49.1 Introduction Performance-based seismic design (PBSD) is a design methodology that aims at achieving predictable structural performances under given levels of seismic attacks. The structural performances are measured by possible damage states that structures may experience during an earthquake [1]. Structures may experience large deformations when subject to severe earthquakes and nonlinear analysis methods are needed. Basically, there are two types of analysis method, the nonlinear pushover analysis and the nonlinear time history analysis (NTHA). While the pushover method is quite easy in implementation, its accuracy relies to a great extent, on whether or not the assumed distribution of seismic loads [2, 3] can precisely represent the actual distribution of seismic forces in the whole structures. Considering the various dynamic characteristics of earthquake motions and of the structures, the accuracy of any given distribution of seismic forces should be carefully examined. On the contrary, NTHA requires no assumptions on the distribution of seismic loads. Earthquakes can be directly applied to the structures by NTHA and precise seismic responses can be acquired. But much computational time can be produced by NTHA, which severely limits its application in PBSD. Designing a cost-efficient structure with high seismic-resistance capacities is quite complicated since many design variables and constraints are involved, which makes it necessary to apply structural optimization techniques when designing structures. Based on the framework of PBSD, the Performance-Based Design Optimization (PBDO) has been developed and much research work has been done over the past decades [4–7]. A great part of these researches on PBDO of structures rely on metaheuristic algorithms for solving PBDO problems. Fragiadakis et al. [7] considered the initial structural cost and the lifecycle cost due to future earthquakes as the optimization objectives and proposed an evolution strategy-based algorithm to solve the multiobjective problem. Farhat et al. [7] applied genetic algorithm (GA) in optimizing the sizes of buckling restrained braces for seismic enhancement of existing structures. GA was also adopted by Apostolakis et al. [5] to find the optimal distribution of buckling restrained braces and friction dampers within steel frames. Other metaheuristic algorithms like particle swarm optimization [4], dolphin echolocation optimization [8, 9], bee colony algorithm [10] and ant colony optimization [11] have also been employed in solving PBDO problems. PBDO problems are usually discrete, nonlinear and complicatedly constrained. Conventional optimization methods such as the gradient-based methods and optimality criteria perform poorly when applied to these problems. Metaheuristic algorithms are capable of handling these problems and therefore are frequently used by researchers. But metaheuristic algorithms are troubled with their slow convergence rates, compared to the gradient-based methods or optimality criteria. This shortcoming becomes a critical problem when expensive structural evaluations are needed in the optimization process.

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To improve the computational efficiency, Kaveh et al. [12] proposed a neural network-based method for estimating the structural responses to earthquake motions and applied this method to the optimal design of steel frames. Similarly, Saeed Gholizadeh [13] utilized a neural network to predict the results of nonlinear pushover analysis and incorporated the neural network into the framework of a firefly algorithm. These two researches took advantage of the neural network for structural evaluation, and the computational time can be considerably reduced. But the neural network requires a careful training and even with a carefully-trained neural network, errors in structural evaluations are still inevitable. The above mentioned methods attempt to reduce the computational time by replacing the expensive structural evaluation with a less expensive one. Another way to cut down the computational time is by accelerating the convergence history of the optimization process, as done in the research by Moghaddam et al. [14]. The uniform deformation method, which belongs to optimality criteria, was proposed which could achieve an optimum distribution of structural properties in a few iterations. This method was found effective for shearing buildings [14, 15], truss-like structures [16] and concrete frames [17]. When this method was applied to steel frames, the computational merit was preserved but its optimum results were inferior to those by metaheuristic algorithms [18]. The shift from truss-like structures to steel frames indicates that constraints on the seismic demands of beams and columns and the inter-story drifts are included in the optimization, making the optimization tasks much more complicated. If several performance levels were considered, the optimization would become much more difficult. Compared to PBDO of steel frames, PBDO of steel braced frames (SBF) also include constraints on the seismic demands of braces. Therefore, PBDO of SBF with several performance levels required becomes a problem incredibly difficult for methods like the uniform deformation method. When designing steel frames and SBF, the structural elements are usually taken from standard profiles defined in design manuals. The standard sections are discretely distributed in the design space, and two common approaches have been developed to deal with the discreteness of the design space. The first approach can be named regression model approach [19–22], which usually establishes a regression relationship between sectional properties and one single sectional property, for example the cross-sectional area (Area) or moment of inertia (Ix), can be used to identify a certain section. In this way the discreteness of sections is removed and gradient-based methods can be applied. But, due to the fact that sections with the same Area may have different sectional sizes, the use of regression models means that sections that cannot fit into the regression relationship are removed from the design space. The other approach leave the discreteness unchanged and sections are directly taken from standard profiles, which implies that methods like the gradient-based methods and optimality criteria cannot be applied and metaheuristic algorithms become a frequent choice [4, 23–25], at the expense of a slow convergence. To overcome this dilemma, a deformation-based method is proposed to speed up the convergence process. The proposed method works directly in a discrete design space defined by Area and Ix of standard sections. In this way no regression models are required and all standard

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sections can be included in the design space. Since the proposed optimization method is similar to optimality criteria, a high computational efficiency can be achieved. To deal with the many constraints involved in PBDO of SBF, in this paper a strategy is proposed to deal with constraints of different types respectively. First, the serviceability constraints are checked and the structure is modified if necessary. Next NTHA is performed and the seismic demands of structural elements are calculated. The structure is then updated using the proposed deformation-based method, in which elements with seismic demands greater than their abilities are amplified and others are reduced. In this way the constraints on the deformations of columns and beams are gradually satisfied. But the deformations of braces are closely affected by the dimensions of beams and columns and a minor change in beams and columns can remarkably change the deformations of braces. Therefore the constraints on the deformations of braces are not strictly treated in the optimization process until the optimization process ends. To verify the effectiveness and efficiency of the proposed algorithm, three SBF are optimized for various performance levels. GA and DE, together with the proposed algorithm, are applied to solving these problems and the results demonstrate that the proposed algorithm can achieve comparable structural designs for simple PBDO problems and better designs for complex PBDO problems. Additionally, the proposed algorithm converges in a much faster speed than GA and DE, proving that the proposed algorithm is efficient in solving PBDO problems of SBF.

49.2 Performance-Based Design Optimization In this paper three performance levels are considered, including Immediate Occupancy (IO), Life Safety (LS) and Collapse Prevention (CP) performance levels, and the corresponding hazardous levels are selected as earthquakes with 20%, 10% and 2% chance of exceedance in 50 years, respectively. The OpenSees [26] platform is used to evaluate the nonlinear time history responses of structures. To achieve the required performance levels several constraints are imposed on the structure, which are explained in detail in the following sections.

49.2.1 Design Constraints The design checks are implemented in the optimization process, which include the serviceability check and the limit state check. For the serviceability check, the nonseismic load combination is considered: T ype1

QG

= 1.2Q D + 1.6Q L

where Q D and Q L are dead loads and live loads, respectively.

(49.1)

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According to the LRFD code [27], the strength constraint is checked for all structural elements for the non-seismic load combination:   Muy Pu Mux Pu −1≤0 (49.2) < 0.2; + + for φc Pn 2φc Pn φb Mnx φb Mny   Muy Pu Pu 8 Mux for −1≤0 (49.3) ≥ 0.2; + + φc Pn φc Pn 9 φb Mnx φb Mny where Pu is the required strength; Pn is the nominal axial strength for tension or compression; Mux and Muy are the required flexural strengths in x and y directions; Mnx and Mny are the nominal flexural strengths in x and y directions; φc and φb are resistance reduction factor. For the limit state check, both the seismic and non-seismic loads are considered. The non-seismic loads can be defined as follows: T ype2

QG

= 1.1Q D + 1.1Q L

(49.4)

The lateral drift constraints at three performance levels can be expressed as follows: gi,δ j =

d ij i dall

− 1 ≤ 0, i = IO, LS, CP, j = 1, 2, . . . , ns

(49.5)

i is the allowable where d ij is the kth story lateral drift at ith performance level; dall values which is chosen to be 0.5%, 1.5% and 2% for IO, LS and CP performance level, respectively; ns is the number of stories. The plastic rotation constraints for columns and beams are as follows:

θ = gi,l

θli − 1 ≤ 0, i = IO, LS, CP, l = 1, 2, . . . , ne i θall

(49.6)

where θl and θall are the plastic rotation and its allowable value for the lth element. In FEMA-356 [1], θall is set θ y , 6θ y , 8θ y for IO, LS and CP performance levels. The axial deformation constraints for braces can be expressed as follows: gl =

ik − 1 ≤ 0, i = IO, LS, CP, k = 1, 2, . . . , nb iall

(49.7)

where ik and iall are the axial deformation of a brace and its allowable value at ith performance level. For braces in tension iall at three performance levels are set 0.25T , 7T and 9T in which T is the axial deformation at expected tensile yielding load, and for braces in compression iall at three performance levels are 0.25C , 5C and 7C in which C is the axial deformation at expected buckling load; nb is the total number of braces.

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49.2.2 Optimization Problem Formulation In this paper the structural cost is to be minimized. The cost of a structure can be measured in various ways, and the most convenient way is through the structural weight. Thus a discrete optimization problem can be formulated as follows:   Find X = x1 , x2 , . . . , xng , xi ∈ D to minimi ze W (X ) =

ne 

ρi Ai L i

(49.8) (49.9)

i=1

subject to g j (X ) ≤ 0, j = 1, 2, . . . , nc

(49.10)

where xi is the design variable for the ith design group and xi is selected from D, which represents the set of standard sections in design manuals; ng is the number of design groups in the structure; W (X ) represents the structural weight; ρi and L i are weight of unit volume and the length of element i, respectively; ne is the number of elements; g j (X ) is the design constraints and nc is the number of constraints.

49.3 Discrete Optimization Algorithm 49.3.1 Discrete Design Space In this study structural elements are chosen from 267 W-shaped sections given in the AISC design manual [28]. As illustrated in Fig. 49.1, the sections are sparsely distributed in the design space and sections with the same Area can have different Ix. If Area is used as the only design variable, only one section with a given Area can exist and many more sections will have to be excluded from the design space. But if Area and Ix are chosen as the design variable, all standard sections can be uniquely determined and thus all standard sections can be included in the design space, which could potentially allow for better structural designs. But the existence of all standard sections definitely increases the difficulty of finding an optimum solution, which in return requires an effective optimization algorithm.

49.3.2 Proposed Optimization Algorithm The proposed optimization algorithm is consisted of two parts, the updating approach and the strategy for tackling constraints. Due to the fact that the deformations of braces are very sensitive to the changes in columns and beams, which is reasonable

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Fig. 49.1 W-shaped sections

since columns and beams bear the majority of seismic forces, constraints on the deformations of braces are not strictly treated until the optimization process ends. The details of the proposed algorithm are explained in the following sections.

49.3.2.1

Updating Approach

Supposed that Aik and I xik denotes the cross-sectional area and moment of inertia of the section assigned to the ith design group at the kth iteration. Before updating the sections, a relationship needs to be assumed between Aik and I xik since I xik cannot take an arbitrary value with a given Aik . This is also the reason why regression models are frequently used [19, 20]. But the use of a fixed regressive relationship is problematic as explained before. Therefore, a variable parameter β is used to control the relationship between Aik and I xik :  β Ixki ∝ Aki

(49.11)

As can be seen in Fig. 49.1, sections with large dimensions are sparsely distributed. As the sectional dimensions get smaller, the distribution of sections in the design space becomes denser. Inspired by the fact that a structural design usually starts with large sections and approaches smaller sections as the optimization proceeds, a fixed β is used at the start of the optimization process and when a better structural solution is found, β is assigned multiple values to cover as many standard sections as possible as the distribution of sections is getting denser. Accordingly, the optimization process can be divided into two stages, the quick exploration stage in which β takes only one value and the refined exploitation stage in which β takes multiple values. In this

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way, the searching efficiency and the searching accuracy is balanced, based on the distribution of standard sections in the design space. The updating approach can be expressed as follows:  ϕi α k = Ai ϕall   ϕi α∗β k = Ixi ϕall 

Ak+1 i Ixk+1 i

(49.12) (49.13)

where ϕi represents the plastic rotation defined in Eq. (49.6) for columns and beams at IO performance level, or the axial deformation for braces defined in Eq. (49.7) at IO performance level; ϕall is the corresponding allowable value at IO performance level; α is the parameter for controlling the searching step size. Similar to the setting of β, α is given only one fixed value in the quick exploration stage and multiple values in the refined exploitation stage. The use of seismic demands at IO performance level is based on the fact that the ratio of seismic demands to seismic capacities at IO performance level is usually larger than ratios at LS and CP performance levels, which means that the ratios at IO performance level are more sensitive to the changes in the design variables. It should be noted that the updating approach in Eqs. (49.12), (49.13) is similar to the uniform deformation method [16, 17], but these two approaches are different in their searching mechanisms. Take a structure with one design group as an example, the search for a next section is done in a one-dimensional space for the uniform deformation method, while for the proposed approach the search for a next section is conducted in a two-dimensional space and the searching direction is to be determined. This is a fundamental different between the uniform deformation method and the proposed algorithm. The updated section using Eqs. (49.12), (49.13) may not belong to the standard sections, and a rounding is performed   , xs ∈ D = min xs − xk+1 xk+1 i i

(49.14)

  where xik+1 denotes the combination Aik+1 , I xik+1 and xs represents (As , I xs ) of a standard section taken from standard profiles D. The steps in the quick search stage can be summarized as follows: (1) Initialization. A randomly selected section is assigned to all structural elements, and the serviceability check is conducted. A re-selection of sections is performed if the serviceability check is failed. (2) Nonlinear time history analysis is performed and the seismic responses at three performance levels are acquired. (3) Check if the lateral drift constraints are satisfied. If not, the optimization process is terminated, otherwise the optimization process continues.

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(4) Update the design using Eqs. (49.12)–(49.14). Here the values of α and β are set 0.05 and 2.0, respectively. (5) Check if the serviceability constraints are reached. If the constraints are violated by any structural element, amplify all structural elements until the structure has the expected serviceability. The amplification coefficient is set 1.01 in a new iteration, and increases in a power law if the serviceability constraints are repeatedly violated in the same iteration. (6) Compare the structural weight in the current iteration to the weight of a previous iteration. If a lighter structural weight is found, the optimization process repeats from Step (2), otherwise the algorithm jumps to the refined exploitation stage. The refined exploitation stage is consisted of the following steps: (1) Re-initialization. The largest section for columns is selected and assigned to all columns. The same is done to beams and braces. Through such a practice the structural design is likely to be moved away from a local optimum. The serviceability check is performed and the design is amplified if necessary. (2) Nonlinear time history analysis is implemented and the seismic responses are evaluated. (3) Verify if the lateral drift constraints are satisfied. Continue the optimization process if satisfied, otherwise the algorithm is terminated. (4) Update the structural design using all values for α and β. Here α is given 0.5, 1.0 and 1.5 times of its original value in the quick search stage, and β is set 1.5, 2.25 and 3.0. (5) The serviceability check is conducted immediately after the design is updated. Whenever the serviceability constraints are violated by a type of elements, for example columns, amplify all columns with an amplification coefficient. If the constraints are violated by more than one type of elements, elements of the same type are amplified. The amplification coefficient is set 1.005 at the start of a new iteration, and increases within the same iteration. (6) Of all the designs derived using different values of α and β, the lightest structural design is used in the next iteration. (7) Compare the updated structural weight to that of a previous iteration. If the updated structure is heavier, amplify α with a coefficient of 0.95, so that the search can be conducted in a more elaborate manner. (8) If the maximum of α is smaller than the value of α in the quick search stage, or if the maximum iteration number is reached, the optimization process is terminated, otherwise the process is continued from Step (2). Throughout the refined exploitation stage, the dimensions of columns and beams keep changing and the distribution of seismic loads within the structure is not fixed. Since the deformations of braces are closely related to the configurations of columns and beams, constraints on the brace deformations may not be satisfied when the optimization process ends. Thus, modifications of the braces are needed to guarantee that the derived structural design satisfies all performance requirements.

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Modifications of Braces

Braces are used to enhance the inter-story stiffness of the frame structure. For example, when only one X-brace is used, of the two braces one is responsible for tension and the other is for compression. Supposed that the deformation ratios of an Xbrace, which is defined as the seismic deformation demand to the corresponding deformation capacity at a given performance level, can be denoted as εi (i = 1, 2), the deformations of these two braces must belong to one of the three cases: (1) ε1 ≥ 1 and ε2 ≥ 1; (2) ε1 ≥ 1 and ε2 < 1; (3) ε1 < 1 and ε2 < 1; For case (49.1) or case (49.3), the braces can be directly enlarged or reduced and the inter-story stiffness is likely to be modified as expected. For case (49.2), if one brace is strengthened and the other is weakened, the inter-story stiffness may be improved or reduced or remain unchanged. If the inter-story stiffness remains unchanged, since the weakened brace can provide less resisting forces, the strengthened brace has to sustain greater seismic loads and the deformation of the braces may still belongs to case (49.2). If the inter-story stiffness is reduced, the brace deformations would increase and still fall into case (49.2). Therefore, a modification is provided by artificially enlarged the deformation ratios of braces ε = max(ε, εmin ), if ε > 0

(49.15)

where εmin is given as a lower bound of deformation ratios, in order to enlarge braces to a certain degree so that the brace deformations are changed to case (49.1). In this study εmin is set 1.5 based on numerous analyses. The updating approach of braces is also modified as = (1 + c*|εi − 1| ∗ (εi − 1))Aki Ak+1 i

(49.16)

Ixk+1 = (1 + c*|εi − 1| ∗ (εi − 1))β Ixki i

(49.17)

where εi is the deformation ratio for braces of the ith design group at IO performance level and c is a coefficient for tuning the brace modification process, which is set 0.2 in this study. β is the same as used in the quick exploration stage. It should be pointed out that Eqs. (49.16), (49.17) are similar to the method proposed for PBDO of steel frames [18]. Again, a fundamental difference exists since the updating of a section is treated as a two-dimensional problem in this paper, instead of a one-dimensional problem. The brace modification process can be summarized into the following steps: (1) The serviceability check is conducted and the design is modified if necessary, which is the same as explained in the refined exploitation stage.

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(2) Nonlinear time history analysis is performed and the seismic responses are evaluated. (3) Check if the inter-story drift constraints are satisfied. Terminate the algorithm if not satisfied, otherwise the process continues. (4) Check if the deformation constraints of columns and beams are violated. Update the columns and beams if necessary. (5) Verify that if the brace deformation constraints are satisfied. If satisfied, keep track of the current design. (6) Check if the terminal condition is reached. In this study, the terminal condition is set as if the number of iteration is greater than a predefined number. If so, terminate the algorithm, otherwise continue this process. (7) The braces are updated using Eqs. (49.16), (49.17) and a rounding of sections is followed. Continue the process from Step (1) until the algorithm satisfies the terminal condition. Before the brace modification process begins, constraints on the serviceability, the inter-story drift and the deformations of columns and beams are all satisfied. The check for these constraints is performed again when the brace modification process is ended.

49.4 Numerical Examples 49.4.1 Problem Description Three SBF are used to investigate the effectiveness of the proposed algorithm. The structural elements are all taken from 267 W-shaped sections in the AISC manual [28]. These frames have rigid connections between columns and beams while the braces are hinge-connected to the frames. A dead load of Q D = 2500 kg/m and a live load of Q L = 1000 kg/m are applied to all beams. The elastic modulus is 204 GPa and the yield stress for columns is 351.53 MPa and for beams and braces is 253.10 MPa. To account for the nonlinear behaviors of structural elements, a bilinear steel material with 3% strain-hardening ratio is applied to beams and columns and for braces, truss elements with a bilinear kinematics hardening rule in tension and a buckling stress in compression are used [24]. The frame structures are assumed to be placed on the site of Class D. To evaluate the structural responses at IO, LS and CP performance levels, earthquakes with 20, 10 and 2% probability of exceedance in 50 years are applied. Usually at least three earthquake records are needed for evaluation of one performance level [29]. Using three earthquake records for a single performance level would result in an extremely great amount of computational burden, especially when three performance levels are considered. In this paper, one synthetic earthquake is used to statistically represent the dynamic characteristics of earthquake records at one performance level. Three synthetic earthquakes of three hazardous levels are produced by SIMQKE [30] and

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Fig. 49.2 Design spectra and response spectra of synthetic ground motions

applied in the optimization process. As can be seen that, the response spectra of the synthetic earthquakes are quite close to the design spectra. Two metaheuristic methods including GA and DE, together with the proposed algorithm are utilized to solve PBDO problems. The proposed algorithm can arrive at different results with different initializations. Therefore, all three methods are run five times for one problem. The mutation scheme for DE is     νi,G+1 = xbest,G + F1 · xr1 ,G − xr2 ,G + F2 · xr3 ,G − xr4 ,G

(49.18)

in which the definitions of νi,G+1 , xbest,G , xr1 ,G , xr2 ,G , xr3 ,G and xr4 ,G can be found in Rainer Storn et al. [31]. The values of F1 and F2 are set 0.5 and 0.3, respectively (Fig. 49.2).

49.4.2 A One-Bay Two-Story SBF The geometry and design groups of the frame are shown in Fig. 49.3. The population size and the maximum number of generations for DE are 20 and 200, and the population size and the maximum number of generations for GA are 20 and 500, since GA converges much slower than DE. The results with the best, average and worst structural weight are presented in Table 49.1. The structural weights achieved by the proposed algorithm are heavier than GA and DE. The reason may be that GA and DE are global optimization methods, while the proposed algorithm is more likely to fall into a local optimum. As can also be seen in the convergence histories that the proposed algorithm converges very fast in the first few iterations and stagnates afterwards. GA and DE present a slightly slower

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Fig. 49.3 One-bay two-story SBF

Table 49.1 PBDO for one-bay two-story SBF

Weight

GA

DE

Proposed algorithm

Best (kg)

1096.00

1104.50

1311.74

Average (kg)

1172.60

1148.46

1505.63

Worst (kg)

1244.15

1197.91

1647.72

Average no. of analysis

8848.0

2396.0

31.0

convergence speed than the proposed algorithm, and acceptable results can also be achieved very fast. After that, better designs can be discovered by DE, but at the cost of a great amount of computational time. The total number of seismic analysis required by the proposed algorithm is only 0.35% of that needed by GA, and 1.29% of that needed by DE (Fig. 49.4).

Fig. 49.4 The convergence history for one-bay two-story SBF

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49.4.3 A Two-Bay Three-Story SBF A two-bay three-story SBF with 27 design variables is shown in Fig. 49.5. The population size and the maximum number of generations for DE are 40 and 300, and the parameters for GA are 40 and 500. The results with the best, average and worst structural weight are listed in Table 49.2. The results achieved by the proposed algorithm are still inferior to those by DE, but better than those by GA. As PBDO problems get more complex, the global searching ability of GA is dampened by the growing number of design variables, and the results achieved at each iteration becomes less competitive, as can be found in the convergence history. Compared to GA, DE is less affected by the growing complexity of problems, and better results can be achieved in less computational time. The convergence rate of the proposed algorithm is the fastest of the three algorithms, and the total number of dynamic analysis needed by the proposed algorithm is 0.2% of that by DE and 0.0875% of that by GA. Obviously, the proposed algorithm becomes more advantageous as PBDO problems get more complex (Fig. 49.6).

Fig. 49.5 A two-bay three-story SBF

Table 49.2 PBDO for two-bay three-story SBF Element group

GA

DE

Proposed algorithm

Best (kg)

5111.08

3426.53

4111.83

Average (kg)

5472.63

3458.71

4350.47

Worst (kg)

5753.05

3508.17

4494.13

7456.0

14.6

Average no. of analysis

16690.0

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Fig. 49.6 The convergence history for two-bay three-story SBF

49.4.4 A Two-Bay Six-Story SBF A two-bay six-story SBF is shown in Fig. 49.7. There are 54 design variables in this problem, and the population size and the maximum number of generations for DE are 80 and 500 respectively. GA is not applied due to its slow convergence rate. The results with the best, the average and the worst structural weight are listed in Table 49.3. As can be seen that the proposed algorithm achieves better results than DE. The convergence speed of the proposed algorithm, as show in Fig. 49.8, is much faster than DE. The computational time required by the proposed algorithm is only 0.0348% of that required by DE. Conclusion can be drawn that, compared Fig. 49.7 A two-bay six-story SBF

582 Table 49.3 PBDO for three-bay six-story frame

X. Wang et al. Element group

DE

Proposed algorithm

Best result (kg)

9277.26

8273.12

Average result (kg)

9620.96

8992.96

Worst result (kg)

10022.13

9546.39

Average no. of seismic analysis

39120.0

13.6

Fig. 49.8 The convergence history for two-bay six-story SBF

to DE, the proposed algorithm can acquire better structural designs with much less computational effort. With a brief comparison of Figs. 49.4, 49.6 and 49.8, it can be observed that the proposed algorithm remains a fast convergence speed for these examples while the convergence speeds of DE and GA gradually slow down, indicating that the searching ability of the proposed algorithm is not remarkably affected by the increasing complexity of PBDO problems while the searching abilities of DE and GA are seriously dampened. Thus the proposed algorithm can be an efficient choice for complex PBDO problems.

49.5 Conclusion In this paper an efficient discrete algorithm is put forward for PBDO of SBF based on a deformation-based updating approach. The optimization searching process is conducted in a discrete design space defined by standard sections. To speed up the optimization process, a two-fold strategy is proposed, which includes a quick exploration and a refined exploitation. The quick exploration is to found an improved structural design in a few iterations and starting from the improved design the refined exploitation continues. Two types of design constraints, which include the service

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ability constraints and the limit state constraints, are considered. Three numerical examples are presented to investigate the effectiveness of the proposed algorithm. Comparison of the results by DE and the proposed algorithm, it can be seen that for the first two examples the structural designs by the proposed algorithm is less satisfactory than DE, but the computational efficiency of the proposed method is far better than DE. And for the third example, the proposed algorithm can arrive at better designs than DE, and with a much higher efficiency. Thus it can be concluded that for simple PBDO of SBF, DE or similar population-based methods are better choices and satisfactory designs can be found using an acceptable computational time. But when the structural complexity grows, the proposed algorithm becomes a preferable choice as better designs can be provided by the proposed method in a much higher efficiency.

References 1. Fema-356: Prestandard and Commentary for the Seismic Rehabilitation of Buildings. Federal Emergency Management Agency, Washington, DC, USA (2000) 2. Ganjavi, B., Hao, H.: Optimum lateral load pattern for seismic design of elastic shear-buildings incorporating soil—structure interaction effects. Earthq. Eng. Struct. Dynam. 42(6), 913–933 (2013) 3. Hajirasouliha, I., Pilakoutas, K.: General seismic load distribution for optimum performancebased design of shear-buildings. J. Earthq. Eng. 16(4), 443–462 (2012) 4. Gholizadeh, S., Moghadas, R.K.: Performance-based optimum design of steel frames by an improved quantum particle swarm optimization. Adv. Struct. Eng. 17(2), 143–156 (2014) 5. Apostolakis, G., Dargush, G.F.: Optimal seismic design of moment-resisting steel frames with hysteretic passive devices. Earthq. Eng. Struct. Dynam. 39(4), 355–376 (2010) 6. Farhat, F., Nakamura, S., Takahashi, K.: Application of genetic algorithm to optimization of buckling restrained braces for seismic upgrading of existing structures. Comput. Struct. 87(1–2), 110–119 (2009) 7. Fragiadakis, M., Lagaros, N.D., Papadrakakis, M.: Performance-based multiobjective optimum design of steel structures considering life-cycle cost. Struct. Multidiscip. Optim. 32(1), 1–11 (2006) 8. Kaveh, A., Farhoudi, N.: Dolphin monitoring for enhancing metaheuristic algorithms: Layout optimization of braced frames. Comput. Struct. 165, 1–9 (2016) 9. Gholizadeh, S., Poorhoseini, H.: Seismic layout optimization of steel braced frames by an improved dolphin echolocation algorithm. Struct. Multidiscip. Optim. (2016) 10. Mansouri, I., Soori, S., Amraie, H., et al.: Performance based design optimum of CBFs using bee colony algorithm. Steel Compos. Struct. 27(5), 613–622 (2018) 11. Kaveh, A., Azar, B.F., Hadidi, A., et al.: Performance-based seismic design of steel frames using ant colony optimization. J. Constr. Steel Res. 66(4), 566–574 (2010) 12. Kaveh, A., Shojaei, I., Gholipour, Y., et al.: Seismic design of steel frames using multi-objective optimization. Struct. Eng. Mech. 45(2), 211–232 (2013) 13. Gholizadeh, S.: Performance-based optimum seismic design of steel structures by a modified firefly algorithm and a new neural network. Adv. Eng. Softw. 81(C), 50–65 (2015) 14. Moghaddam, H„ Hajirasouliha, I.: Fundamentals of optimum performance-based design for dynamic excitations. Sci. Iran. 12(4) (2005) 15. Mohammadi, R.K., Naggar, M.H.E., Moghaddam, H.: Optimum strength distribution for seismic resistant shear buildings. Int. J. Solids Struct. 41(22), 6597–6612 (2004)

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16. Hajirasouliha, I., Pilakoutas, K., Moghaddam, H.: Topology optimization for the seismic design of truss-like structures. Comput. Struct. 89(7–8), 702–711 (2011) 17. Hajirasouliha, I., Asadi, P., Pilakoutas, K.: An efficient performance-based seismic design method for reinforced concrete frames. Earthq. Eng. Struct. Dynam. 41(4), 663–679 (2012) 18. Mohammadi, R.K., Ghasemof, A.: Performance-based design optimization using uniform deformation theory: a comparison study. Lat. Am. J. Solids Struct. 12(1), 18–36 (2015) 19. Changizi, N., Jalalpour, M.: Stress-based topology optimization of steel-frame structures using members with standard cross sections: gradient-based approach. J. Struct. Eng. 143(8) (2017) 20. Gong, Y., Xue, Y., Xu, L., et al.: Energy-based design optimization of steel building frameworks using nonlinear response history analysis. J. Constr. Steel Res. 68(1), 43–50 (2012) 21. Xu, L., Gong, Y., Grierson, D.E.: Seismic design optimization of steel building frameworks. J. Struct. Eng. 132(2), 277–286 (2006) 22. Akbari, J., Ayubirad, M.S.: Seismic optimum design of steel structures using gradient-based and genetic algorithm methods. Int. J. Civ. Eng. 15(2A), 135–148 (2017) 23. Gholizadeh, S., Ebadijalal, M.: Performance based discrete topology optimization of steel braced frames by a new metaheuristic. Adv. Eng. Softw. 123, 77–92 (2018) 24. Gholizadeh, S., Poorhoseini, H.: Performance-Based Optimum Seismic Design of Steel Dual Braced Frames by Bat Algorithm. Springer International Publishing (2016) 25. Talatahari, S., Hosseini, A., Mirghaderi, S.R., et al.: Optimum performance-based seismic design using a hybrid optimization algorithm. Math. Probl. Eng. (693128) (2014) 26. Mazzoni, S., Mckenna, F., Scott, M.H., et al.: OpenSees Command-Language Manual, 2.0 edn. (2009) 27. Manual of steel construction: Load and Resistance Factor Design. American Institute of Steel Construction, Chicago, IL (2001) 28. AISC-360: Specification for Structural Steel Buildings. American Institute of Steel Construction, Chicago, IL (2010) 29. Rezazadeh, F., Mirghaderi, R., Hosseini, A., et al.: Optimum energy-based design of BRB frames using nonlinear response history analysis. Struct. Multidiscip. Optim. 57(3), 1005–1019 (2018) 30. Vanmarke, E.H.: SIMQKE: A Program for Artificial Motion Generation, User’s Manual and Documentation. Massachusetts Institute of Technology, Cambridge (1976) 31. Storn, R., Price, K.: Differential evolution—a simple and efficient heuristic for global optimization over continuous spaces. J. Glob. Optim. (1997)

Chapter 50

Study on Influence of Mechanical Parameters of Cement Plug on Sealing Integrity of Abandoned Wellbore Jiwei Jiang, Jun Li, Jiejing Nie, Gonghui Liu, Tao Huang and Wai Li

Abstract With the continuous exploitation of oil and gas resources, more and more closed offshore oil and gas wells, no oil or gas showed wells and carbon dioxide wells which have no industrial value are facing disposal. Blocking and abandoning operations mainly involve sealing the wellbore through injecting cement plugs into appropriate layers in it preventing the form of fluid turbulence from crossflowing in the wellbore. The key guarantee of the sealing integrity of the wellbore is ensuring the cement plug in the wellbore can’t be destroyed. According to the basic theory of elastic-plastic mechanics, the cement plug-casing-cement sheath-formation combination model was established. The reliability of the model is verified through the utilizing of the finite element method to calculate the sealing capacity of cement plug under different pressure difference, which is consistent with Nagelhout’s experimental results. The effects of the elastic modulus, Poisson’s ratio and thermal expansion coefficient of the cement plug on the integrity of the cement plug seal were analyzed. The results show that the lower the elastic modulus, the lower the risk of cement plug failure; and the Poisson’s ratio has little effect on the cement plug integrity; when the thermal expansion coefficient of the cement plug is less than the thermal expansion coefficient of the stratum, the cement plug is prone to sealing failure; When The coefficient of thermal expansion is greater than the coefficient of thermal expansion of the stratum formation, the possibility of sealing failure of the cement plug is small. According to the construction conditions on site, properly reducing the elastic modulus of the cement plug, and trying to use cement plugs with large thermal expansion coefficient can help cut the possibility of cement plug sealing failure risk down. The J. Jiang (B) · J. Li (B) · G. Liu · T. Huang China University of Petroleum-Beijing, Beijing 102249, China e-mail: [email protected] J. Li e-mail: [email protected] J. Nie Beijing Petroleum Machinery Co., Ltd., CNPC Engineering Technology R&D Co., Ltd., Beijing 102206, China W. Li The University of Western Australia, Crawley, WA 6009, Australia © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_50

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established finite element model and the research results can provide reference for the design of cement plugs during the process of wellbore disposal. Keywords Abandoned well · Cement plug · Seal integrity · Failure risk

50.1 Introduction A complete oil and gas field development cycle generally consists several stages as follows, exploration, development, production, and disposal. As offshore oil exploration steps into the middle and late stages, more and more oil and gas fields will reach its production and economic years and then facing to be abandoned. Oil and gas fields such as the South China Sea, the Gulf of Mexico, the North Sea of the United Kingdom, the Gulf of Thailand, West Africa (Mauritania), and the Middle East all have operation requirements for wellbore disposal. The US Gulf of Mexico is the largest and most active abandonment market in the world. From 2004 to 2015, about 3,000 wells were temporarily abandoned (TA) and 8,000 wells were permanently abandoned (PA) [1]. Statistics up to 2010 reported that between 2013 and 2040, the number of oil and gas fields to be demolished in China will reach 81 (excluding the projects currently under construction), and the number of platforms will be 164. The number of abandoned wells is expected to be 2,310 and the abandoned submarine pipelines will be 3,240 km [2]. The blocking and abandoning operations mainly involve sealing the wellbore through injecting cement plugs into appropriate layers such as the oil, gas and water layers in it preventing the form of fluid turbulence from crossflowing in the wellbore. Ensuring that cement plugs in abandoned wells do not break is the key to ensuring the integrity of the wellbore seal. The study on the integrity of cement plug seals focuses on theoretical analysis (mechanical failure, geometric parameters, elastic parameters, thermal expansion coefficient) [3, 4] and experimental research [5–7], and there are few literatures on numerical simulation studies of cement plug seal integrity. Based on the basic theory of elastic-plastic mechanics, this paper establishes a cement plug-casing-cement sheath-formation combination model, and uses the finite element method to calculate the axial load carrying capacity of cement plugs with different aspect ratios by numerical simulation, and with Nagelhout’s experimental results are compared with the reliability of the model. The effects of the elastic modulus, Poisson’s ratio and thermal expansion coefficient of the cement plug on the integrity of the cement plug seal are analyzed to provide a reference for the design of the cement plug during the wellbore disposal process.

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50.2 Cement Plug-Casing-Cement Sheath-Formation Combination Model 50.2.1 Model Assumptions After the cement well is injected into the wellbore, the cement slurry solidifies under the conditions of down hole temperature and pressure, and the cement plug is consolidated with the casing, the cement ring and the stratum formation as a combination. The following assumptions are made: (1) In the cementing and cementing cement plugging process, the liquid in the wellbore can be replaced completely, and the formed cement ring and cement plug are of good quality; (2) It is assumed that the layers of the composite body are tightly connected without slipping; (3) Cement plugs, casings, cement rings and stratum are isotropic; (4) cement plugs are ideal plastomers, casings, cement rings and formations are ideal elastomers; (5) cement plugs are ideal cylinders the cement ring and casing are ideal rings, and the cement plug, casing and cement ring are concentric with the wellbore. When the cement plug is subjected to the force analysis under non-uniform ground stress, it can be simplified to take the axis-symmetric plane strain problem of a cement plug-casing-cement ring-stratigraphic cross section in the vertical wellbore direction. Based on the above assumptions, the mechanical model of the cement plug-casingcement sheath-formation combination established under the non-uniform ground stress conditions (see Fig. 50.1).

Fig. 50.1 Mechanical model of cement plug-casing-cement sheath-formation combination

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50.2.2 Cement Plug Failure Criterion The mechanical model of cement plug-casing-cement sheath-formation combination is established to obtain the force magnitude and distribution of cement plug under the action of non-uniform ground stress, thus laying a foundation for judging whether the cement plug is damaged or not. Drucker-Prager Strength criterion was introduced as the failure criterion for yielding cement plug [8], and its expression was: f (I1 , J2 ) = α I1 +



J2 − k = 0

(50.1)

2 sin ϕ α= √   3 3 − sin2 ϕ

(50.2)

6c cos ϕ k= √   3 3 − sin2 ϕ

(50.3)

where α and k are constants, c is the cohesion of the material (MPa), ϕ is the internal friction angle of the material (°).

50.2.3 Establishment of Finite Element Model of Cement Plug-Casing-Cement Sheath-Formation Based on the above assumptions and cement sheath failure criteria, the finite element model of cement plug-casing-cement sheath-formation combination established under non-uniform ground stress conditions (see Fig. 50.2).

Fig. 50.2 Finite element model of cement plug-casing-cement sheath-formation combination

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50.3 Model Verification 50.3.1 Cement Plug Experiment Fluid leakage around the cement plug occurs through the micro-annulus where the cement is completely or partially stripped. The experiment to determine the sealing ability of the cement plug is relatively simple in principle, the cement plug is placed in the casing, a pressure difference is applied above and below, and the corresponding fluid flow rate is measured (see Fig. 50.3) [7, 9]. Nagelhout et al. [5] first conducted a systematic study on the sealing capacity of cement plug, and conducted small-scale and large-scale laboratory tests on two different cement systems. The inner diameter of the large test device is 173 mm, and the length is 1.25 m. The 1 m long cement plug can be tested (see Fig. 50.4). The inner

Fig. 50.3 Schematic diagram of typical laboratory setup for determining the sealing capacity of cement plugs [7]

Fig. 50.4 Large scale P&A plug test set-up

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diameter and length of the small test device are 50 mm and 200 mm respectively. It is found that the leakage rate measured depends on the radial scale of cement plug, and the leakage rate increases with the increase of pressure difference between the upper and lower parts of cement plug. For the non-expansive cement system, the “equivalent permeability” in the millidarcy range is much higher than that of high-quality cement, indicating that the leakage measured is through the micro-ring gap around the cement rather than through the cement itself. In addition, the sealing capacity of the expanded cement system is significantly improved.

50.3.2 Model Verification The finite element simulation calculation of the large-size test system B sealing capacity in the Nagelhout cement plug experiment was carried out using the model. The calculation parameters used in the experiment are the same as those in the experiment. The simulation results are shown in Table 50.1. As it can be seen from Table 50.1, when a differential pressure of 0.25 MPa (the lower pressure of the cement plug was 14.75 MPa) was applied, the system B showed no gas leakage. When the pressure difference reaches 0.95 MPa, the micro-annulus size is calculated to be 5.7 μm, and the experimentally measured micro-annulus size is 0 μm. This is because the micro-leakage of less than 5cm3/min flow rate cannot be detected in the system under high absolute pressure. As the pressure decreases, the rate decreases, and once the cement plug-sleeve seal breaks, it will not re-seal, and the calculated result is in good agreement with the experimental results. Table 50.1 Finite element simulation and experimental comparison of sealing ability of large scale test system B Differential pressure/ (MPa·m−1 )

Flow rate/(ml·min−1 )

Effective permeability/md

Modle calculated microannulus width/μm

Experimental microannulus width/μm

0.25

0

No leak

0

0

0.5

0

No leak

0

0

0.75

0

No leak

0

0

0.95

0

No leak

5.7

0

1.2

3.5

0.007

14.8

1.5

0.75 (after leak initiated)

1.5

0.006

18.9

1.4

0.5 (after leak initiated)

0.9

0.007

21.4

1.5

0.25 (after leak initiated)

0.4

0.01

22.3

1.7

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50.4 Model Calculation and Analysis 50.4.1 Selection of Parameters According to Saint-Venant’s theorem, stress concentration occurs near the circular hole in the infinite plate, but the stress concentration is small when the boundary size exceeds 5 times [10, 11]. The model parameters have the following values: boundary size 3,000 mm, maximum horizontal principal stress 49.4 MPa, minimum horizontal principal stress 37.6 MPa, and vertical principal stress 57.1 MPa. The model mechanical parameters are shown in Table 50.2.

50.4.2 Influence of Elastic Modulus It can be seen from Fig. 50.5 that when the modulus of elasticity of the cement is 2.5 GPa, the load path is completely within the surface of the yield failure envelope, and the risk of yield failure of the cement plug is small.

50.4.3 Influence of Poisson’s Ratio It can be seen from Fig. 50.6 that the Poisson’s ratio has little effect on the integrity of the cement plug, and the cement plug with low Poisson’s ratio is beneficial to prevent the tensile failure of the cement plug.

50.4.4 Influence of Thermal Expansion Coefficient It can be seen from Fig. 50.7 that when the thermal expansion coefficient of the cement plug is smaller than the thermal expansion coefficient of the stratum formation, the cement plug is prone to sealing failure. When the thermal expansion coefficient of the cement plug is greater than the thermal expansion coefficient of the formation, the cement plug is less likely to sealing failure.

50.5 Conclusions and Recommendations (1) The cement plug-casing-cement sheath-formation combination model was established by finite element method. The variation of axial bearing capacity of cement plugs with different aspect ratios was obtained by numerical simulation,

Outer diameter/mm

102.72

127

168.28

3000

Name

Cement plug

Casing

Cement sheath

Formation

–

20.64

12.14

–

Wall thickness/mm

22

10

210

8

Elastic modulus/GPa

0.23

0.17

0.3

0.17

Poisson’s

Table 50.2 Geometrical parameter and material properties of model media

2600

3100

7800

2700

1.59

0.98

45

0.98

1256

837

461

837

Density/(kg·m−3 ) Thermal Specific heat/ conductivity/ (J·kg−1 ·°C−1 ) (W·kg−1 ·°C−1 )

10.5

11.0

13.0

11.0

Thermal expansion/ (10−6 ·°C −1 )

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7

Yield Surface 2.5 GPa

6

5.0 GPa

Q/MPa

5

10 GPa

4

20 GPa

3 2 1 0

-2

-1.5

-1

-0.5

0

0.5

P/MPa

Fig. 50.5 Effect of cement plug elastic modulus on wellbore seal integrity 7

Yield Surface 0.1

6

0.2

Q/MPa

5

0.3

4

0.4

3 2 1 0 0

0.2

0.4

0.6

0.8

1

1.2

P/MPa

Fig. 50.6 Effect of cement plug Poisson’s ratio on wellbore seal integrity

1.4

1.6

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7

Yield Surface 1.0e-6/

6

2.0e-6/

Q/MPa

5

3.0e-6/ 4.0e-6/

4 3 2 1 0

0

0.5

1

1.5

2

2.5

P/MPa

Fig. 50.7 Effect of cement plug thermal expansion on wellbore seal integrity

which is consistent with the experimental results of Nagelhout. The reliability of the model was verified. (2) The lower the elastic modulus of the cement plug, the lower the risk of failure; Poisson’s ratio has little effect on the integrity of the cement plug. (3) When the thermal expansion coefficient of the cement plug is less than the thermal expansion coefficient of the stratum formation, the cement plug is prone to sealing failure; when the thermal expansion coefficient of the cement plug is greater than the thermal expansion coefficient of the stratum formation, the cement plug is less likely to sealing failure. (4) According to the site construction conditions, properly reducing the elastic modulus of the cement plug, and trying to use cement plugs with large thermal expansion coefficient can help cut the possibility of cement plug sealing failure risk down. Acknowledgements The authors thank the National Natural Science Funds“Study on optimization of un-uniform clustering perforation for long horizontal inhomogeneous shale formation” (Item No. 51674272), the Key Program of National Natural Science Foundation of China“Basic research on the wellbore pressure control of deepwater oil and gas drilling” (Item No. 51734010), for contributions to this paper.

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References 1. Kaiser, M.J.: Rigless well abandonment remediation in the shallow water U.S. Gulf of Mexico. J. Pet. Sci. Eng. 151, 94–115 (2017) 2. Yongbin, Y.: Offshore platform decommissioning technology and dismantling market prospect forecast. China Offshore Platf. 28(4), 4–7 (2013) 3. Akgün, J.H., Daemen, J.J.K.: Analytical and experimental assessment of mechanical borehole sealing performance in rock. Eng. Geol. 47, 233–241 (1997) 4. Mainguy, M., Longuemare, P., Audibert, A., et al.: Analyzing the risk of well plug failure after abandonment. Oil Gas Sci. Technol. 62(3), 311–324 (2007) 5. Nagelhout, A.C.G., Brunei, S., Bosma, M.G.R., et al.: Laboratory and field validation of a sealant system for critical plug-and-abandon situations. In: SPE 97347-PA (2010) 6. van Eijden, J., Cornelissen, E., Ruckert, F., et al.: Development of experimental equipment and procedures to evaluate zonal isolation and well Abandonment materials. In: SPE/IADC 184640-MS (2017) 7. Opedal, N., Corina, A.N., Vrålstad, T.: Laboratory test on cement plug integrity. In: OMAE2018-78347 (2018) 8. Zoback, M.D.: Reservoir Geomechanics (Lin, S., Zhaowei, C., Yushi, L., trans.). Petroleum Industry Press, Beijing (2011) 9. Vrålstada, T., Saasenb, A., Fjæra, E., et al.: Plug & abandonment of offshore wells: Ensuring long-term well integrity and cost-effciency. J. Petrol. Sci. Eng. 173, 478–491 (2019) 10. Jun, L., Mian, C., Gonghui, L., et al.: Elastic-plastic analysis of casing-concrete sheath-rock combination. Acta Petrolei Sinica 26(6), 99–103 (2005) 11. Shufang, X., Subi, Y.: Rock mass mechanics, pp. 97–98. Geological press, Beijing (1987)

Part XIV

Composite Materials: Modelling, Processing, Design and Application

Chapter 51

Hyperelastic Nonlinear Thermal Constitutive Equation of Vulcanized Natural Rubber Yufei Liao, Chen Li and Weiwei Zhang

Abstract Based on the tensor function, the nonlinear thermal constitutive equation and the corresponding strain energy function of hyperelastic isotropic materials are derived. The equation and the strain energy function are complete and irreducible, They contain 38 independent elastic constants and satisfy the tensor function representation theorem. The constitutive relation of rubber as a typical superelastic incompressible material is studied, The constitutive equation of thermal stress and the corresponding strain energy function of rubber are derived. 28 complete and irreducible elastic constants of rubber are determined. The stress-strain curves of vulcanized rubber at different temperatures (−50, −25, 0, 25, 75, 100 °C) were calculated under static uniaxial loading. The above conclusions and L.R.G. experimental data were fitted by multivariate nonlinear regression. The elastic constants of natural rubber vulcanized at 0 °C, and the specific values of thermoelastic constants. It provides important methods and data for further study of thermal sensitivity of rubber. By fitting the experimental data of vulcanized natural rubber, the constitutive equation can effectively describe the mechanical behavior of vulcanized natural rubber at different temperatures. The calculation process is simple and clear. Keywords Hyperelastic materials · Nonlinear · Thermal constitutive equation

51.1 Introduction Vulcanized natural rubber materials exhibit three different forms at different temperatures: glass-state, high-elastic state and viscoelastic state. Rubber in high elastic state has good mechanical properties, rubber in the other two forms, its mechanical properties change, the application value will be affected. Therefore, temperature has become one of the important factors that affect the mechanical properties of rubber materials [1]. Constitutive relation is one of multidisciplinary core subjects in the deformation mechanics studies, which has aroused great attention of the mechanics, Y. Liao · C. Li (B) · W. Zhang Taiyuan University Science and Technology, Taiyuan, Shanxi, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_51

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materials science, physics and applied mathematics researchers, the thermal constitutive equation of rubber materials and products has always been a hot topic among scholars [2–8]. In this paper, a complete and irreducible nonlinear constitutive equation and strain energy function for isotropic elastic materials are derived from tensor functions, considering the influence of temperature, the constitutive equation of rubber thermal stress and the corresponding strain energy function are derived. The stress-strain curves of vulcanized rubber at different temperatures (−50, −25, 0, 25, 75, 100 °C) were calculated under static uniaxial loading, the deduced results were fitted with the uniaxial tensile test data of vulcanized natural rubber at high temperature by multivariate nonlinear regression. The specific values of elastic constants and thermoelastic constants of vulcanized natural rubber materials with initial temperature of 0 °C were obtained. It provides important methods and data for further study of thermal sensitivity of rubber. By fitting the experimental data of vulcanized natural rubber, the constitutive equation can effectively describe the mechanical behavior of vulcanized natural rubber at different temperatures. The calculation process is simple and clear.

51.2 Nonlinear Constitutive Equations of Hyperelastic Isotropic Materials Wineman and Pipkin have proved that the complete representation of tensor polynomials can be regarded as the complete representation of general tensor functions [9, 10]. Literature [11] in the study of nonlinear constitutive theory, the nonlinear constitutive equation of isotropic hyperelastic materials was given by using constructive proof method, and the polynomial representation method is pointed out to be complete and irreducible. For isotropic hyperelastic materials, it has been proved in reference [11] that the 2n-order isotropic hyperelastic tensor component is equal to zero, so the stress constitutive equation only needs to take the fourth-order polynomial. The constitutive equation of nonlinear isotropic hyperelastic materials represented by conjugate stress-strain tensor K = E can be expressed by a complete quaternion polynomial of three principal traces I¯1 , I¯2 , I¯3 of strain tensor E. As: 3 4 2 2 K = (k1 I¯1 + k3 I¯12 + k4 I¯2 + k6 I 1 + 2k7 I 1 I 2 + k8 I 3 + k10 I 1 + 3k11 I 1 I 2 + k12 I 2

+ 2k13 I 1 I 3 ) 1 + 2(k2 + k4 I 1 + k7 I 1 + k9 I¯2 + k11 I 1 + 2k12 I 1 I 2 + k14 I 3 )E 2

2

+ 3(k5 + k8 I 1 + k13 I 1 + k14 I 2 )E 2

3

(51.1)

1 is unit tensor, K is second Piola-kirchhoff stress tensor, E is Lagrange strain tensor. I¯1 , I¯2 , I¯3 is the three principal trace of strain tensor E, J2 = a • E 2 • a, J2 = a • E 2 • a. From this we can know k1 , k2 , k3 , . . . , k14 has 14 independent

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elastic constants. They are functions of 14 elastic tensor components of isotropic elastic material.

51.3 Nonlinear Thermal Constitutive Equations of Hyperelastic Isotropic Materials The theory [11] of thermoelastic constitutive equations for nonlinear isotropic materials is studied in depth. Considering the stress-strain relationship in general environment, the isothermal condition is not limited, but the temperature variation with time is not considered, the stable value of the state temperature is only considered. At this time, the deformation caused by temperature is only spherical strain, and there is no deviation strain. Considering only the effect of temperature and not the deformation coupling term in the heat conduction equation, the generalized Hooke’s law can be replaced by the Duhamel-Neumann relation. At this time, strain E can be decomposed into the sum of strain E L under external load and strain E T produced by temperature change. As: E = E l + E T = E l + α(T − T0 )1 = E l + αT 1

(51.2)

α is the coefficient of thermal expansion, T is the current temperature, T0 is the reference state of the object temperature, T = T1 − T0 is the object relative to the reference temperature change. The following equation can be derived. As: E l = E − αT 1 (E l )2 = E 2 − 2αT E + α 2 T 2 1 (E l )3 = E 3 + 3α 2 T 2 E − 3αT E 2 − α 3 T 3 1 I 1l = trEl = trE + tr(αT 1) = I1− −3αT   I 2l = trEl2 = tr E 2 − 2αT E + α 2 T 2 1 = I2− −2αT I 1 + 3(αT )2   I 3l = trEl3 = tr E 3 + 3α 2 T 2 E − 3αT E 2 − α 3 T 3 1 = I3− −3αT I2− +3(αT )2 I1− −3α 3 T 3 −

− Ii

Ii l

(51.3)

(i = 1, 2, 3) is three main trace trE l , trE l2 , trE l3 of strain tensor E l .

(i = 1, 2, 3) is three main trace trE, trE 2 , trE 3 of strain tensor E. The thermal constitutive equation can be written as an invariant form. As:

K = K (1, E, E 2 , aa, aa · E + E · aa, a · E · a, a · E 2 · a, trE, trE 2 , trE 3 , T ) (51.4)

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Considering the influence of temperature and strain on the constitutive equation of material, for nonlinear isotropic hyperelastic material, the formulas (51.2), (51.3) are substituted by (1) We can derive the complete irreducible quadratic polynomial formulation of the constitutive equation of thermal stress expressed by the three principal tracks I¯1 , I¯2 , I¯3 of strain tensor E and temperature T . As: 2

3

4

2

2

K = [ k1 I 1 + k3 I 1 + k4 I 2 + k6 I 1 + 2k7 I 1 I 2 + k8 I 3 + k10 I 1 + 3k11 I 1 I 2 + k12 I 2 2

3

2

+ 2 k13 I 1 I 3 + (δ1 + δ3 I 1 + δ7 I 1 + δ8 I 2 + δ16 I 1 + δ17 I 3 )T + (δ2 + δ6 I 1 + δ13 I 1 2

3

+ δ14 I 2 )T 2 + (δ5 + δ15 I 1 )T 3 + δ12 T 4 ] 1 + 2[ k2 + k4 I 1 + k7 I 1 + k9 I 2 + k11 I 1 2

+ 2k12 I 1 I 2 + k14 I 3 + (δ4 + δ10 I 1 + δ20 I 1 + δ21 I 2 )T + (δ9 + δ19 I 1 )T 2 + δ18 T 3 ] E 2

+ 3[ k5 + k8 I 1 + k13 I 1 + k14 I 2 + (δ11 + δ23 I 1 )T + δ22 T 2 ] E 2

(51.5)

k1 , k2 , k3 . . . , k14 are 14 independent elastic constants and δ1 , δ2 , δ3 . . . , δ23 are 23 coefficient of temperature T .

51.4 Thermal Strain Energy Function of Hyperelastic Isotropic Material For isotropic Green elastomers, there is an integral relation between the stress tensor K and the strain energy function W [12, 13]. As:  W (E) =

 K : dE =

ϕ0 d I 1 + ϕ1 d I 2 + ϕ2 d I 3

(51.6)

Since K and E are symmetric affine quantities, there are: ⎧ ⎪ ⎨ 1 : d E = δi j d E i j = d E ii = d I 1 2E : d E = 2E ij d E i j = d(E i j E ji ) = d I 2 ⎪ ⎩ 2 3E : d E = 3E i j E jk d E ki = d(E i j E jk E ki ) = d I 3

(51.7)

Combined with (51.7) formula, the (51.6) formula is brought into (51.5), and the following strain energy function can be obtained: 1 1 1 1 2 3 4 2 2 k1 I 1 + k2 I 2 + k3 I 1 + k4 I 1 I 2 + k5 I 3 + k6 I 1 + k7 I 1 I 2 + k8 I 1 I 3 + k9 I 2 2 3 4 2 1 5 3 2 2 + k10 I 1 + k11 I 1 I 2 + k12 I 1 I 2 + k13 I 1 I 3 + k14 I 2 I 3 + [b1 I 1 + b4 I 2 + b11 I 3 5 1 1 1 1 2 2 3 4 + b3 I 1 + b21 I 2 + b7 I 1 + b16 I 1 + (b8 + b10 )I 1 I 2 + (b17 + b23 )I 1 I 3 2 2 3 4 1 1 2 2 3 + b20 I 1 I 2 ]T + [b2 I 1 + b9 I 2 + b22 I 3 + b6 I 1 + b13 I 1 + (b14 + b19 )I 1 I 2 ]T 2 2 3

W =

51 Hyperelastic Nonlinear Thermal Constitutive Equation …

+ (b5 I 1 + b18 I 2 +

1 2 b15 I 1 )T 3 + b12 I 1 T 4 2

603

(51.8)

In Eq. (51.5), K and E are conjugate stress and strain variables, and K and right Cauchy-Green strain tensor c are conjugate stress and strain variables. Another expression of strain energy function can be deduced by the conjugate stress and strain variables of K and E, which is very effective in the study of incompressible materials [12, 13]. Definition: l1 , l2 , l3 is the first, second and third principal invariants of c, and l 1 , l 2 , l 3 is three main trace tr c, tr c2 , tr c3 of c. Taking into account c = 1 + 2E, there are: c1 e1 + c2 e2 + c3 e3 = (1 + 2E 1 )e1 + (1 + 2E 2 )e2 + (1 + 2E 3 )e3

(51.9)

So: ⎧ ⎧ ⎪ ⎪ ⎨ l1 = 3 + 2I1 ⎨ c1 = 1 + 2E 1 c2 = 1 + 2E 2 and l2 = 3 + 4I1 + 4I2 ⎪ ⎪ ⎩ ⎩ c3 = 1 + 2E 3 l3 = 1 + 2I1 + 4I2 + 8I3

(51.10)

⎧ ¯ ⎪ ⎨ I1 = I1 I¯2 = (I1 )2 − 2I2 ⎪ ⎩¯ I3 = 3I3 + (I1 )3 − 3I1 I2

(51.11)

Because:

According to (51.10), (51.11) formula is available: ⎧ 1 ⎪ I¯1 = (l1 − 3) ⎪ ⎪ 2 ⎪ ⎪ ⎨ 1 1 I¯2 = (l1 − 3)2 + (l1 − 3) − (l2 − 3) ⎪ 4 2 ⎪ ⎪ ⎪ ⎪ ⎩ I¯ = 1 (l − 3)3 + 3 (l − 3)2 − 3 (l − 3)(l − 3) + 3 (l − 3) − 3 (l − 3) + 3 (l − 1) 3 1 1 1 2 1 2 3 8 4 8 8 8 8

(51.12)

Substituting (51.12) into (51.8) formula can be obtained: W = δ11 (l1 − 3)5 + (δ7 + δ15 T )(l1 − 3)4 + (δ4 + δ16 T + δ25 T 2 )(l1 − 3)3 + (δ2 + δ17 T + δ26 T 2 + δ32 T 3 )(l1 − 3)2 + (δ1 + δ18 T + δ28 T 2 + δ31 T 3 + δ34 T 4 )(l1 − 3) + (δ9 + δ22 T )(l2 − 3)2 + (δ3 − δ23 T − δ29 T 2 − δ33 T 3 )(l2 − 3) + (δ6 + δ24 T + δ30 T 2 )(l3 − 1) + (δ5 − δ20 T − δ27 T 2 )(l1 − 3)(l2 − 3) + (δ8 − δ19 T )(l1 − 3)2 (l2 − 3) + (2δ10 + δ21 T )(l1 − 3)(l3 − 1) + δ12 (l1 − 3)3 (l2 − 3) + δ13 (l1 − 3)(l2 − 3)2 + δ14 (l1 − 3)2 (l3 − 1) − δ10 (l2 − 3)(l3 − 1)

(51.13)

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Among them: δ1 , δ2 , δ3 , . . . , δ14 is a linear combination of ki , which is related to the mechanical properties of materials. The δ15 , δ16 , δ17 , . . . , δ34 is a linear combination of bi , which is related to the temperature characteristics of the material. Rubber materials are typical superelastic incompressible materials. For incompressible materials, there are:  l3 = det c = F F = J 2

(51.14)

F is the deformation gradient tensor and J is the Jacobi determinant. Under incompressible conditions, the third invariants that represent the volume change are l3 = 1, so J = 1, So the strain energy function of hyperelastic body can be obtained by using bivariate l1 , l2 expansion function. As: W = δ11 (l1 − 3)5 + (δ7 + δ15 T )(l1 − 3)4 + (δ4 + δ16 T + δ25 T 2 )(l1 − 3)3 + (δ2 + δ17 T + δ26 T 2 + δ32 T 3 )(l1 − 3)2 + (δ1 + δ18 T + δ28 T 2 + δ31 T 3 + δ34 T 4 )(l1 − 3) + (δ9 + δ22 T )(l2 − 3)2 + (δ3 − δ23 T − δ29 T 2 − δ33 T 3 )(l2 − 3) + (δ5 − δ20 T − δ27 T 2 )(l1 − 3)(l2 − 3) + (δ8 − δ19 T )(l1 − 3)2 (l2 − 3) + δ12 (l1 − 3)3 (l2 − 3) + δ13 (l1 − 3)(l2 − 3)2

(51.15)

51.5 Relationship Between Stress and Elongation Ratio of Rubber Materials Because of the particularity of rubber material, the experimental curve of rubber is usually the relationship between nominal stress and elongation ratio. Therefore, it is necessary to establish the constitutive relationship between nominal stress and elongation ratio and temperature. For the most common type of homogeneous strain corresponding to the principal stress, Rivlin gives the following expression [14]: 

∂W 1 ∂W + p (i = 1, 2, 3) − 2 ti = 2 λi2 ∂l1 λi ∂l2

(51.16)

Formula ti is the true principal stress (Cauchy principal stress), p is an arbitrary hydrostatic pressure, and λi is the principal elongation ratio. Under uniaxial tension, the nominal stress is f˜: 

1 ∂W 1 ∂W t1 = 2(λ − 2 ) + f = λ1 λ ∂l1 λ ∂l2

(51.17)

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The relationship between stress and tensile ratio of an isotropic incompressible elastic material considering temperature is obtained by simultaneous Eqs. (51.16) and (51.17). As: f = λ(5) δ1 + λ(4) δ2 + λ(7) δ3 + λ(3) δ4 + λ(8) δ5 + λ(2) δ7 + λ(9) δ8 + λ(6) δ9 + λ(1) δ11 + λ(10) δ12 + λ(11) δ13 + λ(2) T δ15 + λ(3) T δ16 + λ(4) T δ17 + λ(5) T δ18 − λ(9) T δ19 − λ(8) T δ20 + λ(6) T δ22 − λ(7) T δ23 + λ(3) T 2 δ25 − λ(4) T 2 δ26 − λ(8) T 2 δ27 + λ(5) T 2 δ28 − λ(7) T 2 δ29 + λ(5) T 3 δ31 + λ(4) T 3 δ32 − λ(7) T 3 δ33 + λ(5) T 4 δ34

(51.18)

λ(i) (i = 1, 2, . . . , 11) of them are λi (i = 1, 2, . . . , 9) linear combinations.

51.6 Experimental Curve and Analysis Calculation of Vulcanized Natural Rubber Scholars have done a lot of experimental research on rubber materials and obtained many important data. For considering the effects of temperature vulcanization of natural rubber, the British L.R.G compare a typical experiment, draw the vulcanized natural rubber in −50, −25, 0, 25, 50, 75 and 100 °C when the nominal stress and elongation ratio relationship [15]. As Fig. 51.1. According to the experimental data of L.R.G, the specific values of elastic constants of vulcanized natural rubber materials considering the effect of temperature can be obtained by using (51.17) formula and MATLAB software with multi-objective regression fitting method. The experimental data of relationship between nominal stress and elongation ratio of vulcanized natural rubber at different temperatures obtained by L.R.G. Trelawer were compared with the fitting results as follows Fig. 51.2. It can be seen from the Fig. 51.2 that the fitting result of the equation is satisfactory when it is applied to the tensile test of vulcanized rubber at different temperatures −

Fig. 51.1 Stress-stretch ratio cure at different temperature

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Fig. 51.2 Comparison of fitting curves and experimental data of vulcanized natural rubber at different temperatures

50, −25, 0, 25, and 50 °C, but there is an error when the fitting result is at 75, 100 °C. The maximum relative error between the fitting curve and the experimental value is 12%.

51.7 Conclusion 1. The constitutive equation obtained in this paper is complete, irreducible, and the elastic constants are complete and irreducible. 2. The nonlinear thermal constitutive equation of isotropic incompressible materials obtained in this paper is not only applicable to L.R.G. Trelawer experimental vulcanized natural rubber, but also to other isotropic incompressible materials and other types of synthetic rubber. 3. The complete irreducible thermal constitutive equations for isotropic nonlinear hyperelastic materials have 14 independent elastic constants and 23 temperature coefficients. 4. The numerical values of elastic constants and temperature coefficients of specific materials can be determined by experiments. The number of coefficients is generally less than the maximum number of temperature coefficients of independent elastic constants of isotropic constitutive equations. Acknowledgements This study was supported by National Natural Science Foundation of China (11372207).

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References 1. Younan, A.F., Ismail, M.N., Khalaf, A.I.: Thermal stability of natural rubber-polyester short fiber composites. Polym. Degrad. Stab. 48, 103–105 (1995) 2. Jabbari, M., Nejad, M.Z., Ghannad, M.: Thermo-elastic analysis of axially functionally graded rotating thick cylindrical pressure vessels with variable thickness under mechanical loading. Int. J. Eng. Sci. 96, 1–18 (2015) 3. Kim, J., Im, S., Kim, H.G.: Numerical implementation of a thermo-elastic–plastic constitutive equation in consideration of transformation plasticity in welding. Int. J. Plast. 21, 1383–1408 (2015) 4. Safari, A.R., Forouzan, M.R., Shamanian, M.: Thermo-viscoplastic constitutive equation of austenitic stainless steel 310 s. Comput. Mater. Sci. 68, 402–407 (2013) 5. Voyiadjisa, G.Z., Faghihi, D.: Microstructure to macro-scale using gradient plasticity with temperature and rate dependent length scale. Procedia IUTAM 3, 205–227 (2012) 6. Anand, L., Aslan, O., Chester, S.A.: A large-deformation gradient theory for elastic–plastic materials: strain softening and regularization of shear bands. Int. J. Plast. 30–31, 116–143 (2012) 7. Goel, A., Sherafati, A., Negahban, M., Azizinamini, A., Wang, Y.: A thermo-mechanical large deformation constitutive model for polymers based on material network description: Application to a semi-crystalline polyamide 66. Int. J. Plast. 67, 102–126 (2015) 8. Maurel-Pantel, A., Baquet, E., Bikard, J., Bouvard, J.L., Billon, N.: A thermo-mechanical large deformation constitutive model for polymers based on material network description: Application to a semi-crystalline polyamide 66. Int. J. Plast. 67, 102–126 (2015) 9. Pipkin, A.C., Wineman, A.S.: Material symmetry restrictions on non-polynomial constitutive equations. Arch. Ratl. Mech. Anal. 12(1), 420–426 (1963) 10. Wineman, A.S., Pipkin, A.C.: Material symmetry restrictions on constitutive equations. Arch. Ratl. Mech. Anal. 17(3), 184–214 (1964) 11. Chen, L.: The Nonlinear Constitutive Theory of Hyperelastic. National Defense Industry Press, Beijing (2012). (in Chinese) 12. Lu, S.C.H., Pister, K.S.: Decomposition of deformation and representation of the free energy function for isotropic thermoelastic solids. Int. J. Solids Struct. 11(7–8), 927–934 (1975) 13. Lion, A.: On the large deformation behaviour of reinforced rubber at different temperatures. J. Mech. Phys. Solids 45(11–12), 1805–1834 (1997) 14. Rivlin, R.S.: Large elastic deformations of isotropic materials. IV. Further developments of the general theory. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 241(835): 379–397 (1948) 15. Treloar, L.R.G. (ed.): The Physics of Rubber Elasticity (Xiang, Z., trans.). China Light Ind. Pre. (1958)

Chapter 52

Modelling the Edge Crushing Performance of Corrugated Fibreboard Under Different Moisture Content Levels Aiman Jamsari, Andrew Nevins, Celia Kueh, Eli Gray-Stuart, Karl Dahm and John Bronlund Abstract This paper investigates the strength performance of corrugated fibreboard (CFB) under different moisture content levels through edge crush test (ECT) and developing a finite element (FE) model to simulate the response of ECT. The study started by conducting tensile and compressive test at different moisture content levels on the paper components that made up the CFB. Next, the ECT test of the CFB at different moisture levels were conducted. To simulate the response of ECT, an FE model that assumes an orthotropic shell element behavior of CFB was developed. The FE model uses the tensile and compressive data of the paper components at different moisture content levels as the input parameters. The results of the experiment and FE model shows good agreement of the ECT at different moisture content levels. This study proves that modelling the strength of CFB in harsh environment can be done by properly reducing the material properties of its paper components in the input parameters of the FE model. Keywords Corrugated fibreboard · Edge crush · Strength

52.1 Introduction Corrugated fibreboard (CFB) is a well-known packaging material that is used for various goods and products due to being lightweight and cheap while possessing high strength and stiffness [1]. Typical single-walled CFB is made up by a fluted corrugated sheet and two flat linerboards. The fluted structure plays important role in protecting packaged goods from hazards since it acts to resists bending and pressure A. Jamsari (B) · C. Kueh · E. Gray-Stuart · J. Bronlund School of Engineering and Advanced Technology, Massey University, Private Bag 11 222, Palmerston North 4442, New Zealand e-mail: [email protected] A. Nevins Bega Cheese Limited, 123, Bega, NSW 2550, Australia K. Dahm Callaghan Innovation, 69 Gracefield Road, Lower Hutt 5010, New Zealand © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_52

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from any direction when anchored to the linerboards. The two main directions in CFB is known as machine direction (MD) and the cross direction (CD) where MD aligns with the manufacturing flute direction and CD corresponds normal to the flute direction. The design of CFB constantly changes to suit requirements such as strength and storage conditions of the packaged goods. This shows that it is crucial to analyse the structural component of CFB in designing a good packaging box [2]. As emphasized by Markstrom [3], the compressive strength of CFB is a direct measure of the box’s stacking strength. This is also evident as McKee et al. [4] shows that the box crush strength will be dependent by the edge crush test (ECT), bending stiffness and perimeter of the box. In a harsher conditions such as in a cool store, the properties of CFB will decrease much more due to the increase of humidity and moisture content of the material [5]. The impact of increase in humidity has been shown to accelerate the failure of the CFB under compressive loading [6], hence indicating that the reduction of properties due to the humidity levels should be included in designing a good packaging box. This paper investigates the strength performance of CFB and its paper components through ECT, tensile and compressive test at different humidity levels as measured through moisture content. The tensile and compressive material properties of the paper components obtained at different moisture content levels were then used as the input parameters for the finite element (FE) model of the CFB under ECT test.

52.2 Methodology 52.2.1 Tensile and Compressive Test of Paper Components The C-flute CFB used in this research is made up of 250gm−2 Kraft linerboard and 160 gm−2 semi-chemical fluting medium that are manufactured by Carter Holt Harvey, Kinleith, New Zealand (now known as Oji Fibre Solution). Samples of 15 × 250 mm2 were cut from the Kraft linerboards and Semi-chemical fluting medium in the machine direction (MD) and cross-machine direction (CD). Batches of 35 samples (except for Kraft MD with 28 samples) were placed in a sealable plastic container above four different saturated salt solutions at 20 °C (see Fig. 52.1) to obtain the required relative humidity as shown in Table 52.1. In addition to that, one batch perp pulp type and direction was conditioned to 50% RH in accordance with EN 20 187. The samples conditioned above the salt solutions were then sealed in a labelled metalised film bags (Mettler Toledo PG 503-S, Switzerland). The samples were removed from their bags prior to testing and resealed again in their respective bags immediately after testing. The weight before and after testing was measured to determine their moisture content. Moisture content will be used in this study as a measure

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Fig. 52.1 250 gm−2 Kraft linerboard, 160 gm−2 Semi-chemical fluting medium and ECT sample of CFB stored in a beaker above a salt solution at 20 °C

Table 52.1 Relative humidity above saturated salt solutions at 23 °C [10]

Salt

Relative humidity

NaCl

75.5

KCl

85.1

BaCl2

90.0

of moisture absorption as opposed to relative humidity since it is more detailed in terms of how many percent of water intake that the samples absorb. From the batches of 35 (or 28), 15 samples (or 12 for the Kraft MD) were tested using an L & W Compression Strength Tester STFI (AB Lorentzen and Wettre, Sweden) with a clamp speed of 3 mm/min following TAPPI T 826 procedure. The remaining 20 samples (16 for Kraft MD) were tested using an Alwetron TH1 Tensile Tester (AB Lorentzen and Wettre, Sweden). The test span was 100 mm and the clamp speed was 10 mm/min. The procedure follows TAPPI T 494.

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52.2.2 Experimental ECT Test C-flute CFB samples were cut to 100 × 25 mm2 and stored in a sealable container above four different saturated salt solutions (see Table 52.1) at 20 °C for 4 weeks. Each level of RH consists of 10 samples. One batch of samples was conditioned to 50% RH in accordance with EN 20 187. Similarly, the samples were removed from their bags prior to testing and resealed again immediately after testing. The weight before and after the testing was measured to determine their MC. Platens with groves were used to provide support for the ECT test and it was conducted with a platen speed of 12.5 mm/min following the FEFCO No. 8 standard procedure.

52.2.3 Finite Element Model of ECT The ECT model of CFB in this study is assumed as an orthotropic shell element. Each paper component consists of 20 variables with the main variables listed down in Table 52.2. The thickness, density and Poisson’s ratio for each paper components are presented in Table 52.3 in the Appendix 1 while the rest of the main variables are extracted from the tensile and compressive data of the paper components. The remaining variables are computed based on the relations presented in Table 52.4 in the Appendix 1. The geometry of the flute is drawn using the relations presented by Urbanik [7] that compromise of arc-and-tangent curves. This curve was drawn in Solidworks and the liners were also added to create the model of C-flute CFB (100 × 25 mm2 ). The drawing model was then transferred to Ansys to carry out the FE simulation. A combination of static structural and ANSYS Composite PrepPost (ACP) was used to run the FE simulation where static structural performs the crushing simulation and ACP determines the failure criterion plot. Tsai-Wu failure criterion is implemented Table 52.2 The main orthotropic variables of the paper components that made up the CFB

Variables Thickness (mm) Density (kg/m3 ) Elastic modulus, E 11 Elastic modulus, E 22 Poisson’s ratio, ν-12 Tensile strength, σ 11t Tensile strength, σ 22t Compressive strength, σ 11c Compressive strength, σ 22c

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Fig. 52.2 Drawing model of ECT sample of C-flute CFB in Solidworks. This model was transferred to Ansys to perform the FE simulation

in this study where the ultimate failure of CFB is identified once the failed region reaches a full length of the pitch length. The setup of the model was done by assuming a perfect bond between the flute and the liners. Large deformation is included to account for the geometric non-linearities. Figure 52.2 shows the illustration of the ECT model drawn in Solidworks. The mesh was set for a quadratic element with a size of 1 mm. Further refinement of the mesh does not improve the results.

52.3 Results and Discussion 52.3.1 Tensile and Compressive Strength of CFB Figure 52.3 shows the result of the short span compression test conducted on 250 gm−2 Kraft linerboard and 160 gm−2 Semi-chemical fluting medium at different moisture content. The moisture content of 7% db represents the moisture content expected at standard testing conditions (50% RH and 23 °C). It was found that the short span compressive strength measurements decrease with an exponential trend as the moisture content increases up to around 20% db. The compressive strength on MD is higher than CD for both Kraft and Semi-chemical paper. The Kraft linerboard was found to possess stronger compressive strength than Semi-chemical fluting medium.

Short Span Compressive Strength (kN/m)

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10.0

Kraft MD

9.0 8.0

Kraft CD

7.0 6.0

Semi-Chem MD

5.0 4.0

SemiChem CD

3.0 2.0 1.0 0.0 0

5

10 15 Moisture Content (%db)

20

Fig. 52.3 The short-span compressive strength of Kraft and Semi-chemical paper measured at different moisture content levels

Tensile Strength (kN/m)

30.0

Kraft MD

25.0 Kraft CD 20.0 15.0

Semi-Chem MD

10.0

SemiChem CD

5.0 0.0 0

5

10 15 Moisture Content (%db)

20

Fig. 52.4 The tensile strength of Kraft and Semi-chemical paper measured at different moisture content levels

However, on a mass basis (see Fig. 52.7 in Appendix 2), the 160 gm−2 Semichemical fluting medium is stronger than the 250 gm−2 Kraft linerboard. Kline [8] reported that the longer fibres such as in Kraft linerboard possess greater strength but the semi-chemical pulping process leaves more lignin and hemicellulose that contributes to more stiffness in the Semi-chemical fluting medium.

Tensile Stiffness (kN/m)

52 Modelling the Edge Crushing Performance …

615

2500

Kraft MD

2000

Kraft CD

1500

Semi-Chem MD

1000

SemiChem CD

500 0 0

5

10 15 Moisture Content (%db)

20

Fig. 52.5 The tensile stiffness of Kraft and Semi-chemical paper measured at different moisture content levels

Figures 52.4 and 52.5 shows the tensile strength and tensile stiffness of the 250 gm−2 Kraft linerboard and 160 gm−2 Semi-chemical fluting medium at different moisture content. These results were measured from the tensile test conducted on the two types of paper. Similar to the results from short-span compression test, it is observed that Kraft linerboard possess stronger tensile behaviour compared to Semi-chemical fluting medium and the strength on MD is stronger than CD. It is also seen in both the tensile strength and tensile stiffness that the Kraft MD has a steeper decrease slope across the moisture content as opposed to other cases. Unlike the compression strength results, on a mass basis (see Fig. 52.8 in Appendix 2), the Kraft linerboard is stronger than Semi-chemical fluting medium which justifies the explanation made by Kline [8] that longer and higher proportion of softwood fibres provides better fibres bonding and result in a stronger tensile strength. From the short-span compression and tensile test results, exponential and linear equations have been fitted with the data to produce the relation of short-span compression strength, tensile strength and tensile stiffness with moisture content up to 20% db in Appendix 2. These equations will be used to compute the main variables (Table 52.2) of each paper at different moisture content. The main variables were input in the FE model to predict the ECT performance of CFB at different moisture content.

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52.3.2 Experiment and Modelling of ECT Test Figure 52.6 shows the experimental and modelling results of ECT on CFB at different moisture content. It is observed that the ECT reduces as the moisture content increase. This is expected since the absorption of water will weaken the bond of the cellulose fibre of the paper material which affects the mechanical performance of CFB [9]. The reduction of ECT as the moisture content increase also show agreement with the findings from the tensile and compression behaviour of the Kraft and Semi-chemical paper in the previous section. The FE model results show good agreement with the experimental findings where it can predict the reduction of ECT as the moisture content increase. This show that the prediction made by just reducing the material properties of the CFB paper components is enough to capture the strength behaviour at different levels of moisture content without the need of a more detailed model that includes the damages to the paper structure such as delamination and fibre breakage. The ECT value has been shown to be useful in predicting the strength of CFB box through the Mckee formula [4]. Since the CFB boxes typically need to face harsh environments such as low temperature and high humidity during storage and handling, these factors should be considered in designing boxes that can endure these conditions. The outcome from this section shows that these factors can be accommodated in the designs by reducing the tensile and compressive material properties of the paper components in the numerical simulation. 14 12

ECT (kN/m)

10 Experimental ECT

8 6

FE Model 4 2 0 0

5

10 15 Moisture Content (%db)

20

Fig. 52.6 The results of ECT at different moisture content based on the experiment and FE model

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52.4 Conclusion This study investigates the tensile and compressive behaviour at different moisture level of two paper components that made up a C-flute CFB namely 250 gm−2 Kraft linerboard and 160 gm−2 Semi-chemical fluting medium. The different moisture level is achieved through standard conditioning room (7% db) and the use of three saturated salt which are NaCl, KCl and BaCl2 . Then, the strength of CFB at similar moisture level was investigated through ECT test with the inclusion of FE model to evaluate the performance of CFB. The results in the tensile and compression test show reduction as moisture level increase. On a mass basis, the Semi-chemical fluting medium shows better specific compressive performance due to having more lignin and hemicellulose, but weaker specific tensile performance due to having shorter fibres as compared to the Kraft linerboard. Between the direction of the fibres for each paper component, the behaviour along MD is stronger than CD in all cases. The ECT of CFB reduces as the moisture level increase. The FE model that uses the tensile and compression test of the paper components as the input parameters show good agreement with the experimental results. This shows that the strength behaviour of CFB at different moisture level can simply be modelled by using the right input parameters.

Appendices Appendix 1 See Tables 52.3 and 52.4. Table 52.3 The thickness and density of the paper component that made up the C-flute CFB used in this study

Table 52.4 Relations from various literatures on computing the other variables as the input parameters for the FE model of CFB

Paper

250 gm−2 Kraft

160 gm−2 Semi-chemical

Thickness (mm)

0.253

0.204

Density (kg/m3 )

791

784

Poisson’s ratio, v12

0.2

0.22

Variables

Formula

Source

E 33

[11, 12]

G12

E 11 /200 √ 0.387 E 11 .E 22

ν-12 and ν-13

0.01

[15–17]

G13

E 11 /55

[12, 18, 19]

[13, 14]

(continued)

618 Table 52.4 (continued)

A. Jamsari et al. Variables

Formula

Source

G23

E 22 /35

[12, 18, 19]

σ 33t

4 2 σ2 σ11 22



Table 52.5 Best fit exponential equations for the compressive strength and moisture content of Kraft and Semi-chemical paper at 23 °C. (MC is the moisture content in % db)

Table 52.6 Best fit linear equations for the tensile strength and moisture content of Kraft and Semi-chemical paper at 23 °C. (MC is the moisture content in % db)

2

 1 2 σ33

σ 33c

[17]

>

−

1 2 σ11

+

1 2 σ22

[17, 20]

τ 12

0.6 × σ33t √ σ11c .σ22c

τ 13

0.024 MPa

[19]

τ 12

0.024 MPa

[19]

[21]

Sample

Equation

R2 (%)

Kraft MD

SSCS = exp (2.878 – 0.08883*MC)

99.0

Kraft CD

SSCS = exp (2.274 – 0.07914*MC)

97.2

Semi-chem MD

SSCS = exp (2.479 – 0.08135*MC)

97.1

Semi-chem CD

SSCS = exp (2.052 – 0.08300*MC)

90.1

Sample

Equation

R2 (%)

Kraft MD

TSgth = 28.95 – 0.7195*MC

92.4

Kraft CD

TSgth = 11.86 – 0.2244*MC

87.2

Semi-chem MD

TSgth = 14.28 – 0.2854*MC

71.1

Semi-chem CD

TSgth = 6.268 – 0.0838*MC

63.7

Appendix 2 See Tables 52.5, 52.6 and 52.7. See Figs. 52.7 and 52.8. Table 52.7 Best fit linear equations for the tensile stiffness and moisture content of Kraft and Semi-chemical paper at 23 °C. (MC is the moisture content in % db)

Sample

Equation

R2 (%)

Kraft MD

TSstiffness = 2979 – 97.46*MC

95.94

Kraft CD

TSstiffness = 1240 – 47.08*MC

96.21

Semi-chem MD

TSstiffness = 1798 – 60.11*MC

95.12

Semi-chem CD

TSstiffness = 796.2 – 32.10*MC

92.46

52 Modelling the Edge Crushing Performance …

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Fig. 52.7 Calculated specific (per unit basis weight) short span compression strength of 250 gm−2 Kraft linerboard and 160 gm−2 Semi-chemical fluting medium where the short span compression strength was estimated using the equations in Table 52.5

Fig. 52.8 Calculated specific (per unit basis weight) tensile strength of 250 gm−2 Kraft linerboard and 160 gm−2 Semi-chemical fluting medium where the tensile strength was estimated using the equations in Table 52.6

References 1. Association FB: Fibre Box Handbook. Fibre Box Association (1999) 2. Biancolini, M.: Evaluation of equivalent stiffness properties of corrugated board. Compos. Struct. 69(3), 322–328 (2005) 3. Markstrom, H.: Testing Methods and Instruments for Corrugated Boards. Lorentz and Wettre, Kista, Sweden (1999)

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4. McKee, R., Gander, J., Wachuta, J.: Compression strength formula for corrugated boxes. Paperboard Packag. 48(8), 149–159 (1963) 5. Fadiji, T., Berry, T., Coetzee, C.J., Opara, L.: Investigating the mechanical properties of paperboard packaging material for handling fresh produce under different environmental conditions: Experimental analysis and finite element modelling. J. Appl. Packag. Res. 9(2), 3 (2017) 6. Navaranjan, N., Dickson, A., Paltakari, J., Ilmonen, K.: Humidity effect on compressive deformation and failure of recycled and virgin layered corrugated paperboard structures. Compos. B Eng. 45(1), 965–971 (2013) 7. Urbanik, T.: Effect of corrugated flute shape on fibreboard edgewise crush strength and bending stiffness. J. Pulp Pap. Sci. 27(10), 330–335 (2001) 8. Kline, J.E.: Paper and Paperboard: Manufacturing and Converting Fundamentals. Backbeat Books (1991) 9. Allaoui, S., Aboura, Z., Benzeggagh, M.: Effects of the environmental conditions on the mechanical behaviour of the corrugated cardboard. Compos. Sci. Technol. 69(1), 104–110 (2009) 10. Greenspan, L.: Humidity fixed points of binary saturated aqueous solutions. J. Res. Natl. Bur. Stand. 81(1), 89–96 (1977) 11. Åslund, P.E., Hägglund, R., Carlsson, L.A., Isaksson, P.: Modeling of global and local buckling of corrugated board panels loaded in edge-to-edge compression. J. Sandw. Struct. Mater. 16(3), 272–292 (2014) 12. Gooren, L.: Creasing Behaviour of Corrugated Board. Eindhoven University of Technology, Eindhoven, The Netherlands (2006) 13. Baum, G.A., Habeger Jr, C.C., Fleischman Jr, E.H.: Measurement of the Orthotropic Elastic Constants of Paper (1982) 14. Aboura, Z., Talbi, N., Allaoui, S., Benzeggagh, M.: Elastic behavior of corrugated cardboard: experiments and modeling. Compos. Struct. 63(1), 53–62 (2004) 15. Baum, G.A.: The Elastic Properties of Paper: A Review (1985) 16. Harrysson, A., Ristinmaa, M.: Large strain elasto-plastic model of paper and corrugated board. Int. J. Solids Struct. 45(11–12), 3334–3352 (2008). https://doi.org/10.1016/j.ijsolstr. 2008.01.031. [Published Online First: Epub Date] 17. Haj-Ali, R., Choi, J., Wei, B.-S., Popil, R., Schaepe, M.: Refined nonlinear finite element models for corrugated fiberboards. Compos. Struct. 87(4), 321–333 (2009) 18. Allansson, A., Svärd, B.: Stability and Collapse of Corrugated Board-Numerical and Experimental Analysis (2001) 19. Beldie, L.: Mechanics of paperboard packages–performance at short term static loading. Licentiate Dissertation, Lund University, Lund, Sweden (2001) 20. Nordstrand, T.: On buckling loads for edge-loaded orthotropic plates including transverse shear. Compos. Struct. 65(1), 1–6 (2004) 21. Biancolini, M.E., Brutti, C., Porziani, S.: Experimental characterisation of paper for corrugated board. In: Proceedings of Sixth International Symposium: Moisture and Creep Effects on Paper, Board and Containers, Madison, Wisconsin, USA (2009)

Chapter 53

Multiaxial Stress Based High Cycle Fatigue Model for Adhesive Joint Interfaces M. A. Eder , S. Semenov

and M. Sala

Abstract Large utility wind turbine rotor blades (WTBs) comprise of adhesive joints with typically thick bond lines. The dynamic aero-elastic interaction of the WTB with the airflow generates multiaxial non-proportional, variable amplitude stress histories in the adhesive joints. Structural optimization of WTBs employed at an early design stage sets high demands on computationally efficient interface fatigue models capable of accurately predicting the critical locations prone for interface failure. The numerical stress-based interface fatigue model presented in this work uses the Drucker-Prager (DP) criterion to compute three different damage indices corresponding to the two interface shear tractions and the outward normal traction. The DP model was chosen because of its ability to consider shear strength enhancement under compression and shear strength reduction under tension. The model was implemented as Python plug-in for the commercially available finite element code Abaqus. The model was used to predict the interface damage of an adhesively bonded, tapered glass-epoxy composite cantilever I-beam tested by LM Wind Power under constant amplitude compression-compression tip load in the high cycle fatigue regime. Results show that the model was able to predict the location of debonding in the adhesive interface between the webfoot and the cap. Keywords Adhesive · Fatigue · Interface · Multiaxial stress · Failure mode

53.1 Introduction Many glass-epoxy composite structures such as large utility wind turbine rotor blades (WTBs) comprise of adhesive joints with typically thick bond lines used to connect the different components during assembly. M. A. Eder (B) · S. Semenov Technical University of Denmark, 4000 Roskilde, Denmark e-mail: [email protected] M. Sala LM Wind Power Blades, Jupitervej 6, 6000 Kolding, Denmark e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_53

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Performance optimization of rotor blades to increase power output by simultaneously maintaining high stiffness-to-low-mass ratios entails intricate geometries in conjunction with complex anisotropic material behavior. Consequently, adhesive joints in WTBs are subject to multiaxial stress states with significant stress gradients depending on the local joint geometry. Moreover, the dynamic aero-elastic interaction of the WTB with the airflow generates non-proportional, variable amplitude stress histories in the material. Empiricism shows that a prominent failure type in WTBs is high cycle fatigue failure of adhesive bond line interfaces, which in fact over time developed into a design driver as WTB sizes increase rapidly. Structural optimization employed at an early design stage therefore sets high demands on computationally efficient interface fatigue models capable of predicting the critical locations prone for interface failure. The numerical stress-based interface fatigue model presented in this work uses the Drucker-Prager criterion [1] to compute three different damage indices corresponding to the two interface shear tractions and the outward normal traction. The equivalent stress signals are obtained by self-similar scaling of the Drucker-Prager surface whose shape is defined by the uniaxial tensile strength and the shear strength such that it intersects with the stress point at every time step. This approach implicitly assumes that the damage caused by the prevailing multiaxial stress state is the same as the damage caused by an amplified equivalent uniaxial stress state in the three interface directions. The two-parameter Drucker-Prager model was chosen because of its ability to consider shear strength enhancement under compression and shear strength reduction under tension. The damage indices are computed through the well-known linear Palmgren-Miner rule after separate rain flow-counting of the equivalent shear stress history and the equivalent pure normal stress history. The governing interface damage index is taken as the maximum of the triple. Hahne et al. [2] originally suggested the general approach of using a continuum damage failure surface to predict the multiaxial fatigue life of carbon fibre epoxy composite structures follows. Their approach was later implemented into a software by Pörtner [3]. The proposed interface fatigue model was implemented as Python plug-in for the commercially available finite element code Abaqus [4] for its use with solid elements. The model was used to predict the interface damage of an adhesively bonded, tapered glass-epoxy composite cantilever I-beam tested by LM Wind Power under constant amplitude compression-compression tip load in the high cycle fatigue regime. Results show that the model was able to predict the location of debonding in the adhesive interface between the adhesive and the cap. It can be concluded that the fidelity, robustness and computational efficiency of the proposed model makes it especially suitable for rapid fatigue damage screening of large 3D finite element models subject to complex dynamic load histories.

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53.2 Methodology 53.2.1 Sub-component Fatigue Testing Methodology LM Wind Power using structural optimization developed a novel adhesively connected subcomponent. The aim was to represent and test a WTB adhesive cap-joint as realistic as possible with the objective function to maximize the shear stress in the lower bondline of the gauge section. When testing sub-components it is important to initiate failure in the gauge area remote from boundary conditions and load application points [5, 6]. After several design iterations it was found that an hour-glass shaped subcomponent with a non-constant lengthwise geometrical variation of the beam height induces shear dominated bondline failure in the tapered region next to the neck (see Fig. 53.1). Furthermore, the taper chosen in the design is a realistic representation of bondlines in WTBs accounting for taper effects on the stress state. The symmetric subcomponent consists of an I-type cross-section comprising of two caps (tf = 10 mm) and a sandwich shear web (t = 24 mm) itself consisting of a balsa wood core and two Biax material skin layers in the gauge region. Caps and web were produced separately using vacuum infusion with subsequent curing and post curing. Caps and web were adhesively connected with a variable bondline thickness featuring its maximum with 18 mm at the gauge section. The web was reinforced with additional skin layers of Biax material in the clamped region. The cantilever type subcomponent had an overall free length of 700 mm and a max. section height of 185 mm. Figure 53.1b shows that the section height decreased smoothly to a neck height of 120 mm. The subcomponent was clamped on one end between two steel platen using eight pre-stressed steel bolts. The steel clamping device was mounted onto a pair of steel I-beams themselves bolted to the machine frame. The subcomponent was loaded at the tip by a hydraulic actuator from below.

Fig. 53.1 a Taylor made test rig fitted into a 250 kN hydraulic uniaxial servo hydraulic testing machine and b a close-up of the subcomponent readily installed between the clamps featuring a cutout in order to provide lateral stability to the subcomponent

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The force was applied through a cylindrical bearing using contact pressure and simultaneously avoiding the introduction of excessive shear force into the actuator. The subcomponent was tested in compression-compression fatigue applying a constant amplitude at a frequency of 2 Hz avoiding adiabatic heating. The test was performed under force-control with the inertia compensated load cell located on the actuator side. Strains in the bondline were measured with electrical resistance strain gauges surface mounted on the bondline. The subcomponent was considered to have failed after initiation of debonding between the adhesive and the lower cap.

53.2.2 The Multiaxial Fatigue Interface Model Figure 53.2 shows the local Cartesian directions of an infinitesimal interface element whose outward normal stipulated as x3 . Accepting that an interface has zero thickness i.e. the spatial dimension in the 3-direction is zero the interface tractions can be written as ti = σi j .(0, 0, 1)T . Consequently, the interface stress tensor reduces to three components as follows ⎡

⎤ 0 0 σ13 σi j = ⎣ 0 0 σ23 ⎦ σ13 σ23 σ33

(53.1)

where σ12 , σ13 and σ33 are the out-of-plane shear, the transverse shear and the normal traction in the local material coordinate system respectively. For the adhesive interface fatigue model, we seek an approach similar to the multiaxial laminate fatigue model originally proposed by [2]. The adopted model for interface fatigue is the well-known two-parameter Drucker-Prager (DP) criterion [1], which can be written as follows  J2 = A + B I1 (53.2)

Fig. 53.2 Definition of the interface in a bi-material situation with the three principal directions and the associated stress components where the 1-direction in this paper coincides with the fibre direction; in the numerical procedure implemented the nodal stresses are extracted for each side of the interface separately, where stress values of shared nodes are averaged

53 Multiaxial Stress Based High Cycle …

625

where I1 is the first invariant of the stress tensor and J2 is the second invariant of the deviatoric stress tensor and A, B are two material dependent parameters. Specialising the invariants for the interface tensor Eq. 53.1 and substitution into Eq. 53.2 defines the DP failure criterion notably  F(σ ) =

1 2 2 2 σ + σ23 + σ13 − A + Bσ33 3 33

(53.3)

Per definition an interface cannot fail in compression. The two remaining interface failure modes are therefore uniaxial tension and shear. Consequently, the two parameters A and B are associated with the uniaxial tensile strength σ0 and the pure shear strength τ0 . The latter is assumed identical in the 1-direction and 2-direction. The constant A can be determined by calibration of Eq. 53.3 in pure shear in the 1-direction by substituting σ13 = τ0 and setting σ23 = σ33 = 0. The remaining constant B can be determined by calibration of Eq. 53.3 in pure tension in the 3-direction assuming σ13 = σ23 = 0 and substituting σ33 = σ0 . The two constants can be written as follows A = τ0

(53.4)

1 τ0 B=√ − σ 3 0

(53.5)

Upon substitution of A and B into Eq. 53.3 the failure criterion can eventually be written as follows   1 2 1 τ0 2 2 σ33 F(σ ) = + σ13 = τ0 + √ − (53.6) σ33 + σ23 3 3 σ0 Figure 53.3a shows the rotationally symmetric DP surface for F(σ) = 0 for a situation where τ0 > σ0 in which the surface forms a rotational paraboloid. It shows that failure is attained at σ33 = σ0 . For σ33 < σ0 the shear strength increases as the uniaxial tensile stress decreases. The circular contour at a state of pure shear i.e. σ33 = 0 has the radius ρ = τ0 according to the calibration. The shear strength increases for σ33 < 0 showing the dependency of the DP criterion on the hydrostatic stress state enhancing the shear strength with increasing compressive traction stresses. It can also be inferred from Fig. 53.3 that a tensile stress component in conjunction with shear is rather damaging in contrast to a compressive stress. Similarly to the adopted concept of equivalent fatigue stresses obtained by scaling of the material strength properties in the Puck criterion [2, 3], the equivalent interface fatigue stresses will be obtained by scaling of the DP criterion Eq. 53.6. In this case for any arbitrary stress point P(σ33 , σ13 , σ23 ) an equivalent pure shear stress τeq can be obtained by Eq. 53.7 through replacing τeq with τ0 in Eq. 53.6 and isolation (see Fig. 53.3b). The equivalent pure tensile stress σeq can subsequently be obtained through Eq. 53.8.

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Fig. 53.3 a axisymmetric DP failure surface depicted in the interface traction stress-space depicted for σ0 = 1 MPa and τ0 = 1.5 MPa; The circular markers indicate the two material constants namely the uniaxial tensile strength and the shear strength. b A longitudinal section through the failure surface where the black graph represents the failure envelope and the red graph shows the scaled surface which intersects an arbitrary stress point P(σ33 , σ13 , σ23 )

 τeq =

 1 2 1 τ0 2 2 σ33 → 0 ≤ τ f ≤ τ0 σ33 + σ23 + σ13 − √ − 3 3 σ0 σ0 τeq σeq = → 0 ≤ σeq ≤ σ0 τ0

(53.7) (53.8)

Figure 53.4 shows the equivalent shear stress and the equivalent tensile stress of the scaled DP criterion. The underlying idea of this model is that the damage caused by any multiaxial stress state is equivalent to the damage caused by either τeq or by σeq provided that they share the same DP envelop. In this way the multiaxial stress state was decomposed into two equivalent uniaxial stress states. It can be seen from Fig. 53.4 that the DP surface is open towards the compression side as compressive failure of an interface is not physical as pointed out by Lemaitre and Rodrigue [7].

Fig. 53.4 Computation of the equivalent stress components by scaling the DP law such that it intersects the stress point. Depending on the invariants of the current stress state the equivalent stresses are obtained in two separate cases. Where Case A is dominated by tension, Case B is dominated by shear where the equivalent stresses in the latter case are obtained by linear scaling the vector in the shear plane such that its norm corresponds to τeq

53 Multiaxial Stress Based High Cycle …

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Therefore, the equivalent fatigue stresses can be distinguished into two separate cases as follows: Case A I1 ≥ J2 In this case the stress state is governed by tension and the three equivalent Cauchy fatigue stresses are consequently σ33eq = σeq

(53.9)

σ13eq = σ13

(53.10)

σ23eq = σ23

(53.11)

Case B I1 < J2 In this case the stress state is governed by shear and the three equivalent Cauchy shear stresses are consequently σ33eq = σ33

(53.12)

σ13eq = τeq sin α

(53.13)

σ23eq = τeq cos α

(53.14)

α = tan

σ13 σ23

(53.15)

Figure 53.5 shows principle of the multiaxial fatigue model in which the stress histories obtained from any time history analysis are converted into three scaled equivalent stress histories before they are rain flow counted separately. Figure 53.6 shows that a symmetric linear Goodman diagram or constant life diagram are used for the two equivalent shear stresses, whereas a shifted linear Goodman diagram is used for the equivalent normal stress component. A damage index Dˆ is computed for each stress component separately using the well-known linear Palmgren-Miner damage accumulation hypothesis Dˆ =

k

ni Nfi i=1

(53.16)

where ni is the number of cycles in a bin, k represents the number of bins and Nfi denotes the number of cycles to failure of a specific bin. The final nodal damage index is defined as the maximum of the triplet according to

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Fig. 53.5 Converting stress histories obtained from time history analysis into two equivalent shear stress histories and one equivalent normal stress history. The model poses no restrictions on linearity insofar non-linear stress analysis, variable amplitude and non-proportional loading are admissible

Fig. 53.6 Calculation of three separate damage indices using the equivalent stress component histories. Linear Goodman diagrams based on the Basquin power law were used for simplicity where USS, UCS and UTS represent the ultimate shear strength, the ultimate compressive strength and the ultimate tensile strength of the interface respectively

 D = max Dˆ 13 , Dˆ 23 , Dˆ 33

(53.17)

Equation 53.17 allows the distinction between the three failure modes and in this work referred to as state variable. In the current implementation a state index of 1000 (blue), 2000 (green) and 3000 (red) was assigned to Dˆ 13 , Dˆ 23 and Dˆ 33 respectively.

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Fig. 53.7 a Elevation of the numerical model with clamped boundary constraints on the left hand side (triangles) and loaded with a concentrated force (arrow); the adhesive layer of interest is highlighted in red and b half of the cross section showing different materials namely UD dark grey, adhesive (ochre), biax (green) and sandwich core (light grey)

53.2.3 Numerical Model The sub-component was modelled in the commercial finite element package Abaqus [4] and discretized with 8-node solid elements (Abaqus type C3D8I). Figure 53.7a shows that all nodes on the left hand side face were fully constrained. This assumption somewhat deviates from the clamped conditions shown in Fig. 53.1 as it over estimates the rotational stiffness in the support. However it is deemed a good approximation as the difference in rotation is small and does not affect the stress state in the gauge area. The load was applied through a master node itself connected to the lower cap using a kinematic coupling constraint. The mean was applied in a static step, followed by a second step in which a harmonic constant load amplitude was applied. Linear elastic material properties were assumed. Since the displacements involved in fatigue testing are small in comparison to the beam length, the analysis was performed linear (Abaqus type linear perturbation). Figure 53.7b shows that due to symmetry only one half of the cross section was modelled, where the nodes of the symmetry plane were restrained in the global z-direction (horizontal). Perfect bond was assumed between the adhesive and the interface. The multiaxial fatigue model described in Sect. 53.2.2 was implemented as Abaqus plugin in Python. The plugin extracts the stress histories along the predefined interface sets, depending on the modelling technique used for both sides of the interface separately. Since the stress history extraction step is rather time demanding, these are saved in a pickle-file and can be re-used for damage calculation.

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53.3 Results Figure 53.8 shows the failed sub-component with a typical 45º shear crack in the bulk adhesive. The significant speed of failure propagation makes it hard to detect the initiation site exactly and the application of high-speed cameras for fatigue tests is impractical. However, it is rather certain that failure initiated at the location of the shear crack seen in the lower adhesive. It is not abundantly clear whether that shear crack in the bulk initiated interface failure or vice versa. In any case, upon initiation the crack propagated a considerable distance along both interfaces with a distinct fibre bridging situation. Figure 53.9a shows the adhesive of the numerical model (see red part in Fig. 53.7a) separately and slightly rotated as to expose the lower interface. The contours represent the damage index (see Fig. 53.6) predicted for that interface. Comparison of Fig. 53.8 with Fig. 53.9a shows that the numerically predicted failure location agrees well with experiment. It also shows that failure was initiated along the edge of the adhesive due to stress concentration (singularity) effects caused by the corner. Figure 53.9b shows that the failure region depicted in green is dominated by outof-plane shear (σ13 ) as initially designed for. Interestingly, the prevailing potential failure mode in the neck crown is in-plane shear σ12 and that at the transition from the neck into the constant height is dominated by normal traction σ33. The latter is caused by the curvature of the cap in conjunction with high axial stresses leading to normal traction bearing stresses. Note that the colours shown in Fig. 53.9b are representing the state variable only and are therefore not indicative for the magnitude of fatigue damage.

Fig. 53.8 Failed specimen with failure initiation in the lower bond line above the second red tick mark counted from the tip. Typical fibre bridging patterns can be recognised along the debonded interfaces

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Fig. 53.9 a Damage index contour plot of the lower interface between cap and adhesive and b damage state variable contour plot of the same interface with the colour code corresponding to the failure modes schematically depicted. The local interface Material-1 direction is in the general direction of the beam axis

53.4 Discussion and Conclusion The hypothesis of the proposed DP-fatigue model is that the damage caused by an arbitrary multiaxial stress state is equivalent to the damage caused by a uniaxial stress state located on the same failure surface. In this way, it is possible to convert a multiaxial fatigue situation into a set of equivalent uniaxial fatigue cases, which can be handled using the classic stress based fatigue approach. The proposed model is developed for the purpose of fatigue damage screening of large structures subject to complex loading conditions at an early design stage. Since the model does not rely on any superposition principles, geometric and/or material nonlinearities as well as non-proportional do not represent a limitation to its applicability. Due to its computational efficiency the model lends itself for applications in structural optimization. The model does neither account for material stiffness- nor material strength degradation and is therefore strictly speaking only valid for crack initiation prediction. The predicted interface damage agrees well with the experimentally obtained failure location. The predicted state variable associated with the damage is a useful tool to improve the fatigue life of the adhesive joint. Acknowledgements This work was conducted within the industrial research project IMPACT with Journal number 64016-0065 funded by the Danish Energy Technology Development and Demonstration Program (EUDP). The support is gratefully acknowledged. The authors are very grateful for the scientific and technical support provided by Dr. Michael Wenani Nielsen and Dr. Thomas Karl Petersen from LM Wind Power.

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References 1. Drucker, D.C., Prager, W.: Solid mechanics and plastic analysis of limit design. Q. Appl. Math. 10(2) (1952) 2. Hahne, C., Knaust, U., Schürmann, H.: Zur Festigkeitsbewertung Von CFK-Strukturen Unter Pkw-Betriebslasten, Fatigue Evaluation of CFRP Structures under Complex Car Loads, in German, Materialpruefung/Materials Testing 56 (7–8). Carl Hanser Verlag (2014) 3. Pörtner, H.: Multi-axial fatigue models for composite lightweight structures. Master’s Thesis in Applied Mechanics Department of Applied Mechanics Division of Material and Computational Mechanics, Chalmers University of Technology, Göteborg, Sweden (2013) 4. Dassault Systémes: Abaqus Analysis Manual, vol. 6.16 (2016) 5. Belloni, F., Eder, M.A., Cherrier, B.: An improved sub-component fatigue testing method for material characterization. Exp. Tech. 42(5), 533–550 (2018) 6. Zarouchas, D.S., Makris, A.A., Sayer, F., Van Hemelrijck, D., Van Wingerde, A.M.: Investigations on the mechanical behavior of a wind rotor blade subcomponent. Compos. Part B-Eng. 43(2) (2012) 7. Lemaitre, J., Rodrigue, D.: Engineering Damage Mechanics: Ductile, Creep, Fatigue and Brittle Failures. Springer, Berlin (2005)

Chapter 54

Rating of Polymers for Low-Cost Rapid Manufacturing of Individualized Anatomical Models Used in Presurgical Planning ˙ Magdalena Zukowska, Filip Górski and Adam Hamrol Abstract Rapid Prototyping and Manufacturing, especially during last few years, become one of the most common tool in creating objects with complicated geometry and in small production as well. Intensive use of these technologies shall be noted in areas like military, transport and general production but also in medicine. Presurgical support and preparation of a surgeon with use of these technologies, especially in complex cases, can help prepare more precise plan of surgery and perform a simulated operation. The aim of these studies was to classify polymer materials for low-cost Rapid Manufacturing, which have to imitate human soft tissue. Materials in future will be used to manufacture personalized anatomical models for presurgical planning and simulative operations on kidneys. Keywords Rapid manufacturing · Anatomical models · Presurgical planning

54.1 Introduction Rapid Prototyping and Manufacturing, especially during last few years, become one of the most common tool in creating objects with complicated geometry and in small production as well. Intensive use of these technologies shall be noted in areas like military, transport and general production. Besides fields, Rapid Manufacturing become more popular also in medicine. Based on EY’s Global 3D printing Report 2016, Pharma and Medical companies are on 3rd place in using 3D printing to create their own products and components. Authors underline that the main benefits are improved quality (44%) and what is important in modern medicine, customized products (41%) [1]. Thanks to the specificity of rapid prototyping technologies, there is possibility to produce personalized metal implants like rib, jaw or hip, custom prosthetics for children and adults and models for presurgical planning and simulative operations [2–4]. ˙ M. Zukowska (B) · F. Górski · A. Hamrol Chair of Management and Production Engineering, Poznan University of Technology, Poznan, Poland e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_54

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Dissemination of 3D Printing was an effect of opening license for technology named FDM (Fused Deposition Modeling)/FFF (Fused Filament Fabrication). Intensive development open source projects had created low-cost 3D Printing. Costs of devices and materials for printing fall by several dozen percent. Producers constantly invest in upgrading, which results in high quality of manufactured products on lowcost 3D Printers. It affected increasing access by wider group of patients and doctors to personalized medical products. The aim of studies is associated with low-cost Rapid Manufacturing technologies. It is connected with costs of health care in Poland, which is the general area of interest in this studies. Based on report Health and Health Care in 2016 by Polish Central Statistic Office, current expenditure on health care in 2015 in Poland amounted to 6304 PLN/1704 USD for one resident.1 For comparison in the same year in Austria it was 18 870 PLN/5 100 USD [5]. Average patient stay in hospital wards is about 5 days and costs for 24 h per person at surgical cubicle costs 223,49 PLN/60,40 USD which gives 1117,54 PLN/302 USD for whole stay. Moreover, patient stay in hospital is motivated by different types of operations and treatments. An example surgery of cholecystectomy (removal of the gallbladder) costs: 463,00 PLN/125,14 USD (materials) and 261,47 PLN/70,67 USD (pharmaceutical) [6]. In summary for one person costs of hospital stay amount 1842,01 PLN/497,81 USD. In this situation costs of personalized anatomical models manufactured with use of 3D Printers have to be mineralized if hospitals would invest in this type of presurgical support. Performed tests were intended to classify polymer materials for low-cost Rapid Manufacturing (especially Vacuum Casting), which have to imitate human soft tissue. Materials in future will be used to manufacture personalized anatomical models for presurgical planning and simulative operations on kidneys.

54.2 Presurgical Planning—Literature Review 54.2.1 Definition of Presurgical Aims Definition of presurgical planning from perspective of engineers, concerns all tools which helps doctors in their everyday work and also in preparing before operation. These are issues relating to presurgical planning, medical software and medical imaging. In last few years part of it becomes 3D Printed medical models as well. All tools are consequences of implementing programs based on FEM (Finite Element Method) which allowsmodeling and simulating in Three-Dimensional Space [7]. Manufacturing of personalized anatomical models helps to precise familiarize with particular problem and prepare carefully delineated plan of operation. In basis of questionnaire held in group of trainee, complement CT scans on 3D Printed model increase overall confidence in performing operation, ability to make operative plan, 1 *Dollar

exchange rate 1 USD = 3,70 PLN.

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select optimal entry and familiarity with shape, orientation and location of pathology in organ [8].

54.2.2 Rapid Manufactured Anatomical Models Rapid Manufacturing in presurgical support in urology can be divide in two main group: for only planning before operation and models for performing simulative operation. Most cases connect both options—firstly surgeon prepare plan of operation and familiarize with organ and pathology area (location, shape, orientation) with use of medical images and anatomical model. Secondly perform simulative operation (if materials in model are allow t do it) on the basis of earlier collected information. Anatomical models consist of patient’s organ and pathology area. Additional can be other elements for better presentation of problem, like blood vessels and other characteristic structures for presented organ (in kidney it can be renal pelvis/spider pelvis). Rapid Prototyping is based on slicing Three-Dimensional object on TwoDimensional slices and extrude material layer by layer to produce again 3D object. Additive Manufacturing Technologies allows this way to create models with atypical geometry and shape. With use of Rapid Manufacturing technologies it is possible to produce phantoms with a very complicated structures and recreated it with all details. Models can help not only surgeons but also students of medicine, trainees and patients. Doctors can familiarize with difficult problem, other than most cases and it increase patient’s safety and can reduce time of surgery. For students and trainees models can be perfect tool for study different cases and also can be use as models for tutorial operation. It can increase students confidence and practice before the real surgery. For patients models can be more understandable than presenting problem on medical images like CT Scans.

54.2.3 Examples of Models Used in Urology In the available literature presented models are produced with use of different additive technologies, low-costed but also professional printers. One of closely tested model was presented during European Association of Urology by scientists from Kobe University (Fig. 54.1). Transparent model of kidney manufactured by PolyJet Technology include tumor and blood vessels system. Used method of printing allow to create phantom with different materials. It facilitated separated health areas form pathology areas and perform simulative operations of robotic-assisted partial nephrectomy. Detailed knowledge of blood vessels distribution in organ restrict area of operation and reduce time of stopped blood flow in kidney from 22 to 8 min [9]. Second example is medical model of kidney which also was used to perform simulative operation. Phantom was part of studies at Washington University School

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Fig. 54.1 Model of kidney with tumor; manufactured and tested by scientists from Kobe University, Japan [9]

Fig. 54.2 Presentation of operated model (right) and moment of suturing (left) [10]

of Medicine (Fig. 54.2). Model was manufactured by Vacuum Casting Technology. Based on CT images were made segmentation of kidney and crated digital 3D model. It become model mother to designed mold for renal cortex (2 parts) and for tumor (3rd part). Model was made of silicone which was suppose to imitate human tissue and was in similar color to the real organ. Material was selected before in consultation with doctors and comparison to porcine kidney. The final test was simulative operation of partial nephrectomy and was part of training for residents at hospital [10].

54.3 Materials and Methods 54.3.1 Methodology Studies was focused on selection more accurate materials for Additive Manufacturing ˙ Technologies like low-cost FDM, SLA and VC. Earlier studies did by Zukowska et al. [11] on manufacturing anatomical models for simulative operation and presurgical

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planning gives main guidelines for type and parameters of material. After consultation with doctors and effects on model after suturing (major stress in sutures, breaking materials under a stress, problems with suturing), determinants have been: hardness under 45 Shore A, flexibility, susceptibility to plastic deformation and transparency. Most of parameters are related to creating better conditions for simulative operation and the last one is property indicated by surgeons as important in familiarize with location and size of tumor or other pathologies in organ. Besides, before tests fulfilled comparison with porcine kidney and liver (subjective hardness evaluation and suturing assessment). The complementation of parameters are information about bovine kidney Young’s modulus −5.9 ± 0.7 kPa [12] and porcine kidney hardness—about 30 Shore A [13]. Studies were divided in stages. First was analysis of materials for low-cost Rapid Manufacturing (FDM, SLA, VC) or biomedical materials for 3D Printing. Selected materials were grouped and additionally restricted to tested group. Secondly, manufactured two types of test samples which imitate: • renal cortex samples with dimensions: 55 × 55 × 25 mm, • membrane after tumor removal with dimensions: 50 × 14 × 3 mm. The last stage was suturing samples with three different types of surgical suture: unabsorbed 1–0, 3–0 and resorbable 3/0. After suturing, samples were monitored right after procedure, 2 h after and one week after. Based on collected information compiled profile (also financial) of the best material for anatomical models used in simulative operation of partial nephrectomy.

54.3.2 Materials The main information about materials was hardness. In Table 54.1 are presented information about chosen materials for all low-cost technologies. After preliminary analysis of materials costs, group was restricted to dedicated for Vacuum Casting (VC). In table Table 54.2 final group of materials were subjective compared to porcine kidney and shortly described. It gave view on probably scope of hardness and plasticity similar to human kidney. Table 54.1 Chosen materials for different low-cost AMT AMT

Type of materials

Name

Hardness [Shore A]

VC

Silicone

ZA 8 LT

VC

Silicone

ZA 22 Mould

21

White

VC

Silicone

Dragon Skin 10 Very Fast

10

Semitransparent

VC

Silicone

Sorta Clear 12

12

Transparent

VC

Silicone

Sorta Clear 37

37

Transparent

8

Color Transparent

(continued)

˙ M. Zukowska et al.

638 Table 54.1 (continued) AMT

Type of materials

Name

Hardness [Shore A]

Color

VC

Silicone

Encapso K

20

Transparent

VC

Silicone

Body Double Standard

25

Blue

VC

Resin

GM-900-1

13

Yellow

VC

Resin

GM 956-45

45

Amber

DLP

Resin

Resin flexible

–

Transparent

DLP

Resin

SuperFine

–

Purple

SLA

Resin

SuperFine Clear

85

Transparent

FDM

PCL

eSun eMate

–

Red

FDM

Experimental

Lay Fomm 40

40

Beige

FDM

Experimental

Lay Fomm 60

60

Beige

FDM

Experimental

GEL LAY

20–30

Beige

FDM

Modified TPE

Ninja Flex Water

85

Semitransparent

Table 54.2 Description of selected materials in compare to porcine kidney Type

Name

Hardness [Shore A]

Description

Soft tissue

Porcine kidney

~30

Soft, not possible cutting with blunt instrument, possible cutting with sharp objects/instruments high plasticity, low flexibility

Silicone

ZA 8 LT

8

Highly soft, low plasticity, low transparent (about 40%), high dimensional accuracy

Silicone

ZA 22 Mould

21

Soft, flexibly, flexible, good plasticity, not transparent (white) high dimensional accuracy

Silicone

Dragon Skin 10 Very Fast

10

Highly soft, flexible, high plasticity, little transparent (about 5%), high dimensional accuracy

Silicone

Sorta Clear 12

12

Soft, high plasticity, low transparent, needs degassing in vacuum, high dimensional accuracy (continued)

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Table 54.2 (continued) Type

Name

Hardness [Shore A]

Description

Silicone

Sorta Clear 37

37

Low softness, good plasticity, flexible, highly transparent, needs degassing in vacuum (high porosity), high dimensional accuracy

Silicone

Encapso K

20

Highly brittle, low softness, high transparent (100%), good plasticity, high dimensional accuracy

Silicone

Body Double Standard

25

Soft, flexible, low plasticity, blue, biocompatible, high dimensional accuracy

Resin

GM-900-1

13

Soft, highly flexible, skin effect, high plasticity, yellow, high dimensional accuracy

54.3.3 Tests Prepared models were used to manufacturing two types of samples. First one imitate renal cortex. It means that samples were thicker, more solid and stress in sutures must be on lower level than for second group of samples. The dimensions for first group are 55 × 55 × 25 mm besides resin which needs silicone mold. For this sample dimensions were 130 × 20 × 10 mm. Samples were subcised in 40 mm long on top surface with use of scalpel number 11 (type employ during operation on kidneys). For better visibility of nick in transparent material, line was marked on blue. On every sample performed suturing with three different types of surgical suture (Sect. 5.3.1). Effects are presented on Figs. 54.3 and 54.4. Second type of samples imitated membranes sutures after tumor removal. In this case stress inside sutures are higher because necessary to holding parts together. In this tests important parameters of materials are their plasticity and flexibility. Dimensions of samples were 50 × 14 × 3 mm (Fig. 54.5). Mold for casting was specially 3D printed on FDM printers. Because of conflict between Resin GM 9001with ABS material of mold, there was necessary to manufactured model in silicone mold in different shape. Suturing was exact the same like in previous case. Effects of test are presented on Figs. 54.6 and 54.7.

640

Fig. 54.3 Samples of materials before tests (left) and after rupturing (right) Fig. 54.4 Samples after test; resin (top) and silicone Encapso K (bottom)

˙ M. Zukowska et al.

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Fig. 54.5 Samples of materials imitate membranes

Fig. 54.6 Resin sample after suturing

54.3.4 Observation and Financial Analysis Both groups of samples were observed in 3 time interval: during suturing and immediately after procedure, 2 h after and one week after. Main changes in materials have occurred in first and third intervals. For both types of samples changes were similar. Material which was disqualified named Encapso K. It had perfect transparency at 100% level, but was highly brittle and suturing was not possible. In both cases material ZA 8 LT too soft for suturing. Process was not comfortable and sutures start cut material right after procedure. Similar situation was for material Sorta Clear 12. After 2 h changes in material

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Fig. 54.7 Samples after suturing, process of connecting two edges create strain inside material and sutures

(high strain in sutures) named GM 956-45. After one week changes like cutting material by sutures appeared in materials: Body Double and ZA 22 LT. The best effects after tests observed in materials: Sorta Clear 37, Dragon Skin Very Fast and Resin GM 900-1. Having regard to summary in Table 54.2, Sorta Clear 37 need additionally degassing in vacuum because of high porosity. Also material was harder than porcine kidney. Problems with Resin GM 900-1 are connected with type of mold and not transparent color. In conclusion the best material was Dragon Skin 10 Very Fast. Its advantages are especially low hardness and good plasticity. Also the material is easy to prepare (proportion of components are 1A:1B) and Rapid for manufactured ready model (30 min). Main disadvantage is low transparency (~5%) (Fig. 54.8). The price of material is 199 PLN/53 USD for 0,91 kg. Mean volume of kidney’s model is about 0.2–0.3 kg. It gives 43-65 PLN/11-17 USD for material needed to manufactured anatomical model of kidney. Total costs must include also manufacturing of mold (material, time, power) and work of engineers (segmentation from CT scans, modeling). It can be different for many models, it depends of case, engineer’s skills and used material to produce mold.

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Fig. 54.8 Selected information about chosen material [14]

54.4 Conclusion Studies have enabled specify what type of material for Vacuum Casting Technology is the best to produce personalized anatomical models of kidney which can be used in simulative operation. Material was selected in few stages and thanks to that was compared to porcine kidney as subjective test and also was define its behavior after suturing on samples with low and high stress. Furthermore, costs of material necessary to produce model of kidney are low. Model manufactured with use of this material can increase safety of patient and confidence of surgeon (especially resident doctors) during operation. Model helps to familiarize to type of pathology and enable planning before operation. Next studies will be include next material analysis in the base of collected information and parameters. Another aim will be improve the material transparency. In addition, for samples (after removing stitches) will be undertaken strength test on Testing Machine SUNPOC WDW-5D-HS and 3D scanning with 3D Scanner DAVID SLS-2 for definition deformation resulting from suturing and to establish shape and dimension accuracy. In the same time will be conducted interview and test with surgeons, residents ad students of medicine. Results will present requirement for imitation soft tissue by used materials.

References 1. Müller, A., Karevska, S.F.: How will 3D printing make your company the strongest link in the value chain? EY’s Global 3D printing Report 2016. Ernst & Young GmbH (2016) 2. Janik, J.: Tak powstawał tytanowy implant z drukarki 3D, wszczepiony pacjentowi. http://www.rynekzdrowia.pl/Uslugi-medyczne/Tak-powstawal-tytanowy-implant-zdrukarki-3D-wszczepiony-pacjentowi,140160,8.html. Accessed July 2017 3. 3ders.org: Argentinan patient leads normal life with 3D printed cranial implant. http://www. 3ders.org/articles/20150510-argentinan-patient-leads-normal-life-with-3d-printed-cranialimplant.html. Accessed July 2017

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4. BioFabris: 3D printed surgical guides make their Malaysian debut. http://biofabris.com.br/en/ 3d-printed-surgical-guides-make-their-malaysian-debut/. Accessed July 2017 5. Polish Central Statistic Office: Health and health care in 2016. Warsaw (2016) 6. Orli´nski, R.: Cost accounting of patients on the example of hospital. Research Papers of Wrocław University of Economics, no. 442, Wydawnictwo Uniwersytetu Ekonomicznego we Wrocławiu, Wrocław (2016) 7. Tavares, J.M.R.S., Jorge, R.N.: Developments in Medical Image Processing and Computational Vision (2015) 8. Spradling, K.: Preoperative planning of percutaneous nephrolithotomy using three-dimensional (3D) printed models of complex kidney stones, 2015 Endourology Society Summer Student Scholarship Summary Report, Department of Urology, University of California, Irvine 9. European Association of Urology: Surgeons develop personalised 3D printed kidney to simulate surgery prior to cancer operation. http://www.alphagalileo.org/ViewItem.aspx?ItemId= 140891&CultureCode=en. Accessed July 2017 10. John T. Milliken Department of Medicine Division of Nephrology: 3D printed kidney models change the way surgeons prepare for surgery. https://renal.wustl.edu/3d-printed-kidneymodels-change-way-surgeons-prepare-surgery/. Accessed July 2017 ˙ 11. Zukowska, M., Górski, F., Bromi´nski, G.: Rapid Manufacturing and Virtual Prototyping of Pre-surgery Aids, vol. 68/3, pp. 399–403. IFMBE 12. Egorov, V., Tsyuryupa, S., Kanilo, S., Kogit, M., Sarvazyana, A.: Soft tissue elastometer. Med. Eng. Phys. 30(2), 206–212 (2008) 13. Bruyère, F., Leroux, C., Brunereau, L., Lermusiaux, P.: Rapid prototyping model for percutaneous nephrolithotomy training. J. Endourol. 22(1), 91–6 (2008) 14. Smooth-On. https://www.smooth-on.com/products/dragon-skin-10-very-fast/. Accessed 2019

Part XV

Computer Simulation of Dynamic Mass and Heat Transfer in Gas-Liquid-Solid 3-Phase Flow in Harvesting Natural Gas from Subsea Gas Hydrate Depositions

Chapter 55

Research on Annular Pressure Buildup in Deepwater Oil and Gas Well Xueting Wu and Ling Xiao

Abstract Accurate prediction of the annular pressure buildup in deepwater oil and gas well is of great significance for drilling and production of oil and gas. In order to predict the annular pressure buildup, casing annulus temperature distribution is calculated by considering the deep water well structure and wellbore heat transfer process. And the prediction model of annular pressure buildup is established. Newton down-hill method is used to solve the model. The reliability of the model is validated by comparing predicted results of the model with monitoring values in the oil field. The results indicate that the predicted results are in good agreement with monitoring values of casing annular temperature and pressure, and the max relative error is less than 10%. The accuracy can meet the needs of engineering. In addition, compared with the simple iterative method, computing speed of the Newton down-hill method is relatively faster. In words, the model can provide theoretical guidance for the accurate calculation of annular pressure buildup. Keywords Drilling and production of oil and gas · Annular pressure buildup · Prediction model · Newton down-hill method · Theoretical guidance

55.1 Introduction Annular pressure buildup in deepwater oil and gas well refers to the phenomenon that additional pressure is generated because of the fluid inflation for temperature rise in the sealed casing annulus [1–4]. In the land, platform and spar-type wells with dry wellheads, the annular pressure can be managed by bleeding off the annular pressure via wellhead equipment [5]. However, because the deep water oil and gas well is limited by subsea wellhead, the sealed annulus has no access to release the annular pressure buildup [6–9]. So, annular pressure buildup is formed in the process of oil and gas production and it makes the wellbore safety received. In 1999, casing annulus X. Wu (B) · L. Xiao School of Earth Resource Sciences and Engineering, Xi’an Shiyou University, 18 Dianzi 2nd Road, Xi’an 710065, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_55

647

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fluid temperature and pressure increased significantly after a few hours production in Marlin oil field in the Mexico gulf [5]. Meanwhile, casing suffered serious damage. It is a technical challenge to monitor and manage casing annular pressure. So, it is significant to study the prediction and mitigation measures of annular pressure buildup [10]. In recent years, some scholar [11, 12, 13, 14] have given different analytical solutions of annular pressure buildup. In this paper, casing annulus temperature distribution is calculated by considering the deep water well structure and wellbore heat transfer process. And the prediction model of annular pressure buildup is established. In addition, the Newton down-hill method is used to solve the annular pressure buildup model. The reliability of the model is validated by comparing predicted results of the model with measured data in the oil field. The remainder of this paper contains four parts: annular pressure buildup model has been established in Sect. 55.2. The solution of the model is presented in Sect. 55.3. In Sect. 55.4, the validity of the model has been conducted. The major conclusions of this study are drawn in Sect. 55.5.

55.2 Annular Pressure Buildup Model 55.2.1 Physical Model There is often subsea wellhead for deepwater oil and gas well. The annulus temperature will increase and the annular pressure buildup forms in the process of testing and production. Figure 55.1 is the schematic diagram of casing program and wellbore heat transfer for typical deep water well. Annulus A is the annular space between test string and production casing. Annulus B is the annular space between production casing and technology casing and this annulus is not filled with the cementing slurry. Annulus C is the annular space between technology casing and surface casing and this annulus is not filled with the cementing slurry.

55.2.2 Mathematical Model (1) Assumption According the existing literatures [15, 16], there is an assumption that the process of heat transfer from the test string to the outer edge of cement ring is the steady-state, and the process of heat transfer from the cement ring outer edge to the formation is the non-steady-state.

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Fig. 55.1 Schematic diagram of casing program and wellbore heat transfer for typical deep water well

(2) Model of casing annulus temperature In the normal test and production operations, annular pressure of annulus A can be released through subsea wellhead control. However, for the annulus B and C, the control of annular pressure is more difficult. Therefore, casing annulus temperature model is established for annulus B. The same process is for annulus C. The heat gradient from the test string to the outer edge of cement ring for an arbitrary depth is as follow. Qw = 2π rto Uto (Ttf − Two )

(55.1)

where ⎤−1 3 3 3    r ln(r /r ) r r ln(r /r ) to oj ij to to om im ⎦ Uto = ⎣ + + r k λ λ an an cj sm n=1 j=1 m=1 ⎡

(55.2)

where, Qw is the heat gradient from the test string to the outer edge of cement ring, W/m; U to is the total heat transfer coefficient from the test string to the outer edge of cement ring, W/(m2 • °C); r to is the outer radius of test string, m; T tf is fluid temperature inside the test string,°C; T wo is temperature of the outer edge of cement

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ring,°C; r an is the distance from wellbore axis to the outer edge of the nth annulus, m; k an is the heat transfer coefficient of the nth annulus, W/(m2 •°C); r oj , r ij is the outer radius and inner radius of the jth casing, m; λcj is the coefficient of thermal conductivity of the jth casing, W/(m2 •°C); r om , r im is the outer radius and inner radius of the mth cement ring, m; λsm is the coefficient of thermal conductivity of the mth cement ring, W/(m2 •°C). The heat gradient from the outer edge of cement ring to the formation for an arbitrary depth is as follow. Qf =

2π λe (Two − Tei ) f (t)

(55.3)

where Tei = Tsf + ge h

(55.4)

where, Qf is the heat gradient from the outer edge of cement ring to the formation, W/m; f(t) is the dimensionless time function of thermal conductivity of formation; T ei is the initial formation temperature,°C; T sf is the subsea temperature,°C; h is the distance from subsea to arbitrary wellbore position, m; ge is the geothermal gradient,°C/m. According to the principle of conservation of energy, we can get: Qw = Qf

(55.5)

Then, it can be got: Two =

Tei λe + Ttf rto Uto f (t) λe + rto Uto f (t)

(55.6)

Because of the steady-state process from the test string to the outer edge of cement ring, so the fluid temperature in annulus B is as follow. TB = Two +

rto Uto ln(rw /RB )(Ttf − Two ) λl

(55.7)

where, T B is the fluid temperature in annulus B,°C; RB is the outer radius of annulus B, m. It should be stressed that fluid temperature distribution (T tf ) inside the test string at different time can be got by the existing literature [17]. (3) Model of annular pressure buildup According to the above model of annulus fluid temperature, for example, prediction model of annular pressure buildup is derived for annulus B. The same process is for annulus A and C. The PVT state equation is the basis of the prediction of annular

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pressure buildup, and the partial differential equation is as follow. p =

1 αl 1 T − Vf V + kT kT V kT Vf

(55.8)

where, p is the change of annular fluid pressure, MPa; α l is the thermal expansion coefficient of annulus fluid,°C−1 ; k T is the isothermal compressibility coefficient of annulus fluid, MPa−1 ; T is the change of annular fluid temperature,°C; V, V is the annular volume and its variation, m3 ; V f , V f is the annular fluid volume and its variation, m3 . The isothermal compression coefficient (k T ) and thermal expansion coefficient (α l ) are often different for different fluid and casing annulus fluid. There is no change in annular fluid volume because of confined spaces. So, V f is zero. In addition, annular volume changes caused by thermal expansion or compression of annulus fluid, mainly including the radial expansion and compression of casing due to radial heat expansion, and thermal expansion and compression of annulus fluid, totally 4 parts. The concrete calculation formula can be found in the existing literature [11]. The total change of annular volume for annulus B is as follow. V = −V1 + V2 + V3 − V4

(55.9)

where, V 1 , V 2 , V 3 , V 4 is the volume change caused by radial expansion and compression of casing, and thermal expansion and compression of annulus fluid, respectively, m3 .

55.3 Solution of Model After comprehensive analysis, the Newton down-hill method is used to solve the model of annular pressure buildup in this paper. There are two reasons for using this method to solve the model. One is that annular pressure buildup is formed by the coupling process of volume and pressure in annulus, so the Newton down-hill method is used to more satisfy the physical formation process. The other is that the Newton down-hill method can effectively overcome the problem of initial value selection when using the simple iterative method and improve the project feasibility [18]. The solution steps of the Newton down-hill method are as follows: (1) Integrating the formula (55.8) and (55.9), and further get the function relations of annular pressure are as follows.

f (p) = M0 p2 + M1 p + M2

(55.10)

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where, M 0 , M 1 and M 2 are fitting coefficients, respectively. (2) The iterative formula of the Newton down-hill method is as follows. pk+1 = pk − ω

f (pk ) f  (pk )

(55.11)

(3) Using the numerical iterative for formula (55.11) until meet the error requirement, then stop the iteration.

55.4 Validity of Model The monitoring values of annular temperature and pressure of annulus B in West African oil field [14] are used to verify this model in this paper. The casing program of typical deep water well in West African oil field is shown in Fig. 55.2. According to the field actual formation fluid properties, casing program and its parameters such as thermal conductivity, the temperature and pressure of the casing annulus are predicted by the model in part 2. And then comparing predicted results with the actual monitoring values, the results are shown in Tables 55.1 and 55.2. Fig. 55.2 Schematic diagram of casing program of typical deep water well in West African oil field

55 Research on Annular Pressure Buildup in Deepwater Oil … Table. 55.1 The comparison between the predicted results and the monitoring values of annular temperature

Table. 55.2 The comparison between the predicted results and the monitoring values of annular pressure

653

Well

Predicted results (ºC)

Monitoring values (ºC)

Error (%)

No. 1

60.59

60

0.98

No. 2

58.74

58

1.27

No. 3

56.67

55

3.03

No. 4

63.58

62

2.54

No. 5

64.85

61

6.31

No. 6

69.60

67

3.88

No. 7

72.29

70

3.27

Well

Predicted results (MPa)

Monitoring values (MPa)

Error (%)

No. 1

48.67

46

5.80

No. 2

46.95

44

6.70

No. 3

44.98

41

9.70

No. 4

51.53

48

7.35

No. 5

52.74

50

5.48

No. 6

57.24

55

4.07

No. 7

60.18

61

1.34

From Tables 55.1 and 55.2, we can get that the predicted results of casing annular temperature and pressure of 7 wells are in good agreement with the monitoring values. In addition, the maximum relative error is within 10%, and the accuracy can meet the needs of engineering. Therefore, the model in this paper is reliable. Moreover, it needs an average time of 1.8 s to obtain the predicted result for a well by use of the Newton down-hill method, while an average time of 2.5 s when using the simple iterative method. In other words, computing speed of the Newton down-hill method is relatively faster. With the increase of the amount of calculation, the time-saving effect of the Newton down-hill method is more apparent. So, it is more suitable for large amounts of calculation in oil field.

55.5 Conclusions On the basis of previous analysis, the following conclusions are drawn: (1) Based on casing program of deep water well and wellbore heat transfer process, casing annulus temperature model is derived and the prediction model of annular pressure buildup is established (2) The predicted results of casing annular temperature and pressure of 7 wells are in good agreement with the monitoring values. In addition, the maximum relative error is within 10%, and the accuracy can meet the needs of engineering.

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(3) Moreover, compared with the simple iterative method, computing speed of the Newton down-hill method is relatively faster. This method is more suitable for large amounts of calculation in oil field.

References 1. Oudeman, P., Kerem, M.: Transient behavior of annular pressure buildup in HP/HT Wells. In: Paper SPE 88735 Presented at the 11th Abu Dhabi International Petroleum Exhibition and Conference, Abu Dhabi, UAE, 10–13 Oct 2004 2. Oudeman, P., Baccareza, L.J.: Field trial results of annular pressure behavior in a highpressure/high-temperature well. SPE Drill. Complet. 84–88 (1995) 3. Vargo Jr., R.F., Payne, M., Faul, R., LeBlanc, J., Griffith, J.E.: Practical and successful prevention of annular pressure buildup on the marlin prohect. SPE Drill. Complet. 18(3), 228–234 (2003) 4. Sathuvalli, U.B. et al.: Development of a screening system to identify deepwater wells at risk for annular pressure buildup. In: Paper SPE/IADC 92594 Presented at the SPE/IADC Drilling Conference, Amsterdam, The Netherlands, 23–25 Feb 2005 5. Vargo Jr., R.F., Payne, M., Faul, R., et al.: Practical and successful prevention of annular pressure buildup on the Marlin project. In: Paper SPE 77473 Presented at the SPE Annual Technical Conference and Exhibition, San Antonio, Texas, 29 Sept–2 Oct 2002 6. Loder, T., Evans, J.H., Griffith, J.E.: Prediction of and effective preventative solution for annular fluid pressure buildup on subsea completed wells—case study. In: Paper SPE 84270 Presented at the SPE Annual Technical Conference and Exhibition, Denver, Colorado, 5–8 Oct 2003 7. Oudeman, P., Kerem, M.: Transient behavior of annular pressure build-up in HP/HT wells. SPE Drill. Complet. 2(4), 234–241(2006) 8. Pattillo, P.D., Cocales, B.W., Morey, S.C.: Analysis of an annular pressure buildup failure during drilling ahead. SPE 89775 (2004) 9. Williamson, R., Sanders, W., Jakabosky T., et al.: Control of contained annulus fluid pressure buildup. In: Paper SPE 79875 Presented at the SPE/IADC Drilling Conference, Amstredam, The Netherlands, 19–23 Feb 2003 10. Sultan, N., Faget, J.-B., Fjeldheim, M., et al.: Real-time casing annulus pressure monitoring in a subsea HPHT exploration well. In: Paper OTC 19286 Presented at the Offshore Technology Conference, Houston, Texas, 5–8 May 2008 11. Baokui, G.: Practical model for calculating the additional load on casing by high temperature. Oil Drill. Prod. Technol. 24(1), 8–10 (2002) 12. Yuanzhou, D., Ping, C., Huili, Z.: Calculating the pressure in sealed annulus in oil well by iterative method. Offshore Oil 26(2), 93–96 (2006) 13. Gao, D., Qian, F., Zheng, H.: One method of prediction of the annular pressure buildup in deepwater wells for oil & gas. CMES 89(1), 1–15 (2012) 14. Jin, Y., Haixiong, T., Zhengli, L., et al.: Prediction model of casing annulus pressure for deepwater well drilling and completion operation. Petrol. Explor. Dev. 40(5), 616–619 (2013) 15. Hasan, A.R., Izgec, B., Kabir, C.S.: Ensuring sustained production by managing annularpressure buildup. SPE 121754 (2009)

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16. Ramey, H.J.: Wellbore heat transmission. J. Pet. Technol. 14(4), 427–435 (1962) 17. Azzola, J., Tselepidakis, D., Pattillo, P.D., et al.: Application of vacuum insulated tubing to mitigate annular pressure buildup. SPE Drill. Complet. 22(1), 46–51 (2007) 18. Moler, C.: Numerical Computing with MATLAB. SIAM (2008)

Chapter 56

Optimal Drilling Parameters Design Based on Single Drilling Depth Indicator in Controlled Gradient Drilling Jiangshuai Wang, Jun Li, Gonghui Liu, Hongwei Yang and Kuidong Luo

Abstract Controlled gradient drilling technology was put forward to solve the problem of narrow pressure margins in deep water drilling. In this paper, an optimization model for optimal drilling parameters design is established for controlled gradient drilling. The established optimization model aims at achieving the longest single drilling depth, while reducing pressure difference between wellbore pressure and formation pore pressure as much as possible. It is beneficial to increase the ROP and reduce formation damage. The SQP (sequential quadratic programming) algorithm is presented to solve the optimization model. Simulation results indicate that longer single drilling depth and better drilling parameters (control variables) can be obtained by use of optimization model when the range of control variables widens. Furthermore, in case study, small bottom-hole pressure difference can be obtained by use of optimization model. In addition, the established optimization model ensured the dynamic wellbore pressure within narrow pressure margins during optimization process. It is conducive to safe and efficient drilling. Keywords Narrow pressure margins · Controlled gradient drilling · Optimization model · Optimal drilling parameters · Safe and efficient drilling

56.1 Introduction Since the 1970s, operations in deep sea area have been considered as an important frontier for the oil industry. Exploration and development of deep-sea oil in the Gulf of Mexico, Brazil, and West Africa have been greatly developed [1–3]. Meanwhile, drilling in deep water faces many challenges [4], such as narrow pressure margins between pore pressure and fracture pressure, marine geological disasters, cryogenic J. Wang · J. Li (B) · G. Liu · H. Yang · K. Luo College of Petroleum Engineering, China University of Petroleum-Beijing, 18 Fuxue Road, Changping, Beijing 102249, China e-mail: [email protected] G. Liu Beijing University of Technology, 100 Ping Le Yuan, Chaoyang District, Beijing 100124, China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_56

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environment and unstable seabed. Among them, narrow pressure margins is a significant difficulty and it is directly related to the safety and high efficiency of drilling operations in deep water. To solve the problem of narrow pressure margins in deep water drilling, many studies have been conducted. In recent decades, dual-gradient drilling with gas-injected, dual-gradient drilling with hollow spheres, dual-gradient drilling with subsea pump, and control mud cap drilling have been developed [5–9]. Then, the idea of controlled gradient drilling was put forward and is considered as a feasible method to solve the problem of narrow pressure margins in deep water drilling [10]. Controlled gradient drilling is a new drilling technology that drilling fluid containing hollow spheres is injected into the drill pipe at the surface, and hollow spheres are separated into annulus using a separator mounted on the drill string (Fig. 56.1). As a result, multi-gradient are formed in annulus to better fit into the narrow pressure margins. Because of the separation of part hollow spheres, content of hollow spheres above separator is high and content of hollow spheres under separator is low in annulus. In other words, wellbore pressure is composed of two or more parts for the controlled gradient drilling technology. One is the light drilling fluid above separator (green part in Fig. 56.1), and the other is the heavy drilling fluid under separator (purple part in Fig. 56.1). In addition, comparing wellbore pressure of target point under three different status, it can be seen that the length of light drilling fluid section

Status (a)

Status (b)

Status (c)

Fig. 56.1 Controlled gradient drilling and composition of wellbore pressure under different status (From status a to c, well depth is increasing during drilling)

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increases and the length of heavy drilling fluid section decreases from status (a) to (c) during drilling process. This is because that separator moves down as well depth increases. So, wellbore pressure of target point changes in the three status. It may lead wellbore pressure not to be within the prescriptive pressure boundary, lower than formation pore pressure or higher than fracture pressure. Eventually, it may result in the complicated underground situations caused by improper down-hole pressure control, such as overflow and well leakage. The analysis above indicates that wellbore pressure in same depth changes with the increase of well depth during drilling. In addition, the wellbore pressure for controlled gradient drilling proposed in the paper is a function of six drilling parameters (control variables). Therefore, to maximize the single drilling depth, the optimal control variables are necessary to be acquired. In this study, an optimization model for optimal drilling parameters design is established for controlled gradient drilling. The established optimization model aims at achieving the longest single drilling depth, while reducing pressure difference between wellbore pressure and pore pressure as much as possible. It is beneficial to increase the ROP and reduce formation damage.

56.2 Calculation of Dynamic Wellbore Pressure For controlled gradient drilling, two or more liquid columns with different densities in annulus present double or multiple pressure gradients taking separator as the boundary line. Under status (a) in Fig. 56.1, the wellbore pressure of target point can be obtained in Eq. (56.1). Pa = Pcp + ρ1 g H11 + P f 1 H11 + ρ2 g H12 + P f 2 H12

(56.1)

where, Pa represents wellbore pressure of target point, Pa. Pcp represents wellhead back pressure, Pa. ρ 1 , ρ 2 represents density of light drilling fluid and heavy drilling fluid, respectively, kg/m3 . H 11 , H 12 represents the length of wellbore section filled with light drilling fluid and heavy drilling fluid under status (a), respectively, m. Pf 1 , Pf 2 represents frictional pressure drop per meter of wellbore section filled with light drilling fluid and heavy drilling fluid, respectively, Pa/m. According to the existing research results [11], Pf1 , Pf2 can be obtained in Eqs. (56.2) and (56.3), respectively. P f 1 = 5.7503

1.8 ρ10.8 μ0.2 1 Q 3 (Di − D p ) (Di + D p )1.8

(56.2)

P f 2 = 5.7503

1.8 ρ20.8 μ0.2 2 Q (Di − D p )3 (Di + D p )1.8

(56.3)

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where, Q is pump rate, L/s. μ1 , μ2 represents plastic viscosity of light drilling fluid and heavy drilling fluid, respectively, Pa·s. Di is the outer diameter of annulus, cm. Dp is the inner diameter of annulus, cm. The parameters such as density, length, and viscosity in above equations can be represented by six control variables (separation efficiency, volume fraction, density of the pure drilling fluid, distance between the separator and drill bit, wellhead back pressure and pump rate). The expressions are shown in the following.  

ρ1 = V0 ρs + (1 − V0 )ρm μ1 = V0 μs + (1 − V0 )μm

(56.4)

ρ2 = (1 − ε)V0 ρs + (1 − (1 − ε)V0 )ρm μ2 = (1 − ε)V0 μs + (1 − (1 − ε)V0 )μm

(56.5)

H11 = L − H12

(56.6)

H12 = H20

(56.7)

where, ρ s is density of hollow spheres, kg/m3 . ρ m is density of the pure drilling fluid, kg/m3 . μs is viscosity of hollow spheres, Pa·s. μm is plastic viscosity of the pure drilling fluid, Pa·s. V 0 is volume fraction. ε is separation efficiency. H 02 is distance between the separator and drill bit, m. L is the depth of target point, m. So, the relationship between wellbore pressure at arbitrary depth and the six variables has been established. Under status (b) and (c), the wellbore pressure can be obtained in Eqs. (56.8) and (56.9), respectively. Pb = Pcp + ρ1 g H21 + P f 1 H21 + ρ2 g H22 + P f 2 H22

(56.8)

Pc = Pcp + ρ1 g H31 + P f 1 H31

(56.9)

where, Pb represents wellbore pressure of target point under status (b), Pa. Pc represents wellbore pressure of target point under status (c), Pa. H 21 represents the length of wellbore section filled with light drilling fluid under status (b), m. H 22 represents the length of wellbore section filled with heavy drilling fluid under status (b), m. H 31 represents the length of wellbore section filled with light drilling fluid under status (c), m. For the same target depth, ratio of the length of wellbore section filled with light drilling fluid and the length of wellbore section filled with heavy drilling fluid is different under three status. Therefore, wellbore pressure of target point under three different status is unequal. That is Pa = Pb = Pc

(56.10)

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In other words, the wellbore pressure at arbitrary depth in open-hole section is changeable dynamically with the increase of well depth, which is obviously different from the wellbore pressure in conventional drilling. Therefore, to achieve the longest single drilling depth, it is of great significance to establish an optimization model of optimal drilling parameters design for controlled gradient drilling.

56.3 Optimization Model Optimization model for optimal drilling parameters design: aiming at achieving the longest single drilling depth.

56.3.1 Objective Function The purpose of optimization model is to ensure the minimum wellbore pressure difference and increase the ROP under the premise of the longest single drilling depth. What emphasized here is that analysis above indicates, it is impossible to establish a direct relationship between the single drilling depth and six control variables and it is mathematically infeasible to consider the single drilling depth as an objective function. Therefore, the pressure difference between wellbore pressure and pore pressure at the current well depth is set as the objective function in this study. In addition, wellbore pressure in open-hole section and six control variables are set as constraints. As a result, the minimum borehole pressure difference is actually achieved and the longest single drilling depth is ensured within the constraints. For symbolic simplicity, optimization model mentioned above can be formulated as min f (ε, V0 , ρm , H20 , Pcp , Q)

(56.11)

where, f(ε, V 0 , ρ m , H 02 , Pcp , Q) is the objective function, defined as the pressure difference between wellbore pressure and pore pressure at the current well depth. The expression is shown in the following. ∗ − Pp∗ ) f (ε, V0 , ρm , H20 , Pcp , Q) = (P jd

(56.12)

∗ is the where, Pp∗ is the formation pore pressure at the current well depth, Pa. P jd bottom-hole pressure at the current well depth, Pa. It can be formulated as ∗ = Pcp + ρ1 g(L ∗ − H20 ) + P f 1 (L ∗ − H20 ) + ρ2 g H20 + P f 2 H20 P jd

where, L * is the current well depth, which is gradually increased, m.

(56.13)

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56.3.2 Constraints Control variables of the optimization model are separation efficiency, volume fraction, density of the pure drilling fluid, distance between the separator and drill bit, wellhead back pressure and pump rate, i.e., ε, V 0 , ρ m , H 02 , Pcp , Q. All control variables and wellbore pressure in open-hole section are limited by on-site equipment and narrow pressure margins. Then Eq. (56.14) can be obtained. ⎧ εl ≤ ε ≤ ε u ⎪ ⎪ ⎪ ⎪ ⎪ V0l ≤ V0 ≤ V0u ⎪ ⎪ ⎪ l u ⎪ ⎨ ρm ≤ ρm ≤ ρm l 0 H2 ≤ H2 ≤ H2u ⎪ l u ⎪ ⎪ Pcp ≤ Pcp ≤ Pcp ⎪ ⎪ ⎪ ⎪ Ql ≤ Q ≤ Q u ⎪ ⎪ ⎩ i Pp ≤ P i ≤ P if

(56.14)

where, εl , V l0 , ρ lm , H l2 , Plcp , Ql represent lower limit of separation efficiency, volume fraction, density of the pure drilling fluid, distance between the separator and drill bit, wellhead back pressure and pump rate, respectively. εu , V u0 , ρ um , H u2 , Pucp , Qu denote upper limit of separation efficiency, volume fraction, density of the pure drilling fluid, distance between the separator and drill bit, wellhead back pressure and pump rate, respectively. Pi represents wellbore pressure at arbitrary depth in open-hole. Pi can be obtained by calculation method in Sect. 56.2. Pip , Pif represent pore pressure and fracture pressure at arbitrary depth in open-hole, respectively. What emphasized here is that, it is difficult to adjust these variables in real time and automatically to achieve any desired value during actual drilling. In other words, the optimal variable value obtained through the optimization model is constant during single drilling process. Thus, each variable was divided into n groups of data at equal intervals within the range of values in this study. In addition, separation efficiency depends on structure of the separator. Thus, the separation efficiency is set to be constant during optimization process.

56.3.3 Optimization Model Solution The pressure difference between wellbore pressure and pore pressure at the current well depth is set as the objective function. However, the objective function is changeable dynamically during drilling and number of constraints increases with the increase of well depth. In other words, constraints is increasing and changeable with the increase of well depth. So, this engineering problem is a constrained nonlinear dynamic optimization problem. Because of the constraints of control variables, the constrained-nonlinear dynamic optimization problem is divided into finite

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constrained-nonlinear static optimization problem solved by use of the optimization method. Algorithm The sequential quadratic programming (SQP) algorithm is presented to solve the established optimization model. SQP algorithm is considered as one of the most effective methods to solve the nonlinear optimization problem with constraints [12]. Optimization function The SQP algorithm is compiled based on the Matlab [13] platform. Combined with the fmincon function [14], the SQP algorithm can be invoked to solve the established optimization model. Therefore, fmincon function is proposed to solve the established optimization model.

56.4 Case Study In this section, the longest single drilling depth and optimal drilling parameters (control variables) are obtained by use of optimization model. Moreover, optimization results in different range of control variables are analyzed. The simulation parameters are shown in the following. The current well depth is 1200 m and water depth is 1000 m. The surface casing depth is 200 m below the mudline. The outer diameter, the inner diameter for riser is 660.4 mm and 609.6 mm, respectively. The outer diameter, the inner diameter for drill pipe is 127 mm and 101.6 mm, respectively. The inner diameter of surface casing is 457.2 mm and the diameter of drill bit is 342.9 mm. In addition, the limiting well depth set in simulation is 2200 m. When drilling to this depth, the simulation ends. The range of control variables is wider with the increase of n. The range of control variables is shown in Table 56.1. In the case of n is 5, 6 and 7 respectively, the optimization design is conducted when the narrow pressure margins is 1.03~1.05 g/cm3 . The optimization results are presented in Table 56.2, Figs. 56.2, 56.3, and 56.4. In Table 56.2 and Fig. 56.2, in the case of n is 5, 6 and 7, the longest single drilling depth is 1440 m, 1510 m, and 1550 m, respectively. This is because, the range of control variables becomes wider as the value of n increases. Thus, better drilling parameters can be obtained to achieve a longer single drilling depth. Table 56.1 Range of control variables under different n

n

5

6

7

ε

0.4

0.4

0.4

V0

[0.3, 0.38]

[0.3, 0.4]

[0.3, 0.42]

ρ m /(kg/m3 )

[1000, 1200]

[1000, 1250]

[1000, 1300]

H 02 /(m)

[100, 500]

[100, 600]

[100, 700]

Pcp /(Pa)

[0, 2000000]

[0, 2500000]

[0, 3000000]

Q/(L/s)

[30, 42]

[30, 45]

[30, 48]

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Table 56.2 Optimization results under different range of control variables n

5

6

7

ε

0.4

0.4

0.4

V0

0.3

0.34

0.38

ρ m /(kg/m3 )

1150

1250

1300

H 02 /(m)

100

100

100 500000

Pcp /(Pa)

1000000

500000

Q/(L/s)

30

36

42

Longest single drilling depth/(m)

1440

1510

1550 n=5 n=6 n=7

1

Exitflag

0

-1

-2 1200

1400

1600

1800

2000

2200

Depth/(m)

Fig. 56.2 The exitflag to characterize the longest single drilling depth under different range of control variables

Wellbore pressure difference/(Pa)

120000

n=5, longest single drilling depth n=6, longest single drilling depth n=7, longest single drilling depth n=5, exceeds longest single drilling depth n=6, exceeds longest single drilling depth n=7, exceeds longest single drilling depth

100000 80000 60000 40000 20000 0 -20000 1200

1250

1300

1350

1400

1450

1500

1550

1600

Depth/(m)

Fig. 56.3 The wellbore pressure difference distribution under different range of control variables

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Bottom-hole pressure difference/(Pa)

200000

160000

120000

80000

40000

0 1200

1250

1300

1350

1400

1450

1500

1550

1600

Depth/(m)

Fig. 56.4 The bottom-hole pressure difference under different range of control variables

In Fig. 56.3, drilling with optimal drilling parameters can ensure that wellbore pressure difference at arbitrary depth in open-hole section is greater than zero when well depth is the corresponding longest single drilling depth (dashed line) under different range of control variables. In other words, wellbore pressure at arbitrary depth in open-hole section is within narrow pressure margins, effectively avoiding the complicated underground situations caused by improper wellbore pressure. On the other hand, when the drilling depth exceeds the corresponding longest single drilling depth (solid line), a negative pressure differential will occur. In other words, wellbore pressure at some depth in open-hole section is less than the formation pore pressure, and complicated underground situations may occur. It is not conducive to safe and efficient drilling. In Fig. 56.4, the bottom-hole pressure difference is presented when drilling from 1200 m to the corresponding longest single drilling depth under different range of control variables. It can be seen that the pressure difference is all below 0.25 MPa. Small bottom-hole pressure difference obtained by use of optimization model is beneficial to increase the ROP, reduce the formation damage, and reduce drilling costs.

56.5 Conclusions and Further Research (1) An optimization model for optimal drilling parameters design has been established for controlled gradient drilling. The established optimization model aims at achieving the longest single drilling depth. In addition, the established optimization model ensured the dynamic wellbore pressure within narrow pressure margins during optimization process. It is conducive to safe and efficient drilling.

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(2) Simulation results indicate that longer single drilling depth and better drilling parameters (control variables) can be obtained can be obtained by use of optimization model, when the range of control variables widens. Furthermore, in case study, small bottom-hole pressure difference obtained by use of optimization model is beneficial to increase the ROP, reduce the formation damage, and reduce drilling costs. (3) This paper mainly focuses on optimal drilling parameters design for offshore vertical well. Therefore, optimal drilling parameters design for offshore horizontal or deviated well are also an important task to study. Acknowledgements Project supported by the Key Program of National Natural Science Foundation of China (Project No. 51734010).

References 1. Peres, W.E.: Shelf-fed turbidite system model and its application to the Oligocene Deposits of the Campos basin, Brasil. AAPG Bull. 77(1): 81–101 (1993) 2. Deluca, M.: Canada, Mexico escaping shadow of the US Gulf. Offshore 1999(6), 52 (1999) 3. Deluca, M.: Deep discoveries keep West Africa at global forefront. Offshore 1999(2), 32–33 (1999) 4. Charlez, P.A., Simondin, A.A.: Collection of innovative answers to solve the main problematics encountered when drilling deep water prospects. In: OTC 15234 (2003) 5. Gaddy, E.D.: Industry group studies dual gradient drilling. Oil Gas J. 97(33), 32 (1999) 6. Smith, K.L., Gault, A.D., Witt, D.E., et al.: Subsea mud-lift drilling joint industry project: delivering dual gradient drilling technology to industry. In: SPE 71357 (2001) 7. Schumacher, J.P., Dowell, J.D., Ribbeck, L.R., et al.: Subsea Mud Lift Drilling (SMD): planning and preparation for the first subsea field test of a full scale dual gradient drilling system at green canyon 136, Gulf of Mexico. In: SPE 71358 (2001) 8. Maurer, W.C., Medley, G.H., Mcdonald, W.J.: Multi-gradient drilling method and system. US (2003) 9. Fossli, B., Sangesland, S.: Controlled mud-cap drilling for subsea applications: well-control challenges in deep waters. SPE Drill. Complet. 21(2): 133–140 (2006) 10. Deng, S., Fan, H., Tian, D., et al.: Calculation and application of safe mud density window in deepwater shallow layers. In: Offshore Technology Conference (2016) 11. Coleman, N.L.: The drag coefficient of a stationary on a boundary of similar spheres. La Houiile Blanche 1, 17–21 (1972) 12. Temet, D.J., Biegler, L.T.: Recent improvements to a multiplier free reduced Hessian successive quadratic programming algorithm. Comput. Chem. Eng. 22(7), 963–978 (1998) 13. Hiebeler, D.E.: R and MATLAB. CRC Press, Boca Raton (2015) 14. The MathWorks Inc.: Optimization Toolbox 6 User’s Guide Version 6.0 (2011)

Chapter 57

Calculation of Transient Fluctuation Pressure in Deep Water Dual Gradient Drilling Kuidong Luo, Jun Li, Nan Ma, Reyu Gao and Jiangshuai Wang

Abstract Transient fluctuation pressure is extremely harmful during the drilling process. Fluctuation pressure can cause problems such as kicks or blowouts and lost circulation. Mastering and calculating fluctuation pressure is of great significance for reducing drilling accidents, increasing drilling speed and reducing drilling costs. In this paper, a mathematical model of downhole transient fluctuation pressure is established for the deepwater dual gradient drilling process. The mathematical model is solved by the characteristic line method and the finite difference method. The effects of drilling fluid performance (density, consistency coefficient, fluidity index), drilling speed and drilling acceleration on transient fluctuation pressure were analyzed. The results show that the transient fluctuation pressure increases with the increase of drilling fluid density, and the greater the penetration speed, the more obvious the increase. The consistency coefficient K and the fluidity index n have the same effect on the transient fluctuation pressure. The performance is as follows: the maximum transient fluctuation pressure increases with the increase of K(n), and when the acceleration changes greatly, the amplitude of the fluctuation decreases with the increase of K(n) value. The greater the drilling speed or the drilling acceleration, the greater the transient fluctuation pressure. The calculation results have a certain reference function for better control of the drilling speed and drilling fluid performance at the drilling site. Keywords Dual gradient drilling · Unstable flow · Transient fluctuation pressure · Method of characteristics · Finite difference method

57.1 Introduction When the drill pipe is lifted or lowered in the wellbore filled with drilling fluid, the flow in the wellbore is unstable, and the downhole transient fluctuation pressure caused by it is extremely harmful during the drilling process, which may cause a kick K. Luo · J. Li (B) · N. Ma · R. Gao · J. Wang China University of Petroleum, 18 Fuxue Road, Changping, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_57

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or blowout, lost circulation and other issues. Therefore, mastering and calculating the fluctuation pressure is of great significance for reducing drilling accidents, increasing drilling speed and reducing drilling costs. Based on the unstable flow, Lubinski [1] (1977) takes into account the fluid and wellbore properties and establishes a transient fluctuation pressure calculation model. Lal [2] (1983) uses a finite difference method to solve transient models. Zhong et al. [3, 4] (1989) used the hybrid implicit feature line method to numerically solve the governing equations. Based on the unstable flow, Fan [5] (1995) established a prediction model for the downhole fluctuation pressure and compiled related calculation software. Rommetveit et al. [6] (2003) conducted field experiments on the Brazilian deepwater platform, collected a large amount of field data, and verified the transient model. Li et al. [7, 8] (2013) established a wave pressure prediction model that considers the drill pipe joint. Although many scholars have carried out a lot of research on the wellbore fluctuation pressure, the calculation of the downhole transient fluctuation pressure in the deepwater dual gradient drilling process has not been reported. Therefore, a mathematical model of downhole transient fluctuation pressure is established for the deepwater dual gradient drilling process in this paper. The mathematical model is solved by the characteristic line method and the finite difference method. The effects of drilling fluid property (density, consistency coefficient, fluidity index), drilling speed and drilling acceleration on transient fluctuation pressure were analyzed. The calculation results have a certain effect on the drilling site to better control the drilling speed and drilling fluid property.

57.2 Physical Model The hollow sphere dual gradient drilling system is shown in Fig. 57.1. At the surface of the sea, the carrier fluid and the hollow sphere are injected into the injection line. At the mud line, the hollow sphere enters the riser and the carrier fluid returns to the surface through the return line. The drilling fluid is injected into the drill pipe at the surface of the sea and circulated into the casing annulus through the drill bit. When the drilling fluid reaches the mud line, it mixes with the injected hollow sphere and returns to the sea through the riser. The mud and the hollow sphere are separated by the separation device on the sea surface and reused, thereby completing the circulation process of the entire system.

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Fig. 57.1 Physical model of HGS dual gradient system

57.3 Mathematical Model and Solution 57.3.1 Mathematical Model There are two basic equations describing the one-dimensional unstable flow [9–13], namely the continuity equation and the dynamic equation. The continuity equation is as follows: ∂ ∂ (ρvA) + (ρA) = 0 ∂z ∂t

(57.1)

The dynamic equation is as follows:   ∂v 4τ0 1 ∂p 1 ∂v + +v + =0 1+ ρg ∂z g ∂t ∂z ρgD

(57.2)

According to Eqs. (57.1) and (57.2), the transient fluctuation pressure control equation describing the unstable flow is obtained 

Q ∂p + A ∂z ∂p ρ + ∂z A

∂p ∂t ∂Q ∂t

+ ρCA ∂Q =0 ∂z ∂Q + ρQ + Pf = 0 2 A ∂z 2

(57.3)

where, ρ is the fluid density, g/cm3 ; z is the depth of a certain position, m; v is the flow rate of the fluid, m/s; A is the sectional area, m2 ; t is time, s; p is the average

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pressure of the total flow section, MPa; τ 0 is the wall shear stress, MPa; D is the diameter of the tube, m; Q is the fluid flow, m3 /s; C is the propagation speed of the pressure wave, m/s; Pf is the pressure loss per unit weight of liquid per unit distance, MPa.

57.3.2 Mathematical Model Solution The characteristic equation of transient pressure equation. 

dQ + CPf dt = 0 ±dp + ρC A dz = v ± C dt

(57.4)

The finite difference scheme of characteristic equation (Fig. 57.2). ⎧ ρC ⎪ (Qw − QR ) + CtPfR = 0 ⎨ Pw − PR + A ⎪ ⎩ −P + P + ρC (Q − Q ) + CtP = 0 w s w s fs A

(57.5)

where, Pw , PR , Ps are the average flow section average pressures of w, R, and S nodes, respectively, MPa; Qw , QR , Qs are the fluid flows of the w, R, and S nodes, respectively, m3 /s; PfR , Pfs are the pressure loss of unit weight liquid in unit distance of R and S nodes, respectively, MPa.

Fig. 57.2 Schematic diagram of meshing

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Determination of boundary points. Wellhead boundary point of the annular flow channel and the inner flow channel of the drill pipe: ⎧ ⎨ pw = 0 ⎩ Qw =

pR +

ρCR QR AR −CR tpfR ρCR AR

(57.6)

Boundary point of the bottom hole flow channel. 

QW = 0 pw = pR −

ρCR QR AR

(57.7)

− CR tpfR

Junction. The treatment is carried out according to the formula (57.5). Meeting point. The drill pipe is a blocked pipe and there are only two flow channels. The displacement of the drill pipe is the sum of the flow rates of the two flow channels near the junction point. The following three equations are established: ⎧ ⎪ ⎨ QP = Qw1 + Qw2 s1 pw = ps1 + ρC − Qs1 ) + Qs1 tpfs1 As1 (Qw1 ⎪ ⎩ p = p + ρCs2 (Q − Q ) + Q tp w s2 w2 s2 s2 fs2 As2

(57.8)

Solved by Eq. (57.8): ⎧



si Asi ⎪ QP − 2i=1 Qsi − 2i=1 pρC + 2i=1 ⎪ si ⎪ ⎪ pw =

2 Asi ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

pfsi Asi t ρ

i=1 ρCsi

Qwi

pfsi Asi t Asi = (pw − psi ) + Qsi − ρCsi ρ

(57.9)

57.4 Case Study 57.4.1 Parameter Settings In this example, the hollow sphere is injected into the riser using a lightweight carrier fluid carrying a hollow sphere. The basic data of the well [14] is as follows (Tables 57.1, 57.2, 57.3, 57.4 and 57.5).

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Table 57.1 Drilling tool combination, pipeline and bore diameter data Basic data

Top depth/m

Drill pipe

Bottom depth/m

Inner diameter/cm

Outer diameter/cm 12.7

0

2850

10.861

2850

3000

7.315

0

1000

Boost line Riser

17.145

11.43

–

0

1000

48.895

53.34

Casing

1000

2850

27.36

–

Open hole section

2850

3000

26.987

–

Table 57.2 Seawater properties data Seawater

Density (g/cm3 )

Specific heat (J/kg °C)

Viscosity (mPa s)

Thermal conductivity (W/m °C)

Sea surface temperature (°C)

Sea water depth (m)

Value

1.03

4182

0.001

0.6

21

1000

Table 57.3 Drilling fluid data Drill fluid

Density (g/cm3 )

Specific heat (J/kg °C)

Viscosity (mPa s)

Thermal conductivity (W/m °C)

Displacement (L/s)

Valve

1.548

1658.9

32

1.6218

20

Table 57.4 Hollow sphere data Hollow sphere

Volume concentration (%)

Density (g/cm3 )

Displacement (L/s)

Valve

45

0.35

34

Table 57.5 Other basic data Inner diameter of high pressure pipeline/cm

12

Length of high pressure pipeline/m

15

Inner diameter of standpipe/cm

13

Length of standpipe/m

30

Inner diameter of the hose/cm

15

Length of the hose/m

18 20

Inner diameter of the kelly/cm

15

Length of the kelly/m

Maximum pump pressure/MPa

31.5

Pump power/kw

1470

Pump efficiency/Dimensionless

0.9

Mud inlet temperature/°C

21

Nozzle diameter/mm

3 × 12.7

Nozzle flow coefficient/dimensionless

0.9

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57.4.2 Transient Fluctuation Pressure Calculation Result Figure 57.3 shows the variation of the drilling speed with time. In 0–9 s, the drilling speed is accelerated from 0 m/s to 0.5 m/s. In 9–60 s, the drilling speed is maintained at 0.5 m/s. In 60–69 s, the drilling speed is reduced from 0.5 m/s to 0 m/s. The drilling speed after this is 0 m/s. It can be seen from Fig. 57.4 that in 0–9 s, the fluctuation pressure increases with the increase of the drilling speed, and reaches the peak (1.3720 MPa) at the end of 9 s. This is because the flow rate is increasing, resulting in an increase in pressure consumption. In 9–50 s, the fluctuation pressure showed a tendency to fluctuate up and down, and this trend gradually becomes smaller and is basically stable at 50 s (1.14 MPa). In 60–68.5 s, the fluctuation pressure gradually decreased

Fig. 57.3 Drilling speed changes with time

Fig. 57.4 The fluctuation pressure of the bottom hole changes with time

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Fig. 57.5 Influence of drilling fluid density on downhole surge pressure

and decreased to 0 MPa at 68.5 s, after which the surge pressure became a swab pressure. In 68.5–70.5 s, the swab pressure gradually increased and peaked at the end of 70.5 s (0.2556 MPa), after which the swab pressure decreased and stabilized at around 0.1 MPa in about 78 s.

57.4.3 Sensitivity Parameter Analysis Drilling fluid density ρ. As shown in Fig. 57.5, the absolute value of the fluctuation pressure increases with the increase of the drilling fluid density, and the greater the drill speed, the more obvious the increasing amplitude. Therefore, when the drilling fluid density is large, the maximum drilling speed should be strictly controlled. Consistency coefficient K and the fluidity index n. As shown in Fig. 57.6, the impact of the drilling fluid consistency coefficient K on downhole fluctuation pressure is mainly manifested in two aspects: (1) The maximum fluctuation pressure increases with the increase of K. This is mainly because K increases and the flow friction increases, and the flow friction has a great relationship with the flow velocity. (2) When the speed changes greatly and stops, that is, when the acceleration changes greatly, the amplitude of the fluctuation is reduced as the K value increases. This is mainly because the K value is large, the drilling fluid has a large viscosity, and the damping is large. Through the above analysis, it can be concluded that in order to reduce the drilling time under the premise of strictly controlling the maximum drilling speed, the acceleration of the drilling can be increased when the drilling fluid is thicker (the K value

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Fig. 57.6 Influence of consistency coefficient on underground surge pressure

is larger). The effect of the change in the fluidity index n on the downhole fluctuation pressure is similar to the effect of the consistency coefficient K, as shown in Fig. 57.7. Drilling speed. As shown in Fig. 57.9, the fluctuation pressures of the three cases are the same in 0–5.4 s; In 5.4 s, the fluctuation pressure becomes smaller as the speed becomes smaller; At 12.6 s, the fluctuation pressure also becomes larger as the speed becomes larger. It is concluded that the greater the drilling speed, the greater the fluctuation pressure (Figs. 57.8 and 57.9). Drilling acceleration. As shown in Fig. 57.11, the fluctuation pressure is large at the beginning of the start and the end, and the larger the acceleration, the larger the absolute value of the

Fig. 57.7 Influence of flow index on underground fluctuation pressure

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Fig. 57.8 Drilling speed changes with time

Fig. 57.9 Influence of drilling speed on underground surge pressure

fluctuation pressure. It is concluded that the brakes and slamming operations should be strictly controlled when trip out and trip in (Figs. 57.10 and 57.11).

57.5 Conclusions In this paper, the mathematical model of transient fluctuation pressure during dual gradient drilling is established and solved. The influence of drilling parameters on the transient fluctuation pressure is analyzed. The following conclusions are obtained: (1) The transient fluctuation pressure increases with the increase of drilling fluid density, and the greater the drilling speed, the more obvious the increasing

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Fig. 57.10 Drilling acceleration changes with time

Fig. 57.11 Influence of drilling acceleration on underground surge pressure

amplitude. When the drilling fluid density is high, the maximum drilling speed should be strictly controlled. (2) The consistency coefficient K and the fluidity index n have the same effect on the transient fluctuation pressure in the well, which is expressed as: The maximum transient fluctuation pressure increases with the increase of K(n), and when the acceleration changes greatly, the amplitude of the fluctuation decreases with the increase of K(n) value. (3) The greater the drilling speed or the drilling acceleration, the greater the transient fluctuation pressure. When trip out and trip in, the brakes and slamming operations should be strictly controlled.

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Acknowledgements The authors wish to acknowledge the Key Program of National Natural Science Foundation of China (Contract No. 51734010) for the financial support.

References 1. Lubinski, A., Hsu, F.H., Nolte, K.G.: Transient pressure surges due to pipe movement in an oil well. Revue de l’Institut Français du Pétrole 32(3), 307–348 (1977) 2. Lal, M.: Surge and Swab modeling for dynamic pressures and safe trip velocities. In: IADC/SPE Drilling Conference. Society of Petroleum Engineers (1983) 3. Zhong, B., Zhou, K., et al.: In-well transient wave pressure analysis. J. Southwest. Pet. Inst. 11(2), 33–41 (1989) 4. Zhong, B., Zhou, K., Hao, J., et al.: Dynamic simulation of in-well wave pressure and its influencing factors. Nat. Gas. Ind. 10(5), 43–47 (1990) 5. Fan, H., Chu, Y.: Prediction of dynamic fluctuating pressure in wellbore during drilling. J. Univ. Pet. 19(5), 36–41 (1995) 6. Rommetveit, R., Bjorkevoll, K.S., Gravdal, J.E., et al.: Ultra-deepwater hydraulics and well control tests with extensive instrumentation: field tests and data analysis. SPE Drill. Complet. 20(04), 251–257 (2005) 7. Li, J., Huang, Z., Li, K., et al.: The influence of collar on surge pressure caused by the drilling fluid viscous force under pumping condition. Res. J. Appl. Sci. Eng. Technol. 5, 1781–1785 (2013) 8. Li, J., Bi, S., Zou, J., et al.: The influence of collar on surge pressure caused by the drill string inertia force under closed pipe condition. Res. J. Appl. Sci. Eng. Technol. 16, 4155–4157 (2013) 9. Guo, Y.: Calculation Model and Experimental Study of Wellbore Pressure Fluctuation under Drilling Conditions. China University of Petroleum, Beijing (2014) 10. Fan, H.: Practical Drilling Fluid Hydrodynamics, pp. 302–317. Petroleum Industry Press, Beijing (2014) 11. Li, Q.: Research on a New Prediction Model of Surge Pressure Appropriated to Horizontal Well. Northeast Petroleum University, Daqing (2011) 12. Crespo, F., Ahmed, R.: A simplified surge and swab pressure model for yield power law fluids. J. Petrol. Sci. Eng. 101, 12–20 (2013) 13. Fontenot, J.E., Clark, R.K.: An improved method for calculating swab and surge pressures and circulating pressures in a drilling well. Soc. Petrol. Eng. J. 14(05), 451–462 (1974) 14. Wu, J.: Hydraulics Calculation of Dual Gradient Drilling. China University of Petroleum, Dongying (2007)

Chapter 58

Design and Analysis of the Composite Hollow Glass Spheres Separator for Dual-Gradient Drilling in Deep Water Ruiyao Zhang and Jun Li

Abstract Aiming at improving the HGS injection and separation issue in HGS dualgradient drilling(DGD) in deep water, the underwater two-stage composite separator with axial vane to guide flow and no smooth straight cone section has been designed. The traditional tangential inlet is designed to the type of axial vane guiding flow in axial direction. The types of Hyperbola and parabola sections are used to replace the smooth straight cone ones in two stages respectively. The composite separator is provided with a hollow thin-walled pipe with a wall holes from the first overflow outlet to the second bottom outlet in the axial direction. The three dimensional modeling of the composite separator has been established by SolidWorks. The RNG κ-ε is adopted to describe the turbulence model in internal flow field and solid-liquid two phase flow is depicted by Euler model equation. At last, the is numerical simulation is directed by fluent software. The results show that the composite separator can achieve high separation efficiency and overflow ratio. To some extent, the study can promote the development of HGS dual-gradient drilling in deep water. Keywords Dual-Gradient drilling · The hollow glass spheres · Axial vane · Two stage · Fluent

58.1 The Introduction With the rapid development of the oil and gas industry, the demand for oil and gas resources keeps increasing. Therefore, offshore drilling develops rapidly and gradually moves towards deep water and ultra-deep water [1]. However, the pressure window is narrow and the pressure control is difficult in deep water drilling. Therefore, in response to this problem, dual gradient drilling system was developed abroad in the 1990s [2]. According to different principles, dual gradient drilling can be divided into riser free drilling, submarine pump lifting drilling system and dual density drilling system [3]. R. Zhang · J. Li (B) College of Petroleum Engineering, China University of Petroleum, Beijing, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_58

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Dual density drilling can be divided into hollow glass microspheres injection, low density fluid injection and gas injection according to the different fluids injected [4]. Compared with several other dual-gradient drilling systems, the hollow glass spheres drilling system requires less auxiliary equipment and the hollow glass spheres can be recovered and reused, which greatly reduces the cost when meeting the operational requirements. Hollow glass sphere refers to hollow glass sphere, whose diameter varies from dozens of microns to several millimeters and whose density is less than 1 g/cm3 [5]. According to the physical characteristics of the HGS, the drill pipe— ball injection system, seawater (carrier fluid)—ball injection system and HGS—gas mixture injection system were developed [6]. At present, the research on the injection and separation of HGS in China is still in its infancy, so it is a difficult problem to realize the injection and separation of HGS efficiently and economically [7]. This paper mainly studies the injection and separation of HGS in the drill-ball injection system. The HGS can be effectively separated and smoothly entered into the annulus through the composite separator installed in the drill pipe, without adding other auxiliary equipment, which has low cost and simple operation process. The research of this paper will promote the further development of this technology.

58.2 Principle Analysis of Injection and Separation 58.2.1 Analysis of Drill-Ball Injection System As shown in Fig. 58.1, drill pipe—ball injection system is used. The drilling fluid mixed with HGS is injected through the drill pipe through the inner cavity of the drill pipe to the inlet of the separator through the pump. Since axial guide vanes are installed at the inlet of the composite separator, solid-liquid two-phase flow entering the separator will generate rotation in the separator. Due to the density difference

Fig. 58.1 Schematic diagram of hollow ball dual gradient drilling system

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Fig. 58.2 Schematic diagram of double gradient drilling pressure window

between the hollow ball and drilling fluid, the centrifugal force of different sizes is generated in the separator cavity due to rotation, thus the separation of the two phases is realized. The separated HGS will be carried by some drilling fluid from the overflow pipe of the separator into the upper part of the annulus, and the drilling fluid will be diluted. The rest of the drilling fluid flows out of the bottom hole to the bit and into the bottom of the annulus, creating a density difference between the upper and lower annulus. Because the drilling fluid in the upper annulus is diluted by HGS, the density of the drilling fluid decreases close to the density of sea water. In this way, the reference plane for calculating the hydrostatic pressure of drilling fluid in conventional drilling can be changed from sea level to seabed [8]. As shown in Fig. 58.2, the pressure window is narrow in deep water drilling. If the hydrostatic pressure system of conventional drilling is still used, the pressure range that can be controlled is very narrow, and it is very difficult to control pressure when encountering complex strata.

58.2.2 Analysis of the Separation Principle of the Composite Separator The composite separator is mainly composed of overflow pipe, drainage pipe, first and second axial guide vanes, hyperbolic separation section and parabolic separation section. The composite separator is installed between the platform and the seabed and is connected up and down to the drill pipe by thread or flange [9]. The separation efficiency of hyperbola segment is higher than that of normal smooth straight conical

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segment [10]. Parabolic segments have a higher shunt ratio than straight cone segments [11]. By adding a drainage pipe, the separation and overflow of the HGS from the external cyclone and the internal cyclone can be realized, while in the smooth straight cone, the HGS can only overflow through the overflow pipe in the process of passing through the internal cyclone. The drilling fluid mixed with HGS is pumped into the drill pipe and its axial flow is converted into rotary motion by the first axial guide vane of the composite separator. Then the rotating solid-liquid two-phase fluid enters the hyperbolic separation section, and the drilling fluid and the hollow ball generate an external cyclone in the hyperbolic cavity. Due to the difference in density between HGS of solid phase and drilling fluid, radial force of the two will produce radial resultant force of different sizes under the combined action of centrifugal inertia force, hydraulic resistance and centripetal buoyancy [12], so that the two phases will separate in radial direction. After that, the HGS will do the top-down external swirling motion near the axis. Because the separated HGS is close to the axis, a part of the HGS will enter the inner part of the drainage pipe directly along the drainage pipe wall from the outlet of the pipe wall during the outer cyclone movement. Since radial pressure gradient exists from the center to the outer wall of the hyperbolic section, and the pressure gradually increases from the inside to the outside, and the annulus is a low-pressure area, HGS entering the drainage tube will enter the annulus from the upper end of the drainage tube under the pressure difference. The remaining HGS, on the other hand, follow the drilling fluid along the hyperbola segment to the downstream. Since the fluid carrying the HGS in this part cannot completely go out from the bottom flow outlet [13], some of them will generate bottom-up internal swirl in the axial direction. In this process, a part of the separated HGS will enter the drainage pipe to reach the annulus, and the others will go up through the overflow pipe to enter the annulus, thus achieving the first separation process of the HGS. In addition, the unseparated HGS will enter the second guide vane, then generate rotation motion, and enter the parabolic segment. The separation process of parabola segment is the same as the hyperbola segment in the first stage. Due to the sealing of the lower end of the drainage pipe in the parabolic separation section, the pressure gradient in the radial direction and the pressure difference with the annulus combined, the HGS finally separated will overflow into the annulus through the two processes of external cyclone and internal cyclone through the drainage pipe (Fig. 58.3).

58.3 Structure Parameters of the Composite Separator According to the comprehensive factors, such as 125 mm drill pipe for deep water drilling and connection mode, the inner diameter of the separator is designed to be 128 mm, the outer diameter is 168 mm, and the total length is 1850 mm. The three-dimensional structure is shown in Fig. 58.4. According to the design manual of the hydraulic cyclone and relevant technical data [14], the other structural parameters of the separator are obtained: The inner

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Fig. 58.3 Two-dimensional structure diagram of composite separator. 1-drainage tube, 2-the first guide vane shell, 3-hyperbolic separation section, 4-second guide vane shell, 5-parabolic separation segment, 6-drain pipe sealing, 7-diversion hole, 8-blade, 9-blade, 10-overflow tube

Fig. 58.4 Three-dimensional structure of the composite separator

diameter of the overflow pipe is d0 = 37.5 mm, the length is 200 mm, the diameter of the drainage pipe is d1 = 16.25 mm, and the total length is 1850 mm. The total length of the axial guide blade segment is 150 mm, the length of hyperbola segment and parabola segment is 750 mm, and the diameter in the middle of the hyperbola segment is 18.75 mm. According to the rules of the design of the guide vane, the entry Angle generally is taken between 80° and 90° [15], the entry angle of the guide vane in this paper is chosen as the beta 3 = 80°, and blade is H = 150 mm in height. Generally, the rotation effect of the fluid will be better if the outlet Angle decreases or the ratio of the inlet and outlet cross-sectional areas increases. However, when the Angle is too large, the pressure drop will increase significantly [16]. So according to the principle of the good rotation effect as well as not too high pressure drop, it is reasonable to take exit angle of the neat line within as beta 1 = 23°, meanwhile take exit angle of the neat line outside as beta 2 = beta 1 + (5° to 10°) is advisable [17]. Therefore, in order to avoid too much loss of pressure drop, so take the exit angle of the neat line outside for 28°. According to the known conditions and the geometric relationship shown in Fig. 58.5, the radius of the basic segment shape, the length of the straight segment and the basic segment length of the inner and outer directrix can be calculated by the following formula.

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Fig. 58.5 Geometric parameter diagram of guide vane

The calculation formula of internal alignment correlation is shown as follows. Similarly, the relevant parameters of external alignment can be obtained. The horizontal length of the straight line segment and the basic segment of the internal alignment line is 12.1 mm and 13.15 mm respectively. To the same token, the horizontal length of the straight line segment and the basic segment of the external alignment line is 13.9 mm and 18.1 mm respectively. Draw the corresponding blade graph according to the function and the known data. R=

H l l =  = 1 2sinβ cosβ1 − cosβ2 2sin β2 −β 2

(58.1)

Basic segment: x=ρ− ρ=



ρ2 − y2

L 1−sinβ1

(58.2) (58.3)

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Fig. 58.6 three-dimensional structure of the blade

Straight line segment: x=

y − y0 + x0 tanβ1

y0 = 

R L Beta Beta 1 Beta 3 ρ

ρ K1

(58.4) (58.5)

1 + K 12

radius of type line, mm; chord length of the basic segment, mm; half of the central Angle corresponding to the base segment; the exit Angle of the inner directrix; the exit Angle of the outer directrix; the curvature of the basic segment (Fig. 58.6).

58.4 Analysis of Mechanical Model The drilling fluid and the HGS is a mixture of two phase flow, When operating numerical analysis to observe the distributed situation of the two phases, water and the HGS can be defined as the first phase and the second phrase respectively. It is assumed that both solid and liquid phases are continuous media in a composite separator [18]. Therefore, the continuity equation and momentum equation of solid phase and liquid phase can be obtained respectively from euler model. Continuity equation for the liquid phase:   ∂ → vl = 0 (αl ρl ) + ∇ · αl ρl − ∂t

(58.6)

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Continuity equation for the solid phase:   ∂ → vs = 0 (αs ρs ) + ∇ · αs ρs − ∂t

(58.7)

Momentum equations of liquid phase and solid phase are as follows:    ∂ (αl ρl vl ) + ∇ · (αl ρl vl vl ) = −αl ∇ p + ∇ · αl μl ∇ vl + ∇ vlT + αl ρl g + Fl ∂t (58.8)    ∂ (αs ρs vs ) + ∇ · (αs ρs vs vs ) = −αs ∇ p + ∇ · αs μs ∇ vs + ∇ vsT + αs ρs g + Fs ∂t (58.9) αl αs t vl vs ρs ρl Fs Fl

volume fraction of liquid; volume fraction of solid; time, s; velocity of Liquid phase, m/s; velocity of solid phase, m/s; The density of solid phase, kg/m3 ; The density of liquid phase, kg/m3 ; The force of the liquid phase on the solid phase, N; The force of the liquid phase on the solid phase, N.

58.5 Numerical Simulation On the one hand, the solid phase is HGS with a density of 650 kg/m3 , a size of 0.45, 0.6 and 1 mm, and the liquid phase is water. The viscosity of the liquid phase was appropriately increased, and an example analysis of solid-liquid two-phase was carried out. On the other hand, the inlet velocity is set as 3 m/s, 4 m/s, 5 m/s, 6 m/s and 7 m/s respectively, and then the velocity size of the overflow outlet and the bottom outlet is monitored respectively. Therefore, the influences of different diameters and sizes of hollow balls on the separation efficiency under different injection velocities were calculated respectively, and the results as shown in the following table were finally obtained. Finally, the distribution of the HGS in the liquid phase was checked. It can be seen from Fig. 58.7 that most of the HGS are clustered around the drainage pipe and show a spiraling upward trend. Finally, the volume fraction at the overflow pipe reaches the highest. In order to judge the drainage effect of the drainage pipe, the velocity vector diagram was viewed, and the overall velocity vector diagram of the HGS can be seen from Fig. 58.8. Based on the local observation of the vector diagram at the

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Fig. 58.7 Volume content distribution of HGS

Fig. 58.8 Overall velocity vector diagram of HGS

overflow pipe, the HGS starts to separate from the liquid phase under the action of the external cyclone, and the approximate circulation flow is formed locally and finally enters the drainage pipe. For one thing, as the pressure from the inner wall of the separator to the center gradually decreases, under the action of radial pressure difference, after entering the drainage pipe, the HGS will not circulate out of the pipe and spiral upward in the pipe, and finally enter the overflow port. For another, the HGS that has not been separated will form an internal swirl after it reaches the bottom flow outlet, and then spiral around the flow from the bottom up, and finally drain again in the drainage pipe and reach the overflow outlet. Therefore, the biggest advantage of the composite separator not only improves the separation efficiency. But more importantly, the HGS, after the two-stage separation reaches the central axis of the separator and generate direct overflow in the drainage pipe by the process of the external cyclone and the internal cyclone. However, on one side, it avoids only one stage separation, low efficiency and only overflow through the overflow pipe of the traditional separator. On the other side, for the traditional separator, after the HGS is separated, the HGS overflow is achieved only by the single axial force provided by the internal cyclone (Figs. 58.9, 58.10, 58.11, 58.12, 58.13 and 58.14 and Tables 58.1, 58.2 and 58.3).

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Fig. 58.9 Distribution of HGS at the drainage pipe

Fig. 58.10 Velocity vector diagram of HGS at the drainage pipe

Fig. 58.11 Inlet and outlet content of 0.45 mm HGS

As can be seen from the figure, with the increase of injection speed, the content of the HGS with different diameters shows an overall trend of increasing within the overflow and bottom flow. Because the density of HGS is relatively small, and there is also the effect of viscosity, there will be the aggregation and adsorption of HGS in the compound separator. Therefore, there will be certain calculation errors, but the

58 Design and Analysis of the Composite Hollow Glass Spheres … Fig. 58.12 Inlet and outlet content of 0.6 mm HGS

Fig. 58.13 Inlet and outlet content of 1 mm HGS

Fig. 58.14 Separation efficiency under different diameters and velocities

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Table 58.1 Separation efficiency of 0.45 mm HGS at different injection speeds Inlet velocity (m s−1 )

0.45 mm (HGS diameter) Inlet mass (kg s−1 )

Overflow mass (kg s−1 )

Underflow mass (kg s−1 )

Separation efficiency (%)

3

2.5093

0.7452

1.5464

48.18

4

3.3546

1.0413

2.2136

47.04

5

4.1820

1.5329

2.9492

51.97

6

5.0185

1.5219

3.2073

47.45

7

5.8549

1.9572

3.6870

53.08

Table 58.2 Separation efficiency of 0.6 mm HGS at different injection speeds Inlet velocity (m s−1 )

0.6 mm (HGS diameter)

Separation efficiency (%)

Inlet mass (kg s−1 )

Overflow mass (kg s−1 )

Underflow mass (kg s−1 )

3

2.5093

0.7901

1.6189

48.80

4

3.3546

1.2178

2.3263

52.34

5

4.1820

1.2885

2.8930

44.54

6

5.0185

1.6455

3.2929

49.97

7

5.8549

1.9874

3.6671

54.19

Table 58.3 Separation efficiency of 1 mm HGS at different injection speeds Inlet velocity (m s−1 )

0.6 mm (HGS diameter) Inlet mass (kg s−1 )

Overflow mass (kg s−1 )

Underflow mass (kg s−1 )

Separation efficiency (%)

3

2.5093

0.8290

1.6501

50.23

4

3.3546

1.1973

2.3407

51.15

5

4.1820

1.3292

2.8513

46.61

6

5.0185

1.6051

3.1961

50.22

7

5.8549

1.8347

3.5661

51.44

overall separation efficiency fluctuates between 48 and 50%, when the separation efficiency varying with the injection speed and the diameter of the HGS.

58.6 Conclusion (1) According to the separation principle of cyclone centrifugal force, an underwater composite separator with two stages in series was designed. The traditional tangential inlet mode was designed as axial vane flow guide, which reduced the pressure consumption and enhanced the cyclone effect.

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(2) By using hyperbola and parabola instead of the existing smooth straight cone, the overflow ratio and separation efficiency can be improved synthetically, and the overflow efficiency can be improved by adding a drainage pipe. (3) Numerical simulation was carried out by fluent software, and it was found that the separation efficiency was significantly improved with the increase of injection speed. (4) The increase of viscosity will lead to the aggregation of HGS, which is not conducive to separation. Therefore, it is necessary to further optimize the structure of the composite separator to improve the separation efficiency as much as possible under the condition of close to the actual drilling fluid viscosity.

References 1. Ziegler, R., Paul Ashley, P.: Successful application of deep water dual gradient drilling. Presented at the IADC/SPE Managed Pressure Drilling and Underbalanced Operation Conference and Exhibition Held in San Antonio, Texas, USA, 17–18 Apr 2013 2. Li, S., Ren, H.: Design and analysis of hollow ball dual density drilling underwater injection valve. Petr. Mach. 45(5), 57–60 (2017) 3. Yin, Z.: Principle, method and application of new deep water dual-gradient drilling system. China University of Petroleum, Beijing (2007) 4. Yang, S.: Research on hollow ball injection and separation technology for dual-gradient drilling in deep water. China University of Petroleum, East China (2007) 5. Halkyard, J.: Hollow glass microspheres: an option for dual gradient drilling and deep ocean mining lift. Presented at the Offshore Technology Conference Asia held in Kuala Lumpur, Malaysia, 25–28 Mar 2014 6. Fang, H.: Underwater equipment for dual-gradient drilling in deep water. Oil Field Mach. 37(11), 1–6 (2008) 7. Yin, Z., Chen, G., Sheng, L., Jiang, S., Xu, L.: Dual-gradient drilling technology for deep-water hollow ball. China Shipbuild. 46(11), 71–76 (2005) 8. Stave, R.: Implementation of dual gradient drilling. Presented at the Offshore Technology Conference held in Houston, Texas, USA, 5–8 May 2014 9. Li, Z., Sun, H.: Performance simulation experiment of downhole centrifugal cyclone high efficiency oil and gas separator. Pet. Mach. 35(12), 5–8 (2007) 10. Sun, Q.: Separation Machinery. Chemical Industry Press (1993) 11. Li, Y.: Numerical calculation and optimization design of downhole oil-water cyclone separator. China University of Petroleum, East China (2008) 12. Chen, S.: Flow field simulation and experimental study of compact axial flow guide vane deoilcyclone. Daqing Petroleum University (2010) 13. Cui, R.: Simulation of flow field and particle motion in a double-cone cyclone and its industrial application. Wuhan University of Science and Technology (2015) 14. Bradley, D.: The Hydrocyclone, pp. 65–67. Pergamon Press, London (1965) 15. Guan, X.: Axial Flow Pump and Oblique Flow Pump: Hydraulic Model Design Test and Engineering Application. China Aerospace Press (2009)

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16. Ding, C.: Centrifugal Pump and Axial Flow Pump. Machinery Industry Press (1981) 17. Jin, Y., Fan, C., Mao, Y., Shi, M.: Study on blade parameter design method of guide vane cyclone tube. Chem. Mach. 26(1), 21–23 (1999) 18. Chen, G.: Improving performance of low density drilling in fluids with hollow glass sphere. Presented at the SPE European Formation Damage Conference held in The Hague, The Netherlands, 13–14 May 2003

Chapter 59

Methane Hydrate Generation Model and Software Development Based on P. Englezos Method Nan Ma, Jun Li, Kuidong Luo, Shujie Liu and Min Wen

Abstract At present, most scholars judge the hydrate formation conditions based on the thermodynamic prediction model, but more accurate prediction methods should be based on hydrate formation kinetics, while kinetic studies mostly focus on the qualitative interpretation of the generation mechanism. Based on this, this paper quantitatively studied the kinetics of hydrate formation. Combined with van der Waals and Platteeuw prediction model, thermodynamic equilibrium theory and P. Englezos method, the methane hydrate formation model was established. The generation time of hydrate was calculated under different thermodynamic generation conditions, and the data were used for verification. At the same time, using the model to calculate the approximate time required for clogging of the wellbore by hydrate formation. The results show that the hydrate formation conditions obtained by thermodynamic methods are only threshold conditions, which will not cause wellbore blockage within a certain time range. The methane hydrate formation model established in this paper can be calculated for hydrate formation time, wellbore clogging time and hydration volume. Keywords Hydrate · Generation mechanism · Phase equilibrium · Hydrate generation time In 1934, thermodynamic studies and kinetic studies on hydrate formation were carried out since the day when American scientist Hammerschmidt [1] discovered that gas pipelines were blocked by natural gas hydrates. At present, most scholars believe that in the production process of deepwater oil and gas wells, as long as the hydrate thermodynamic formation conditions are met, hydrates will form. Commercial software (PIPE sim software, PVT sim software and PETRO sim software) are mostly compiled based on van der Waals and Platteeuw (vdW-P) thermodynamic prediction model [2] or its derivative model [3–6]. Tohidi [7] studied a set of hydrate formation N. Ma (B) · J. Li · K. Luo China University of Petroleum, Beijing 102200, China e-mail: [email protected] S. Liu · M. Wen CNOOC Research Institute, Beijing 100029, China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_59

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monitoring and early warning systems based on the thermodynamic characteristics of hydrate formation. Gao [8] builds a predictions of hydrate formation based on the vdW-P prediction model. Although the thermodynamic method of hydrate formation can accurately determine the temperature-pressure conditions of hydrate formation, the thermodynamic method does not study the quantitative problems such as the formation time and volume of hydrate, and it is impossible to judge the actual formation state of hydrate. Based on this, combined with thermodynamic phase equilibrium theory [9] and vdWP model, the thermodynamic process of hydrate formation was analyzed in detail, and its applicability was evaluated. At the same time, based on the hydrate formation kinetics method proposed by Englezos [10], a methane hydrate formation model was established. For different hydrate formation conditions, the model can calculate the parameters such as the volume of methane hydrate formation and the generation time. For different sizes of tubing, the model calculates the time that the wellbore is blocked by hydrate formation. Such parameters can be used as a basis for judging the actual production of hydrates. Combining the characteristics of thermodynamic methods and kinetic methods, two different methods are combined to provide a systematic new idea for hydrate prediction research.

59.1 Thermodynamic Process of Hydrate Formation The thermodynamic equilibrium state [9] refers to the phenomenon that the system can maintain its state for a long time without being affected by external influences. That is to say, in the thermodynamic equilibrium state, the number of molecules in the system, macroscopic thermodynamic properties, etc. do not change with time. Under normal temperature and pressure conditions, the gas phase and the liquid phase form a solution at the interface of the two phases. The gas molecules and the water molecules in the solution system are mixed with each other, and there is no clear boundary between the two types of molecules, as shown in Fig. 59.1. Subsequently, the environmental conditions begin to change, and when the temperature and pressure values reach the thermodynamic equilibrium conditions, hydrate molecules begin to form in the solution system. At this time, the solution system is composed of gas molecules, water molecules, and hydrate molecules. During the subsequent period of time, water molecules and gas molecules continue to be consumed, and hydrate molecules continue to be formed. When the number of gas molecules, the number of water molecules and the number of hydrate molecules reach a certain amount, respectively, the entire solution system will be in a thermodynamic equilibrium state, and the number of the three types of molecules will remain constant, as shown in Fig. 59.2. It should be noted that in the above process, the hydrate molecules are all dissolved in the solution system, and there is no natural gas hydrate present in the form of solid crystal. In fact, in the process of achieving the thermodynamic equilibrium of the entire solution system, the gas phase and the liquid phase supplement the gas

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Fig. 59.1 Schematic diagram of the solution system under normal conditions

Fig. 59.2 Schematic diagram of the solution system under thermodynamic equilibrium conditions

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Fig. 59.3 Supplementary process of gas molecules and water molecules

molecules and water molecules in the solution system, causing the molecular number of the two types of molecules to increase sharply, thereby causing the thermodynamic equilibrium to be destroyed, as shown in Fig. 59.3. In order to regain the thermodynamic equilibrium, the solution system will continue to consume the added gas molecules and water molecules. Through this process, the number of gas molecules and the number of water molecules will return to the number of molecules required for thermodynamic equilibrium, but the number of hydrate molecules in the system is much higher than the number of molecules required for thermodynamic equilibrium, as shown in the Fig. 59.4. As a result, excess hydrate molecules will precipitate out of the solution system in solid crystalline form, and the number of hydrate molecules will return to the number of molecules required for thermodynamic equilibrium, allowing the entire solution system to again reach thermodynamic equilibrium. The hydrate crystals that are precipitated and aggregated are the real “culprits” that cause the wellbore to clog, as shown in Fig. 59.5. It can be seen from the entire hydrate formation process that the hydrate formation conditions obtained by the thermodynamic method are threshold conditions for hydrate formation. Under these conditions, the hydrate molecules only begin to be removed from the solution system in the form of solid crystals, and their volume and quantity are far from harmful to normal production. Therefore, thermodynamic methods can judge the external environment in which hydrates are formed. That is to say, this method can be used to determine which hydrates can be formed under temperature-pressure conditions and which hydrates cannot be formed under temperature-pressure conditions. However, the limitation

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Fig. 59.4 Consumption of gas molecules and water molecules and formation of hydrate molecules

Fig. 59.5 Molecular precipitation process of excess hydrate molecules

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of this method is that the actual state of hydrate formation cannot be described, and quantitative problems such as hydrate formation time and volume generation cannot be solved. However, in the production process, only the quantitative problem of hydrate formation can be solved, and the hydrate hazard can be more accurately prevented and treated.

59.2 Derivation and Verification of Methane Hydrate Kinetic Model 59.2.1 Derivation and Verification of Methane Hydrate Formation Model Based on P. Englezos Method According to Karpinski’s (1980) modelling method for crystal growth in solution, Englezos [10] proposed the following hydrate crystal growth steps, as shown in Fig. 59.6: (1) Forming a thin hydrate film with pores on the surface of the droplet; (2) The water molecules inside the droplet continuously diffuse to the surface to form a hydrate, which thickens the hydrate film; (3) The droplets are eventually completely converted to hydrate particles. The P. Englezos method considers that the hydrate particles are spherical, the outer surface of the shell has the same area as the inner surface layer, and there is no cumulative effect, and the gas consumption rate is equal to the hydrate formation rate. This gives the average rate of hydrate formation:

Fig. 59.6 Formation process of hydrate film

59 Methane Hydrate Generation Model and Software Development …

 G avg =

G avg M ρ L fg fb f eq

K∗M ρL



699

  f g + f b − 2 f eq (coshγ − 1) + (L − y L )( f b − f eq ) yL · γ sinhγ (59.1) 

Hydrate average production rate, m/s; molecular weight of the hydrate of the form X · n w H2 O; hydrate density, kg/m3 ; distance between the (g-l) interface and the bottom of the reactor, m; the gas fugacity at the gas-liquid interface, MPa; the gas fugacity at the bottom of the reactor, MPa; Gas fugacity at three-phase equilibrium, MPa.

Among them, the comprehensive rate parameter is: 1 1 1 = + K∗ kr kd

(59.2)

K ∗ Comprehensive rate parameter, mol/m2 MPa s; kr Reaction rate constant, mol/m2 MPa s; kd Mass transfer coefficient around hydrate particles, mol/m2 MPa s. Among them, the liquid phase thickness is: yL = yL D kL a kL a

D Da = kL (k L a)

(59.3)

Liquid phase thickness, m; Gas diffusion coefficient, m2 /s; Mass transfer coefficient in liquid phase, m/s; Volume of liquid on the unit interface area, m2 /m3 ; Constant, which can be calculated from the dissolution experiment.

Among them, the Hatta number is:  μ2 γ = y L 4π K ∗ ∗ D D∗ =

D · cw0 H

γ Hatta number; cw0 Initial concentration of water molecules, mol/m3 ; H Henry constant, can be obtained experimentally, MPa.

(59.4) (59.5)

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Among them, the distribution of the crystal at the second moment is: μ2 (t) = μ2 G α2 t

2G 2 t 2 μ00 + 2Gtμ01 + μ02 1 − 2Gα2 t 3

(59.6)

the distribution of crystals at the second moment, m2 /m3 ; Hydrate production rate, m/s; Nucleation constant, Nucleus/m2 s; time, s.

Among them, the number of crystal particles per unit volume of liquid at the initial moment is:   3M n tb − n eq NP = (59.7) μ00 = 3 VL 4π VL ρ rcr μ00 n tb n eq VL ρ

Initial crystal grain number; the number of moles of gas dissolved at the cloud point, mol; The number of moles of gas dissolved in the three-phase equilibrium, mol; liquid phase volume, m3 ; hydrate density, kg/m3 .

Among them, the number of initial hydrate crystal particles at the first moment is: μ01 = 2rcr μ00

(59.8)

μ01 Initial hydrate crystal grain number at the first moment. Among them, the number of initial hydrate crystal particles at the second moment is: 2 0 μ0 μ02 = 4rcr

(59.9)

μ02 The number of initial hydrate crystal particles at the second moment. Among them, the critical size of the nucleus is: rcr = −

2σ g

(59.10)

rcr Nucleus critical dimension, m σ Hydrate—the surface energy of a free water system equal to the surface energy of ice in water, J/m2 .

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Among them, the amount of change in free energy is:  

  n w vw Pex p − Peq RTex p fb g = − + ln vm f eq RTex p g R Tex p Pex p Peq vm vw nw

(59.11)

the amount of change in free energy when forming a new phase per unit volume, J/m3 ; State parameter constant, take 8.31434 J/mol; Experimental temperature, K; Experimental pressure, MPa; Three-phase equilibrium pressure, MPa; Molar volume of hydrate, m3 /mol; Molar volume of water, m3 /mol; the number of water molecules per gas molecule is 5.75 for methane hydrate and 7.67 for ethane hydrate.

The simultaneous Eqs. (59.1), (59.6), (59.10), and (59.11) provide the formula for the average hydrate formation rate: √ G avg =

K∗M ρL 



⎛

   √ √

f g + f b − 2 f eq (coshγ − 1) Dcw0 2π VL ρ 1 − 2G 2 α2 t 3 ⎝  ln

·

⎛

⎜ 2 3 2 · sinh r · π N ⎝G t + 2  P 

⎞3 

fb f eq

⎛  ln

fb f eq

 RTex p

2σ vm +n w vm ( Pex p −Peq )

 Gt

+ 2⎝  ln



fb f eq

 RTex p

2σ vm

⎠

+n w vm ( Pex p −Peq )

⎞2 ⎞ ⎟ 2σ vm  ⎠ ⎠H  RTex p +n w vm ( Pex p −Peq )

(59.12)

It has been measured that when the hydrate size reaches about 2 cm, the presence of hydrate is clearly observed. Thus, the hydrate formation time calculation formula is: tgenerate =

0.02 G avg

(59.13)

tgenerate Hydrate generation time, s; Hydrate average production rate, m/s. G avg The time of formation of methane hydrate under different temperature and pressure conditions was obtained by consulting the literature [10–14]. Combined with the above calculation method, the methane hydrate formation model was compiled by Matlab software. The hydrate formation time was calculated under different temperature and pressure conditions. The results are shown in Table 59.1. The generation time calculated by the model does not exceed the generation time of the experiment by more than 1 h.

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Table 59.1 Hydrate generation time under different conditions Serial number

Temperature (K)

Pressure (MPa)

Calculating time (s)

Experimental time (s)

Error (±s)

1

275.34

4.5

3032.6

3060

27.4

2

276

4.86

3486.3

3480

6.3

3

278.5

10

11746

14400

2654 (44 min)

4

295

7

7793.9

8100

306.1 (5 min)

5

286.2

13.16

19670

21143

1473 (24.55 min)

59.2.2 Calculation Method of Methane Hydrate Blocking Wellbore Time It is assumed that methane hydrate can cause blockage of the wellbore when the size of the methane hydrate reaches 1/2 of the tubing size. Check the relevant data, the calculation method of hydrate blocking the wellbore under different tubing sizes is as follows:  (1) 11 2 inch tubing blockage time calculation formula: t 1  = 1 /2

0.02015 G avg

(59.14)

 t 1  time that caused 11 2 inch tubing blockage, s; 1 /2 Hydrate average production rate, m/s. G avg  Note: The 11 2 inch tubing has an inside diameter of 40.3 mm.  (2) 21 2 inch tubing blockage time calculation formula: t 1  = 2 /2

0.02515 G avg

 t 1  time that caused 21 2 inch tubing blockage, s; 2 /2 Hydrate average production rate, m/s. G avg  Note: The 21 2 inch tubing has an inside diameter of 50.3 mm.

(59.15)

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 (3) 27 8 inch tubing blockage time calculation formula: 0.031 t 7  = 2 /8 G avg

(59.16)

 t 7  time that caused 27 8 inch tubing blockage, s; 2 /8 Hydrate average production rate, m/s. G avg  Note: The 27 8 inch tubing has an inside diameter of 62 mm.  (4) 31 2 inch tubing blockage time calculation formula: t 1  = 3 /2

0.03795 G avg

(59.17)

 t 1  time that caused 31 2 inch tubing blockage, s; 3 /2 Hydrate average production rate, m/s. G avg  Note: The 31 2 inch tubing has an inside diameter of 75.9 mm. (5) 4 inch tubing blockage time calculation formula: t4 =

0.0443 G avg

(59.18)

time that caused 4 inch tubing blockage, s; t4 G avg Hydrate average production rate, m/s. Note: The 4 inch tubing has an inside diameter of 88.6 mm.  (6) 41 2 inch tubing blockage time calculation formula: t1 = 4 /2

0.05015 G avg

(59.19)

 t1 time that caused 41 2 inch tubing blockage, s; 4 /2 G avg Hydrate average production rate, m/s.  Note: The 41 2 inch tubing has an inside diameter of 100.3 mm. Based on the data provided by Yu [15], Matlab software was used to compile the wellbore blockage time prediction model, and the time required for methane hydrate  to cause 41 2 in tubing blockage under different temperature-pressure conditions was calculated, as is shown in Table 59.2.

1

275.40

1.0

246.8189 (4 min)

Serial number

Temperature (K)

Pressure (MPa)

Blockage time (s)

1704.1 (28 min)

2.0

280.15

2

4024.5 (67 min)

3.0

284.86

3

7019.1 (117 min)

4.0

287.94

4

 Table 59.2 Hydrate blockage of different conditions 41 2 in tubing time 5

10536 (175.6 min)

5.0

289.66

6

34105 (568 min)

10.0

292.77

7

65542 (1092 min)

15.0

293.94

8

104090 (1734 min)

20.0

295.39

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59.3 Conclusion and Understanding The hydrate formation conditions obtained by the thermodynamic method are the threshold conditions for hydrate formation. The volume of hydrate formed under this condition is far from harmful to normal production; According to the kinetic method, the methane hydrate formation model established in this paper can explain the quantitative problems such as hydrate formation time and volume generation, and also calculate the time of clogging caused by hydrate in the production process. On the basis of summarizing the respective characteristics of thermodynamic methods and kinetic methods, this paper believes that the two methods can be combined. In the actual production process, the thermodynamic method is first used to determine whether the external environment has reached the hydrate formation requirements. If the requirements are not met, the temporary anhydrate formation risk. If the requirements are met, the kinetic method is used to judge the formation of the hydrate in the environment. The parameters calculated by the model are specific indicators, and the hydrate formation status is judged to determine the severity of the hydrate hazard. The combination of two different methods provides a comprehensive and systematic approach to hydrate prediction.

References 1. Wu, M., Wang, S., Liu, H.: A study on inhibitors for the prevention of hydrate formation in gas transmission pipeline. J. Nat. Gas Chem. 16(1), 81–85 (2007) 2. Van Der Waals, J.H., Platteeuw, J.C.: Clathrate solutions. Adv. Chem. Phys. (2), 2–57 (1958) 3. Du, Y., Guo, T.: Prediction of gas hydrate formation conditions I: system without inhibitor. Acta Pet. Sin. (Petroleum Processing Section) 4(3), 82–91 (1988) 4. Chen, G., Guo, T.: Thermodynamic study of hydrate formation process. Acta Pet. Sin. (Natural Science Edition) 19(2), 88–91 (1995) 5. Chen, G., Ma, Q., Guo, T.: Establishment of hydrate model and its application in salt-containing systems. Acta Pet. Sin. 21(1), 64–70 (2000) 6. Madsen, J., Pedersen, K.S.: Modeling of structure H hydrates using a Langmuir adsorption model. Ind. Eng. Chem. Res. 39(4), 1111–1114 (2000) 7. Tohidi, B., Chapoy, J., Yang, F.: Developing hydrate monitoring and early warning systems. OTC 19247 (2008) 8. Gao, Y.: Study on Multiphase Flow and Well Control of Deepwater Oil and Gas Drilling Wellbore. China University of Petroleum (East China), Shandong Dongying (2007) 9. Huang, X., Xu, G.: Engineering Thermodynamics. China Electric Power Press, Beijing (2015) 10. Englezos, P., Kalogerakis, N., Dholabhai, P.D., Bishnoi, P.R.: Kinetics of formation of methane and ethane gas hydrates. Chem. Eng. Sci. 42(11), 2647–2658 (1987) 11. Li, M., Fan, S., Zhao, J.: Experimental study on formation of natural gas hydrate in porous media. Nat. Gas Ind. 26(5), 27–28 (2006) 12. Dai, M., Zhou, L.: Experimental study on synthetic natural gas hydrate. Chem. Prog. 16(5), 747–750 (2004) 13. Chen, M., Cao, Z., Ye, Y.: Simulation experiment study on synthesis of marine gas hydrate. Chin. Soc. Oceanogr. 28(6), 39–43 (2006)

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14. Wu, H., Lu, Y., Lv, X.: Experimental study on gas hydrate formation rate. Exp. Technol. Manag. 31(1), 36–40 (2014) 15. Yu, X., Zhao, J., Guo, J.: Comparison of prediction models for gas hydrate formation conditions. Oil Gas Storage Transp. 21(1), 20–24 (2002)

Part XVI

New Horizon of Computational Science and Engineering with Heterogeneous Many-Core Processors

Chapter 60

Implementation of SPH and DEM for a PEZY-SC Heterogeneous Many-Core System Natsuki Hosono

and Mikito Furuichi

Abstract Particle-based simulation strategies such as the smoothed particle hydrodynamics (SPH) and discrete-element method (DEM) approaches are useful in industry and disaster-prevention applications. One of the key problems for a successful large-scale particle-based simulation is energy (electricity) costs. To mitigate this problem, using PEZY-SC many-core microprocessors should provide an effective solution because their power efficiency is good. We have implemented parallelized SPH and DEM codes based on Framework for Developing Particle Simulator (FDPS). Our strategy involves using the PEZY-SC microprocessors as an external accelerator device in an off-load-type implementation in which the interaction kernel is allocated to these microprocessors and a pre/post-processing is performed on the host CPU. Performance tests show good scalability of the computing interaction kernel per number of threads, although a breakdown of calculation costs shows a room for further improvements. Water and granular dam-break tests were also performed on the PEZY-SC system to validate the SPH and DEM codes, respectively. Keywords SPH · DEM · PEZY-SC · Off-load

60.1 Introduction The Lagrangian nature of particle-based simulation such as smoothed particle hydrodynamics (SPH) and discrete element method (DEM) offers great advantages in the applications to the disaster prevention and industrial problems, for example, dealing with non-diffusive material transport with large surface deformations. For these particle-based methods, a number of particles is essential to improve the resolution N. Hosono · M. Furuichi (B) Japan Agency for Marine-Earth Science and Technology, 3173-25 Showa-machi, Kanazawa-ku, Yokohama, Kanagawa 236-0001, Japan e-mail: [email protected] N. Hosono RIKEN Center for Computational Science, 7-1-26 Minatojima-minami-machi, Chuo-ku, Kobe, Hyogo 650-0047, Japan © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_60

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and to increase the system size [1, 2]. For example, reproducing the geotechnical centrifuge experiment using over hundred million particles in a large-scale DEM simulation is quantitatively a huge challenge [3]. Numerical simulations provide an alternative tool to the expensive real experiments to model the behavior of geotechnical structures and predict safety performance and reliability. However, the energy expended in performing the numerical simulation is enormous when we handle a large number of particles. In industrial scenes, using large-scale simulations is difficult when there is no reduction in energy costs. One solution to this energy problem is to use an energy-efficient or “green” supercomputer (SC) system. Ubiquitous “green” SC systems have recently begun to exploit the many-core architecture. An example of such a system is the PEZY-SC accelerators, which won first place in GREEN500 of 2018 [4]. Efficient implementation of SPH and DEM for the PEZY-SC2 is however non-trivial. In this paper, we introduce our implementations of them using the Frame-work for Developing Particle Simulator (FDPS) [5, 6]. We rate their performance and present results of validation tests performed on them.

60.2 Off-load Implementation of SPH and DEM with FDPS We developed SPH and DEM applications based on the FDPS, which is a highperformance library for parallelizing particle-based methods over distributed and shared memory systems. The FDPS supports SPH by default, but not DEM. Both SPH and DEM simulations commonly incorporate short-range interaction forces along the normal direction, that is, parallel to the vectors of the relative particle position. In addition, DEM particles are acted upon by tangential forces at the contacting plane; these forces are determined by integrating the particle displacements at each contact site. Because the contacting pair list varies in time, an efficient implementation of the parallelized DEM is non-trivial. Here, we customized the use of the FDPS to deal with data allocated for each contacting pair list by assuming a maximum contact number for each particle. Although the current approach is simple, it is not robust and consumes a lot of memory. Further improvements customized for parallel DEM are discussed for example in [7]. Our target machine is the PEZY-SC system consisting of host Intel Xeon CPUs and PEZY-SC devices (Fig. 60.1). For programming on this heterogeneous manycore system, we off-load portions of the SPH and DEM tasks to run on many-core devices. With the off-load implementation, speed-ups in computation by the accelerators can be obtained with relatively small changes in the original SPH and DEM codes de-signed for the CPUs. Our implementation of the SPH and DEM comprises three steps: pre-conditioning, kernel calculation, and synchronization. In the preconditioning stage, lists of potentially interacting particle pairs are constructed using the tree method. This task is performed on the host CPU. Then, the pre-conditioned particle data set is sent to a PEZY-SC processor, where in the kernel calculation, the particles are updated using the equation of motion. The host waits for the off-loaded

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Fig. 60.1 Architectural design of PEZY-SC system

Fig. 60.2 Schematic process of off-load implementation

tasks to receive the updated particles information. Afterwards, the synchronization tasks among the different distributed memory nodes are performed using a messagepassing-interface (MPI) application. The preconditioning and synchronization parts are conducted within the FDPS. The source code written from scratch contains only those parts of the kernel calculation and data transfer between the host and PEZY-SC device, for which we use PZCL—an OpenCL like language. Figure 60.2 shows a schematic of the processes of this off-load implementation.

60.3 Results Figure 60.3 show the results of the dam-break test with 400 K and 1 K particles for the SPH and DEM simulations, respectively. The particles are initially placed on a cubic lattice. Our off-load implementation for both SPH and DEM tasks were successfully handled by the PEZY-SC devices in simulating the complex fluid and granular motions. In the following subsection, we present our results of performance and validation tests of the SPH simulation. The computer systems were Shoubu and Shoubu System B installed at RIKEN with PEZY-SC1 and PEZY-SC2 devices, respectively.

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Fig. 60.3 Snapshots of the dam break test performed on PEZY-SC2 for a SPH and b DEM

60.3.1 Performance Test Figure 60.4 presents a breakdown of the calculation cost for preconditioning, execution kernel, and data transfer to offload the tasks. The results using different numbers of threads are plotted. This simulation was performed using a single PEZYSC1 device, which has 1,024 cores of 8 threads with simultaneous multi-threading designed to handle 8,192 threads. The kernel computation dominates the elapsed time cost with a small number of 128 threads but has been greatly reduced by increasing the number of threads on the accelerator, indicating that the kernel computation is

Fig. 60.4 Breakdown of the calculation cost (elapsed time) of a SPH simulation with PEZY-SC1

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Fig. 60.5 Scaling performances for a a number of threads with a single PEZY-SC device and for b multiple PEZY-SC device

off-loaded to the PEZY-SC1. With 8,192 threads, the SPH kernel computation no longer dominates the cost. Moreover, the other processes are almost independent of the number of threads because they are managed in the host CPU. For further performance improvements, off-loading pre/post processes to the PEZY-SC device would be important, especially in reducing the data transfer cost. The challenge of a fully native implementation on the accelerator is left for future work. To see the details of the computational performance improvement of SPH kernel calculation, we present scaling-up test results for PEZY-SC1 and SC2 (Fig. 60.5a). In both architectures, good strong scaling-up behaviors are attained. Because PEZYSC2 uses 1,936 active cores, and designed to handle 15,488 threads, a successful scaling-up performance is obtained with the larger number of threads than for PEZYSC1. We also checked speed-up obtained by handling multiple PEZY-SC2 devices (Fig. 60.5b). These results justify the use of PEZY-SC chips as accelerators to speed up the SPH simulation.

60.3.2 Validation Test The water dam-break test (Fig. 60.3) demonstrates that our application qualitatively captures the correct fluid motion. This test is one of the standard benchmark problems for numerical hydrodynamic methods involving the motion of a free surface [8]. Here, we compare quantitatively our SPH simulation on PEZY-SC with laboratory experiments. From plots of the height of the water surface and pressure at certain points (Fig. 60.6), the numerical simulation is found to reproduce the experimental data quantitatively. These results support the claims that our SPH simulations can be applied to various engineering problems.

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Fig. 60.6 Comparison of results from numerical simulations and laboratory experiments for the height of water surface at points a H2 and b H4, and c pressure at P7. The locations of the monitoring points are given in Fig. 60.3. For further details, see [8]

60.4 Concluding Remark We demonstrated that PEZY-SC, designed for green computing, is reliable as a general-purpose platform for multiple-particle simulations using SPH and DEM. The off-loading implementation does not require a large change in source code. Good scaling-up performances are shown for the off-loaded SPH kernel calculation for the number of threads and multiple chips. Validation tests also show that our SPH application reproduces quantitatively the experimental data. There is still room for further improvement in the speed-up of simulations, for example, increasing the workload for the accelerator to reduce the load of the host CPU. Other topics of interest include optimizing the SPH and DEM algorithms for PEZY-SC. For example, one of the characteristic features of PEZY-SC is the low byte/FLOPS (B/F), which requires high arithmetical intensity against data access for the high efficiency of the computations. We found that the sharing technique of the interaction-pair list is useful in speeding up the SPH simulation although this technique is not useful for high B/F machines such as the K-computer. Moreover, this technique may not work well for DEM because the number of interacting particles is small. The details of this analysis will be discussed in a forthcoming paper. We believe that these results justify the use of PEZY-SC devices in achieving successful green applications of large-scale particle simulations in various engineering and scientific fields.

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Acknowledgements The authors would like to thank Ryutaro Himeno and Toshikazu Ebisuzaki for using Shoubu system B at RIKEN. Funding The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: Computing with General Purpose Computers (Research and development in the next-generation area, Large Scale Computational Sciences with Heterogeneous Many-Core Computers) and Post-K Issue 3 supported by MEXT (Ministry of Education, Culture, Sports, Science and Technology-Japan), and a Grant-in-Aid for Scientific Research (JP18K03815). Declaration of Conflicting Interests The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

References 1. Nishiura, D., Furuichi, M., Sakaguchi, H.: Computational performance of a smoothed particle hydrodynamics simulation for shared-memory parallel computing. Comput. Phys. Commun. 194, 18–32 (2015) 2. Furuichi, M., Nishiura, D., Kuwano, O., Bauville, A., Hori, T., Sakaguchi, H.: Arcuate stress state in accretionary prisms from real-scale numerical sandbox experiments. Sci. Rep. (2018). https://doi.org/10.1038/s41598-018-26534-x 3. Nishiura, D., Sakaguchi, H., Yamamoto, S.: Multibillion particle DEM to simulate centrifuge model tests of geomaterials. In: Proceedings of the 9th International Conference on Physical Modelling in Geotechnics 2018, vol. 1, pp. 227–232 (2018) 4. https://www.top500.org/green500/ 5. Iwasawa, M., Tanikawa, A., Hosono, N., Nitadori, K., Muranushi, T., Makino, J.: Implementation and performance of FDPS: a framework for developing parallel particle simulation codes. Publ. Astron. Soc. Jpn. 68, 54 (2016) 6. Iwasawa, M., Tanikawa, A., Hosono, N., Nitadori, K., Muranushi, T., Makino, J.: FDPS: a novel framework for developing high-performance particle simulation codes for distributed-memory systems. In: Proceedings of the 5th International Workshop on Domain-Specific Languages and High-Level Frameworks for High Performance Computing, WOLFHPC ‘15, pp. 1:1–1:10. ACM, New York, NY, USA (2015) 7. Nishiura, D., Sakaguchi, H.: Parallel-vector algorithms for particle simulations on sharedmemory multiprocessors. J. Comput. Phys. 230(5), 1923 (2011) 8. Kleefsman, K.M.T., Fekken, G., Veldman, A.E.P., Iwanowski, B., Buchner, B.: A Volume-ofFluid based simulation method for wave impact problems. J. Comput. Phys. 206, 363–393 (2005)

Chapter 61

pzqd: PEZY-SC2 Acceleration of Double-Double Precision Arithmetic Library for High-Precision BLAS Toshiaki Hishinuma and Maho Nakata

Abstract We implemented pzqd, a high precision arithmetic library for the PEZYSC2 that is based on Hida et al.’s QD library. PEZY-SC2 is an MIMD (multiple instruction stream, multiple data stream) -type many-core processor. We optimized matrix-matrix multiplication (Rgemm) in double-double precision (DD) on the PEZY-SC2. Porting the CPU code to PEZY-SC2 code is relatively easy because PEZY-SC2 is a MIMD-type processor; it runs all the threads independently. As a proof of concept, we ported pzqd with minimal modifications to the original QD library; pzqd can treat a DD type variable in a unified way on the host CPU and the PEZY-SC2. The performance of our implementation of Rgemm in DD (DDRgemm) on the PEZY-SC2 attained 75% of the peak performance of DD, which is 20 times faster than an Intel Xeon E5-2618L v3, even including the communication time between the host CPU and the PEZY-SC2. The most important technique for optimizing the DD-Rgemm on the PEZY-SC2 is to make use of the high-speed scratch-pad memory (local memory) installed in each core. We stored the 2 × 2 DD block matrices and other temporary variables in local memory by reducing the number of threads to increase the local memory size per thread as they occupy local memory even for this block size. Keywords PEZY-SC2 · Double-double precision · MIMD · Many-core · Matrix-matrix multiplication

T. Hishinuma (B) PEZY Computing, 5F Chiyoda Ogawamachi Crosta, 1-11, Tokyo 101-0052, Japan e-mail: [email protected] M. Nakata RIKEN, Wako, Saitama 351-0198, Japan e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_61

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61.1 Introduction The binary64 [1] (so-called double-precision floating-point numbers) of IEEE Std 754-2008, is a kind of floating point number commonly used for numerical simulations in computational science. It has finite precision and the calculation time and error may increase due to rounding errors when using it to solve problems requiring higher precision than 16 decimal digits. Various high-precision calculation methods are used to reduce the influence of errors on floating point operations in problems in physics, chemistry, mathematics, etc. [2]. High-precision arithmetic has become relatively more straightforward to perform because of the dramatic increase in numbers of floating point operations per second of computers as a result of Moore’s law. Therefore, we expect an increase in demand for high-precision arithmetic to solve numerically difficult problems that are ill-conditioned or so huge that they require enormous numbers of floating-point operations. Demand will also grow for using high-precision arithmetic for numerical verification [3] and examining numerical reproducibility [4]. High-precision calculations using software are very costly, because they need a lot of computations and memory, and they are not supported in general programming languages. In any case, we should reduce the computational and implementation costs. Double-double precision (DD) arithmetic is frequently used because of its relatively high speed and low implementation cost. DD arithmetic does not need any special hardware and runs on general-purpose processors, which only use doubleprecision operations for DD operations. DD floating point numbers are cheap versions of the binary128 (quadruple precision floating point numbers) of IEEE Std 754-2008 and are defined as two non-overlapping double precision floating numbers [5]. Hida et al.’s QD library [5] significantly reduces the implementation cost of DD, as it implements DD floating point numbers as a class of C++. This essentially overcomes the first obstacle. To remove the second obstacle, we can use accelerators such as GPUs or SIMD (single instruction multiple data) processors [6–8]. One of the problems that currently limits the performance of computers is power consumption. In particular, it is important to find ways how to increase power efficiency and how to dissipate the heat generated by a computer efficiently. To solve these problems, PEZY Computing and ExaScaler have been developed from the PEZY-SC2 many-core processor [9] and ZettaScaler 2.2 series of PEZYSC2-based supercomputers that use a liquid immersion cooling system [10]. In the Green 500 supercomputer energy conservation ranking [11], One of our supercomputers “Shoubu system B” [9] has been certified for three consecutive terms from 2017 to 2018. Presently, we are working on new hardware and are conducting numerical simulations on it [12, 13]. In this paper, we describe the pzqd library, which is a port of the high-accuracy arithmetic library QD [5] on the PEZY-SC2 processor for high-precision BLAS. We implemented and evaluated the DD matrix-matrix product (DD-Rgemm) on the PEZY-SC2 using the pzqd library.

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We obtained about 74% of the peak performance of DD for the DD-Rgemm. The key to attaining high execution performance in this case was to launch four threads per processor element (the processor element is the minimal computation unit of the PEZY-SC2) so that we could effectively use the local memory space. The peak performance was 59 GFlops in DD, or equivalently, 1297 GFlops in double precision. The rest of the paper is organized as follows. Section 61.2 describes related work, and Sect. 61.3 shows the architecture and programming model of the PEZY-SC2 processor. An overview of DD arithmetic is given in Sect. 61.4. We describe the pzqd library, how we optimized DD-Rgemm using it, and how we measured its performance in Sect. 61.5. Section 61.6 summarizes the paper.

61.2 Related Work Some hardware implements binary128 [14, 15]; however, most hardware can use only double precision. Recently, the GNU Compiler Collection and Intel C/C++ and Fortran Compiler implemented “__float128” or “_Quad” binary128 in software. Double-double precision, sometimes referred to as double-word arithmetic, is widely used. It is used by the IBM XL FORTRAN compiler, IBM XL C compiler, and gcc for the Power series (RS6000), including the MacOSX until 10.6 implemented “long double” and “REAL*16” as double-double precision [16–19]. (Note: doubledouble precision and binary128 are not compatible.) Hida et al.’s QD library is available for other systems [5]. By using the QD library, users can use dd_real and qd_real (octuple precision) like float or double in C++ or Fortran 90. A most critical application of DD would be T. Aoyama et al.’s numerical evaluation of the electron anomalous magnetic moment from quantum electron dynamics [20]. They used DD and even quad-double precision for calculating certain Feynman diagrams. There is a lot of research on accelerating DD operations. Because DD operations can only be done using double operations, we can use traditional optimization techniques for DD, i.e., thread parallelization, distributed parallelization, SIMD, and so forth. Mukunoki et al. [7] reported an acceleration on an NVIDIA GPU and accelerated Krylov subspace methods on GPUs [21] for matrix-matrix multiplication in doubledouble precision. Mukunoki et al. [22] also reported an implementation and optimization of double-double and triple-double precision GEMM, GEMV, and AXPY on GPUs. Nakata et al. implemented a similar acceleration for double-double precision to replace the CPU implementation of MPACK (multiple precision arithmetic BLAS and LAPACK) [23] and applied it to semidefinite programming [6, 24]. In CAMPARY, M. Joldes et al. implemented double-double, triple-double and quad-double precision [25]. They also accelerated matrix-matrix multiplication and

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other routines on GPU to apply semidefinite programming solver for these precisions [26]. For solving sparse linear equations, Kotakemori et al.implemented and accelerated quadruple precision for the Lis library [27]. Hishinuma et al. accelerated sparse matrix-vector multiplication for CPUs accelerated by SIMD AVX/AVX2 [8, 28, 29] for the DD iterative solver library.

61.3 Architecture of the PEZY-SC2 Processor 61.3.1 Overview Figure 61.1 shows the block diagram of the PEZY-SC2 processor, and Table 61.1 shows the specifications of the PEZY-SC2 processor. The PEZY-SC2 processor is a MIMD (multiple instruction, multiple data) type many-core processor, and the calculation core of the PEZY-SC2 is called the Processing Element (PE). The PEZY-SC2 processor has in total 2048 PEs in a three-layer hierarchical structure called “Prefecture,” “City,” and “Village.” Each Village has four PEs, each City has four Villages, and each Prefecture has sixteen Cities. The PEZY-SC2 has eight Prefectures. The inside of each PE has eight register files and eight program counters for running eight threads independently.

Fig. 61.1 Block diagram of the PEZY-SC2 processor

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Table 61.1 Specifications of the PEZY-SC2 processor Process Clock freq. L1 cache L2 cache LLC Local memory PCIe I/F DDR I/F # of PEs (cores) SIMD vector length Peak performance (DP) Peak performance (SP) Peak performance (HP) Power consumption

16 nm 1 GHz 4 MB (D), 8 MB (I) 8 MB (D), 4 MB (I) 40 MB (X-bar connection) 40 MB (20KB/PE) PCIe Gen4 8 Lane 4 port (64 GB/s) DDR4 64 bit 3200 MHz 4 port (100 GB/s) 2048 MIMD cores 64 bit 4.1 TFlops 8.2 TFlops (x2 SIMD) 16.4 TFlops (x4 SIMD) 200 W (Peak)

Fig. 61.2 Thread control mechanism of the PEZY-SC2 processor. THxF means “front threads,” THxB means “back threads.” The front and back threads can be switched by using I.chgthread or I.actthread instructions. Threads are controlled by fine-grained multi-threading

Figure 61.2 shows the thread control mechanism, and Fig. 61.3 shows the latency hiding mechanism. The PEZY-SC2 can launch eight threads (=4 × 2) per PE. There are four active threads, called “front threads”; the remaining four inactive threads are called “back threads.”

Fig. 61.3 Pipeline architecture of the PEZY-SC2 processor

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These front and back threads can be switched sequentially in each cycle (i.e., by using fine-grained multi-threading [30]). When the original front threads stall (e.g., when loading data from memory), we can hide some of the latency by switching back threads to the front and front threads to the back. Each PE has arithmetic units (an adder and a multiplier). To process an instruction, we need four cycles to read from memory, perform operations, and write to registers or memory. Consequently, the processor needs at least four threads to fully occupy the arithmetic unit in the pipeline of PE. Next, let us focus on the instructions and computing unit of the PEZY-SC2. The PEZY-SC2 can use the MAD (Multiply-Add; d = a + b × c) instruction; MAD is addition and multiplication. Unlike the FMA (Fused-Multiply-Add) instruction, it rounds the result of the multiplication. The PE performs the MAD by running the adder and multiplier at the same time. If we run eight MAD instructions on eight threads on one PE, each MAD instruction is processed in one cycle on average. The PEZY-SC2 processor supports 64-bit SIMD instructions. It can compute one double-precision operation, two single-precision operations, or four half-precision operations simultaneously. One special function unit (SFU) is installed in each City to calculate division, modulo, square root, and inverse of the square root. Thus, the PEZY-SC2 processor has 128 SFUs (=16 (Cities) × 8 (Prefecture)) in total. Finally, let us focus on the cache memory. Each PE has a 2 KB L1 data cache, and each City has a 64 KB L2 data cache. Each Prefecture has a 2.5 MB LLC (Last Level Cache), and the LCCs of each Prefecture are connected by an X-bar. PEZY-SC2 has 20 KB worth of small and fast scratchpad memory called “local memory.” Figure 61.4 shows the address layout of the local memory. The PEZY-SC2 processor allocates local memory as a stack, and a user can use the remainder. The local memory can load or store data in one cycle.

Fig. 61.4 PEZY-SC2 address layout of the local memory. The size of the local memory is 20 KB. This area is shared by the stack and the userspace

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61.3.2 PEZY-SC2 Programming Model The PEZY-SC2 processor supports PZCL, an OpenCL-like programming interface. To run a program on the PEZY-SC2 processor, it needs two types of program: a kernel program and a host program. A host program runs on the host CPU. A host program is written in C/C++, and the PZCL API compiles it by using an ordinary C compiler (e.g., GNU C compiler). The PZCL API allocates the PEZY-SC2 memory, transfers data between the CPU and PEZY-SC2, launches the kernel program, and so forth. In addition, the SDK (Software Development Kit) for the kernel programs provides mathematical function libraries and atomic operations libraries. A kernel program runs on the PEZY-SC2 device. It is written in “PZCL C” and compiled with LLVM, which is almost the same as OpenCL C. PZCL C has built-in functions for the PEZY-SC2 architecture, such as thread control using the thread ID (tid) and process ID (pid), synchronization, flushing data, switching between front and back threads, and so forth.

61.4 Implementation of pzqd Library 61.4.1 Double-Double Precision Arithmetic A DD number is represented by two double-precision numbers as a = (ahi , alo ) (see Fig. 61.5). It consists of a sign part of 1 bit, an exponent part of 11 bits, and a significant part of 104 (52 × 2) bits. For comparison, the binary128 of IEEE Std 7542008 is composed of a 1-bit sign part, 15-bit exponent part and 112-bit significant part; the DD number has four fewer bits in its exponent part and eight fewer bits in its significant part.

Fig. 61.5 Schematic view of a double-double precision number in comparison with IEEE 754 quadruple precision. DD has a 1-bit sign part, 11-bit exponent part, and 104-bit significant part

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Next, we show how we realize addition and multiplication of two DD numbers. First, we use the fact that we can add, subtract, and multiply floating point numbers with numerical round offs. Still, these round off errors can correctly be evaluated [5, 31, 32]. Here, let us calculate exactly the addition s = a ⊕ b and its error e = a + b − (a ⊕ b) for a floating point number a, and b, where ⊕ denotes addition as a floating point operation. When |a| ≥ |b|, the addition s = a ⊕ b and its error e = a + b − (a ⊕ b) can be evaluated with the following Quick-Two-Sum (a, b) algorithm: Quick-Two-Sum (a, b): 1. s ← a ⊕ b 2. e ← b  (s  a) 3. return(s, e). When the relation of a and b is not known, we can use the Two-sum (a, b), although it requires more floating point number operations: Two-Sum (a, b): 1. 2. 3. 4.

s ←a⊕b v ←sa e ← (a  (s  v)) ⊕ (b  v) return(s, e),

where  denotes subtraction as a floating point operation. The Two-Sum algorithm requires six double-precision operations compared with the three of QuickSum. Next, let us evaluate the multiplication p = a ⊗ b and its error e = a × b − (a ⊗ b) exactly; multiplication as a floating point operation is denoted by ⊗. Using the sub-function Split (a), the double-precision number a is divided into two double-precision numbers ahi , alo and a = ahi + alo as follows: Split (a): 1. 2. 3. 4.

t ← (227 + 1) ⊗ a ahi ← t  (t  a) alo ← a  ahi return(ahi , alo )

Moreover, we use Two-Prod (a, b) to calculate p and e above, as follows: Two-prod (a, b): 1. 2. 3. 4. 5.

p ←a⊗b (ahi , alo ) ← Split(a) (bhi , blo ) ← Split(b) e ← ((ahi ⊗ bhi  p) ⊕ ahi ⊗ blo ⊕ alo ⊗ bhi ) ⊕ alo ⊗ blo return( p, e)

61 pzqd: PEZY-SC2 Acceleration of Double-Double Precision … Table 61.2 Flop count of double-double precision arithmetic Algorithm Add Mult. Two-Sum Split Two-Prod QuadAdd-IEEE QuadMul

6 3 10 20 15

0 1 7 0 9

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Sum 6 4 17 20 24

Finally, let us define addition (QuadAdd-IEEE) and multiplication (QuadMul) for two arbitrary DD numbers a and b: QuadAdd-IEEE (a, b): 1. 2. 3. 4. 5. 6. 7.

(shi , ehi ) = Two-Sum(ahi , bhi ) (slo , elo ) = Two-Sum(alo , blo ) ehi = ehi ⊕ slo (slo , elo ) = Quick-Two-Sum(shi , ehi ) ehi = ehi ⊕ slo (shi , elo ) = Quick-Two-Sum(shi , ehi ) return(c)

and QuadMul (a, b): 1. 2. 3. 4.

( phi , plo ) = Two-Prod(ahi , bhi ) plo = plo ⊕ (ahi ⊗ blo ⊕ alo ⊗ bhi ) (chi , clo ) = Quick-Two-Sum( phi , plo ) return(c)

Table 61.2 shows the number of flops (FLOating-Point operationS) of the doubleprecision arithmetic operations that constitute the DD arithmetic operation. Note that it is also possible to reduce the number of calculations by lowering the operation precision and using the FMA instruction [5, 6]. We ran a simple benchmark of calculating the dot product of a vector of length 105 . We compared the elapsed time of FORTRAN REAL*16 in the Intel FORTRAN compiler 13.0.1 (this is software-implemented binary128) and the DD on Intel XeonE5-2618L v3. By quadruple precision calculation (REAL*16) took 3.5 [ms], while the DD calculation took about 0.45 [ms]. The DD operations were approximately 7.7 times faster than the FORTRAN REAL*16 quadruple-precision operations. Finally, let us point out another feature of the DD arithmetic operation; the data request amount in bytes (Byte/Flop) for the one floating point operation is smaller

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than in the double-precision operation. For example, the MAD (d = a × b + c) operation for DD numbers requires 44 floating operations using double-precision numbers, whereas the memory requirement is 48 = 16 × 3 Bytes. Thus, Byte/Flop ratio is about 0.91. The double-precision operation requires 24 bytes to be read from memory and performs two floating operations; its Byte/Flop ratio is therefore 12.

61.4.2 Implementation Details of the pzqd Library We developed a definition of the DD type, DD mathematical functions, and DD arithmetic operations for the kernel program by using the syntax of PZCL C++ (under development). All of the operations in the QD library are single-threaded programs. As mentioned above, the kernel program supports C syntax, and the program is parallelized with tid (0–7) and pid (0–1983) as indices; each thread can be operated independently. Therefore, we could port most of the functions without modification. An exception was that the pzqd library does not support I/O functions. We implemented operator overloading of DD arithmetic operations by using PZCL C++. The kernel program can handle a DD-type variable that is transferred from the CPU in the same way as the host program using the QD library. We confirmed that the pzqd library and the QD library give bit-wise the same result for sample codes of the QD library. We deleted the I/O parts on the PEZY-SC2 and run only one thread. Figures 61.6 and 61.7 show examples of a host program using the QD library and kernel program using the pzqd library. The code of Fig. 61.6 runs the following flow: 1. 2. 3. 4.

It declares dd_real type array textbia, textbib, and textbic, transfers textbia and textbib to the PEZY-SC2 from the CPU, calls the “pzc_Add_dd” function written in the kernel program, and receives the answer array textbic from the PEZY-SC2. The code of Fig. 61.7 runs the following flow:

1. It gets its own thread ID (tid) and PE ID (pid), 2. computes its own global thread ID by using tid and pid, 3. computes the locations of the arrays textbia and textbib corresponding to the global thread ID, and 4. computes textbic[index] += textbia[index] × textbib[index]. As shown above, pzqd treats the dd_real type in the PEZY-SC2 kernel program in the same way as the CPU code using the QD library.

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#include ”qd . h” ... i n t main ( ) { d d r e a l ∗a , ∗b , ∗ c ; ... s i z e = s i z e o f ( d d r e a l ) ∗ N; cl mem mem a = c l C r e a t e B u f f e r ( . . . , N) ; cl mem mem b = c l C r e a t e B u f f e r ( . . . , N) ; cl mem mem c = c l C r e a t e B u f f e r ( . . . , N) ; ... e n q u e u e W r i t e B u f f e r ( mem a , t r u e , 0 , s i z e , a ) ; e n q u e u e W r i t e B u f f e r ( mem b , t r u e , 0 , s i z e , b ) ; ... Add dd . s e t A r g ( 0 , mem a ) ; Add dd . s e t A r g ( 1 , mem b ) ; Add dd . s e t A r g ( 2 , mem c ) ; Add dd . s e t A r g ( 3 , N) ; ... enqueueNDRangeKernel ( Add dd , ThreadsNum , . . . ) ; enqueueReadBuffer (mem C , t r u e , 0 , s i z e , C) ;

Fig. 61.6 Example of host program using QD library written in C++ with PZCL API #include ” p z q d r e a l . h” void pzc Add dd ( d d r e a l ∗ a , d d r e a l ∗ b , d d r e a l ∗ c , i n t N) { int t i d = g e t t i d ( ) ; int pid = g e t p i d ( ) ; i n t i n d e x = p i d ∗ get maxpid ( ) + t i d ; i n t maxid = get maxpid ∗ g e t m a x t i d ( ) ; for ( ; i n d e x < N ; i n d e x += maxid ) { a [ index ] = ” 3.14159265358979323846264338327950288 ” ; b [ index ] = ” 2.249775724709369995957 ” ; c [ i n d e x ] += a ∗ b ; } flush () ; }

Fig. 61.7 Example of kernel program using pzqd library written in PZCL C

61.5 Experimental Results We conducted two experiments. The first evaluated DD elementary functions in pzqd; the second evaluated an implementation of matrix-matrix multiplication in DD using pzqd.

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We used a PEZY-SC2 system that operated 1984 PEs at 700 MHz and used an Intel Xeon D-1571 as the host CPU (with DDR4 at 2400 MHz, 64GB of memory and a memory bandwidth of 76 GB/s). Initially, four Cities were disabled to improve the yield. The peak performance of the PEZY-SC2 in double precision was 2777 GFlops (≈ 0.7 [GHz] × 1984 [cores] × 2 [MAD operations]). The operating system was CentOS 7.2, the host program compiler was gcc 4.8.5, and the kernel program compiler was pzSDK 4.1 + LLVM 3.6.2. We verified that the LLVM compiler issued the MAD instructions for the kernel program at the assembly level. For comparison, we used an Intel Xeon E5-2618L [email protected] GHz with eight cores and 64 GB of memory. In this experiment, we did not explicitly use the FMA or the SIMD AVX2 instructions, i.e., the SIMD extension instruction set of the Xeon CPU. Without the SIMD AVX2 instructions, the theoretical peak performance in double precision is 73.6 GFlop for Intel Xeon E5-2618L [email protected] GHz eight cores. Note that when we used the SIMD AVX2 instructions, it reached 294.4 GFlops. The Turbo Boost feature was disabled. While we did not write SIMD or FMA instructions, we did not suppress FMA instructions generated by the compiler.

61.5.1 Performance of Elementary Operations in One Thread on the PEZY-SC2 We roughly estimated the ratio of the speeds of the CPU and the PEZY-SC2 in one thread. For the PEZY-SC2 processor, the peak performance dropped to 1/4 when only one thread was running, because the PEZY-SC2 operates one thread every four clocks. Usually, we fill up the pipeline of the PEZY-SC2 processor by executing every four threads sequentially in the PE to hide latency. Therefore, the performance ratio of the PEZY-SC2 processor between Intel Xeon E5-2618 [email protected] GHz at peak performance in one thread is 13.1 = (2.3 [GHz]/(0.7 [GHz]/4 )). Thus, we estimate that the PEZYSC2 is least 13.1 times slower than Xeon E5-2618 when we operate both with only one thread. The implementations of the elementary functions of pzqd are similar to those of the QD library; we used a Taylor expansion to obtain them. We verified that our implementation gave the same bit-wise results as the QD library by inputting random DD values. Consequently, we found that we can run the same math functions of the QD library on the host machine and on the PEZY-SC2. Table 61.3 shows the results of benchmarking the elementary functions by using one thread of the Xeon E5-2618Lv3 CPU and one thread of the PEZY-SC2 106 times. In these cases, we fixed the input to 0.5 in order to obtain more systematic results because some functions vary in performance depending on their input. The elapsed time of the PEZY-SC2 was about 15–25 times longer than that of the CPU. Addition and multiplication were 14.7 and 14.5 times slower, respectively; these values are very reasonable. However, the results for other functions (sin, cos,

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Table 61.3 Elapsed times of executing elementary functions 106 times in one thread [second] (ratio) on E5-2618L v3 and the PEZY-SC2 Xeon E5-2618L v3 PEZY-SC2 add mult div sin cos pow(x,2) sqr sqrt asin acos sinh cosh log exp

0.007 (1.00) 0.010 (1.00) 0.422 (1.00) 0.394 (1.00) 0.411 (1.00) 0.038 (1.00) 0.022 (1.00) 0.005 (1.00) 0.588 (1.00) 0.583 (1.00) 0.444 (1.00) 0.466 (1.00) 0.425 (1.00) 0.433 (1.00)

0.11 (14.7) 0.15 (14.5) 6.96 (16.4) 8.44 (21.3) 8.51 (20.7) 0.73 (18.7) 0.37 (16.6) 0.09 (15.3) 14.3 (24.2) 14.5 (24.8) 7.45 (16.8) 7.32 (15.7) 5.91 (13.9) 6.98 (16.1)

acos, and asin) were much slower. We suspect this was due to the differences in the special function unit and compiler between the CPU and PEZY-SC2. Nevertheless, the performance losses were not too serious.

61.5.2 DD-Rgemm in PEZY-SC2 Here, we explain the details of the matrix-matrix multiplication in DD (DD-Rgemm). The DD-Rgemm routine calculates C = α AB + βC, where A, B, and C are DD square dense matrices of size N × N , and α and β are scalar values in DD. This routine is a straightforward extension of DD to the BLAS Level 3 gemm. The core operation of the DD-Rgemm is the DD multiply-add operation. This operation consists of 35 double-precision additions and nine multiplications. We defined the peak performance of DD as 1745 GFlops = 2777/35 × 22, where 2777 is the peak performance of the PEZY-SC2. We defined the peak performance of DD because the numbers of additions and multiplications are not uniform for double-precision arithmetic; the DD multiply-add operation cannot be processed in (35 + 9)/2 = 22 cycles on the MAD unit. It takes 35 cycles even if we use the MAD instruction for all double operations; however,

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the numbers of additions and multiplications of DD multiply-add are not equal. Therefore, the peak performance of DD-Rgemm on the PEZY-SC2 should be 1745 GFlops. The peak performance of the Intel Xeon E5-2618L v3 is 73.6 GFlops, when we issue the FMA instructions. Therefore, the peak performance of the DD calculation is 73.6/35 × 22 = 46 GFlops, if we consider the unequal numbers of additions and multiplications in DD arithmetic. To speed up DD-Rgemm, we made 2 × 2 blockings [33]. The block size can be small, and we set the block size to 2 × 2 since the amount of data requested to be sent to memory per operation is small in DD arithmetic. In this way, we can store all the blocking matrix and temporary variables of DD arithmetic in the local memory. Also, the DD addition and multiplication operations are inline expanded to eliminate the overhead of function calls. As the matrix size increases, the data size increases and exceeds the capacity of the local memory space. We reduced the number of threads and in turn the usage of the stack area, so that all the data fit in the local memory.

61.5.3 Performance of DD-Rgemm in the PEZY-SC2 Figure 61.8 shows the results of DD-Rgemm of several implementations, with various matrix sizes and with/without communications between the CPU and the PEZY-SC2. The matrices used in the experiment were dense square matrices, and we filled them with random numbers. The horizontal axis is the matrix size N , and the vertical axis is performance, which is equal to the floating point operations per second divided by 44 × N 3 . We performed the calculation six times for each size N . We discarded the first result and averaged the remaining five. We verified the resultant matrices against the QD library; the pzqd library was equal bit-wise to the QD library. The results for the cases including communications included (i) the time to allocate memory to the matrices A, B, and C on the PEZY-SC2, (ii) the time to transfer the data of each of the matrixes from the CPU to the PEZY-SC2, (iii) the time to compute DD-Rgemm, and (iv) the time to transfer the resultant matrix C from the PEZY-SC2. We tested the following implementations: PEZY-SC2 (2 threads/PE) Launch total 3968 threads (= 1984 ×2), two threads per PE. 2 × 2 blocking, and temporary variables stored in local memory. PEZY-SC2 (4 threads/PE) Launch total 7936 threads (= 1984 ×4), four threads per PE. 2 × 2 blocking, and temporary variables stored in local memory. PEZY-SC2 (8 threads/PE, no localmem.) Launch total 15872 threads (= 1984 ×8), eight threads per PE. 2 × 2 blocking, all variables stored in global memory because the stack overflowed local memory. Intel Xeon E5-2618L v3 16 threads in total launched using OpenMP on the Intel Xeon. 2 × 2 blocking, the same program except for the PEZY-SC2-specific stuff.

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Fig. 61.8 Performance of DD-Rgemm on the PEZY-SC2 (right: includes the communication overhead between the CPU and the PEZY-SC2, left: pure kernel execution time; i.e., the communication overhead is not included). The horizontal axis is the matrix size N , and the vertical axis is performance, which is equal to the floating point operations per second divided by 44 × N 3

We could not perform 4 × 4 blocking because the stack size overflowed the local memory for any number of threads, and we could not obtain correct results. Using four threads per PE, the performance reached 1111 GFlops counting communications and 1297 GFlops not counting communications when all data were stored in local memory. These values represent 40 and 47% of peak performance in double precision (2777 GFLOPS) and 64 and 74% of peak performance in DD (1745 GFLOPS). Using two threads per PE, performance reached 656 GFlops counting communications and 674 GFlops not counting communications when all data were stored in local memory. The performance of four threads without counting communications was 1.97 times faster than that of two threads, i.e., almost two times faster. Since the PEZY-SC2 fills up the pipeline with four threads per PE (as shown in Fig. 61.3), when we use only two threads per PE, the performance is halved, and there are no operations for the remaining two clocks. The memory bandwidth is not a bottleneck because the performance of four threads is twice that of two threads. The amount of data required for the calculation is small enough for the DD arithmetic to fill up the instruction pipeline. Using eight threads per PE, the performance reached 167 GFlops counting communications and 168 GFlops not counting communications when all data were stored in global memory. These values are about 10% of peak performance in DD. It seems that overflow of the local memory caused a performance degradation. The blocking matrices and temporary data of DD arithmetic were too large.

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From the above results, it turns out that the most effective implementation is one with four threads per PE. Even including communications, its performance was the highest. We obtained 43 GFlops at maximum on the CPU, which was about 93% of peak performance in DD. The PEZY-SC2 runs were 4.3 (13.8) times faster than the CPU with (without) communication for a matrix size of 100, and 19.5 (22.5) times faster for a matrix size of 2000. When the matrix is small, the kernel startup overhead time and data communication overhead are large on the PEZY-SC2; in this case, it was only four times faster than the CPU. Next, let us focus on the overhead of the kernel startup time and evaluate the performance when the matrix is small. We measured the time for kernel launch. It was 30 [ms] on average and 50 [ms] at maximum. The elapsed time when for a matrix size of 100 was 220 [ms] without communications and 740 [ms] with communications. The communication time, kernel startup time, and calculation time were 520, 50, and 170 [ms], respectively. As the kernel startup time amounted to about 30% of the calculation time, including communications caused the PEZY-SC2 to be only about four times faster than the CPU. On the other hand, when the matrix size was 200, the total calculation took 650 [ms] without communications and 1350 [ms] with communications. The communication time, kernel activation time, and calculation time were 700, 50, and 600 [ms] in this case, so the speedup when including communications was 11 times relative to the times of the CPU. Even if the kernel startup time is assumed to be the maximum, it occupies 10% or less of the total elapsed time. Thus, the kernel startup time is not a big problem; we can expect a performance speedup even when the matrix is small. Looking at the case of even smaller matrixes, the performance for a matrix size of 60 was 50.9 GFlops without communications and 13.6 GFlops with communications, whereas the CPU ran at 14.2 GFLOPS. Consequently, the PEZY-SC2 performs DDRgemm faster than the CPU when the matrix size is larger than 60. From these results, the performance reached 168 GFlops when we did not use the local memory, and it reached a ceiling of 1297 GFlops, or 74% of peak performance in DD, when we used the local memory. This value is equivalent to 59 GFlops by DD. All of the results were faster than those of the CPU for a matrix size of 60 or more.

61.5.4 Efficiency of Thread Parallelization of DD-Rgemm on the PEZY-SC2 We analyzed the causes of the performance degradation of DD-Rgemm on the PEZYSC2. To analyze the parallelization efficiency, we increased the number of Cities and evaluated PEZY-SC 2’s three-layer hierarchical cache structure.

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Table 61.4 Efficiency of thread parallelization (matrix size N = M = K = 2, 000, no communication, four threads per PE). Performance means 44 × N 3 /time # of threads (PE) Time [s] Perf. (ratio) Peak of DD ratio (%) 64 (16, 1 City) 128 (32, 2 Cities) 256 (64, 4 Cities) 512 (128, 8 Cities) 1024 (256, 1 Prefecture) 2048 (512, 2 Prefectures) 4096 (1024, 4 Prefectures) 7936 (1984, 8 Prefectures)

25.4 13.0 6.8 3.5 1.9

13.4 (1.0) 26.7 (2.0) 52.7 (3.9) 102.2 (7.6) 194.3 (14.5)

95 95 94 91 86

1.0

356.1 (26.6)

79

0.5

712.9 (53.2)

79

0.3

1234.0 (92.1)

71

Table 61.4 shows the efficiency of thread parallelization for a matrix size of 2000 in four threads per PE and increasing the number of PEs to be activated. The total size of the DD matrices A, B, and C comes to 192 MB, too big to fit in the LLC (40 MB). We obtained the highest performance for 64 threads (128 PE, 1 City), i.e., 95% of peak performance. The efficiency of thread parallelization was more than 90% of the theoretical value, even when we increased the number of threads to 1024 (256 PE, 1 Prefecture), and performance increased linearly as we increased the number of threads. For a matrix size of 2000, the CPU performed at about 41.3 GFlops. The PEZYSC2 exceeded the performance of the CPU when we used 64 PEs and 256 or more threads. The peak performance ratio of the CPU and the PEZY-SC2 was about 24 times. The PEZY-SC2 needed to use 83 or more PEs (=1984/24) to beat the CPU. However, 95% of peak performance was high enough to exceed the performance of the CPU with 64 PEs or more. Although the efficiency of thread parallelization was less than 90% when the number of threads was 2048 or more (512 PE or more, 2 Prefecture or more), the efficiency of thread parallelization was still very high (75%) even with 7986 threads (i.e., when we used all of the PEs); this amounts to 71% of peak performance. The performance loss when we used a large number of threads (especially when we used more than 1024 threads) may be due to LLC misses. One Prefecture is up to 1024 threads, and it completely occupies the LLC. If we had used more than one Prefecture, thrashing of the LLC may have occurred. When we look at the performance counter, the cache hit rate of the LLC at 1024 threads is about 95%, whereas for 7936 threads it drops to about 88%. For 1024

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threads, one Prefecture exclusively occupies the LLC, and when the number of threads is 2048 or more, accesses across the LLCs start to occur; thus, LLC thrashing may occur. We expect that the LLC thrashing would resulted in a substantial loss in performance.

61.6 Summary We developed the pzqd library; a double-double precision arithmetic library based on Hida et al.’s QD library for the PEZY-SC2 processor. The features of PEZY-SC2 are: (i) 2048 processor elements (PEs) in total, (ii) a three-layer hierarchical cache structure consisting of Village, City, and Prefecture, and (iii) fast local memory (20 KB per PE). It can load or store in one cycle; it has (iv) an efficient threading mechanism with eight threads using fine-grained multi-threading, and (v) a MIMD-type processor, making it easy to port conventionally threaded CPU codes. We also implemented DD matrix-matrix multiplication (DD-Rgemm) using pzqd and evaluated its performance. To make use of these features, we reduced the number of threads from eight to four to increase the remaining area of local memory. This allowed us to store the intermediate variables for DD arithmetic and the small blocking matrix in local memory, so that it became possible to calculate matrix-matrix multiplications of any size. Our optimized Rgemm routine attained 74% of peak performance at maximum, not counting the communication time between the CPU and the PEZY-SC2. This level of performance is equivalent to 59G Flops in DD operations, or 1297 GFlops in double-precision operations. Moreover, it is faster by 23 times than the Intel Xeon CPU. Even when we included the communication time between the CPU and the PEZY-SC2, the PEZY-SC2 outperformed the CPU when the matrix size was 60 or more. Thus, we also demonstrated the usefulness of the PEZY-SC2 even in comparatively small problems. The execution efficiency of our implementation for the PEZY-SC2 was 91% of peak performance when running in 256 PEs, i.e., one Prefecture. This value is quite good; it is 14.5 times faster than that of running 16 PEs. Using all 1984 PEs was 92 times faster than using 16 PEs. Our future tasks will be (i) to improve the parallelization efficiency by optimizing the data management in the cache memory and (ii) to implement quad-double precision and BLAS functions other than matrix-matrix multiplication. Acknowledgements We used the Shoubu System B installed at RIKEN in collaboration with RIKEN, PEZY Computing, and ExaScaler Inc. A subsidy supporting this research for advanced use of high-performance general-purpose computers was provided by the Ministry of Education, Culture, Sports, Science, and Technology. A Grant-in-Aid for Scientific Research (B) (KAKENHI: 18H03206) also supported this research.

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References 1. IEEE: IEEE standard for floating-point arithmetic, IEEE Std 754-2008, 1–70 (2008). https:// doi.org/10.1109/IEEESTD.2008.4610935 2. Bailey, D.H.: High-precision floating-point arithmetic in scientific computation. Comput. Sci. Eng. 7, 54–61 (2005). https://doi.org/10.1109/MCSE.2005.52 3. Rump, S.M.: Verification methods: rigorous results using floating-point arithmetic. Acta Numer. 19, 287–449 (2010). https://doi.org/10.1017/S096249291000005X 4. Demmel, J., Nguyen, H.D.: Numerical reproducibility and accuracy at ExaScale. In: IEEE 21st Symposium on Computer Arithmetic, Austin, TX, 2013, pp. 235–237 (2013). https://doi.org/ 10.1109/ARITH.2013.43 5. Hida, Y., Li, X.S., Baily, D.H.: Library for Double-Double and Quad-Double Arithmetic. http:// crd-legacy.lbl.gov/~dhbailey/mpdist/ and reference therein 6. Nakata, M.: Numerical evaluation of highly accurate multiple-precision arithmetic version of semidefinite programming solver: SDPA-GMP, -QD and -DD. In: 2010 IEEE International Symposium on Computer-Aided Control System Design. IEEE Press, New York (2010). https:// doi.org/10.1109/CACSD.2010.5612693 7. Mukunoki, D., Takahashi, D.: Implementation and evaluation of quadruple precision BLAS functions on GPUs. In: PARA 2010: Applied Parallel and Scientific Computing. LNCS, vol. 7133, pp. 249–259. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-281518_25 8. Hishinuma, T., Tanaka, T., Hasegawa, H.: SIMD parallel sparse matrix-vector and transposedmatrix-vector multiplication in DD precision. In: VECPAR2016: 12th International Meeting on High Performance Computing for Computational Science. LNCS, vol. 10150, pp. 21–34. Springer, Heidelberg (2016). https://doi.org/10.1007/978-3-319-61982-8_4 9. PEZY Computing: PEZY-SC2 module & processor. https://www.pezy.co.jp/products/pezysc2module-processor/ (in Japanese) 10. Torii, S., Ishikawa, H., Kimura, Y., Saitoh, M.: Technologies and future prospects of green supercomputer ZettaScaler. IEICE Trans. C J100-C, 537–544 (2017) (in Japanese) 11. Green500. https://www.top500.org/green500/ 12. Tanaka, H., Ishihara, Y., Sakamoto, R., Nakamura, T., Kimura Y., Nitadori, K., Tsubouchi, M., Makino, J.: Automatic generation of high-order finite-difference code with temporal blocking for extreme-scale many-core systems. In: ESPM2 2018: Fourth International Workshop on Extreme Scale Programming Models and Middleware, Dallas, pp. 1–8 (2018) 13. Hishinuma, T., Kurosawa, N.: Development and Evaluation of OpenFOAM for PEZY-SC series toward to PEZY-SC3. In: OpenCAE Symposium 2018, Tokyo, no. A25, pp. 1–6 (in Japanese) 14. Lichtenau, C., Carlough, S., Mueller, S.M.: Quad precision floating point on the IBM z13. In: 2016 IEEE 23nd Symposium on Computer Arithmetic (ARITH). IEEE Press, New York (2016). https://doi.org/10.1109/ARITH.2016.26 15. Patterson, D., Waterman, A.: The RISC-V Reader: An Open Architecture Atlas, 1 edn. Strawberry Canyon, pp. 1–200 (2017). ISBN: 0999249118 16. 128-bit long double floating-point data type. https://www.ibm.com/support/knowledgecenter/ en/ssw_aix_71/com.ibm.aix.genprogc/128bit_long_double_floating-point_datatype.htm 17. IBM XL Fortran for AIX, V16.1.0, Language Reference. https://www.ibm.com/support/ knowledgecenter/SSGH4D_16.1.0/com.ibm.compilers.aix.doc/langref.pdf?view=kc 18. IBM XL Fortran for Linux, V16.1.1, Language Reference. https://www.ibm.com/support/ knowledgecenter/SSAT4T_16.1.1/com.ibm.compilers.linux.doc/langref.pdf?view=kc 19. Libm source. https://opensource.apple.com/source/Libm/Libm-315/Source/PowerPC/ 20. Aoyama, T., Hayakawa, M., Kinoshita, T., Nio, M.: Tenth-order electron anomalous magnetic moment: contribution of diagrams without closed lepton loops. Phys. Rev. D 91, 033006 (2015). https://doi.org/10.1103/PhysRevD.91.033006 21. Mukunoki, D., Takahashi, D.: Using quadruple precision arithmetic to accelerate Krylov subspace methods on GPUs. In: PPAM2013: Proceedings of the 10th International Conference

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Chapter 62

Efficient GPU Integration for Multi-loop Feynman Diagrams with Massless Internal Lines Elise de Doncker, Fukuko Yuasa and Ahmed Almulihi

Abstract We compute Feynman loop integrals or expansion coefficients for sets of self-energy diagrams with massless internal lines and which give rise to either finite integral values or UV-divergences. In case of UV-divergence, dimensional regularization can be implemented using a linear extrapolation as the dimensional regularization parameter tends to zero. The numerical integration is performed with lattice and composite lattice rules combined with a transformation to alleviate boundary singularities, and implemented in CUDA C. The GPU results are accurate and efficient in execution time compared to other numerical methods and architectures. Keywords Feynman loop diagrams · Massless internal lines · Numerical integration · Extrapolation · GPU/CUDA

62.1 Background The precise prediction of higher-order radiative corrections based on the perturbation method in quantum field theory is indispensable for future physics experiments in the International Linear Collider and the Future Circular Collider. For the calculation of higher-order corrections in a diagrammatic approach, we have to evaluate a huge number of multi-loop Feynman diagrams. For one-loop Feynman integrals, several computer programs based on an analytic solution by t’Hooft and Veltman are available. For the multi-loop case, various approaches have been proposed and their implementations have been actively developed. We have proposed and are further

E. de Doncker (B) · A. Almulihi Western Michigan University, Kalamazoo, MI 49008, USA e-mail: [email protected] URL: http://www.cs.wmich.edu/elise F. Yuasa High Energy Accelerator Research Organization (KEK), 1-1 OHO, Tsukuba, Ibaraki 305-0801, Japan © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_62

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developing fully numerical approaches based on numerical multivariate integration and extrapolation methods for multi-loop Feynman integrals. In previous work we applied adaptive integration with the multivariate ParInt package, layered over MPI (the Message Passing Interface parallel environment), for 3-loop Feynman diagrams with massless propagators [8–10, 12]. Results were also obtained with repeated one-dimensional adaptive integration using programs from Quadpack [24] (see, e.g., [11, 15, 16]). These methods use region partitioning guided by the importance of subregions of the integration domain as indicated by the estimated local error. While very effective in lower dimensions, they are generally too time-consuming and hence impractical in higher dimensions. Using non-adaptive integration we reported results in [10] based on a DE method, which involves a trapezoidal rule type approximation and a double-exponential transformation [31, 32]. We further applied non-adaptive Quasi-Monte Carlo (QMC) integration with lattice rules, combined with transformations to alleviate boundary singularities. Apart from being suitable for higher dimensions, the regular structure of QMC methods allows for their implementation on many-core architectures. We gave efficient parallel results using CUDA C on GPUs [6, 7] for some classes of loop integrals with massive internal lines. Other recent GPU implementations have been performed after sector decomposition, as the resulting Feynman integrals become relatively easy and can be handled with lattice rules [3, 20]. In this paper we compute Feynman loop integrals for sets of self-energy diagrams with massless internal lines and which give rise to either finite integral values or UV-divergences. In Sect. 62.2, the diagrams and general loop integral representation are given. Section 62.3 covers the basic numerical methods including lattice and composite lattice rules, and the extrapolation approach for dimensional regularization. Subsequently, numerical results and conclusions are presented in Sects. 62.4 and 62.5.

62.2 Loop Diagrams and Integrals A Feynman integral for a diagram with L loops and N internal lines can be represented by I = (4π)−ν L/2 I L ,N with I L ,N = (−1) N Γ (N −

νL ) 2



N  CN

j=1

d x j δ(1 −



xj)

C N −ν(L+1)/2 (62.1) (D − i C) N −ν L/2

written as an integral over the unit cube C N = {(x1 , . . . , x N ) | 0 ≤ xj ≤ 1, 1 ≤ j ≤ N}, where C and D are homogeneous polynomials in x1 , . . . , x N ; the term in  is present to prevent the denominator from vanishing; and the space-time dimension is

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Fig. 62.1 4-loop diagrams with finite integral value: a M44 diagram, N = 9, Baikov and Chetyrkin [1]; b M45 diagram, N = 9, Baikov and Chetyrkin [1]; c BH4b diagram, N = 11, Binoth and Heinrich [2], Fig. 4b

Fig. 62.2 3- and 4-loop diagrams with UV-divergence: a P3, Baikov and Chetyrkin [1] (3-loop sunrise-sunset diagram); b M01, Baikov and Chetyrkin [1] (4-loop sunrise-sunset diagram); c M36, Baikov and Chetyrkin [1] (Shimadzu diagram)

ν = 4 − 2ε, expressed in terms of the dimensional regularization parameter ε. After  eliminating the δ-function in view of Nj=1 x j = 1, Eq. (62.1) gives rise to the integral

I L ,N

νL ) = (−1) Γ (N − 2



N

S N −1

C N −ν(L+1)/2 dx. (D − i C) N −ν L/2

(62.2)



over the d-dimensional unit simplex (d = N − 1), Sd = {x ∈ Cd | 0 ≤ dj=1 xj ≤ 1}. We will target the diagrams in Figs. 62.1 and 62.2 where the polynomials C and D, which depend on the topology of the Feynman diagram and physical parameters, are evaluated for zero masses m 1 = · · · = m N = 0. The integrals for the diagrams in Fig. 62.1 satisfy expansions in (integer powers of) ε of the form (62.3) I(ε) = a0 + a1 ε + a2 ε2 + · · · . corresponding to a finite integral value. The diagrams in Fig. 62.2 exhibit a UVdivergence yielding an expansion I(ε) = a−1

1 + a 0 + a 1 ε + a 2 ε2 + · · · . ε

(62.4)

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62.3 Numerical Methods 62.3.1 Rank-1 Lattice Rules and Transformations The integral approximations in this paper are computed with rank-1 lattice rules, of the form n−1 1 j Q(z, n) f = f ({ z}) (62.5) n j=0 n where z is an integer generator vector with components z ∈ Zn = {1 ≤ z < n, gcd(z, n) = 1},

(62.6)

and {x} denotes the vector in the half-open unit cube {x ∈ Cd | 0 ≤ xj < 1, 1 ≤ j ≤ d} obtained by taking the fractional part of each component of x. Classically n is prime (i.e., Zn = {1, 2, . . . , n − 1}); this has been relaxed, cf. [22, 23]. Rank-1 lattice rules corresponding with a good lattice point (or z with optimal coefficients) as defined by Korobov [17, 18] enjoy favorable properties with respect to bounds of the error (Q(z, n) f − I f ) given for certain classes of functions that are periodic with period 1 in all components (see, e.g., [13, 22, 29]). Various periodizing transformations can be incorporated, such as Korobov’s transformation [5, 19]. We further used the tanh transformation [25] and Sidi’s sinm transformations [27, 28] to smoothen the function behavior at the boundaries of the domain [6, 7]. For the results in this paper we applied Sidi’s Ψ6 transformation, Ψ6 (t) = t − (45 sin(2πt) − 9 sin(4πt) + sin(6πt))/(60π), 16 6 Ψ6 (t) = sin (πt). 5

(62.7)

We implemented the lattice rule summations and integrand functions in CUDA C to allow accelerating the computations on GPUs [6, 7].

62.3.2 Composite Lattice Rules To increase the number of sample points for some integrals, we have made use of the composite rules defined and illustrated in [29]. Consider the d-dimensional unit cube Cd divided into m equal parts in each of r coordinate directions. Then the rule sum Q r applies a scaled version of a basic rank-1 rule Q 0 (with generator vector z) within each of the m r subregions,

62 Efficient GPU Integration for Multi-loop Feynman Diagrams with Massless …

Qr f =

  n−1 m−1 m−1  1  j 1 . . . f , . . . , k , 0, . . . , 0) , z + (k 1 r m r n k =0 k =0 j=0 n n r

741

(62.8)

1

for 0 ≤ r ≤ d, and n and m relatively prime. Q r has m r n points and is of rank r for 1 ≤ r ≤ d; Q d is the m d -copy rule of Q 0 . Furthermore the successive rules form an embedded sequence, i.e., the points of Q r +1 include the points of Q r for 0 ≤ r < d. An error estimate is calculated for Q d using a sequence of rules of order m d−1 n embedded in Q d . We utilize composite rule approximations for the M36 diagram in Sect. 62.4.2 below.

62.3.3 Linear Extrapolation Asymptotic expansions are derived in [1, 30] for loop integrals of the form I L ,N in Eq. (62.1), with respect to the dimensional regularization parameter ε. These expansions will be at the basis of the numerical extrapolation technique to solve for the coefficients. In general the underlying expansion is of the form I(ε) ∼ C0 ϕ0 (ε) + C1 ϕ1 (ε) + C2 ϕ2 (ε) + · · · , as ε → 0.

(62.9)

For linear extrapolation it is assumed that the ϕ(ε) functions are known (for example, they could be integer powers of ε). Then the extrapolation can be performed by solving a linear system (see, e.g., [4]). We create an approximating sequence of I (ε ) ≈ I (ε) such that I(ε ) ≈ C0 ϕ0 (ε ) + C1 ϕ1 (ε ) + · · · + C K ϕ K (ε ),  = 0, . . . , K ,

(62.10)

and solve the resulting linear systems of orders (K + 1) × (K + 1), for increasing values of K and decreasing ε = ε . We use linear extrapolation to approximate the leading expansion coefficients for the UV-divergent diagrams in Sect. 62.4.2. When the ϕ(ε) functions are unknown, it may be possible to use nonlinear extrapolation techniques as we have done commonly in other work, with the -algorithm [26, 33].

62.4 Results The GPU computations presented in this section are performed with a Kepler-20m NVIDIA GPU in a node of the thor cluster at WMU, where the host process runs sequentially on dual Intel Xeon E5-2670, 2.6 GHz processors with 128 GB of mem-

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ory. The Kepler-20m GPU has 2496 CUDA cores and 4.8 GB of global memory. Some comparisons are given below with the ParInt package executed with 64 MPI processes on four thor cluster nodes. We also draw comparisons with the FIESTA package in [30] even though FIESTA, based on sector decomposition with Mathematica [21] and integration with Cuba Vegas [14], is not fully numerical and thus belongs to a different category of methods. It is noted that the expansion coefficients in [30] are different from those of [1] (except for the first coefficient), in view of the different normalization coefficients in the integral representation—the relationship is given by the factor given as an expansion in powers of ε in Eq. (5) of [30]. We can, however, compare relative errors of the coefficients. The execution times are also difficult to compare because of the different architectures. It is stated that the times tabulated in [30] are for FIESTA (multi-threaded) in full parallel mode (both Mathematica and C code parts fully parallelized) on 8-core Intel Xeon E5472 3.0 GHz, 4 GB/core RAM, 4.6 TB disk/node computers.

62.4.1 Finite Diagrams Tables 62.1 and 62.2 show integration results for the diagrams M44 and M45, respectively. The lattice rules (LR) use between n = 200M and 500M points (M = million). The actual number of samples evaluated by the lattice generator is the first prime greater than n. The execution times (Time [s]) are below 2 s in Table 62.1 for M44, notwithstanding the moderate integration dimension (d = 8). The results are compared with those of ParInt from [10], where the number of function evaluations ranges between 100B and 300B (B = billion), with running times of about 185– 554 s. Table 62.2 presents a similar comparison for the diagram M45. The Exact values are from [1, 30]. The 4-loop BH4b diagram gives rise to a 10-dimensional integral. The GPU results listed in Table 62.3 are obtained in less than 2 s and show good accuracy compared with the Exact value from [2].

Table 62.1 Figure 62.1a M44, 4-loop finite, Results LR (left) and comparison with ParInt (right) LR(n) Result Time [s] ParInt Result Time [s] 200M 350M 500M Exact:

55.5947 55.5782 55.5809 55.5853

0.75 1.31 1.87

100B 200B 300B Exact:

55.5947 55.5868 55.5852 55.5853

185.1 370.0 554.3

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Table 62.2 Figure 62.1b M45, 4-loop finite, Results LR (left) and comparison with ParInt (right) LR Result Time [s] ParInt Result Time [s] 200M 400M Exact:

52.0105 52.0145 52.0179

1.59 3.18

100B 300B Exact:

52.0264 52.0191 52.0179

Table 62.3 Figure 62.1c BH4b, 4-loop finite, Results LR LR Result 200M 350M 400M Exact:

239.5 479.9

Time [s]

35.63 35.53 35.16 35.10

1.04 1.82 1.91

62.4.2 UV-Divergent Diagrams The underlying expansions for the coefficients of Tables 62.4, 62.5 and 62.6 are of the form  Cjεj, (62.11) η(ε) L I(ε) ∼ j≥−1

where η(ε) L is a normalization factor and η(ε) =

Γ (2 − 2ε) . Γ (1 + ε)Γ 2 (1 − ε)

(62.12)

Expressions for the C j coefficients are listed in [1]. Tables 62.4 and 62.5 give linear extrapolation results for the sunrise-sunset diagrams P3 and M01, showing high accuracy obtained in execution times of a fraction Table 62.4 Figure 62.2a P3, 3-loop UV-divergent, LR 10M, Extrapolation ε = 2−  Time [s] C−1 C0 C1 8 9 10 11 12 13

0.032 0.032 0.032 0.032 0.032 0.032 Exact:

0.027771824173 0.027777797878 0.027777777737 0.027777777771 0.027777777771 0.027777777777¯

0.1665888789 0.1620010731 0.1620371668 0.1620370367 0.1620370369 0.1620370370

0.782985520 0.764505542 0.764661001 0.764660486 0.764660494

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Table 62.5 Figure 62.2b M01, 4-loop UV-divergent, LR 10M, Extrapolation ε = 2−  Time [s] C−1 C0 C1 8 9 10 11 12 13

0.038 0.038 0.038 0.038 0.038 0.038 Exact:

−0.0017351792554 −0.0017361158823 −0.0017361111004 −0.0017361111116 −0.0017361111116 −0.0017361111111¯

−0.01763788653 −0.01691855709 −0.01692712629 −0.01692708325 −0.01692708335 −0.01692708333

−0.02276555702 −0.01183781275 −0.01184295409 −0.01184293013 −0.01184293017

Table 62.6 Figure 62.2c M36, 4-loop UV divergent, LR 27 ×350M, Extrapolation ε = 2−  Time [s] C−1 C0 C1 C2 10 11 12 13 14 15

247.5 247.5 247.5 247.5 247.5 247.5 Exact:

5.18460507 5.18463852 5.18463853 5.18463853 5.18463853 5.18463878

−2.47956842 −2.58230783 −2.58243549 −2.58243569 −2.58243609

70.1367688 70.3982136 70.3991841 70.3991515

−15.298 −15.469

of a second per iteration. The input sequence for the extrapolation procedure approximates the left-hand side of Eq. (62.11) for a sequence of ε = ε = 2− with  ≥ 8, using a lattice rule (LR) with 10M points. In comparison, the times incurred by ParInt as reported in [10] range from 0.37 to 1.01 s for P3, and from about 30 to about 40 s for M01, while the ParInt accuracy is better for the C−1 coefficient but the GPU/LR accuracy is better for the C1 coefficient. For the M36 diagram, Table 62.6 displays extrapolation results of a composite lattice rule based on the 7-dimensional rule with 350M points and a composition according to Eq. (62.8) with m = 2, r = d = 7. The extrapolation is performed on the m 7 -copy results for ε = 2− and  ≥ 10. Each iteration thus takes 27 × 350M function samples and is timed at just over 4 min. Note that, for example, the result for C−1 has an absolute accuracy of 2.5 × 10−7 at the third iteration (second linear system solved). Results for this diagram were given in [10] using a double exponential (DE) method [31, 32] and multi-threading on a former computing system at KEK. For comparable accuracy the time per iteration was reported to be below 20 min per iteration. The M36 diagram is also treated in [30], and we can compare the relative errors (|(Exact − Result)/Exact|) of the more accurate results for (C−1 , C0 , C1 ) given here and the FIESTA results in [30]. For the row corresponding to ε = 2−15 in Table 62.6 these are (4.8 × 10−8 , 1.5 × 10−7 , 4.6 × 10−7 ), and for FIESTA/Cuba

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Vegas 1.5M these are (1.2 × 10−6 , 3.2 × 10−6 , 2.3 × 10−6 ). The corresponding times for FIESTA in full parallel mode are (375.47 s, 704.23 s, 959.94 s) for (C−1 , C0 , C1 ) (and 1442.97s for C2 ).

62.5 Conclusions We proposed an approach for handling some important classes of loop integral calculations for diagrams with massless internal lines, using lattice rule approximations accelerated on a GPU. Results are included for finite and UV-divergent diagrams having between 4 and 11 propagators, and showing promise for the higher-dimensional applications. The method is fully numerical and thus does not require symbolic manipulations tailored to specific problems. The ability to gain good accuracy in short execution times is important in view of the huge number of diagrams required for an interaction. While the implementation was done for GPUs in this paper, it can be ported to other many-core architectures. The kernels are currently written for one GPU, but can be extended to accomodate multiple GPUs on a node.

Acknowledgements The equipment used for the computations in this paper was funded by the National Science Foundation under Award Number 1126438. We further acknowledge the Grantin-Aid support for Scientific Research (17K05428) of JSPS. This work was also supported in part by the Large Scale Computational Sciences with Heterogeneous Many-Core Computers, Grant-inAid for High Performance Computing with General Purpose Computers from MEXT (Ministry of Education, Culture, Sports, Science and Technology-Japan).

References 1. Baikov, B.A., Chetyrkin, K.G.: Four loop massless propagators: an algebraic evaluation of all master integrals. Nucl. Phys. B 837, 186–220 (2010) 2. Binoth, T., Heinrich, G.: Numerical evaluation of multi-loop integrals by sector decomposition. Nucl. Phys. B 680, 375 (2004). hep-ph/0305234v1 3. Borowka, S., Heinrich, G., Jahn, S., Jones, S.P., Kerner, M., Schlenk, J.: A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec (2018). arXiv:1811.11720v1 [hep-ph], arXiv:1811.11720 4. Brezinski, C.: A general extrapolation algorithm. Numer. Math. 35, 175–187 (1980) 5. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York (1984) 6. de Doncker, E., Almulihi, A., Yuasa, F.: High-speed evaluation of loop integrals using lattice rules. J. Phys. Conf. Ser. (JPCS) IOP Ser. 1085(052005) (2018). http://iopscience.iop.org/ article/10.1088/1742-6596/1085/5/052005 7. de Doncker, E., Almulihi, A., Yuasa, F.: Transformed lattice rules for Feynman loop integrals. J. Phys. Conf. Series (JPCS) IOP Ser. 1136(012002) (2018). https://doi.org/10.1088/1742-6596/ 1136/1/012002

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8. de Doncker, E., Yuasa, F.: Feynman loop numerical integral expansions for 3-loop vertex diagrams. Procedia Comput. Sci. 108, 1773–1782 (2017). https://doi.org/10.1016/j.procs.2017. 05.253 9. de Doncker, E., Yuasa, F., Ishikawa, T., Kapenga, J., Olagbemi, O.: Adaptive integration and singular boundary transformations. Procedia Comput. Sci. 80, 1428–1438 (2016). https://doi. org/10.1016/j.procs.2016.05.462 10. de Doncker, E., Yuasa, F., Kato, K., Ishikawa, T., Kapenga, J., Olagbemi, O.: Regularization with numerical extrapolation for finite and UV-divergent multi-loop integrals. Comput. Phys. Commun. 224, 164–185 (2018). https://doi.org/10.1016/j.cpc.2017.11.001 11. de Doncker, E., Yuasa, F., Kato, K., Ishikawa, T., Olagbemi, O.: Automatic numerical integration methods for Feynman integrals through 3-loop. J. Phys. Conf. Ser. (JPCS) IOP Ser. 608 (2015). https://doi.org/10.1088/1742-6596/608/1/012071 12. de Doncker, E., Yuasa, F., Olagbemi, F.: Adaptive integration for 3-loop Feynman diagrams with massless propagators. Procedia Comput. Sci. 51, 1333–1342 (2015). https://doi.org/10. 1016/j.procs.2015.05.318 13. Disney, S.A.R., Sloan, I.H.: Error bounds for the method of good lattice points. Math. Comput. 56, 257–266 (1991) 14. Hahn, T.: Cuba—a library for multidimensional numerical integration. Comput. Phys. Commun. 176, 712–713 (2007). https://doi.org/10.1016/j.cpc.2007.03.006 15. Kato, K., de Doncker, E., Ishikawa, T., Kapenga, J., Olagbemi, O., Yuasa, F.: High performance and increased precision techniques for Feynman loop integrals. J. Phys. Conf. Ser. (JPCS) IOP Ser. 762(012070) (2016). http://www.iopscience.iop.org/article/10.1088/1742-6596/762/ 1/012070 16. Kato, K., de Doncker, E., Ishikawa, T., Yuasa, F.: Direct numerical computation and its application to the higher-order radiative corrections. J. Phys. Conf. Ser. (JPCS) IOP Ser. 1085(052002) (2018). http://iopscience.iop.org/article/10.1088/1742-6596/1085/5/052002/pdf 17. Korobov, N.M.: The approximate computation of multiple integrals. Dokl. Akad. Nauk SSSR 124, 1207–1210 (1959). (Russian) 18. Korobov, N.M.: Properties and calculation of optimal coefficients. Dokl. Akad. Nauk SSSR 132, 1009–1012 (1960). (Russ.). Eng. Trans. Soviet Math. Dokl. 1, 696–700 19. Korobov, N.M.: Number-Theoretic Methods in Approximate Analysis. Fizmatgiz, Moscow (Russian) (1963) 20. Li, Z., Wang, J., Yan, Q.-S., Zhao, X.: Efficient numerical evaluation of Feynman integral. Chin. Phys. C 40(033103) (2015). https://doi.org/10.1088/1674-1137/40/3/033103, Preprint at arXiv:1508.02512v1 [hep-ph] 21. Mathematica. http://www.wolfram.com/mathematica 22. Niederreiter, H.: Existence of good lattice points in the sense of Hlawka. Monatshefte für Math. 86, 203–219 (1978) 23. Nuyens, D., Cools, R.: Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points. J. Complex. 22, 4–28 (2006) 24. Piessens, R., de Doncker, E., Überhuber, C.W., Kahaner, D.K.: QUADPACK, A Subroutine Package for Automatic Integration. Springer Series in Computational Mathematics, vol. 1. Springer (1983) 25. Sag, T.W., Szekeres, G.: Numerical evaluation of high-dimensional integrals. Math. Comput. 18(86), 245–253 (1964) 26. Shanks, D.: Non-linear transformations of divergent and slowly convergent sequences. J. Math. Phys. 34, 1–42 (1955) 27. Sidi, A.: A new variable transformation for numerical integration. Int. Ser. Numer. Math. 112, 359–373 (1993) 28. Sidi, A.: Extension of a class of periodizing transformations for numerical integration. Math. Comput. 75(253), 327–343 (2005) 29. Sloan, I.H., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press, Oxford (1994)

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Part XVII

Structural Uncertainty Quantification and Reliability Analysis

Chapter 63

Structural Damage Detection with Uncertainties Using a Modified Tree Seeds Algorithm Zhenghao Ding, Jun Li and Hong Hao

Abstract This paper proposes a novel structural damage identification approach by modifying the original Tree Seeds Algorithm (TSA), termed as M-TSA. To enhance the performance of the standard TSA, a one-step K-means clustering mechanism is conducted before starting the seeds search, which turns out to be beneficial to improve the standard algorithm’s global search capacity. The objective function is formulated by using the frequency do-main data. The first several natural frequencies and the incomplete mode shape are employed to build up the objective function and the proposed M-TSA is adopted to obtain the optimal solution and reflect the structure’s health status. Numerical studies on a 26-bar truss are conducted to investigate the accuracy of using the proposed approach for structural damage identification. Two types of uncertainties including the finite element modelling errors and measurements noise, are considered simultaneously. The final identification results show that with the assistance of the K-means clustering, the proposed M-TSA is able to identify the damages accurately and the performance is better than the standard TSA. Keywords Structural damage identification · Modelling errors · Frequency domain data · Noises · Tree seeds algorithm · K-means

63.1 Introduction With various unforeseen reasons, structures usually accumulate damages inevitably during the service, therefore it is important to detect these damages to assure safety of structures. Many different types of methods have been developed to solve structural damage identification problem, which can be found in the reviews [1, 2]. Among these approaches, vibration-based methods, due to their convenient and efficient operation, have gained much attention.

Z. Ding (B) · J. Li · H. Hao School of Civil and Mechanical Engineering, Centre for Infrastructural Monitoring and Protection, Curtin University, Kent Street, Bentley, WA 6102, Australia e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_63

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Structural damage identification can be boiled down to an inverse problem by defining an objective function relevant to structural parameters, swarm intelligence approaches [3] have been applied to tackle the identification problem. The advantage of the swarm intelligence approaches is that they do not need a good initial value and gradient information, which are quite essential for the traditional techniques [4, 5]. However, the initial values and the gradient information are sometimes difficult to obtain when applying the techniques to complex civil structures. Therefore, swarm intelligence methods that are used to solve structural damage identification problem gain significant attentions. Sun et al. [6] formulated the model updating of structures as an inverse problem, and dealt it by using a modified Artificial Bee Colony (ABC) algorithm, in which a nonlinear factor for convergence control is employed for the basic ABC to enhance its global search ability. In addition, the Birds Mating Optimizer (BMO) [7], the Particle Swam Optimizer (PSO) [8], and the Imperialist Competitive Algorithm (ICA) [9] etc. are also employed for structural damage identification. Although good identification results are achieved, challenges still exist, i.e. the above-mentioned methods [4–9] do not consider the modelling errors and other uncertainties effect. Lately, a novel swarm intelligence method, named as Tree Seeds Algorithm (TSA), has been proposed [10]. The optimization abilities among the Genetic Algorithm (GA), differential evolution (DE), PSO and the developed TSA are compared in optimizing the classical benchmarks, and the results show that TSA equips with a better global optimization ability. In this study, TSA is extended to structural damage identification with considering the modeling errors and a higher measurement noise. In addition, to enhance the performance of the original TSA, a one-step K-means clustering mechanism is applied before individuals’ updating to form a modified version, termed M-TSA. The mechanism is quite simple but beneficial to enhance the colony’s diversity and thus amplify the algorithm’s global search ability. To remedy the above mentioned shortcomings, both the modelling uncertainties and the measurement noise are considered, which would make the identification process more difficult, since more uncertainties are introduced. However, the modeling uncertainties are inevitably avoided in the real situation. A Gaussian distribution model is introduced to describe the modeling uncertainties [11]. Numerical example of a 26-bar truss is employed to illustrate the identification accuracy. The final results show that M-TSA has a better performance than the standard TSA in structural damage identification with uncertainties.

63.2 Problem Formulation In this study, the damage model is assumed to be only related to the stiffness reduction. Therefore, the damage of a structure can be expressed by a series of scalar variables for each element αi (i = 1, 2, . . . , nel) with the value in a range from 0 to 1, as described in the Eq. (63.1)

63 Structural Damage Detection with Uncertainties …

Kd =

nel 

(1 − αi )kei

753

(63.1)

i=1

where kei denotes the ith elemental stiffness matrix with the undamaged status; nel is the total number of the elements of a structure; Kd means the structural stiffness matrix under the damaged state; αi denotes the elemental stiffness reduction parameter to be identified. Optimization techniques can be conducted to minimize the discrepancies between the analytical modal data and the measured ones, and the frequency domain-based objective function is given as f (αi ) =

NF 

ωi2 +

i=1

NM 

(1 − M ACi )

(63.2)

i=1

in which   c ω − ωm  i i ωi = ωim

(63.3)

(cT · m )2 M ACi =  i 2  i 2 c  m  i i

(63.4)

where ωic and ci represent the ith calculated natural frequencies and mode shape from the finite element model, respectively; ωim and m i are the corresponding measured frequency and mode shape. When the identified values are close to the real ones, the objective function value of Eq. (63.2) would be close to zero. The following mission is to apply the TSA to optimize the defined objective function and perform the structural damage identification.

63.3 The Proposed M-TSA 63.3.1 The Standard TSA TSA is constructed based upon the natural phenomenon of trees propagation. In the real world, many trees usually spread their seeds to generate the next generations. In the algorithm, these locations of trees and seeds are regarded as the search spaces for the optimization issues. The core for the TSA is the seeds search phase, since it would create the new feasible solutions in this phase. It offers two updated equation to realize the seeds search Si, j = Ti, j + γi, j × (B j − Tr, j )

(63.5)

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Si, j = Ti, j + αi, j × (Ti, j − Tr, j )

(63.6)

where Si, j is the jth dimension of the ith seed that would be created by the ith tree, Ti, j is the jth dimension of the ith tree, B j is the jth dimension of best-so-far solution in the colony, Tr, j is the jth dimension of the rth solution randomly selected from the colony size. The coefficient αi, j is a uniform random number, γi, j is the scaling factor arbitrarily generated in the range of [−1, 1], and i and j are different indices. It could be found that Eq. (63.5) concentrates on the local search while Eq. (63.6) focuses on the global search. To achieve the balance between the two search modes, a parameter named ‘ST’ is introduced. If the random number is larger than ST, Eq. (63.5) is chosen as the update equation. Otherwise Eq. (63.6) will be selected. Therefore, for every tree seed, a higher ST value makes it have a higher chance of local search while a lower ST value renders it possess likely for global search. When applying TSA, the initial possible solutions for the optimization problem are produced as Ti, j = L j,min + ri, j × (H j,max − L j,min )

(63.7)

where L j,min and H j,max present the lower and upper bounds of the search space, respectively, ri, j is a random number produced for every dimension and location, which is in the range of [0, 1]. The seeds number can be determined approximately as 10 and 25% of the colony size [10]. The exact number of the seed generation is random in TSA.

63.3.2 M-TSA To improve the performance of the standard TSA, a one-step clustering mechanism is conducted before the commencement of the seeds search. Through using the K-means clustering, the colony information could be effectively utilized, which is quite beneficial to accelerate the algorithm’s convergence. The procedure of conducting the K-means clustering is described as follows [12]: Step 1: Choose k initial cluster centers c1 , c2 , . . . , ck randomly from the n points {X 1 , X 2 , . . . , X n }. Step 2: Assign the point X i ,(i = 1, 2, . . ., n) to a clusterC j , j = 1, 2, . . . , k, only if the distance satisfies  X i − c j  ≤  X i − c p , the point X i belongs to the cluster with the center point c j . In this paper, the Euclidean distance is introduced to calculate the distance between the clustering center and the normal point,   D    (X i, p − X j, p )2 d(X i , X j ) =  X i − X j  =  p=1

(63.8)

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where X i and X j represent any two individuals from the colony and D denotes the dimension number of the individual. Step 3: Calculate new cluster centers c1 , c2 , . . . , ck based on the following equation ci =

1  X j , i = 1, 2, . . . , k n i X ∈c j

(63.9)

i

where n i is the number of individuals within the cluster ci . In the proposed M-TSA, one-step clustering mechanism is conducted before the √ seeds search. Specifically, it assumed that K ∈ [2, C S] clusters would be randomly created and K individuals are chosen from the colony as the initial clustering points. Then K-means clustering mechanism is applied to generate the new cluster centers, as mentioned in Steps 1–3. K individuals are randomly selected from the colony again and these individuals are combined with new clustering centers as a sample P. The objective values are calculated for individuals in the sample P, and the best K individuals are picked and put into the colony again. Therefore, the elite preservation is enabled [12]. Afterwards, the procedures of the standard TSA are operated.

63.4 Numerical Studies To demonstrate the effectiveness of the proposed M-TSA, a 26-bar truss is employed as the numerical example. The Young’s modulus, mass density and Poisson ratio of the truss structure member are respectively E = 70 Gpa, ρ = 2.7×103 kg/m3 and μ = 0.33. The boundary supports are modeled with three springs with a large stiffness of 2 × 1010 N/m, as shown in Fig. 63.1. To simulate the modelling uncertainties, 1% Gaussian distribution is introduced into the elemental stiffness parameters [12]. Two damage cases including a singe damage and a multiple damage, are considered. The first four natural frequencies and four mode shapes are employed as the input. The mode shape is measured alone the vertical direction. The standard TSA and the developed M-TSA are used for identification. As regards parameters setting, the colony size, the search tendency and the maximum iteration number are set as

Fig. 63.1 A 26-bar truss model

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C S = 50, ST = 0.4 and 200, respectively. Each damage case would calculate 20 times and the mean value as well as the standard deviations are recorded. 1% white noise and 10% white noise are introduced to the natural frequencies and the mode shapes to simulate the measurement noise.

63.4.1 Identification Results of the Single Damage Case The first case is assumed that there appears 15% stiffness reduction in the 6th element. Figure 63.2 shows the evolutionary process of the damage index α6 . It is observed that the whole iteration process of using M-TSA is quite stable and it only costs 60 iterations to convergence to the neighbourhood of the preset value. In contrast, the iteration situation of the TSA is relatively fluctuated. Figure 63.3 and Table 63.1 present the final identified result of this case. It can be found the results acquired by M-TSA are more competitive. It is worth noting that for a more specific understanding of the identification results, average results and its plus and minus deviation are presented in Fig. 63.3. M-TSA is capable of providing a more reliable damage identification result.

Fig. 63.2 The iteration process of the damage index based on TSA and M-TSA for the single damage case

63 Structural Damage Detection with Uncertainties …

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(a)

(b)

Fig. 63.3 Damage identification results in damage case 1. a TSA, b M-TSA

Table 63.1 The identification results for a 26-bar truss based on TSA and M-TSA TSA

M-TSA

Single damage α6 = 0.15

Mean

Std.

Mean

Std.

0.1281

0.0434

0.1489

0.0027

α6 = 0.1

Mean

Std.

Mean

Std.

0.1059

0.0021

0.1049

0.0049

α11 = 0.2

Mean

Std.

Mean

Std.

0.2251

0.022

0.2084

0.0093

Mean

Std.

Mean

Std.

0.2778

0.0311

0.2933

0.0066

Multiple damage

α16 = 0.3

63.4.2 Identification Results of the Multiple Damage Case The second case is assumed with 10%, 20% and 30% stiffness reductions in the 6th element, 11th element and 16th element, respectively. Figure 63.4 presents the iteration process of the mentioned three damage indexes. Similar as the single dam-

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Fig. 63.4 The iteration process of the damage indexes based on M-TSA for the multiple damage case

age case, several iterations are cost for the three damage indexes converging to the neighborhood of the assumed values, which further demonstrates the quick convergence rate of the developed M-TSA. Figure 63.5 and Table 63.1 present the final identification results obtained from TSA and M-TSA. The maximum identification errors when using TSA and M-TSA are 2.51% and 0.84%, respectively. The superiority of M-TSA is illustrated, owing to the conduction of the K-means clustering mechanism, which enhance the global optimization ability of the standard TSA.

63.5 Conclusions A new type of swarm intelligence algorithm called M-TSA is presented for structural damage identification. To balance the algorithm’s exploration and exploitation ability further, the K-means clustering is introduced before the seeds search phase. From the numerical studies on a truss structure, it can be found that M-TSA is a potential good tool to deal with the structural damage identification, even when both the measurement noise and the modeling uncertainties are considered.

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(a)

(b)

Fig. 63.5 Damage identification results in the multiple damage case. a TSA, b M-TSA

Acknowledgements The first author acknowledges China Scholarship Council Postgraduate Scholarship (CSC 201606380106) and Curtin top-up scholarship to conduct his Ph.D. study in the School of Civil and Mechanical Engineering at Curtin University.

References 1. Li, Y.Y.: Hypersensitivity of strain-based indicator for structural damage identification: a review. Mech. Syst. Signal Process. 24(3), 653–664 (2010) 2. Fan, W., Qiao, P.: Vibration-based damage identification methods: a review and comparative study. Struct. Health Monit. 10(1), 83–111 (2011) 3. Diptangshu, P., Li, Z., Samiran, C., Chee, P.L., Liu, C.Y.: A scattering and repulsive swarm intelligence algorithm for solving global optimization problems. Knowl. Based Syst. 156, 12–24 (2018) 4. Lu, Z.R., Law, S.S.: Features of dynamic response sensitivity and its application in damage detection. J. Sound Vib. 303(1), 305–329 (2007) 5. Wang, L., Liu, J.K., Lu, Z.R.: Incremental response sensitivity approach for parameter identification of chaotic and hyperchaotic systems. Nonlinear Dyn. 89(1), 153–167 (2017) 6. Sun, H., Lu¸s, H., Betti, R.: Identification of structural models using a modified artificial bee colony algorithm. Comput. Struct. 116, 59–74 (2013) 7. Zhu, J.J., Huang, M., Lu, Z.R.: Bird mating optimizer for structural damage detection using a hybrid objective function. Swarm Evolut. Comput. 35, 41–52 (2017) 8. Chen, Z.P., Yu, L.: A novel PSO-based algorithm for structural damage detection using Bayesian multi-sample objective function. Struct. Eng. Mech. 63(6), 825–835 (2017) 9. Ding, Z.H, Yao, R.Z., Huang, J.L., Huang, M., Lu, Z.R.: Structural damage detection based on residual force vector and imperialist competitive algorithm. Struct. Eng. Mech. 62(6), 709–717

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10. Kiran, M.S.: TSA: tree-seed algorithm for continuous optimization. Expert Syst. Appl. 42, 6686–6698 (2015) 11. Xia, Y., Hao, H., Brownjohn, J.M.W., Xia, P.Q.:, Damage identification of structures with uncertain frequency and mode shape data. Earthq. Eng. Struct. 31, 1053–1066 (2002) 12. Cai, Z.H., Gong, W.Y., Ling, C.X., Zhang, H.: A clustering-based differential evolution for global optimization. Appl. Soft Comput. 11, 1363–1379 (2011)

Part XVIII

Multi-field and Multi-scale Modeling of Advanced Materials

Chapter 64

Tensile Properties of Carbon Fiber Reinforced Polymer Matrix Composite Eva Kormanikova , Milan Zmindak

and Peter Sabol

Abstract The paper deals with determination of material characteristics of carbon/epoxy composite due to numerical homogenization and experimental investigation. Within numerical homogenization there is used periodic microstructure model without and with pores. The numerical homogenization and simulation of experiment are provided in FEM program ANSYS. In this study, unidirectional quasi-static tensile test on carbon/epoxy composite laminate is carried out to investigate its material properties. The quasi-static tensile tests were conducted by INOVA FU 160 machine. The extensometer EPSILON 3542 was installed to measure the deformation of the specimen. The experiment was controlled also using video extensometer measuring system ARAMIS. The results obtained from numerical and experimental investigation were compared. Keywords Tensile properties · Carbon fiber · Polymer matrix · Composite material

64.1 Introduction Fiber reinforced polymer (FRP) composite is a material made of polymer matrix reinforced with high strength fibers. The quality of fibers, their orientation, shape and fiber volume fraction determine the material properties. These material properties can be obtained from analytical or numerical microstructure fictitious models and experimental investigation.

E. Kormanikova (B) · P. Sabol Technical University of Kosice, Vysokoskolska 4, 042 00 Kosice, Slovakia e-mail: [email protected] P. Sabol e-mail: [email protected] M. Zmindak University of Zilina, Univerzitna 8215/1, 010 26 Žilina, Slovakia e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_64

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The microstructure models are based at the class of so-called unit cell methods. The unit cell methods provide valuable information on the local microstructure fields as well as the effective material properties [1, 2]. These properties are generally determined by fitting the averaged microscopic stress-strain fields, resulting from the analysis of a microstructure representative cell subjected to a certain loading. In recent years, a promising alternative approach for the homogenization of engineering materials has been developed, i.e. multi-scale computational homogenization, also called global-local analysis. The basic ideas of this approach have been presented in [3–8]. These micro-macro modeling procedures do not lead to closed-form overall constitutive equations, but compute the stress-strain relationship at every point of interest of the macro-component by detailed modeling of the microstructure attributed to that point [9, 10]. From experiment it can be obtained real mechanical properties such as for example strength, failure strain, Young´s modulus and others. The experimental designation of quasi-static and dynamic tensile properties of FRP composite is clarified in works of [11, 12].

64.2 Microscopic Stress-Strain Field Most fiber reinforced composites have a random arrangement of the fibers at the micro-scale. A simpler alternative is to assume that the random microstructure is well approximated by the periodic microstructure (Fig. 64.1). With the initiation of the computer technology the concept of the representative volume element (RVE) in combination with a finite element analysis gets more and more importance.

Fig. 64.1 A periodic microstructure model

fiber matrix

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The stiffness tensor of microstructure is written in following form ⎧ ⎫ ⎡ ⎪ C σ¯ ⎪ ⎪ ⎪ ⎢ 11 ⎪ 1⎪ ⎪ ⎪ ⎪ ⎪ C12 ⎪ ⎪ ⎢ ⎪ σ¯ 2 ⎪ ⎢ ⎬ ⎨ σ¯ 3 ⎢ C12 =⎢ ⎪ 0 ⎪ ⎢ ⎪ σ¯ 4 ⎪ ⎢ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎣ σ ¯ 0 ⎪ ⎪ 5⎪ ⎪ ⎩ ⎭ σ¯ 6 0

C12 C22 C23 0 0 0

C12 C23 C22 0 0 0

0 0 0 C44 0 0

0 0 0 0 C66 0

⎤⎧ ⎫ ε¯ 1 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ 0 ⎥⎪ ε ¯ 2 ⎪ ⎨ ⎪ ⎬ ⎥⎪ 0 ⎥ ε¯ 3 . ⎥ 0 ⎥⎪ γ¯4 ⎪ ⎪ ⎪ ⎥⎪ ⎪ ⎪ ⎪ 0 ⎦⎪ γ¯5 ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ γ¯6 C66

(64.1)

In order to evaluate the tensor C of the composite, the RVE is subjected to an average strain. The volume average of the strain in the RVE equals the applied strain ε¯ i j =

1 V

 εi j d V.

(64.2)

V

The coefficients in C are found by setting a different problem for each column of C. The components C ij i, j = 1,2,3 (z, x, y) of the tensor C are determined by solving three elastic models of RVE with parameters (a1 , a2 , a3 ) subjected to the different boundary conditions [13, 14]. By using a unit value of applied strain, it is possible to compute the stress field, whose average gives the required components of the elastic matrix as  1 1 σi d V, ε0j = 1. (64.3) Ci j = σ¯ i = σiV = V V V

For calculation of the component C 44 , C 66 = C 55 , the following strain is applied to the RVE 0 0 0 0 + ε31 = 1, γ60 = ε12 + ε21 = 1. γ40 = ε23

(64.4)

The coefficient C 44 , C 66 is calculated as   1 1 1 1 C44 = σ¯ 4 = σ4V = σ4 d V C66 = σ¯ 6 = σ6V = σ6 d V. V V V V V

(64.5)

V

64.3 Experiment The performance of the proposed model is demonstrated on tensile tests of three specimens (Fig. 64.2 (left)). The fiber volume fraction and fiber diameter were found from electron microscope digital shot (Fig. 64.2 (right)). The specimens were cut out from the unidirectional composite plate 250 × 25 × 2 mm made of 16 autoclaved prepreg carbon/epoxy layers, V f = 0.6. Each layer of the laminate has the same thickness n h = 0.125 mm.

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Fig. 64.2 The specimens (left), the electron microscope digital shot (right)

The quasi-static tensile specimens were prepared as per guidelines STN EN ISO 527-4 [15], that stipulate the requirements for the tensile testing of plastics. Each specimen was equipped with aluminum tabs at the ends. The gauge length is 80 mm long and the aluminum tabs are 60 mm long. The tabs with thickness of 1 mm were glued onto the specimens to avoid any possible damage when gripping the specimen. All layers of specimens have fiber orientations θ = 0°. The dimension of the specimen was adjusted to satisfy the specific requirements by the INOVA FU 160 machine. As shown in Fig. 64.3 (left) an extensometer was installed to measure the deformation of the specimen during the quasi-static test. The machine has an inbuilt load cell to measure the load during the test. An example of tested specimen with attached extensometer is shown in Fig. 64.3 (right). Three specimens were tested also with the help the videoextensometry by ARAMIS system.

Fig. 64.3 The INOVA FU 160 (left), the axial extensometer Epsilon 3542 (right)

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Fig. 64.4 The load-displacement diagram (left), the strain-time diagram (right)

64.4 Results The strains were investigated at gauge length. The measured load-displacement and strain-time curves are displayed in Fig. 64.4 (left) and 64.4 (right), respectively. The displacement u in the graphs corresponds to values from the extensometer with initial length 80 mm. The specimens contain certain amount of primary interfiber damage and the final rupture is visible as non-uniform cracking across the specimen width (Fig. 64.5 (left)). Strain distribution obtained from by GOM Aramis software is shown on the Fig. 64.5 (right). Tensile strength and ultimate strain were obtained by FEM in program ANSYS (Fig. 64.6). The summary of results are written in Tables 64.1 and 64.2.

64.5 Conclusion The numerical homogenization within periodic microstructure model without and with pores is considered in the paper. The periodic microstructure model without pores was solved by the FEM. The computational finite element model of unidirectional fiber-reinforced composite plate is presented in program ANSYS. The performance of the proposed model is demonstrated on example of tensile test. The original supplier material data are adjusted during the quasi-static numerical analysis. The comparison of the results from experiment and simulation shows satisfactory agreement.

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crack

Fig. 64.5 The photograph of fractured specimen (left), the strain distribution obtained by GOM Aramis software (right)

Fig. 64.6 The contour plot of longitudinal displacements (right)

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Table 64.1 Summary of results of microstructure model Material characteristics

Periodic FEM model without pores V f = 0.6

Periodic model with ellipsoidal pores V p = 0.1

E 1 [GPa]

139.2

112.12

E 2 [GPa]

14.219

11.633

E 3 [GPa]

14.219

8.896

ν 12

0.3

0.272

ν 13

0.3

0.372

ν 23

0.202

0.228

G23 [GPa]

3.182

2.734

G12 [GPa]

4.738

3.835

G13 [GPa]

4.738

3.734

Table 64.2 Summary of FEM and experimental results

Material characteristics

Experiment

FEM

Periodic model with pores

Longitudinal modulus [GPa]

109.13

109.13

112.12

Tensile strength [MPa]

1746.89

1760

–

Ultimate strain [–]

0.01676

0.01613

–

Acknowledgements This work was supported by the Scientific Grant Agency of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences under Projects VEGA 1/0374/19 and 1/0078/16. The authors declare that they have no conflict of interest.

References 1. Sladek, J., Novak, P., Bishay, P.L., Sladek, V.: Effective properties of cement-based porous piezoelectric ceramic composites. Constr. Build. Mater. 190, 1208–1214 (2018) 2. Murcinkova, Z., Novak, P., Kompis, V., Zmindak, M.: Homogenization of the finite-length fibre composite materials by boundary meshless type methods. Arch. Appl. Mech. 88(5), 789–804 (2018) 3. Suquet, P.M.: Local and global aspects in the mathematical theory of plasticity. In: Sawczuk, A., Bianchi, G. (eds.) Plasticity Today: Modelling, Methods and Applications, pp. 279–310. Elsevier Applied Science Publishers, London (1985) 4. Feyel, F., Chaboche, J.-L.: FE2 multiscale approach for modeling the elastoviscoplastic behaviour of long fiber SiC/Ti composite materials. Comput. Methods Appl. Mech. Eng. 183, 309–330 (2000) 5. Terada, K., Kikuchi, N.: A class of general algorithms for multi-scale analysis of heterogeneous media. Comput. Methods Appl. Mech. Eng. 190, 5427–5464 (2001)

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6. Kouznetsova, V., Brekelmans, W.A.M., Baaijens, F.P.T.: An approach to micro-macro modeling of heterogeneous materials. Comput. Mech. 27, 37–48 (2001) 7. Miehe, C., Koch, A.: Computational micro-to-macro transition of discretized microstructures undergoing small strain. Arch. Appl. Mech. 72, 300–317 (2002) 8. Barbero, E.J.: Finite Element Analysis of Composite Materials. CRC Press, USA (2007) 9. Barretta, R., Luciano, R., Willis, J.R.: On torsion of random composite beams. Compos. Struct. 132, 915–922 (2015) 10. Lapcik, L., et al.: Effect of filler particle shape on plastic-elastic mechanical behavior of high density poly(ethylene)/mica and poly(ethylene)/wollastonite composites. Compos. Part B 141, 92–99 (2018) 11. Chen, W., Meng, Q., Hao, H., Ciu, J., Shi, Y.: Quasi-static and dynamic tensile properties of fiberglass/epoxy laminate sheet. Constr. Build. Mater. 143, 247–259 (2017) 12. Chen, W., Hao, H., Jong, M., Ciu, J., Shi, Y., Chen, L.: Quasi-static and dynamic tensile properties of basalt fibre reinforced polymer. Compos. Part B 125, 123–133 (2017) 13. Kormanikova, E., Kotrasova, K.: Multiscale modeling of liquid storage laminated composite cylindrical tank under seismic load. Compos. Part B 146, 189–197 (2018) 14. Vorel, J., Urbanová, S., Grippon, E., Jandejsek, I., Maršálková, M., Šejnoha, M.: Multi-scale modeling of textile reinforced ceramic composites. Ceram. Eng. Sci. Proc. 34(10), 233–245 (2014) 15. STN EN ISO 527-4: plastics. Determination of tensile properties. Part 4: test conditions for isotropic and orthotopic fibre-reinforced plastic composites (ISO 527-4:1997)

Chapter 65

Increasing of Fluid Effect on Liquid Storage Laminated Composite Tank During Seismic Excitation Kamila Kotrasova

and Eva Kormanikova

Abstract This paper presents influence of fluid filling on the increasing of fluid seismic effect on solid of liquid storage tank. The liquid inside of the tank exerts hydrodynamic effect on tank walls and bottom in addition to hydrostatic effect during earthquake. The knowledge of forces acting onto walls and bottom of the container and total hydrodynamic effects of liquid on storage tank during ground motion plays fundamental role in the design of an earthquake resistance of fluid storage container. The seismic response of a fluid filling laminated composite container was solved for region of Slovak Republic respecting recommendations of Eurocode 8 Part. 4. and the laminated representative volume element for obtaining of the of the laminated composite effective material properties. The effective material properties of liquid storage laminated composite tank were obtained by using of a fictitious hexagonal microstructure and the laminated representative volume element at the meso-scale. Keywords Fluid effect · Laminated composite cylindrical tank · Earthquake

65.1 Introduction Liquid-containing tanks are using in industries, in liquid distribution systems or for storing of variety of liquids. Seismic analysis of liquid-containing tanks has different approaches from seismic analysis of typical structures and the special considerations are necessary [1]. The liquid storage tank is available in range of suitable materials including of combination of different kind of materials. The storage function of container and the kind of storage medium influence affect type of tank construction, its shape and size, the construction technology and the used tank materials. The toxic and flammable liquids may be stored usually in concluded fixed-roof tanks [2]. Using of K. Kotrasova (B) · E. Kormanikova Technical University of Košice, Vysokoškolská 4, 04200 Košice, Slovakia e-mail: [email protected] E. Kormanikova e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_65

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appropriate material for the container construction and the knowledge of hydrodynamic effect of fluid on liquid storage tanks of fluid-structure interaction problem during earthquake contributes to reliability and earthquake resistance of fluid filled facilities-containers especially in the case of storage of flammable liquids, where the possibilities damages can have environmental consequences [3]. The walls and bottom of liquid storage containers may be made from traditional materials: steel, reinforced concrete, metals, plastic, polyethylene or by combination of the two or more materials, or up-to-date composite material. The circular containers in shape are used for the most common storage tanks for the optimal exploitation. The solution of interaction of fluid-composite tank and investigation of earthquake vibration problems of anisotropic shells is more difficult [4–7]. Laminated composites belong to well-known class of composite materials [8].

65.2 Seismic Analysis of Liquid Storage Cylindrical Tank By seismic loading, liquid-storage cylindrical tank, its walls, bottom and fluid filling, is subjected to horizontal acceleration. The motion of contained fluid in a cylindrical container can by expressed as the sum of two separate contributions, which are called “impulsive” and “convective”, respectively [9]. The dynamic analysis of a liquid-filled tank may be carried out using the concept of generalized single-degree-of freedom (SDOF) systems that represents the impulsive mode and convective modes of vibration of the tank-liquid system [8], see Fig. 65.1. The impulsive mode is represented by impulsive mass of fluid mi that is attached rigidly to the container wall at height hi (or h*i ). The convective masses mcn are connected to the tank walls at heights hcn (or h*cn ) by springs with stiffness k cn [5].

Fig. 65.1 a Liquid-storage cylindrical tank and b single degree of freedom systems for fluid filled cylindrical tank

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The natural period of the convective response T cn in [s] of cylindrical tank with the walls rigidly connected with the foundation slab is given in Eq. (65.1) Tcn = √

2π , gλn tanh(λn γ )/R

(65.1)

where λn are the roots of the first-order Bessel function of the first-order Bessel function of the first kind, g is acceleration due to gravity, γ = H/R is the dimensionless tank slenderness parameter, R is inner tank radius and H is height of fluid filling of tank [8]. The first oscillating, or sloshing, mode and frequency of the oscillating liquid for n = 1 is significantly dominant [5]. The natural period of the impulsive response T i , in [s], is taken √ H ρ Ti = Ci √ √ . s/R E

(65.2)

The results of the dynamic analysis of a liquid-filled container considered only horizontal ground motion are the base shears and moments. Total base shear Q immediately at the bottom of the tank wall can be also obtained by base shear in impulsive mode and base shear in convective mode, Eq. (65.3) [6]. Q = (m i + m w + m r )Se (Ti ) +

∞ 

m cn Se (Tc ),

(65.3)

n=1

The bending moment M of the tank wall immediately at the wall bottom can be also obtained by bending moment in impulsive mode and in convective mode, Eq. (65.4). The overturning moment M * immediately below of the bottom plate of the tank is dependent on the hydrodynamic effect on the tank wall as well as that on the tank bottom, is given by Eq. (65.5). ∞      m cn h cn Se (Tc ), M = m i h i + m w h w + m b h b + m r h r Se (Ti ) +

(65.4)

n=1 ∞      m cn h ∗cn Se (Tc ). M ∗ = m i h i∗ + m w h w + m b h b + m r h r Se (Ti ) +

(65.5)

n=1

m is the total liquid mass in [kg], mi [kg] is the impulsive mass of fluid, mc [kg] is the convective mass of fluid, mw [kg] is the mass of the tank wall, mb [kg] the mass of the tank base plate with foundation and mr [kg] the mass of the tank roof. hi and hc in [m] are the heights of the centroids on the impulsive and convective hydrodynamic wall pressures from tank bottom. h*i [m] is height of the centroid on the impulsive hydrodynamic tank wall pressures as well as that on the tank bottom and h*c [m] is height of the centroid on the convective hydrodynamic tank wall pressures as well as that on the tank bottom. hw [m] is the height of the centre of gravity of tank

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wall, hb [m] and hr [m] are the heights of the centres of gravity of tank base plate with foundation and roof, respectively. S e (T i ) is the impulsive spectral acceleration obtained from a 2% damped elastic response spectrum and S e (T c ) is the convective spectral acceleration obtained from a 0.5% damped elastic response spectrum [7].

65.3 Numerical Homogenization of Laminated Shell Most composites have random arrangement of the fibers. A random microstructure results in transversely isotropic properties at the meso-scale. The analysis of composites with random microstructure can be done by using of a fictitious periodic microstructure. A simple alternative is to assume that the random microstructure is well approximated by the hexagonal microstructure (Fig. 65.2) [10]. Analysis of microstructure yields a transversely isotropic stiffness tensor. A transversely isotropic material is described by five constants. When the axis of symmetry is the fiber direction, 3D Hooke’s law reduces to ⎧ ⎫ ⎡ ⎤⎧ ⎫ ⎪ C11 C12 C12 σ¯ 1 ⎪ 0 0 0 ⎪ ⎪ ε¯ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ε¯ ⎪ ⎢C C C ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ ¯ 0 0 0 2⎪ 2⎪ ⎪ ⎢ 12 22 23 ⎥⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎥⎨ ⎪ ⎬ ⎨ ⎬ ⎢ σ¯ 3 0 0 0 ⎥ ε¯ 3 ⎢ C12 C23 C22 =⎢ . (65.6) ⎥ ⎢ 0 0 0 21 (C22 − C23 ) 0 0 ⎥⎪ ⎪ σ¯ 4 ⎪ γ¯4 ⎪ ⎪ ⎪ ⎪ ⎪ ⎢ ⎪ ⎥ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ σ¯ 5 ⎪ γ¯5 ⎪ 0 C66 0 ⎦⎪ ⎪ ⎪ ⎣ 0 0 0 ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎭ ⎭ ⎩ ⎪ σ¯ 6 γ¯6 0 0 0 0 0 C66 The elastic properties of the homogenized material can be computed by [11]. In order to evaluate the elastic matrix C of the composite, the RVE is subjected to an average strain. The volume average of the strain in the RVE equals the applied strain  1 ε¯ i j = εi j d V. (65.7) V V

Fig. 65.2 a Hexagonal microstructure model and b laminated representative volume element

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The coefficients in C are found by setting a different problem for each column of C. The components of the tensor C are determined by solving three elastic models of RVE with parameters (a1 , a2 , a3 ) subjected to the boundary conditions. By using a unit value of applied strain, it is possible to compute the stress field, whose average gives the required components of the elastic matrix as Ci j = σ¯ i =

1 V

 σi d V ε0j = 1, i, j = 1, 2, 3.

(65.8)

V

A simile procedure to that used to obtain the RVE at the micro-scale can be used to analyse laminates on the meso-scale [12]. In this case the RVE represents a laminate. Therefore, the through-thickness direction should remain free to expand along the thickness. In general, the laminated RVE (Fig. 65.2) must include the whole thickness. For symmetrical laminates subjected to in-plane loads, the RVE can be defined with half the thickness using symmetry boundary conditions. From laminated representative volume element can be calculated effective material characteristics for laminate E eff .

65.4 Numerical Experiment and Conclusion A ground supported vertical cylindrical container, made of composite laminate, has diameter D = 0.47 m and wall height H w is 1.68 m. The tank wall thickness is uniform 0.022 m. The height of water full filling (H2 O) is 1.6 m. The base slab of tank is 0.02 m thick composite laminate and it is grounded on the 0.15 m thick circular concrete base slab with diameter 0.53 m. The tank has laminated composite roof slab 0.02 m thick. An unidirectional composite layer consists of isotropic fibres E = 270 GPa, ν f = 0.3, and isotropic matrix E m = 5 GPa, ν m = 0.3. The fiber volume fraction V f = 0.4. Each layer of the laminate has the same thickness n h = 0.125 mm. The tank is made of the laminate [0/0/90/90]NS , where N = 22 for the wall and for the base slab and N = 11 for the roof slab. We consider only horizontal seismic loading. Used elastic response spectrum is determined for Slovak republic region, ag = 1.5 ms−2 , B category of subsoil [13] and we consider of partial fulfilment of tank with fluid filling heights 0.4, 0.8, 1.2, and 1.6 m. This paper summarizes the results of the seismic analysis of water filling ground supported cylindrical composite laminate tank subject to the horizontal component of earthquake ground motion. Increasing hydrodynamic fluid effect-total base shears and total bending moments during seismic excitation was shown in Fig. 65.3. The values of total base shears, total bending and overturning moments as results of seismic response liquid storage laminated composite tank growth with tank fluid filling, and the maximal seismic effect was given by complete fluid fulfilment of tank.

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Fig. 65.3 Increasing fluid effect immediately at the wall bottom during earthquake in depending from fluid filling, a the base shear, b the bending moment

Acknowledgements This work was supported by the Scientific Grant Agency of the Ministry of Education of Slovak Republic and the Slovak Academy of Sciences under Project VEGA 1/0374/19.

References 1. Kotrasová, K., Kormaníková, E.: The study of seismic response on accelerated contained fluid. Adv. Math. Phys. 2017, 1–9 (2017) 2. Moˇcilan, M., Žmindák, M., Pastorek, P.: Dynamic analysis of fuel tank. Procedia Eng. 136, 45–49 (2016) 3. Farahani, H., Barati, F., Batmani, H.: Vibration analysis of composite horizontal cylindrical tank with different layering using the finite element method. Indian J. Sci. Technol. 8(S7), 213–219 (2015) 4. Kotrasová, K., Kormaníková, E.: A case study on seismic behavior of rectangular tanks considering fluid—structure interaction. Int. J. Mech. 10, 242–252 (2010) 5. Vajir, D., Jadhav, R., Jain, K.: Analytical and numerical analysis of composite material storage tank under seismic loading. SSRG-IJME 4(5), 33–38 (2017) 6. Lau, K.T., Hung, P.Y., Zhu, M.H., Hui, D.: Properties of natural fibre composites for structural engineering applications. Compos. Part B Eng. 136, 222–233 (2018) 7. Malhotra, P.K., Wenk, P., Wieland, M.: Simple procedure for seismic analysis of liquid-storage tanks. Struct. Eng. Int. 3, 197–201 (2000) 8. Kotrasová, K., Grajciar, I., Kormaníková, E.: A case study on the seismic behavior of tanks considering soil-structure-fluid interaction. J. Vib. Eng. Technol. 3(3), 315–330 (2015) 9. Kareem, M.Ch., Abdullah, M.Q., Wasmi, H.R.: Structural behaviour for composite cylindrical tank under seismic load. IJISET Eng. Technol. 3(12), 37–42 (2016) 10. Gürdal, Z., Haftka, R.T., Hajela, P.: Design and Optimization of Laminated Composite Material. Wiley (1999) 11. Altenbach, H., Altenbach, J., Kissing, W.: Structural Analysis of Laminate and Sandwich Beams and Plates. Lublin (2001) 12. Luciano, R., Barbero, E.J.: Formulas for the stiffness of composites with periodic microstructure. Int. J. Solids Struct. 31(21), 2933–2944 (1995) 13. Eurocode 8—Design of structure for earthquake resistance—part 4: silos, tanks and pipelines (2006)

Chapter 66

Bending of Piezo-Electric FGM Plates by a Mesh-Free Method V. Sladek, L. Sator and J. Sladek

Abstract Unified formulation for bending of elastic piezoelectric plates is derived with incorporating the assumptions of three plate bending theories, such as the Kirchhoff-Love theory, 1st order and 3rd order shear deformation plate theory. The functional gradation of material coefficients in the transversal as well as in-plane direction is allowed and the plate thickness can be variable. Both the governing equations and the boundary conditions are derived from the variational formulation of 3D electro-elasticity. For numerical solution a mesh-free method is developed with using the Moving Least Square approximation for spatial variations of field variables. The high order derivatives of field variables are eliminated by decomposing the original governing partial differential equations (PDE) into the system of PDEs with not higher than 2nd order derivatives. The numerical simulations are presented for illustration of coupling effects and verification of the developed theoretical and numerical formulations. Keywords Unified formulation · Strong form mesh-free method · Moving least square approximation

66.1 Introduction The direct piezoelectric effect was discovered in 1880 by the brothers Pierre and Jacques Curie, when they applied mechanical stress on certain crystals (tourmaline, quartz, topaz, Rochelle salt) and observed voltage of produced electrical charges. This voltage was proportional to the applied stress. The discovery of the converse effect followed later in 1881 in accordance with the proposal of Gabriel Lippmann resulting from the mathematical aspects of the theory. This illustrates how the theoretical study of phenomena and coupled effects can results into proposals and subsequent experimental confirmation of such discoveries. The development of new V. Sladek (B) · L. Sator · J. Sladek Institute of Construction and Architecture, Slovak Academy of Sciences, 845 03 Bratislava, Slovakia e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_66

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piezoelectric materials gave rise to their applications as actuators and sensors for the controlled performance of engineering structures called as intelligent or smart structures [1–4]. Such structures are usually a laminated original made by ceramic slice. Owing to discontinuities of material coefficients on layer interfaces, however, there are concentrations of gradient fields (stress, electric field) which promote the growth of interfacial micro-cracks and finally limit the application and development of such structures. This problem can be solved by using functionally graded piezoelectric materials (FGPM) [5–7], which are micro-composites exhibiting continuously varying material properties according to the volume contents of micro-constituents. Certain review of the works devoted to plates made of FGPMs can be found in works [3, 8–10]. It is known that functional gradation of material coefficients gives rise to coupling effects, e.g. transversal gradation of Young’s modulus yields coupling between the bending and in-plane deformation modes [14–16]. Besides the electro-elastic coupling in piezoelectric plates, one could expect some additional coupling effects in the FGMP plates. In this paper, we present the derivation of the unified theory for bending of FGMP plates based on the assumptions of three plate bending theories, such as the classical Kirchhoff-Love theory, 1st and 3rd order shear deformation plate theories. Governing equations as well as the boundary conditions are derived for these coupled problems from the variational formulation. The attention is paid also to numerical implementation by using the mesh-free Moving Least Square (MLS) approximation for field variables. Finally some numerical simulations are presented for illustration of coupling effects and verification of the developed theoretical and numerical formulations.

66.2 Three Plate Bending Theories. Unified Formulation for FGM Piezoelectric Plates 66.2.1 Constitutive Equations The stress formulation constitutive equations in linear phenomenological theory of piezoelectric solids [11–13] are given as σi j (ε, E) = ciEjkl skl − eki j E k ,

Dk (ε, E) = eki j si j + χkiε E i

(66.1)

with strain skl and electric field E i being the natural variables and the thermodynamic potential being the enthalpy H = H (ε, E), d H = σi j dεi j − Dk d E k . Furthermore, the strains and electric field are expressed in terms  of the primary field variables, displacements vi and scalar potential φ, as si j = vi, j + v j,i /2, E i = −φ,i . The piezoelectric materials in this work will be, at most, orthotropic, including isotropic and transversely isotropic. The choice of the mm2 crystal structure is general

66 Bending of Piezo-Electric FGM Plates by a Mesh-Free Method

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enough to cover the piezoelectric materials often associated with smart structures. Using the Voigt notation, the elasto-piezo-dielectric material matrix that correspond to the mm2 crystal structure is ⎞ ⎛ ⎞ ⎛ E σ1 c11 σ11 ⎜ σ22 ⎟ ⎜ σ2 ⎟ ⎜ c E ⎟ ⎜ ⎟ ⎜ 12 ⎜ ⎜ σ ⎟ ⎜ σ ⎟ ⎜ cE ⎜ 33 ⎟ ⎜ 3 ⎟ ⎜ 13 ⎜σ ⎟ ⎜ σ ⎟ ⎜ 0 ⎜ 23 ⎟ ⎜ 4 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ σ13 ⎟ = ⎜ σ5 ⎟ = ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ σ12 ⎟ ⎜ σ6 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎜ ⎜ D1 ⎟ ⎜ D1 ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎜ ⎜ ⎝ D2 ⎠ ⎝ D2 ⎠ ⎝ 0 D3 D3 e31 ⎛

E c12 E c22 E c23 0 0 0 0 0 e32

E c13 E c23 E c33 0 0 0 0 0 e33

0 0 0 E c44 0 0 0 e24 0

0 0 0 0 E c55 0 e15 0 0

0 0 0 0 0 0 0 0 0 0 0 −e24 0 −e15 0 E 0 0 c66 ε 0 χ11 0 ε 0 0 χ22 0 0 0

⎞⎛ ⎞ s1 −e31 ⎜ ⎟ −e32 ⎟⎜ s2 ⎟ ⎟ ⎜ ⎟ −e33 ⎟ ⎟ ⎜ s3 ⎟ ⎜ ⎟ 0 ⎟ ⎟ ⎜ s4 ⎟ ⎟⎜ ⎟ 0 ⎟⎜ s5 ⎟ (66.2) ⎟⎜ ⎟ 0 ⎟ ⎜ s6 ⎟ ⎟⎜ ⎟ ⎜ ⎟ 0 ⎟ ⎟⎜ E 1 ⎟ ⎝ E2 ⎠ ⎠ 0 ε χ33 E3

    with s1 s2 s3 s4 s5 s6 = s11 s22 s33 2s23 2s13 2s12 . In the case of hexagonal mm6 class of symmetry, the material becomes transversely isotropic, if the poling axis coincides with the material symmetry axis (directions perpendicular to the symmetry axis are equivalent). Then, the material matrix is ⎞ ⎛ E E E c11 c12 c13 0 0 0 0 0 −e31 ⎜ cE cE cE 0 0 0 0 0 −e31 ⎟ ⎟ ⎜ 12 11 13 ⎜ cE cE cE 0 0 0 0 0 −e33 ⎟ ⎟ ⎜ 13 13 33 ⎟ ⎜ 0 0 0 c E 0 0 0 −e 0 ⎟ ⎜ 15 44  1 E ⎟ E ⎜ E E . (66.3) = c11 − c12 0 ⎟, c66 ⎜ 0 0 0 0 c44 0 −e15 0 ⎟ ⎜ 2 E ⎜ 0 0 0 0 0 c66 0 0 0 ⎟ ⎟ ⎜ E ⎜ 0 0 0 0 e15 0 χ11 0 0 ⎟ ⎟ ⎜ ⎝ 0 0 0 e15 0 0 0 χ E 0 ⎠ 11 E 0 χ33 e31 e31 e33 0 0 0 0

66.2.2 Governing Equations and Boundary Conditions In the unified formulation [14–16] for all three plate bending theories (KLT— Kirchhoff-Love theory; FSDPT—1st order shear deformation plate theory; TSDPT—3rd order shear deformation plate theory), the displacements can be expressed in terms of the in-plane displacements u α (x), transversal displacements (deflections) w(x) and rotations of the normal to the mid-surface ϕα (x) by

 vi (x, x3 ) = δiα u α (x) + [c1 ω(x3 ) − x3 ]w,α (x) + c1 ω(x3 )ϕα (x) + δi3 w(x).

(66.4)

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 2 where ω(x3 ) := x3 − c2 ψ(x3 ), ψ(x3 ) := 43 xh3 x3 , x3 ∈ [−h/2, h/2], x ∈ , 0, KLT 0, FSDPT, KLT c1 = , c2 = . 1, SDPT 1, TSDPT The assumed expansion of displacements with respect to x3 -coordinate is admissible, because L  h, where L is the characteristic linear dimension of the mid-plane domain . Now, in view of (66.4), one can obtain the strains and from the constitutive equations also stresses and electric displacements. The variational formulation for piezoelectric plate (with absent body forces and free electric charges) is given as

h/2

h/2 H dd x3 − δWe =

δ −h/2 

  σi j δvi, j + Dk δφ,k d x3 d

 −h/2

−

  t¯3 δw + σ¯ δφ d = 0

(66.5)



where t¯3 and σ¯ denotes the tractions and density of free electric charge, respectively, ¯ x3 ) = σ¯ (x)δ(x3 − on the upper plate surface, i.e. t¯i (x, x3 ) = δi3 t¯3 (x)δ(x3 −h/2), q(x, h/2). Owing to plate bending assumptions, the dependence of all mechanical fields on the in-plane and transversal coordinates is factorized, but it is not in electrical fields. In the development of inherently consistent theory, we assume the expansion of the electric potential with respect to the dimensionless coordinate z = x3 / h up to the 2nd order φ(x, x3 ) ≈ φ0 (x) + zφ1 (x) + z 2 φ2 (x)

(66.6)

with φa (x)(a = 0, 1, 2) being new field variables. Two of them can be determined from the boundary conditions on the bottom and top surfaces of the plate. For simplicity, let us consider the Dirichlet electrical boundary condition:  2 1 1 φ2 (x) = φ ± (x) φ(x, ±h/2) = φ0 (x) ± φ1 (x) + 2 2 ⇒ φ1 (x) = φ + (x) − φ − (x),   φ2 (x) = 2 φ + (x) + φ − (x) − 4φ0 (x)

(66.7)

Then, in view of (66.7), Eq. (66.6) results in   φ(x, x3 ) ≈ 1 − 4z 2 φ0 (x) + z − + 2z 2 + , ± := φ + (x) ± φ − (x)

(66.8)

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Assuming σ¯ (x) = 0, after integration with respect to the x3 -coordinate, we obtain

h/2 δ

H dd x3 − δWe =

−h/2 



(ϕ)

Tαβ (x)δu α,β (x) + Mαβ (x)δϕα,β (x)



  (wϕ) (w) − Mαβ (x)δw,αβ (x) + T3α (x) δw,α (x) + δϕα (x)  +Bα (x)δφ0,α (x) + B3 (x)δφ0 (x) − t¯3 δw d = 0 (66.9)

in which the integral fields are defined as

h/2 Tαβ (x) :=

σαβ (x, x3 )d x3 −h/2

(ϕ) Mαβ (x)

h/2 := c1

ω(x3 )σαβ (x, x3 )d x3 ,

(w) Mαβ (x)

−h/2 (wϕ) T3α (x)

h/2 := c1

h/2 :=

[x3 − c1 ω(x3 )]σαβ (x, x3 )d x3 −h/2



 [(1 − c2 )κ + c2 ] − c2 ψ,3 (x3 ) σ3α (x, x3 )d x3

−h/2

h/2 Bα (x) := −h/2



h/2  x 2  x3 8 3 1−4 Dα (x, x3 )d x3 , B3 (x) := − D3 (x, x3 )d x3 (66.10) h h h −h/2

where the shear correction factor [(1 − c2 )κ + c2 ] is introduced in the FSDPT according to the Reissner modification of shear stresses. In order to get the governing equations and boundary conditions expressed in terms of primary fields, we need to accomplish the integrations in the definitions of the semi-integral fields by Eq. (66.10). Let us consider FGPM plates with unspecified in-plane gradation of material coefficients and the power-law gradation in the transversal direction as p 1 ± z , a, b ∈ {1, 2, 3, 4, 5, 6} 2 r  1 ±z eab (x, x3 ) = eab(0) e(H ) (x)e(V ) (x3 ), e(V ) (x3 ) = 1 + ξ (66.11) 2 

E E E E E cab (x, x3 ) = cab(0) c(H ) (x)c(V ) (x 3 ), c(V ) (x 3 ) = 1 + ζ

χiεj (x, x3 )

=

ε ε χiεj (0) χ(H ) (x)χ(V ) (x 3 ),

ε χ(V ) (x 3 )



1 ±z =1+λ 2

s , i, j ∈ {1, 2, 3}.

After certain rearrangements, we obtain from (9) the system of governing equations in  and boundary restrictions on the boundary edge ∂. In linear theories, it is convenient to utilize dimensionless formulations. Let us introduce the dimensionless coordinates

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xβ∗ :=

xβ x3 , x3∗ := = h ∗ (x)z, L h0

(66.12)

and the dimensionless fields u ∗β (x) :=

u β (x) w(x) φ(x) , ϕβ∗ (x) := ϕβ (x), w∗ (x) := , φ ∗ (x) := , h0 h0 0 − (x) + (x) −∗ (x) := , +∗ (x) := . (66.13) 0 0

The governing equations in the dimensionless form can written as ∗ Tαβ,β =0 (w)∗ + Mαβ,αβ



L h0

2

(ϕ)∗ Mαβ,β

(66.14a)

(wϕ)∗ T3α,α = −t¯3∗ , t¯3∗ =

 −

L h0

2

L4 t¯3 D0 h 0

(wϕ)∗ T3α =0

(66.14b) (66.14c)

∗ Bα,α − B3∗ = 0

(66.14d)

and the possible boundary conditions become   ∗  = 0 or u ∗α ∂ is prescribed n β Tαβ ∂  ∂w∗  = 0 or is prescribed ∂ ∂n ∂   V ∗ ∂ = 0 or w∗ ∂ is prescribed

(66.15a)





(w)∗  n α n β Mαβ 

 (ϕ)∗  n β Mαβ 

∂

 = 0 or ϕα∗ ∂ is prescribed

  n β Bβ∗ ∂ = 0 or φ0∗ ∂

(66.15b) (66.15c) (66.15d) (66.15e)

in which V ∗ is the generalized shear force  (w)∗ Mαβ,β (x)

∗

V (x) := n α (x) −

  c

 +

L h0

2

 (wϕ)∗ T3α (x)

+

∂ T (w)∗ (x) ∂t

 L 2 (w) T (w)∗ (xc ) δ(x − xc ), T (w)∗ = T D0 h 0

(w) with T (w) := tα n β Mαβ being the twisting moment.

(66.16)

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66.3 Numerical Implementation Inserting the expressions for the semi-integral fields in terms of primary fields into (14), one can find that the governing equations are the PDE of the 4th order for the deflections w∗ and 3rd order for in-plane displacements u ∗α and rotations ϕα∗ . In general, the accuracy of approximation of derivatives of field variables is decreasing with increasing the order of derivatives. In what follows, we shall consider transversely isotropic piezoelectric material without  gradation of material coefficients,  E in-plane E E E E E E /2, e32 = e31 . Then, it can be shown = c11 , c23 = c13 ,c66 = c11 − c12 when c22 that introducing new field variables m ∗ (x) := ∇ 2 w∗ (x), sα∗ (x) := ∇ 2 u ∗α (x),

f α∗ (x) := ∇ 2 ϕα∗ (x)

(66.17)

we achieve the decomposition of the original system of the PDE into the system of the PDE with not higher than 2nd order derivatives [17]. The enhancement of DOFs is the price which should be paid for decreasing the order of derivatives.  For approximation of field variables g ∗ ∈ w∗ , m ∗ , u ∗α , sα∗ , ϕα∗ , f α∗ , we shall use the MLS (Moving Least Square) approximation [18–20] around the node xq N  q

g ∗ (x) ≈

gˆ ∗a¯ φ (q,a) (x),

a¯ = n(q, a)

(66.18)

a=1 N  q

∗ (x) g,α

≈

N  q

gˆ

∗a¯

(q,a) φ,α (x),

∗ g,αβ (x)

a=1

≈

(q,a)

gˆ ∗a¯ φ,αβ (x)

a=1

where a¯ is the global number of the a-th node from the influence domain of xq , N q is the number of nodal points in the influence domain (which is smaller than the total number of nodes), and φ (q,a) (x) is the shape function associated with the node n(q, a). The MLS approximation belongs to mesh-free methods, when the mesh of nodal points is sufficient instead of discretisation elements. Since the MLS approximation does not satisfy the Kronecker delta property [18, 21], the nodal unknown gˆ a¯ is different from the nodal value g(xa¯ ). Note that the evaluation of shape functions is time consuming. Therefore the strong formulation (collocation of both the governing equations and boundary conditions at nodal points) leads to more efficient computation than the weak formulation, where the evaluation of shape functions at each integration point is required. Finally, the implementation of boundary conditions should be discussed with bearing in mind introduction of new field variables. Firstly, let us consider the plate with traction-free and clamped edges (CE):  ∗ (xb )∂ = 0, n β (xb )Tαβ

 w∗ (xb )∂ = 0,

 ∗ n α (xb )w,α (xb )∂ = 0,

 ϕα∗ ∂ = 0 (66.19)

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with Dirichlet and/or Neumann electrical b.c. on ∂, respectively ∂  φ0∗ (xb )∂ = φ¯ 0 (xb ),

 n α (xb )Bα∗ (xb )∂ = 0.

(66.20)

Note that the Dirichlet b.c. (192 ), (194 ) and (201 ) can be easily implemented by collocating the approximation of the prescribed deflections and/or electric potential at the boundary node xb ∈ ∂. However, it is difficult to collocate the Neumann b.c. at a corner point, where the normal vector is not defined uniquely. This problem can be solved by consideration of the governing equation (171 ) in weak sense on the local subdomain b around the node xb ∈ ∂. Similarly, consideration of the case of electrical Neumann b.c. (202 ) and the traction-free b.c. in weak sense (191 ) could solve the difficulty associated with nonunique definition of the normal vector at corner. Now, let us consider the plate with traction-free and simply supported edges (SSE):   ∗ (xb )∂ = 0, w∗ (xb )∂ = 0, n β (xb )Tαβ    (ϕ)∗ (w)∗ b  (x ) = 0, n β (xb )Mαβ (xb ) n α (xb )n β (xb )Mαβ ∂

∂

=0

(66.21)

with Dirichlet and/or Neumann electrical b.c. on ∂, respectively ∂  φ0∗ (xb )∂ = φ¯ 0 (xb ),

 n α (xb )Bα∗ (xb )∂ = 0.

(66.22)

Also now, the boundary conditions on rectangular plate can be expressed in terms of field variables and their derivatives with order being not higher than 2.

66.4 Numerical Examples Let us consider a square plate L × L (L = 1) with constant thickness h = h 0 = L/50. For such a thin plate the KLT is applicable. As regards the material composition, we assume either homogeneous or FGPM plates with power-law gradation of material coefficients according Eq. (66.11) and constant Poisson ratio ν = 0.3. The plate is subjected to stationary voltage loading between the top and bottom surfaces +∗ (x) = Φ¯ + = const and t¯3 (x) = 0. The plate edges are traction-free (TF) with either clamped (CE) or simply supported (SSE) boundary conditions, while the electric boundary conditions are assumed to be either Dirichlet (DE) or Neumann (NE) electric b.c. In numerical computations, we have used uniform grid of nodes for MLS approximation (with δ being the shortest distance between nodes) with Gaussian weights with the shape parameter ca = δ, and the radius of the interpolation support domain equals to 3.001 δ. Firstly, consider a homogeneous plate. The deflection responses with using 4 different regular grids of nodes are shown in Fig. 66.1. The finite deflection response to pure electric loading even in the homogeneous plate is due to electro-elastic coupling.

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Fig. 66.1 Deflections in the homogeneous plate subjected to uniform voltage between the top and bottom surfaces

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The response in plate with SSE is weaker than in the plate with CE. Convergence of numerical results with increasing the number of nodes can be observed. As regards the results for the electric potential (Fig. 66.2), they are the same for both the plates with CE and/or SSE. It means that the influence of deflections on the electric field is negligible. The comparison of the deflection responses of the homogeneous and FGPM plates on the electric loading is shown in Fig. 66.3. The gradation of piezoelectric coefficient increases the deflection response as compared with the homogeneous plate. On the other hand, lower deflections are observed in plates with graded elastic coefficient (plate becomes stiffer) as well as with graded dielectric permittivity (more supplied energy is consumed for polarization). Since the deflection response in the plate with SSE is much weaker than in the plate with CE, the gradation of the Young modulus and/or dielectric permittivity can results into negative deflections. Owing to the electro-elastic coupling, there is finite response also for in-plane displacements even in the homogeneous plate (Fig. 66.4). It can be seen that the gradation of dielectric permittivity does not affect the in-plane displacements. Moreover, the same in-plane displacement response is achieved also in the plate with SSE. Recall that the u-w coupling is present only if ζ = 0 and the u-w coupling is proportional to ε2 , if the electro-elastic coupling is proportional to ε. Finally, the response of the

Fig. 66.2 The electric potential in the homogeneous plate subjected to uniform voltage between the top and bottom surfaces

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Fig. 66.3 Comparison of deflection responses on voltage loading in homogeneous and FGPM plates

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Fig. 66.4 In-plane displacements response on the voltage loading in homogeneous and/or FGPM plates

electric field potential on the voltage loading is shown in Fig. 66.5. The gradation of elastic coefficients does not affect the distribution of electric field potential, because the influence of elastic fields on electric field is negligible in case of voltage loading. Therefore the difference between the electric field potentials in the plates with CE and SSE is not observable.

66.5 Conclusions The unified mathematical formulation has been developed for bending of FGPM plates which allow us to get results within the assumptions of three plate bending theories. The original 3D problem is converted completely to 2D problem for a plate structure. The power-law gradation of material coefficients in transversal as well as in-plane directions are allowed. The in-plane gradations make the plate bending problems much more difficult because the governing equations are given by PDE with variable coefficients. Combining the strong formulation with compactly supported mesh-free approximation, we developed an effective element-free method for bending of piezoelectric plates. The difficulties with higher order derivatives of field variables have been resolved by decomposing the original PDE into the PDEs with

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Fig. 66.5 Electric potential field response on the voltage loading in homogeneous and/or FGPM plates

lower-order derivatives. The developed formulation as well as the numerical implementation have been verified in test examples with assuming stationary and uniform voltage loading between the top and bottom surfaces of the plate. Acknowledgements The financial support by the Slovak Research and Development Agency under the contract No. APVV-14-0440 is greatly acknowledged.

References 1. Rao, S.S., Sunar, M.: Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey. Appl. Mech. Rev. 47, 113–123 (1994) 2. Tani, J., Takagi, T., Qiu, J.: Intelligent material systems: application of functional materials. Appl. Mech. Rev. 51, 505–521 (1998) 3. Carrera, E., Brischetto, S., Nali, P.: Plates and shells for smart structures. Wiley & Sons, Chichester (2011) 4. Pohanka, M.: Overview of piezoelectric biosenzors, immunosensors and DNA sensors and their applications. Materials 11, 448 (2018) 5. Zhu, X.H., Meng, Z.Y.: Operational principle, fabrication and displacement characteristic of a functionally graded piezoelectric ceramic actuator. Sen. Actuators A Phys. 48, 169–176 (1995) 6. Wu, C.C.M., Kahn, M., Moy, W.: Piezoelectric ceramics with functional gradients: a new application in material design. J. Am. Ceram. Soc. 79, 809–812 (1996)

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7. Shelley, W.F., Wan, S., Bowman, K.J.: Functionally graded piezoelectric ceramics. Mater. Sci. Forum 308–311, 515–520 (1999) 8. Li, X.Y.: Axisymmetric problems of functionally graded circular and annular plates with transverse isotropy. Ph.D. Thesis, Zhejiang University, Hanghzhou, China (2017) 9. Behjat, B., Salehi, M., Armin, A., Sadighi, M., Abbasi, M.: Static and dynamic analysis of functionally graded piezoelectric plates under mechanical and electrical loading. Sci. Iran. B 18, 986994 (2011) 10. Yang, Z.X., He, X.T., Li, X., Lian, Y.S., Sun, J.Y.: An electroelastic solution for functionally graded piezoelectric circular plates under the action of combined mechanical loads. Materials 11, 1168 (2018) 11. Alik, H., Hughes, T.J.R.: Finite element method for piezoelectric vibration. Int. J. Num. Meth. Eng. 2, 151–168 (1979) 12. Tzou, H.S., Tseng, C.I.: Distributed piezoelectric sensor/actuator design for dynamic measurement/control of distributed parameter systems: a piezoelectric finite element approach. J. Sound Vibr. 138, 17–34 (1990) 13. Tzou, H.S.: Piezoelectric Shells, 2nd edn. Springer Nature B.V, Dordrecht (2019) 14. Sator, L., Sladek, V., Sladek, J.: Coupling effects in elastic analysis of FGM composite plates by mesh-free methods. Compos. Struct. 115, 100–110 (2014) 15. Sator, L., Sladek, V., Sladek, J., Young, D.L.: Elastodynamics of FGM plates by mesh-free method. Compos. Struct. 140, 309–322 (2016) 16. Sator, L., Sladek, V., Sladek, J.: Multi-gradation coupling effects in FGM plates. Compos. Struct. 171, 515–527 (2017) 17. Sladek, V., Sladek, J., Sator, L.: Physical decomposition of thin plate bending problems and their solution by mesh-free methods. Eng. Anal. Bound. Elem. 37, 348–365 (2013) 18. Lancaster, P., Salkauskas, K.: Surfaces generated by moving least square methods. Math Comput. 37, 141–158 (1981) 19. Sladek, V., Sladek, J., Zhang, Ch.: Computation of stresses in non-homogeneous elastic solids by local integral equation method: a comparative study. Comput. Mech. 41, 827–845 (2008) 20. Sladek, V., Sladek, J.: Local integral equations implemented by MLS approximation and analytical integrations. Eng. Anal. Bound. Elem. 34, 904–913 (2010) 21. Atluri, S.N.: The meshless method, (MLPG) for domain and BIE discretizations. Tech Science Press, Forsyth (2004)

Part XIX

Functional Materials

Chapter 67

Temperature-Dependent Raman Spectroscopy of Graphitic Nanomaterials Prabhakar Misra, Daniel Casimir and Raul Garcia-Sanchez

Abstract The sp2 carbonaceous molecules possess a single atomic type per unit cell, which makes these materials very good candidates for quantum mechanical studies associated with their vibrational and electronic energy levels. Significant findings, such as the Kohn anomaly, electron-phonon interactions, and other exciton-related effects, associated with these molecules can be transported to other 2-D materials. Information derived from the distinctive Raman bands from a single layer of carbon atoms also aids in gaining insight into new physics from such materials and other graphitic nanomaterials. The present paper focuses on our investigations of the G, D, and G bands of graphene and graphite, and the specific information provided by each. The G-band peak located at ~1586 cm−1 , shared by all sp2 carbons, has been used by us extensively in the estimation of thermal conductivity and thermal expansion characteristics linked to single-walled carbon nanotubes. In addition, we have investigated functionalized graphene nanoplatelets. For all three materials (graphene, graphite, and functionalized graphene nanoplatelets), we made use of the relationship discovered by Tuinstra and Koenig based on the relative intensities of the D and G Raman bands. In addition to the analysis based on Raman spectroscopy of the nanomaterial samples, SEM visualization/dimensional analysis was also performed on the graphene nanoplatelet samples. The bulk macroscopic 3-D character of graphite was clearly apparent, in contrast to the 2-D nature of graphene. However, the graphene nanoplatelets exhibited both 2-D and 3-D characteristics, without one dimension dominating the other. Keywords Temperature-dependence · Raman spectroscopy · Graphitic nanomaterials

P. Misra (B) · D. Casimir · R. Garcia-Sanchez Laser Spectroscopy Laboratory, Department of Physics & Astronomy, Howard University, DC, WA 20059, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_67

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67.1 Graphitic Nano-carbons Overview Graphene is a two-dimensional carbon nanomaterial with a single layer of sp2 hybridized carbon atoms arranged in a crystalline structure of six-membered rings [1, 2]. Figure 67.1 illustrates the hexagonal lattice of a perfectly flat graphene sheet and the resulting nanotube after it’s rolled along the vector labeled Ch . The shaded portion of the nanotube in Fig. 67.1b represents one unit cell of the resulting armchair nanotube in this case, and it results from rolling the initial planar sheet in Fig. 67.1a, so that points A and C coincide with points B and D, respectively. Ch is known as the chiral vector and is constructed from the vector addition of the graphene basis vectors a1 and a2 . The integer number of each of the basic lattice vectors used in the construction, n and m, designated for a1 and a2 respectively, is arbitrary with the only provision that √ (0 ≤ |m| ≤ n).√The Cartesian components of the lattice vectors a are (a 3/2, a/2) and (a 3/2, −a/2) respectively, where the quantity a = a1 and √ 2 aC-C 3 = 2.46 Å. The quantity aC-C is the bond length between two neighboring carbon atoms in the hexagonal lattice equal to 1.42 Å. The chiral vector Ch is usually written in terms of the two integers n and m as C h = na1 + ma2

(67.1)

 |C h | = a n2 + m2 + nm

(67.2)

and has a magnitude of

Fig. 67.1 a Unit cell of a (3, 3) carbon nanotube, b resulting (3, 3) armchair carbon nanotube

67 Temperature-Dependent Raman Spectroscopy …

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which equals the carbon nanotube’s circumference. In a fashion similar to applying the above rolling operation on the graphene unit cell in Fig. 67.1 for the construction of single walled carbon nanotubes, graphite is also described in terms of stacking multiple graphene layers one atop the other. The newest graphitic nanomaterial in our investigation of the thermo-mechanical properties are functionalized graphene nanoplatelets, which are comprised of platelet-shaped graphene sheets, identical to those found in SWCNT but in planar form. All of the nanoplatelet samples we used (functionalized oxygen, nitrogen, argon, ammonia, carboxyl and fluorocarbon) have similar shape. Graphene nanoplatelet aggregates (aggregates of sub-micron platelets with a diameter of 190 cm height), so it will not get longer than 30 h total.

97.4 Conclusions The presented work was an attempt at using intelligent, automated CAD modelling for a customized medical product. The approach was proven to be successful in the initial tests of the wrist orthosis—it was possible to gather data and obtain a correct digital and physical model of an orthosis for different patients, without making any changes in the CAD model. However, certain aspects of the presented approach must be improved and the other parts of the whole process must be also partially or fully automated for the concept to be economically justified. The intelligent model should be less prone to errors generated by slightly wrong measurement technique. The measurement itself should be automated in order to not require a qualified 3D scanner operator each time. On the other hand, the 3D printing process should be optimized, to obtain a working orthosis in shorter time, which will translate into lower costs. The authors continue on development and studies of all the partial processes mentioned above and throughout the paper, also for different products (leg orthosis, hand prosthesis), up until a fully automated and low-cost system will be obtained, to bring 3D printed customized orthotic and prosthetic supplies to more patients at lower price, what is especially important regards children patients.

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Acknowledgements The studies were realized with a support from Polish National Center for Research and Development, in the scope of the “LIDER” program (grant agreement no. LIDER/14/0078/L-8/16/NCBR/2017).

References 1. Marro, A., Bandukwala, T., Mak, W.: Three-dimensional printing and medical imaging: a review of the methods and applications. Curr. Probl. Diagn. Radiol. 45, 2–9 (2016). https://doi. org/10.1067/j.cpradiol.2015.07.00 2. Tukuru, N., Gowda, S.K.P., Ahmed, S.M., Badami, S.: Rapid prototype technique in medical field. Int. J. PharmTech Res. 1(4), 341–344 (2008) 3. Chua, C.K., Leong, K.F., Lim, C.S.: Rapid Prototyping: Principles and Applications. World Scientific Publishing Co. Pte. Ltd., Singapore, pp. 25–35 (2010) 4. Otawa N., Sumida T., Kitagaki H., Sasaki K., Fujibayashi S., Takemoto M., Nakamura T., Yamada T., Mori Y., Matsushita T.: Custom-made titanium devices as membranes for bone augmentation in implant treatment: modeling accuracy of titanium products constructed with selective laser melting. J. Craniomaxillofac. Surg. 43(7), 1289–1295 5. Banaszewski, J., Pabiszczak, M., Pastusiak, T., Buczkowska, A., Kuczko, W., Wichniarek, R., Górski, F.: 3D printed models in mandibular reconstruction with bony free flaps. J. Mater. Sci. Mater. Med. 29, 23 (2018) 6. Górski F., Wichniarek R., Kuczko W., Banaszewski J., Pabiszczak M., Application of low-cost 3D printing for production of CT-based individual surgery supplies. In: World Congress on Medical Physics and Biomedical Engineering 2018, June 3–8, Prague, Czech Republic (2018) 7. Kelly S., Paterson A., Bibb R. J.: A review of wrist splint designs for additive manufacture. In: Rapid Design, Prototyping and Manufacture Conference, Loughbrough, Great Britain, p. 12 (2015) 8. Huotilainen, E., Jaanimets, R., Valasek, J., Marcian, P., Salmi, M., Tuomi, J., Makitie, A., Wolff, J.: Inaccuracies in additive manufactured medical skull models caused by the DICOM to STL conversion process. J. Craniomaxillofac. Surg. 42, e259–e265 (2014). https://doi.org/ 10.1016/j.jcms.2013.10.001 9. Górski, F., Wichniarek, R., Zawadzki, P., Hamrol, A.: Computation of mechanical properties of parts manufactured by fused deposition modeling using finite element method. In: Advances in Intelligent Systems and Computing. Springer, 368, pp. 403–413. https://doi.org/10.1007/9783-319-19719-7_35 10. Cha, H.Y., Lee, K.H., Ryu, H.J., Joo, I.W., Seo, A., Kim, D., Kim, S.J.: Ankle-Foot Orthosis Made by 3D Printing Technique and Automated Design Software. Republic of Korea, Hindawi (2017) 11. Verhagen, W.J.C., Bermell-Garcia, P., Van Dijk, R.E.C., Curran, R.: A critical review of knowledge-based engineering: an identification of research challenges. Adv. Eng. Inform. 26(1), 5–15 (2012) 12. Burdzinska, M.: Prototyping of customized wrist orthesis using additive manufacturing technologies, Diploma thesis, Poznan University of Technology (2015) 13. Evill, J.: Cortex (cited 28 Oct 2015). Available from: http://jakevilldesign.dunked.com/cortex 14. Fitzpatrick, A., Mohanned, M., Collis, P., Gibson, I.: Design of a patient Specific, 3D printed Arm Cast. In: DesTech Conference Procedings, The International Conference on Design and Technology, Geelong (2017) 15. Wierzbicka, N., Wesołowska, I.: Design automation of individualized limb orthoses. Diploma thesis, Poznan University of Technology (2018)

Part XXVI

Impact Loading and Failure of Structures and Materials

Chapter 98

Research on the Permeability Model of Fractal Fractured Media in 3D Coordinate System Huan Zhao, Wei Li, Lei Wang, Xin Ling, Dandan Shan, Bing Li and Zhan Su

Abstract The fractured media is the important storage and migration space of underground oil and gas in petroleum engineering. Because of the complex spatial distribution of fractured media, it brings enormous difficulties to numerical analysis of fractured formation permeability. Based on the fractal geometry theories, regarding fractal fracture as the basic model of describing the fractured medium distribution, building the fractal description model of multiple fracture permeability system whose strike angle and dip angle are different, according to the relationship between fracture occurrences (strike angle, dip angle) and ground stress field, building the calculation model of fractal fractured permeability tensors in the local principal stress coordinate system, and realizing the projection mapping of fractal fracture permeability in the whole geographic coordinates. Accounting for the fractal fracture permeability in different coordinate systems by taking Budate Group in Beier Depression of Hailare Oilfield as an example. Keywords Fractured formation · Fractal fracture · Permeability · Ground stress

98.1 Introduction The fractured media is one of the important storage space of oil and gas. It’s main factor for controlling oil-gas concentration and production capacity in low permeability reservoir. Some researchers, such as Zhou et al. [1], Yangsheng et al. [2], Hao et al. [3] found that the spatial distribution and anisotropic of fractured media are so quite complicated and obvious that it’s difficult to make a quantitative analysis for fractured media. Some researchers, such as Heping and Zhida [4], Chang and Yortsos [5], Acuna et al. [6], Fu Xiaofei et al. [7], Fangfeng et al. [8] showed that fractured media whose spatial distribution is high-unordered has fractal feature.

H. Zhao (B) · W. Li · L. Wang · X. Ling · D. Shan · B. Li · Z. Su Department of Petroleum Engineering, Northeast Petroleum University, Daqing, Heilongjiang, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_98

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The fractured media system model has always been the focus and difficulty in oil-gas field development engineering. There are three main kinds of models: The equivalent continuous model, discrete model and comprehensive model. Ganzer [9], Zhandong et al. [10] found there were three main kinds of models: the equivalent continuous model, discrete model and comprehensive model. The equivalent continuous model distributes the fracture property into every unit averagely. Bourne et al. [11], Olson et al. [12] found the discrete model simulates fractured media system by deterministic modeling or random modeling in the base of fracture computational formulas of Mathematics and Geomechanics. Cacas et al. [13], Swaby and Rawnsley [14] found the comprehensive model reflects the anisotropy of fractured system by combining deterministic modeling with random modeling and Artificial Intelligence Technology synthetically. This paper discretized the fractured media system which is three-dimensional distribution into limited units by discrete model. And described the fractured media distribution of unit by the fractal fracture of fractal geometry, and given the nonlinear and quantitative model of the fractal fractured permeability in every unit. And built the permeability tensor calculation model of fractured media system in local principal stress coordinate system by the relationship between fracture occurrence and stress field, and achieved the conversion of fractal fractured permeability between local principal stress coordinate system and global geographic coordinate system.

98.2 The Fractal Fractured Media Description Model of Discrete Unit According to fractal geometry theory, there is a set of fractal fracture media in DT (DT ≤ 3) euclidean space, then the measurement scale r, accumulated number N and accumulated length L have the power-law relationship: N = N0 r −D

(98.1)

L = L 0 r −D

(98.2)

where N 0 is the original number of fractal fracture media; L 0 is the original scale of fractal fracture media; D is the factuality dimension of fractal fracture media. In 3-dimension Euclidean space (DT = 3), there are some discrete units whose side length is L T (L Tmin ≤ L T ≤ L Tmax ). There are m fractal fracture medias in one discrete unit, as shown in Fig. 98.1. For fractal fracture media i: Ni = N0i r −Di

(1 ≤ i ≤ m)

(98.3)

L i = L 0i r −Di

(1 ≤ i ≤ m)

(98.4)

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Fig. 98.1 The fractal fractured media description of discrete unit

where L Tmax is high limit of discrete unit; L Tmin is low limit of discrete unit; N 0i is the original number of fractal fracture media i; L 0i is the original scale of fractal fracture media i; N i is the accumulated number of fractal fracture media i; L i is the accumulated length of fractal fracture media i; Di is the factuality dimension of fractal fracture media i.

98.3 The Permeability of Fractal Fractured Media For the 3D fractured media whose distribution is random, the permeability tensor of fractal fractured media i is: K fi =

bi3 L i (δ jk − n i j n ik ) 12 A

(98.5)

Then the permeability tensor of fractal fractured media (group m) is: K fm =

m  b3 L i i

i=1

12 A

(δ jk − n i j n ik )

(98.6)

where δ jk is Kronecker Delta, when j = k, δ jk = 1 or else δ jk = 0; nij and nik are the projection of permeability in coordinate axis; A is seepage area of unit, A = L 2T . The linear density ρ l , areal density ρ s and volume density ρ v are the description parameters of fractured media distribution density. For fractured media in strict mathematical meaning: ρl = ρs = ρv

(98.7)

Then the areal density of fractal fractured media i could be calculated by formula (98.4): ρsi =

L i (r < L T ) L 2T

(98.8)

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where ρ si is the areal density of fractal fractured media i. Bringing formula (98.8) into formula (98.5): K fi =

bi3 ρsi (δ jk − n i j n ik ) 12

(98.9)

Bringing formula (98.8) into formula (98.6): K fm =

m  b3 ρsi i

i=1

12

(δ jk − n i j n ik )

(98.10)

The permeability tensor formulas of fractal fractured media of discrete unit are corresponding equivalent forms, such as formulas (98.5), (98.9), (98.6) and (98.10).

98.4 The Permeability of Fractal Fractured Media in Local Principal Stress Coordinate System For the fractal fractured media(group m) in principal stress coordinate system, the strike angle of fracture i is θ spi , the dip angle of fracture i is θ sti . Then the relationship between fracture and principal stress coordinate system in group i is shown in Fig. 98.2. → The normal unit vector of fractal fracture i is n σ i = [n σ 1i , n σ 2i , n σ 3i ]T . According to geometrical relationship between θ spi and θ sti ⎡

⎤

⎡

⎤ cos θsti ⎢ ⎥ ⎣ n σ 2i ⎦ = ⎣ − sin θsti cos θspi ⎦ sin θsti sin θspi n σ 3i Fig. 98.2 The fracture area in principal stress coordinate system

n σ 1i

(98.11)

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Then the permeability tensor formulas of fractal fractured media i in local principal stress coordinate system could be calculated by formulas (98.9) and (98.11): ⎡ ⎤ ⎤ 1 − n 2σ 1i K f σ 1i 3 ⎥ b ρsi ⎢ ⎢ 2 ⎥ = ⎣ K f σ 2i ⎦ = i ⎣ 1 − n σ 2i ⎦ 12 K f σ 3i 1 − n 2σ 3i ⎡

K f σi

(98.12)

Then the permeability tensor formulas of fractal fractured media (group m) is: ⎡ K f σi =

m 

K f σ 1i

⎡

⎤

⎢ ⎥ ⎣ K f σ 2i ⎦ = i=1 K f σ 3i

m 

bi3 ρsi

i=1

12

1 − n 2σ 1i

⎤

⎢ 2 ⎥ ⎣ 1 − n σ 2i ⎦ 1−

(98.13)

n 2σ 3i

98.5 The Permeability of Fractal Fractured Media in Global Geographic Coordinate System The direction is different between local principal stress coordinate system and global geographic coordinate system, there are coordinates translation. The angles between principal stress axes σ 1 , σ 2 , σ 3 and geographic coordinate system x, y, z are α 11 , α 12 , α 13 and α 21 , α 22 , α 23 and α 31 , α 32 , α 33 respectively. Then the matrix transformation of projected coordinate system by formula (98.11) is: ⎡

⎤ ⎡ ⎤ n t xi cos α 11i cos θsti − cos α 21i sin θsti cos θspi + cos α 31i sin θsti sin θspi ⎢ ⎥ ⎢ ⎥ ⎣ n t yi ⎦ = ⎣ cos α 12i cos θsti − cos α 22i sin θsti cos θspi + cos α 32i sin θsti sin θspi ⎦ n t zi cos α 13i cos θsti − cos α 23i sin θsti cos θspi + cos α 33i sin θsti sin θspi (98.14) Then the permeability tensor formulas of fractal fractured media i in local principal stress coordinate system could be calculated by formulas (98.9), (98.14): ⎡

K f t xi

⎡

⎤ bi3 ρsi

1 − n 2t xi

⎤

⎢ ⎥ ⎢ 2 ⎥ K f ti = ⎣ K f t yi ⎦ = ⎣ 1 − n t yi ⎦ 12 K f t zi 1 − n 2t zi

(98.15)

Then the permeability tensor formulas of fractal fractured media (group m) is:

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⎡ K f tm =

m 

K f t xi

⎡

⎤

⎢ ⎥ ⎣ K f t yi ⎦ = i=1 K f t zi

m  i=1

bi3 ρsi 12

1 − n 2t xi

⎤

⎢ 2 ⎥ ⎣ 1 − n t yi ⎦ 1−

(98.16)

n 2t zi

98.6 The Model Application The parameters of fractured media, the occurrence, slit width, distribution fractal dimension and stress direction, should be identified before the model application. This paper takes Budate Group in Bill Pit of Hailer Oilfield as the object. The maximal horizontal crustal stress of Budate Group formation distributes from NE92° to NE117°, the average value is NE103°. The vertical in situ stress gradient is from 0.0237 to 0.0262 MPa/m, the average value is 0.0257 MPa/m. Take Budate Group Well B30 for instance, the centre is well site and analysis area is one quarter area, the analysis size is 1000 m; formation depth is 1800 m, the analysis thickness is 100 m. The initial parameters of the fracture which is near Budate Group Well B30 are shown in Table 98.1. The relationship between the dimension of fractal fracture and the scale distribution is shown in the Fig. 98.3, the dimension of fractal fracture ranges from 1.5 to 2.0 (Fig. 98.4). Take the above parameters into the permeability tensor formulas, then achieve the permeability tensor transformations and anisotropy solutions of fractal fracture in different coordinate systems. The fractal fracture permeability distributions of Budate Group Well B30 in z axi of principal stress coordinate system and geographic coordinate system are shown in Figs. 98.5 and 98.6. The permeability of the above two figures distribute from 0.8 × 10−3 to 1.5 × −3 10 µm2 . Because of the angle variation which is between in principal stress coordinate system and global geographic coordinate system, the permeability distribution of the above two figures are consistent, but their species distributions are different. Table 98.1 The initial parameters of natural fracture in well B30

Parameter

Value

Parameter

Value

fractal dimension

1.749

Initial size (m)

320

fracture dip angle (°)

56

dip angle variation (°)

30

azimuth angle (°)

107

azimuth angle variation (°)

40

Initial numbers (pieces)

162

Numbers variation (pieces)

40

Initial width (mm)

3

average width (mm)

0.6

98 Research on the Permeability Model of Fractal Fractured Media … Fig. 98.3 The fractal dimension distribution of fractures

Fig. 98.4 The vertical in situ stress gradient distribution

Fig. 98.5 The fractal fracture permeability distributions in z axi of principal stress coordinate system

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Fig. 98.6 The fractal fracture permeability distributions in z axi of geographic coordinate system

98.7 Conclusions (1) Discretized the fractured media system which is three-dimensional distribution into limited units, and given the nonlinear and quantitative model of the fractal fractured permeability in every unit by by the fractal fracture of fractal geometry. (2) Built the permeability tensor calculation model of fractured media system in local principal stress coordinate system by the relationship between fracture occurrence and stress field, and achieved the conversion of fractal fractured permeability between local principal stress coordinate system and global geographic coordinate system. (3) Made the model simulation of Budate Group in Beier Depression of Hailare Oilfield by The permeability tensor formulas of fractal fractured media, and achieved the conversion of fractal fractured permeability in different coordinate systems. Acknowledgements National Natural Science Foundation of China (Nos. 51490650 and No. 51774093) is gratefully acknowledged.

References 1. Zhou, X., Cao, C., Yuan, J.: The research actuality and major progresses on the quantitative forecast of reservoir fractures and hydrocarbon migration law. Adv. Earth Sci. 18(3), 398–404 2. Yangsheng, Z., Xiaohai, W., Kanglian, D., et al.: Unsymmetry of scale taransformation of rock mass anisotropy. Chin. J. Rock Mech. Eng. 21(11), 1594–1597 (2002) 3. Hao, M., Hou, J., Hu, Y., et al.: Scale effects of anisotropy in fractured low permeability reservoirs. Pet. Explor. Dev. 34(6), 724–727 (2007)

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4. Heping, X., Zhida, C.: Fractal geometry and fracture of rock. Acta Mech. Sinica 20(3), 264–275 (1988) 5. Chang, J., Yortsos, Y.C.: Pressure-transient analysis of fractal reservoirs. SPEFE 5(1), 31–38 (1990) 6. Acuna, J.A., Ershaghi, I., Yortsos, Y.C.: Practical application of fractal pressure-transient analysis in naturally fractured reservoirs. SPEFE 10(3), 173–179 (1995) 7. Fu Xiaofei, S., Yuping, L.Y., et al.: Fractal characteristic and geological meaning of fault and fracture. Earth Sci. J. China Univ. Geosci. 32(2), 227–234 (2007) 8. Fangfeng, X., Guiyan, H., Le, L., et al.: Fractal study on oil and gas field size distribution of the world. J. Bacis Sci. Eng. 19(1), 95–103 (2011) 9. Ganzer, L.J.: Simulating fracturing reserviros using adaptive dual continuum. SPE75233 (2002) 10. Zhandong, F., Junsheng, D., Xiatian, W., et al.: Quantitative calculation of the tectonic fracture permeability in different coordinate systems. Acta Petrolei Sinica 32(1), 135–149 (2011) 11. Bourne, S.J., Rijkels, L., Stephenson, B.J., et al.: Predictive modeling of naturally fracturingd reservoirs using geomechanics and flow simulation. Geoarabia 6(1), 27–42 (2001) 12. Olson, J.E., Qiu, Y., Holder, J., et al.: Constraining the spatial distribution of fracturing networks in naturally fracturingd reservoirs using fracturing mechanics and core measure ments. SPE71342 (2001) 13. Cacas, M.C., Daniel, J.M., Letouzey, J.: Nested geological modeling of naturally fracturingd reservoirs. Pet. Geosic. 7, S43–S52 (2001) 14. Swaby, P.A., Rawnsley, K.D.: An interactive 3D fracturing modeling environment. In: Society of Petroleum Engineers Petroleum Computer Conference, Texas, USA, SPE36004, pp. 62–65 (1996)

Chapter 99

Modelling and the FEM Analysis of the Effects of the Blast Wave on the Floor of a Vehicle According to the AEP-55 Vol. 2 Methodology ´ Marek Swierczewski, Grzegorz Sławinski ´ and Piotr Malesa Abstract The article presents the results of numerical simulations regarding the effects of the blast wave on a wheeled armoured vehicle additionally equipped with external multi-layer energy-absorbing panels, the aim of which is to minimise the risk of significant destructions of the vehicle’s construction and thus to increase the safety of vehicle’s occupants. In order to assess the effectiveness of the used energyabsorbing structures, the results obtained for a vehicle with the protective system and for a vehicle without the protective system have been compared. Numerical simulations have been conducted using the finite element method and the LS-DYNA software, which enables the modelling of the effects of the blast wave thanks to the CONWEP option implemented in it. The scope of conducted numerical analyses included checking the capabilities of absorbing the energy of the blast wave by energy-absorbing panels mounted to the bottom of the vehicle. The mass of an explosive amounting to 6 kg and its location towards the vehicle have been assumed based on the assumptions included in the AEP-55 vol. 2 test methodology. The effectiveness of multi-layer energy-absorbing panels have been assessed on the basis of the comparison of changes in acceleration values and courses in characteristic construction points of the vehicle. Keywords FEM analysis · Improvised explosive device · Blast wave

99.1 Introduction Land forces are the biggest military potential of every country. Their constant modernisation and development at the end of the 20th and beginning of the 21th century show that heavy weapons (tanks or self-propelled howitzers) are being replaced by vehicles which are characterised by high mobility (Fig. 99.1). The use of such military equipment renders it possible to conduct quick military missions with the possibility of a surprise attack or a fast retaliatory action. ´ M. Swierczewski (B) · G. Sławi´nski · P. Malesa Military University of Technology, Gen. Sylwestra Kaliskiego St. 2, 00-908 Warsaw, Poland e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_99

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Fig. 99.1 Military potential of European countries

Conducting quick actions requires the use of vehicles which are able to quickly move soldiers together with the equipment to the military activity zone. Therefore, a fast development of wheeled armoured vehicles has been observed since the beginning of the Second World War. The design of such vehicles enables also retrofitting depending on the army’s needs, without the necessity to introduce changes in the supporting structure or wheels and the suspension system [1]. Current armed conflicts, which are caused by territorial, social and political, ideological or religious reasons or by a political upheaval, have forced the introduction of further significant modernisation of armoured vehicles. Such changes have been caused by the way of conducting fights called asymmetric conflicts, in which one of the parties has a significantly higher military potential in terms of the number of soldiers, technology and equipment [2, 3]. The asymmetric enemy usually aims at such a resolution of the conflict so that the other party does not decide to use its dominance, even if it is tactically and technically possible [4]. Attacks conducted most often in order to achieve such a purpose are surprise attacks with Improvised Explosive Devices (IED) mounted in bomb cars or buried under or next to public roads [5, 6]. Explosives placed this way are detonated at the moment when an enemy’s vehicle passes by the blast zone. The above-described way of conducting fights using armoured military vehicles forced significant design changes (Fig. 99.2) [7]. Such changes usually concern: – increasing clearances (i.e. increasing the distance between the explosive and the floor plate); – changing the armour protection of a traditional (ballistic) vehicle to a system which enables the dissipation of the blast wave energy created as a result of the IED detonation; – using a modular design; – using deflectors [10].

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Fig. 99.2 BRDM2 manufactured between 1963 and 1989 [8], Cougar 4 × 4 MRAP [9]

In order to check the passive protection level of new designs before they are launched in the military service, normative documents which describe the way of conducting experimental tests have been developed. One of such documents is STANAG 4569 applicable in the NATO. The document describes the way of checking the resistance of the supporting structure to the effects of the detonation wave created as a result of a detonation under a wheel or track of a military vehicle (Figs. 99.3 and 99.4). Table 99.1 presents protection levels dependent on the mass of the detonated explosive. As part of the DOBR-BIO4/022/13149/2013 project entitled “Improving the Safety and Protection of Soldiers on Missions Through Research and Development in Military Medical and Technical Areas”, the team from the Department of Mechanics and Applied Computer Science suggested the increase of passive protection in military vehicles by using an additional multi-layer protective system. The main aim

Fig. 99.3 Location of the explosive under the wheel in steel pot for level 2a, 3a and 4a (front and side view) [11]

Fig. 99.4 Geometrical specifications of the explosive and steel pot [11]

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Table 99.1 Protection levels for occupants of armoured vehicles for grenade and blast mine threats [11] Level

Grenade and blast mine threat

4

4b

Mine explosion under belly

4a

Mine explosion pressure activated under any wheel or track location

3b

Mine explosion under belly

3a

Mine explosion pressure activated under any wheel or track location

2b

Mine explosion under belly

2a

Mine explosion pressure activated under any wheel or track location

3

2

1

10 kg (explosive mass) blast AT mine

8 kg (explosive mass) blast AT mine

6 kg (explosive mass) blast AT mine

Hand grenades, unexploded artillery fragmenting sub-munitions, and other small anti personnel explosive devices detonated anywhere under the vehicle

of the system is to dissipate the blast wave created as a result of the detonation of an improvised explosive device under a vehicle. In order to check the suggested design solution, numerical simulations have been conducted in accordance with the STANAG 4569 document (level 2). The only deviation from the normative requirement concerns the location of the explosive towards the vehicle. During tests, the explosive has been placed centrally under the vehicle. The change of the explosive’s location resulted from possible bigger destructions of the vehicle in such a situation compared to the detonation under a wheel.

99.2 Test Object Armed conflicts and stabilisation missions conducted in Libya, South Sudan, Pakistan, Somalia, Afghanistan or Syria show that a wheeled armoured vehicle is a military vehicle which is used most frequently. Those vehicles are used i.a. as transport, combat, medical and communication vehicles. Due to their significant engagement in conducted actions, they are exposed the most to firing and effects of improvised explosive devices. In order to cause the biggest losses on the part of the enemy, IEDs are detonated at a small distance from vehicles. Therefore, IEDs are usually placed in bomb cars and buried close to roads. Due to the above-mentioned reasons, it has been decided to use a light armoured vehicle in order to test the effectiveness of the suggested protective structure. The explosive has been detonated directly under the vehicle’s bottom (this location is the most unfavourable for the supporting structure). The geometry of the test object,

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Fig. 99.5 Geometrical model of the object—the axonometric view of the light wheeled armoured vehicle

which has been used to test the effectiveness of energy-absorbing panels, is presented in Fig. 99.5.

99.3 Test Methodology The scope of numerical analyses included checking energy-absorbing capabilities of energy-absorbing panels mounted on the vehicle’s body in the case of the detonation of the explosive with the mass of 6 kg under the vehicle (Fig. 99.6). Two identical vehicle bodies constituted the subject matter of the analyses—a referential body with mass equivalents of energy-absorbing panels made of steel and a principal body with mounted energy-absorbing panels. The test has been conducted in a limited scope on the basis of the AEP-55 vol. 2 methodology. Two variants of loading the vehicle have been simulated:

Fig. 99.6 Location of the explosive towards the vehicle for the variant with the protective system (on the left) and for the referential system (on the right)

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– the referential system: a body with mass equivalents of energy-absorbing panels made of steel—Fig. 99.7; Fig. 99.7 The vehicle with the mounted referential system

– the principal system: a body with mounted energy-absorbing panels—Fig. 99.8. Fig. 99.8 The vehicle with the mounted protective system

In order to ensure the same conditions of loading the vehicle, i.e. the same distance of the surface loaded with the blast wave from the detonation place, two variants of mounting the systems, both the referential and energy-absorbing one, have been used. They are presented in Figs. 99.7 and 99.8.

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Furthermore, in order to level the influence of mass differences of two tested systems, the thickness of an armoured plate mounted to the vehicle’s bottom and used in the referential system has been chosen in a way so that its mass corresponds to the mass of energy-absorbing panels.

99.4 Numerical Modelling and Simulations The model of a simplified military vehicle has been prepared using the following software: – – – –

CATIA V5—preparing the surface model, HyperMesh—division into finite elements, LS_PrePost—pre- and post-processing, LS_Dyna—numerical calculations.

ELFORM16—“Full integrate point” surface elements, used for the vehicle’s plating, have been used for the discretisation of the model. Other vehicle’s parts have been divided into finite elements using solid elements with one ELFORM1 integration point. The material model of a modified Johnson-Cook model has been used to describe the properties of the armoured metal plate (Table 99.2). The components made of S355 structural steel have been modelled using the linear and plastic model (Table 99.3).

Table 99.2 Material properties for Armstal MAT_MODIFIED_JOHNSON_COOK material model [12] Hardness

Yield stress

Strain hardening

HB

σ 0.2

A (MPa)

488–566

1707

1875

B (MPa) 415

500

steel

Strain rate hardening n 0.98

C 0.001

the

Temperature softening

ε˙ 0 (s−1 ) 2×

for

10−4

m 1.0

Table 99.3 Material properties for S355 steel for the MAT_PIECEWISE_LINEAR_PLASTICITY material model [12] ρ (t/mm3 )

E (GPa)

(–)

SIGY (MPa)

7.8 × 10−9

2.1 × 105

0.3

355

EPS1 (MPa)

EPS2 (MPa)

ES1 (MPa)

ES2 (MPa)

0

0.8

355

550

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99.5 Results of Numerical Analyses The location and description of measuring points are presented in Fig. 99.9 and in Table 99.4. Figures 99.10, 99.11, 9.12 and 9.13 present the diagrams of acceleration courses for the tests carried out using the explosive underneath the vehicle for the referential system as well as the system with energy-absorbing panels. Tables 99.5 and 99.6 present the maximum and minimum acceleration values together with the moment of time when those values appeared, registered in the measuring points for the referential system as well as the system with energy-absorbing panels.

Fig. 99.9 Orientation of the coordinate system of the acceleration sensor mounted on the floor plate

Table 99.4 Description of measuring points for the central explosion (the referential system and the system with the energy-absorbing panel)

Measuring point number

Measuring point description

Remarks

1

Casing of the reducer

See Fig. 99.10

2

Floor, centre

See Fig. 99.11

3

Mounting of the middle seat

See Fig. 99.12

4

Mounting of the side seat

See Fig. 99.13

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Fig. 99.10 Characteristic of changes in acceleration in the point on the reducer casing

Fig. 99.11 Characteristic of changes in acceleration in the point on the vehicle’s floor

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Fig. 99.12 Characteristic of changes in acceleration in the mounting point of the middle seat

Fig. 99.13 Characteristic of changes in acceleration in the mounting point of the side seat

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Table 99.5 Values of maximum and minimum accelerations in particular measuring points for the variant with the referential metal plate Measuring point

Acceleration amax (g)

Time tmax (ms)

Acceleration amin (g)

Time tmax (ms)

Casing of the reducer

3832.54

2.99

−4789.10

4.98

Floor, centre

12758.83

14.12

−15431.35

25.74

Mounting of the middle seat

2144.66

2.47

−2439.76

4.57

Mounting of the side seat

1460.20

10.71

−1523.30

29.41

Table 99.6 Values of maximum and minimum accelerations in particular measuring points for the variant with the protective panel Measuring point

Acceleration amax (g)

Time tmax (ms)

Acceleration amin (g)

Time tmax (ms)

Casing of the reducer

3149.47

28.70

−2689.82

27.49

Floor, centre

7761.00

10.39

−9072.35

14.52

Mounting of the middle seat

2048.39

14.12

−1959.50

21.52

Mounting of the side seat

1088.97

67.83

−1171.8

74.09

99.6 Conclusions A significant reduction of acceleration pulses has been observed in the conducted numerical calculations regarding the effects of the blast wave, created as a result of the detonation of 6 kg TNT under the vehicle, on the supporting structure of the vehicle in two variants. In the mounting place of the acceleration sensor located closest to the explosive and directly above it, a 39% reduction of acceleration, compared to the referential panel, has been obtained. The reduction of the acceleration pulse value can decrease the occurrence of lower limb injuries which are placed on the floor plate when soldiers are inside the vehicle. When analysing other acceleration values in the points where seats are mounted to the supporting structure, it can be stated that protective panels had a positive influence on the reduction of accelerations in the structure. The reduction of those values will decrease the values of forces transferred directly to the occupant’s seat and thus will contribute to increasing passive protection. In order to precisely imitate all phenomena, further works will be conducted using the MMALE approach. Acknowledgements The research was carried out within Project No. DOBRBIO4/022/13149/2013 “Improving the Safety and Protection of Soldiers on Missions Through

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Research and Development in Military Medical and Technical Areas”, supported and co-financed by NCR&D, Poland.

References 1. Baranowski, P., Małchowski, J.: Numerical investigations of terrain vehicle tire subjected to blast wave. J. KONES Powertrain Transp. 18(1) (2011) 2. Gawliczek, P., Pawłoski, J.: Zagro˙zenia asymetryczne (p. 18). Akademia Obrony Narodowej. Warszawa (2003) 3. Ambroziak, R., Ambroziak S.J., Katulski R.J.: Metody walki z prowizorycznymi urz˛adzeniami wybuchowymi w s´wietle doktryny AJP-3.15, Zeszyty Naukowe WSOWL 4(162) (2011) 4. Zastosowanie zaawansowanych technik komputerowych do opracowania systemów ochronnych dla pojazdów wojskowych przed wybuchem miny lub IED, II Konferencja Naukowa „Bezpiecze´nstwo wojsk w aspekcie zagro˙ze´n wynikaj˛acych z u˙zycia improwizowanych urz˛adze´n wybuchowych (IED)” 31.03.2017 Wojskowy instytut Medyczny WARSZAWA 5. Motrycz, G.: Cases of using improvised explosive devices, Szybkobie˙zne Pojazdy G˛asienicowe 44(2) (2017) 6. Borkowski, W., Motrycz, G.: Analysis of IED charge explosion on carrier road safety. J. KONES Powertrain Transp. 19(4) (2012) 7. Barnat, W.: Wybrane zagadnienia ochrony z˙ ycia I zdrowia załóg pojazdów przed wybuchem, Militarykor (2011) 8. muzeumgryf.pl/eksponaty/brdm-2/ 9. www.military.com 10. Saska, P., Pietrusiak, D., Czmochowski J., Iluk A.: Badanie wpływu kształtu deflektora na przebiegi przyspiesze´n wymuszonych eksplozj˛a małego ładunku wybuchowego Zeszyty Naukowe WSOWL 3(165) (2012) 11. AEP-55, Volume 2, August 2011, Procedures for evaluating the protection level of armoured vehicles ´ 12. Sławi´nski, G., Swierczewski, M., Malesa, P.: Risk assessment regarding the injuries of the lower limbs of the driver of a military vehicle in the case of an explosion under the vehicle. AISC 831, 1–15 (2019). ISSN 2194-5357

Part XXVII

Modeling and Simulation of Chemical Processes: Optimizations and Applications

Chapter 100

Dynamic Impacts of Catalyst Management on Self-fluidized Pump-Free Ebullated-Bed Reactor Bo Chen, Zhaohui Meng, Hailong Ge, Jordy Botello, Tao Yang and Xiangchen Fang Abstract Ebullated-bed residual oil hydroprocessing is advantageous at reaction efficiency and operation stability, and the fluidization state of catalyst bed enables the reactor to manage bed activity by on-stream withdrawal and addition operations. In this study, the catalyst management process for pump-free ebullated-bed reactor was studied; the flow field, catalyst profile, catalyst uniformity, and bed holdup were simulated by multi-phase computational fluid dynamic simulation. The results showed that, the mass fraction of low-activity spent catalyst could reduce by a quick withdrawal operation; one single operation cycle could increase the average bed activity by 1.5%, which was able to compensate the deactivation rate of catalyst. This study developed a theoretical tool for studying the dynamics of ebullated-bed reactor, and provided operational guidance on the design and optimization of pumpfree ebullated-bed residual oil hydroprocessing process. Keywords Hydroprocessing · Process engineering · Hydrocracking · CFD

100.1 Introduction Petroleum oil is the source of about 1/3 of the world’s total primary energy consumption, ranking first place among other energy sources. The world’s oil consumption reached 4418.2 million tons in 2016, and kept a growth rate of 1.8% per annum, indicating petroleum will still be a mainstream energy source for recent future. However, although the proved oil reserves kept growing, the fact that crude oils became heavier and sourer raised the requirements of oil refining technologies, especially when fuel regulations and specifications are strictly implemented in most countries.

B. Chen · Z. Meng · H. Ge · T. Yang · X. Fang (B) Dalian Research Institute of Petroleum and Petrochemicals, SINOPEC, Dalian 116045, LN, China e-mail: [email protected] J. Botello Department of Chemistry, University of Florida, Gainesville, FL 32611, USA © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_100

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To increase margins of refining process, the plants need heavy oil conversion technology to improve light product yield [1–4]. The hydroprocessing and hydrocracking for residual oils, are one of the most competitive technology owing to its high product quality and high sulfur/metal removal efficiency. Ebullated-bed reactor (EBR) is one of the most competitive configurations for residual oil hydroprocessing due to high on-stream factor. EBR is a three-phase fluidized reactor, in which the up-flowing gas bubbles and oils carries catalysts to reach the fluidization state [5, 6], preventing the catalyst bed from blockade by coke and metal sediments [7, 8]. The conventional ebullated-bed residual oil hydroprocessing market is dominated by H-Oil [9] and LC-Fining, both of which apply a high-temperature and high-pressure pump to recycle partial product back to the reactor [10], sustaining a high flow velocity to maintain the fluidization state of catalyst bed. The EBR could directly process heavy residual oil with 200 ppm metal and 10 wt% Conradson carbon residue (CCR), and achieve 3–4 years of cycle length [11, 12]. Besides conventional EBR technologies, SINOPEC has developed a pump-free ebullated-bed residual oil hydroprocessing technology, denoted STRONG [13]. The catalyst shape of STRONG is microsphere (H-Oil and LC-Fining catalyst are cylindrical), which reduces the minimum fluidization velocity significantly; the structure of STRONG reactor improves internal fluid recycle and hence increases oil velocity, enabling catalyst to be totally fluidized [14]; the three-phase separator inside the reactor cuts gas and oil from the fluid, and reject catalyst back to bed, prohibiting catalyst carriage problem even at fast fluidized regime. With all those advantages, a STRONG reactor could be operated without expensive high-temperature and highpressure pumps. The main difference between STRONG and other technologies is the catalyst distribution in the reactor bed. The H-Oil and LC-Fining reactor could be divided into two zones according to catalyst hold-ups. The downside (below recycle cup) contains the catalyst bed, which is denoted catalytic cracking zone; the upper (above recycle cup) contains no catalyst, denoted the thermal cracking zone. In a STRONG reactor, the catalyst is totally fluidized and carried by oil, and above the three-phase separator only gas phase exists, therefore two primary zones are divided, which are gas phase zone and reaction zone. The flow patterns of gas, oil and catalyst in pump-free EBR are illustrated in Fig. 100.1. The catalyst management is the key aspect to maintain bed catalytic activity [15]. The fluidization of catalyst bed enables catalyst particles to move freely; with proper withdrawal and addition operation, low-activity spent catalysts could be purged along with oil (the catalyst withdrawal outlet is positioned near the feedstock distribution grid), and high-activity fresh catalysts could be added to the top of the reactor. With a daily withdrawal and addition routine, the bed activity could be maintained at constant level. Few studies focused on the dynamics of the catalyst management in EBR, especially for pump-free EBR. In this work, the catalyst withdrawal and addition process in EBR was investigated. The effects of catalyst management on catalyst hold-up, phase fraction, catalyst distribution and spent catalyst withdrawal efficiency were studied through computational fluid dynamic (CFD) method. This study aimed to

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Fig. 100.1 Schematic of pump-free ebullated-bed reactor

provide insights on the operation of pump-free EBR, and provide guidance on catalyst management strategy of large scale STRONG ebullated-bed process.

100.2 Theory 100.2.1 CFD Modeling Industrial scale EBR is 4–5 m in diameter and 20–40 m in height, and the oil capacity is about 100–300 million tons per year (single tray). The large scale and nontransparent nature of residual oil, bubbles and catalyst bed keep the flow field and bed dynamics unobservable from conventional equipment. The CFD method provides an alternative way to investigate the three-phase transport phenomenon, and enables them to be presented in detail, gaining insights for reactor design. This work employed Euler-Euler multiphase method to simulate three-phase mass transport phenomenon in EBR. The continuity equation is ∂ρ + ∇ · (ρv) = 0 ∂t

(100.1)

where ρ is fluid density; t is time; v is the velocity vector. The momentum conservation equation is ∂(ρv) + ∇ · (ρvv) = −∇p + ∇ · (τ) + ρg + F ∂t

(100.2)

where p is pressure field; τ is stress tensor; F is the force that is applied on the field.

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The turbulent phenomenon was modeled by k-ε model. Considering the symmetrical nature of a cylindrical reactor shape, this work applied 2D CFD simulation to investigate the catalyst management process to save computational resources. The phase interaction effects are governed by Schiller-Naumann and SyamlalObrien models. The liquid-gas interaction is K sl =

3αs μl Res CD 4ds2

(100.3)

where Ksl is exchange coefficient; α is phase fraction; μ is viscosity; Res is particle Reynold number; CD is drag coefficient, which is defined in Schiller-Naumann model  CD =

24 ,if Res (1+0.15Res0.687 )

Res < 1000

0.44, i f Res < 1000

(100.4)

The liquid-particle granular effect interaction is governed by Syamlal-Obrien model   3αs αl ρl Res |Vs − Vl |C D (100.5) K sl = 4vr2s ds2 vr s where drag coefficient is defined 2   C D = 0.63 + 4.8/ Res /vr s

(100.6)

100.2.2 Catalyst Management in STRONG Ebullated-Bed Reactor The reactor is the core unit in EBR process. Pump-free EBR employs internal structure to promote the circulation and fluidization of catalyst. The reactor could be treated as two zones according to phases: above the three-phase separator is the gas phase zone, in which no liquid or solid exists; below the three-phase separator is the reaction zone, in which the bulk flow of fluidized catalyst, hydrogen bubbles and oils are flowing upward along the reactor as Fig. 100.1 illustrated. The diameter and height of the reactor in this study were 5 m and 40 m. The catalyst hold-up in EBR was set as 50–55 thousand ton. In EBR operation, the main course for catalyst deactivation is sediment formation, including coke, ashes and metals (Ni and V). The sediment formation will increase the density of catalyst particle, which could be used to distinguish the activity of catalyst [16]. STRONG catalyst is spherical, and its multi-scale pore structure promotes ultra-high metal formation capacity; the industrial long-term test showed that the metal content could

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easily reach 80 wt%, while in further pilot scale test, the metal formation reached almost 100 wt%, leading to significant increase in catalyst density. To mimic the change in catalyst bed activity, two kinds of catalyst with different density were used to represent real catalyst bed behavior. The density of spent catalyst (low activity) was set as twice of fresh catalyst. Although in realistic residual oil hydroprocessing process, the density distribution of catalyst was in wide range, using two catalysts with distinguished densities could observe bed dynamics explicitly with minimum model complexity. The catalyst management in EBR includes two steps: catalyst withdrawal and addition. Since catalyst is in fluidization regime inside EBR, the withdrawal process could be achieved by purging oil and catalyst directly from the reactor. The addition process of catalyst is more complicated. The catalyst should be carried by gas oil or diesel, and fed to the reactor with several pretreatments. In this work, only the simplest injection step was considered, i.e., the catalyst was directly fed to EBR carried by liquid. The simulation of this study employed pressure outlet and velocity inlet as the boundary conditions for catalyst withdrawal and addition process, respectively. The initial condition of catalyst composition was 80% fresh catalyst, 20% spent catalyst; the catalyst was loaded with 50% height of the reactor at 0.6 packing fraction. A typical operational cycle is composed by 3–5 min catalyst withdraw, 5–10 min catalyst addition and followed by 60 min equilibration. Although industrial scale EBR withdraws and add catalyst on a daily basis, it’s not possible for a simulation to achieve same time scale; this work ran the catalyst management cycle continuously after equilibration, which could reveal intrinsic reactor dynamics for catalyst withdrawal and addition process.

100.3 Results and Discussion 100.3.1 Flow Field and Phase Distribution in Pump-Free Ebullated-Bed Reactor The flow field of pump-free EBR is distinct from other ebullated-bed reactors. The liquid and catalyst phase are circulating in pump-free EBR, forming a central upflow with down-flow near the reactor wall. However, when catalyst is withdrawn or feed into the reactor, the outlet/inlet flow will disturb the original flow field, and thus causing asymmetric flow field. The effect was shown in Fig. 100.2, in which it could be observed that the catalyst had internal circulation inside the reactor: the central flow direction was upward, while the flow direction near the reactor wall was downward; but the flow filed was not perfectly symmetric due to the ongoing catalyst management. In Fig. 100.2, it could be observed that the internal flow circulation in pump-free EBR was opposite to the ones in H-Oil or LC-Fining reactors (in which the liquid near reactor wall is transporting upward, while the central flow inside the recycle tube is downward); it

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Fig. 100.2 Catalyst distribution and velocity profile in pump-free ebullated-bed reactor

also illustrated that pump-free EBR could form internal circulation without a recycle cup. The fraction of catalyst phase was also shown in contours in Fig. 100.2, it could be observed that the catalyst volume fraction was significantly lower at oil outlet position than other positions, demonstrating the phase separation ability of the three-phase separator. The investigation of catalyst phase distribution and velocity field provided a glimpse at the pump-free EBR; with the understanding of flow filed and phase distribution, the catalyst management routine could be investigated and discussed in detail in the following sections.

100.3.2 Flow Field and Phase Distribution in Pump-Free Ebullated-Bed Reactor Although the catalyst was totally fluidized in pump-free EBR, the phase fraction of catalyst phase still had variance along axial direction. The catalyst distribution will directly affect the efficiency of catalyst management. Figure 100.3 showed a snap shot of catalyst mass fraction distribution along reactor height after a few catalyst management cycles. It could be observed that the mass fraction of fresh and spent catalyst was higher at downside of reactor, and decreased to a constant level at about 12 m height. In Fig. 100.3, the ratio at 0–10 m was higher than the ratio at 10–30 m. This effect indicated that, the density of catalyst could provide slight impact on distribution under the operating condition, i.e., the spent catalyst with higher density tended to accumulate at the bottom of reactor, increasing the mass fraction of spent catalyst there.

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Fig. 100.3 Catalyst mass fraction axial distribution

100.3.3 Dynamic Impacts of Catalyst Addition and Withdrawal on Catalyst Distribution The catalyst management process provided great impacts on catalyst distribution. To observe the influences of catalyst management on the distribution of both fresh and spent catalyst, the profile of catalyst mass fraction was plotted against operation time in Fig. 100.4. In Fig. 100.4a, the distribution of fresh catalyst was presented. The mass fraction of fresh catalyst rapidly decreased or increased during withdrawal and addition process

(a)

(b)

Fig. 100.4 Dynamic evolution of catalyst fraction. a Fresh catalyst; b Spent catalyst

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Fig. 100.5 Catalyst mass fraction ratio of spent/fresh catalyst

respectively; the fraction distribution became steady after a short range of time in the equilibration process. Similar effects could be observed in Fig. 100.4b. Although no spent catalyst was added into the reactor in the addition process, the distribution of spent catalyst still showed similar evolution comparing to the fresh catalyst. The internal circulating flow field was the reason to this effect. During the withdrawal process, the purge flow vacuumed the solid phase at lower height of the reactor; when the purge flow stopped, the internal circulation of the liquid phase compensated catalyst fraction, which lead to rapid increase in spent catalyst distribution. However, after the increase, the mass fraction of the spent catalyst at lower position started to decrease, and reached the steady state after about 30 min. In the next figure, the dynamics of the spent/fresh catalyst ratio was investigated. The mass fraction ratio of spent/fresh catalyst could directly reflect the efficiency of catalyst management. In Fig. 100.5, the ratio decreased while a catalyst management cycle was performed, indicating the addition of fresh catalyst diluted the concentration of spent catalyst, and hence increased the bed activity.

100.3.4 Impacts of Catalyst Management on Catalyst Uniformity The catalyst management routine of EBR includes withdrawal, addition and equilibration steps (in this study, we focused on the dynamics of reactor, therefore other supportive catalyst handling processes, such as catalyst transfer from storage, pressurization etc., are not discussed). All the procedures significantly affect the catalyst distribution due to additional outlet or inlet flow, and thus the catalyst uniformity should be investigated to observe the effects of catalyst withdrawal and addition. The uniformity index was introduced to represent the distribution of catalyst, which was defined

100 Dynamic Impacts of Catalyst Management …

  γ=1−

(u¯ − u)2 dA 2 Au¯

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(100.7)

In this study, the three steps of catalyst management were denoted a cycle. Before the first cycle, an initial cycle was performed to relax the system; between the withdrawal and addition step, a short equilibration step was introduced to balance the system. The uniformity indices of fresh and spent catalyst were shown in Fig. 100.6. According to the figure, the uniformity indices of fresh and spent catalyst were above 0.90 during almost all the simulation time, indicating the catalyst occupied the whole available reactor volume (gas phase zone contained no solid, therefore the maximum value of uniformity index was about 0.9–0.95), despite of catalyst density. At the equilibration operations, the catalyst uniformity indices were at constant level with little fluctuation. Figure 100.6 also showed the dynamics of catalyst uniformity index while being managed. When the reactor was under normal operation condition, the uniformity indices of fresh and spent catalyst were almost identical, indicating the catalyst was under total fluidization scheme, even though the density of spent catalyst is 2 times greater than fresh one. When catalyst withdrawal operation was performed, the uniformity indices of fresh and spent catalyst deviated: the uniformity index of spent catalyst decreased significantly, while the one of fresh catalyst increased. The catalyst addition pipe was located at the top of the reactor, therefore when fresh catalyst was injected, the uniformity index of fresh catalyst was increased significantly. The purge

Fig. 100.6 Dynamic evolution of catalyst uniformity index during catalyst management

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flow in withdrawal operation disturbed the flow field, and increased the oil and solid velocity at the far side of catalyst withdrawal outlet by promoting internal circulation, and hence the catalyst concentration at upper location was raised while the bottom concentration being decreased. The catalyst concentration was higher at the bottom of EBR than upper location, and because of the density variance, the spent catalyst concentration at lower location was slightly higher than at upper, although this wouldn’t cause great change in uniformity index. When catalyst was withdrawing from the bottom of EBR, the spent and fresh catalyst was purged out of the reactor, due to the difference in concentration distribution, more spent catalyst was purge out, leading to great decrease in spent catalyst uniformity index.

100.3.5 Impacts of Inter-equilibration In the previous figure, it could be observed from the last cycle that the uniformity still had fluctuation after long time of equilibration. The inter-equilibration operation was the reason to that. In this study, three catalyst management cycle were performed, in which a 10 min inter-equilibration operation (equilibration between withdrawal and addition) was performed for the first cycle; while in the following two cycles, the addition operation was performed just when the withdrawal operation ended. In this study, the catalyst management process was simulated by manipulating the pressure of withdrawal outlet. The impacts of the operations on the relative pressure (the pressure difference between catalyst withdrawal outlet and gas outlet) of catalyst withdrawal outlet was shown in Fig. 100.7, in which the pressure showed significant fluctuation with the operation of withdrawal and addition. In the withdrawal process, the relative pressure decreases sharply due to purge operation; with the catalyst injecting from the top of reactor, the pressure quickly increased. In the first cycle, after the withdrawal, equilibration and addition operation, the pressure became steady. However, in the following two cycles, the pressure still decreased in the equilibration operation. This effect demonstrated the importance of inter-equilibration: after the purge flow stopped, the flow field was significantly affected, additional momentum was applied to the fluid; if catalyst addition was performed right after withdrawal, additional momentum carried by injected catalyst and fluid flow was applied again to the fluid at the same direction (the direction of gravity), which disturbed the flow field of EBR, and would take more time to reach equilibrium. The inter-equilibration had longer impacts on EBR flow field as Fig. 100.7 implied. In the last cycle (cycle 3), the pressure of withdrawal outlet was more fluctuated comparing to the previous two cycle. This effect implied that the absence of interequilibration operation not only affected the second operation cycle, it also provided impacts on the following cycle. However, this effect could not happen in real-world industrial equipment. In industrial hydroprocessing processes, the catalyst management process was scheduled as

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Fig. 100.7 Relative pressure of withdrawal outlet

daily event, and the withdrawal and addition operation were performed separately by hours of inter-equilibration. From Fig. 100.7 we learned that 10–20 min interequilibration was enough for the EBR to reach steady state, implying the effects observed in the last two cycles in Fig. 100.7 will not happen in industrial equipment.

100.3.6 Effects of Catalyst Management on Withdrawal Stream Composition The catalyst management operation was expected to purge as much spent catalyst as possible, i.e., high mass fraction of spent catalyst in the catalyst withdrawal stream was expected. The efficiency of catalyst management was studied in Fig. 100.8, in which the mass fraction of both kind of catalyst at the position of withdrawal outlet was chosen as monitor to represent the withdrawal efficiency. In Fig. 100.8, it could be observed that the fresh and spent catalyst mass fractions were about 0.29 and 0.07. When catalyst was withdrawing from the reactor, both mass fraction of fresh and spent catalyst decreased sharply, indicating the withdrawal operation almost drained the catalyst holdup near the withdrawal pipe. The purge flow of catalyst was suspended mixture of solid and liquid, indicating that the feedstock and fresh catalyst (high activity catalyst) were purging out of the reactor along with spent catalyst. This effect was an indispensable trade-off since the fluid was the key to provide fluidity to catalyst particles. In the two last operation cycles, the catalyst fraction almost reached 0.1 wt%, implying the withdrawal process

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Fig. 100.8 Impacts of catalyst management on withdrawal stream composition

started to purge liquid rather than catalyst. The effect was partially caused by limited CFD method in this work: the withdrawal operation was performed by manipulating outlet pressure; in industrial reactor, the catalyst management process is controlled by PID controller and the process will be smoother, giving the reactor more time to restore the catalyst holdup at the position of withdrawal outlet, and thus, the depleting effect could not be as intense as Fig. 100.8 showed. Despite of the limitation in simulation, Fig. 100.8 still provided a guidance for catalyst management operation, that the withdrawal process should be operated steadily to improve the efficiency.

100.3.7 Influences of Inter-equilibration on Bed Holdup The dynamics of bed holdup was studied in Fig. 100.9 to investigate the influences of inter-equilibration in pump-free EBR. The catalyst holdup was decreased and increased in the withdrawal and addition process. After the management of catalyst, the bed holdup was at steady state. During the withdrawal process, the mass flow rate ratio of spent/fresh catalyst was about 0.25, which was above the average mass fraction of spent catalyst (0.18–0.22), indicating that the removal efficiency on spent catalyst was higher than simply purging the bulk content of reactor. When the withdrawal, inter-equilibration and addition operation were finished, an equilibration operation was performed to balance the system. During the process, the bed holdup became steady. According to the data, about 6.6% of spent catalyst was

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Fig. 100.9 Dynamics of bed holdup

removed from the reactor during withdrawal operation, while the addition operation injected about 7.6% of fresh catalyst. The overall catalyst management efficiency could be evaluated by comparing the estimated bed activity. This work didn’t consider the influence of coking and metal deformation on catalyst activity; instead, the highdensity catalyst was arbitrarily used to represent low activity catalyst. Therefore, the average bed activity could be estimated through bed holdups. It could be calculated that the average bed activity was improved by 1.5% after one catalyst management cycle. According to the catalyst demetallization experiments, the average catalyst deactivation was about 0.8 wt% per day, which implied that the catalyst management process could maintain the bed activity.

100.4 Conclusion In this study, the catalyst management process for pump-free EBR, including withdrawal, equilibration and addition operation, was studied. The catalyst distribution and flow field were investigated to illustrate the effects of the distinguish structure of pump-free EBR; the dynamic catalyst distribution profile was studied to present the bed behavior during catalyst management process, through which it could be observed that the catalyst phase fraction was significantly affected by withdrawal or addition operation, and equilibration operations were needed to balance the system. The impacts of inter-equilibration were studied by uniformity index; withdrawal outlet pressure and withdrawal composition were calculated to explain the necessity

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of inter-equilibration at balancing the flow field of EBR. Although the absence of inter-equilibration provided great impact for following operation cycles, such effect was unrealistic and could not happen in real-world scenario. The efficiency of catalyst management was investigated. The catalyst management could easily upgrade the bed activity by 1.5%, which was higher than the average deactivation rate of catalyst, indicating that the daily catalyst management process was enough for commercial EBR daily operations. Acknowledgements The authors would like to thank the financial supports from Natural Science Foundation of China (Grant Nos. 61671319 and 61627803) and National Key R&D Program of China (Grant Nos. 2018YFA0209400 and 2018YFA0209404).

References 1. Dudukovi´c, M.P., Larachi, F., Mills, P.L.: Multiphase catalytic reactors: a perspective on current knowledge and future trends. Catal. Rev. 44(1), 123–246 (2002) 2. Rana, M.S., Samano, V., Ancheyta, J., Diaz, J.: A review of recent advances on process technologies for upgrading of heavy oils and residua. Fuel 86(9), 1216–1231 (2007) 3. Xiang, H., Wang, T.: Kinetic study of hydrodesulfurization of coker gas oil in a slurry reactor. Front. Chem. Sci. Eng. 7(2), 139–144 (2013) 4. Wang, H., Farooqi, H., Chen, J.: Co-hydrotreating light cycle oil-canola oil blends. Front. Chem. Sci. Eng. 9(1), 64–76 (2015) 5. Pjontek, D., McKnight, C.A., Wiens, J., Macchi, A.: Ebullated-bed fluid dynamics relevant to industrial hydroprocessing. Chem. Eng. Sci. 126, 730–744 (2015) 6. McKnight, C.A., Hackman, L.P., Grace, J.R., Macchi, A., Kiel, D., Tyler, J.: Fluid dynamic studies in support of an industrial three-phase fluidized bed hydroprocessor. Can. J. Chem. Eng. 81(3–4), 338–350 (2003) 7. Dinkov, R., Kirilov, K., Stratiev, D., Sharafutdinov, I., Dobrev, D., Nguyen-Hong, D., Chapot, S., Le-coz, J.-F., Burilkova, A., Bakalova, D.: Feasibility of bitumen production from unconverted vacuum tower bottom from H-Oil ebullated-bed residue hydrocracking. Ind. Eng. Chem. Res. 57(6), 2003–2013 (2018) 8. Stratiev, D.S., Russell, C.A., Sharpe, R., Shishkova, I.K., Dinkov, R.K., Marinov, I.M., Petkova, N.B., Mitkova, M., Botev, T., Obryvalina, A.N.: Investigation on sediment formation in residue thermal conversion based processes. Fuel Process. Technol. 128, 509–518 (2014) 9. Abid, M.F., Ahmed, S.M., Hassan, H.H., Ali, S.M.: Modeling and kinetic study of an ebullated bed reactor in the H-Oil process. Arab. J. Sci. Eng. (2017) 10. Manek, E., Haydary, J.: Investigation of the liquid recycle in the reactor cascade of an industrial scale ebullated-bed hydrocracking unit. Chinese J. Chem. Eng. (2018) 11. Stratiev, D., Shishkova, I., Nedelchev, A., Nikolaychuk, E., Sharafutdinov, I., Nikolova, R., Mitkova, M., Yordanov, D., Belchev, Z., Rudnev, N.: Impact of oil compatibility on quality of produced fuel oil during start-up operations of the new residue ebullated-bed H-Oil hydrocracking unit in the LUKOIL Neftohim Burgas refinery. Fuel Process. Technol. 143, 213–218 (2016) 12. Stratiev, D., Chavdarov, I., Nikolaychuk, E., Shishkova, I., Sharafutdinov, I., Tankov, I., Mitkova, M.: Investigation of the fluid catalytic cracking of different H-Oil vacuum gas oils and their blends with hydrotreated vacuum gas oil. Pet. Sci. Technol. 34(24), 1939–1945 (2016) 13. Liu, J., Fang, X., Yang, T.: Novel ebullated bed residue hydrocracking process. Energy Fuels 31(6), 6568–6579 (2017)

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14. Cheng, Z.-M., Huang, Z.-B., Yang, T., Liu, J.-K., Ge, H.-L., Jiang, L.-J., Fang, X.-C.: Modeling on scale-up of an ebullated-bed reactor for the hydroprocessing of vacuum residuum. Catal. Today 220, 228–236 (2014) 15. Kim, S.-H., Kim, K.-D., Lee, D., Lee, Y.-K.: Structure and activity of dispersed Co, Ni, or Mo sulfides for slurry phase hydrocracking of vacuum residue. J. Catal. 364, 131–140 (2018) 16. Parisien, V., Pjontek, D., McKnight, C.A., Wiens, J., Macchi, A.: Impact of catalyst density distribution on the fluid dynamics of an ebullated-bed operating at high gas holdup conditions. Chem. Eng. Sci. 170, 491–500 (2017)

Part XXVIII

Theme: Methods for Computer Modeling in Engineering and Sciences

Chapter 101

A Fine Simulation Analysis of Rock Fragmentation Mechanism of TBM Disc Cutter with DEM Yadong Xue, Jie Zhou, Feng Zhao and Hanxiang Zhao

Abstract Research on rock fragmentation mechanism is significant for TBM projects. To study the influence of penetration on the disc cutter force, a numerical simulation was conducted using the software MatDEM based on Discrete Element Method (DEM). A 19 in. disc cutter model was established. Four rock materials characterised by different strength properties and two layered rock combinations were assigned to the large-scale rock specimen. The rock breaking processes of single cutter were simulated at penetrations of 2, 4, 6, 8 and 10 mm respectively. Through the monitoring and analysis of cutter forces and specific energy (SE) changes, it is found that for the same type of rock material, the average normal force and rolling force both increase as the penetration increases, but the growth trends have shown a significant difference due to the rock strength properties. The SE can be significantly affected by penetrations and there is an optimal penetration, respectively, to minimize the SE value for different types of rock. In addition, the dissimilar rock combinations have a significant impact on the cutter forces. To verify the simulation law of disc cutter forces with respect to penetrations, the linear cutting machine (LCM) test has been carried out. The consistency of the comparison results of the two methods indicated that combining the simulation of MatDEM with the LCM test can explore the rock fragmentation mechanism of the disc cutter more deeply and comprehensively. Keywords TBM · Rock fragmentation mechanism · Special energy · Discrete element · MatDEM Y. Xue (B) · J. Zhou Key Laboratory of Geotechnical and Underground Engineering, Ministry of Education, Tongji University, Shanghai 200092, China e-mail: [email protected] Department of Geotechnical Engineering, College of Civil Engineering, Tongji University, Shanghai 200092, China F. Zhao China State Construction Silk Road Construction Investment Group Co., Ltd, Xian 710075, China H. Zhao Shanghai Municipal Engineering Design Institute Co., Ltd, Shanghai, China © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_101

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101.1 Introduction Tunnel Boring Machine (TBM) is a large-scale machine that can realize one-time forming of a tunnel section by rotating the cutterhead to break rock and advance. Compared with the traditional drilling and blasting method, TBM has the advantages of high efficiency, rapidity, high quality and safety, environmental friendliness and low labour intensity. In recent years, it has been widely used in the construction of water conservancy, railway and subway tunnels around the world [1]. When tunnelling, TBM disc cutters interact with the tunnel face, and the cutter-rock interaction will affect the rock breaking efficiency of the machine directly. The research study on the rock fragmentation mechanism of TBM disc cutter helps to optimize the excavation parameters, improve the capacity of breaking rock and advancing rate. And for projects, the research is also of great significance for accelerating the construction speed, improving the utilization rate of the TBM, and shortening the construction period reasonably [2]. Currently, several methods, including construction site measurement, indoor model test, theoretical analysis, and numerical simulation are the most often used in rock fragmentation mechanism study. However, when conducting on-site measurement, due to complicated conditions, it is difficult to obtain reliable and quantitative experimental data as well as time-consuming and costly. The indoor LCM test can artificially control the test parameters and is capable of accommodating a full range of cutter loads and penetrations. For example, Gong et al. [3] applied the mechanical rock breaking test platform to study the influence of different penetrations on rock breaking efficiency. Though, the results can be directly applied to the performance assessment of TBM in real practice, he also faced the problems of high instrument cost and difficult model making. In the theoretical analysis methods, many well-known prediction models, e.g., CSM [4], NTNU [5], and QTBM [6] have been developed to predict the performance of TBM. But based on ideal assumptions, these prediction models are unable to overcome the fundamental limitations associated with indirect estimation methods [7]. With the rapid development of computer technology, the numerical simulation method can accurately simulate the cutter-rock interaction process under different ground conditions, and the application has been more and more extensive [8]. One of the main methods used for the numerical analysis of geotechnical engineering problems is discrete element method (DEM). Compared with the finite element method (FEM), DEM does not need to re-divide the units in the process of simulating rock fracture propagation. The simulation results of large deformation motion of discontinuous medium are more reasonable [9]. Regarding the rock fragmentation mechanism of the TBM disc cutter, researchers have done some research work with DEM. Onate and Rojek [10] used the DEM to analyze the dynamic behaviour of cutting rock. Gong et al. [11] studied the fracture mode of rock under the action of disc cutter by establishing a discrete element model of cutter intrusion. Xu and Xue [12] studied the fracture process of the rock and the force characteristics of the cutter during the cutter intrusion using the particle flow code

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(PFC 2D). Tan et al. [13] used the DEM to obtain the law of the structural parameters of the cutter and the rock fracture characteristics. Through the exploration by researchers and it is recognized that the DEM is good for geotechnical engineering. But there are still many simulation difficulties such as complex modelling, low computational efficiency, and difficult analysis of results. The size of the rock specimen model in existing simulations based on particle discrete element, due to the limitation of the calculation efficiency on the number of particle units, is usually less than 0.5 m, and most of them are two-dimensional models. Thus it is difficult to study the rock fragmentation mechanism of the TBM disc cutter systematically. In addition, in the previous studies, the rock specimen model is usually assumed to be homogeneous. Some studies have considered structural faces (joints) but have not considered different rock types, which is inconsistent with the actual situation. In this paper, a linear cutting simulation model of full-size TBM disc cutter was established using the Fast GPU Matrix Computing of Discrete Element Method (MatDEM) based on 3D contact algorithm. The detailed analysis of the rock fragmentation mechanism was performed under the conditions of different rock materials, different penetrations, and dissimilar rock combinations. A comparative analysis was also conducted by the LCM tests to verify the numerical simulation results.

101.2 The Principle of MatDEM MatDEM is a three-dimensional discrete element simulation software applied for geotechnical engineering. An innovative GPU matrix calculation method and a threedimensional contact algorithm are employed in the software to realize the dynamic calculation speed of 14 million three-dimensional units per second. MatDEM features automatic stacking modelling, assigning layered material, setting joint surface, arbitrary load settings, and powerful post-processing. It can complete large-scale discrete element simulation of geology and geotechnical engineering problems in a more efficient and faster way.

101.2.1 Discrete Elemental Mechanical Properties As shown in Fig. 101.1a, a tightly packed stack consist of a series of particle units that follow Newton’s law of motion is used in MatDEM to construct a discrete element model [14]. The elements are connected to each other by a rupturable spring and the force can only occur at the contact points between adjacent elements, as shown in Fig. 101.1b, c. The units interact by spring force, and the normal force (Fn ) between the two units is given by Eq. (101.1) [15, 16]:

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Fig. 101.1 Discrete element particle model: a three-dimensional particle packing model of the discrete element; b normal spring force between particles; c tangential spring force between particles

 Fn =

Kn Xn Xn ≤ Xb , Connection (1a) 0 Xn > Xb , Fracture (1b)

(101.1)

where Kn is the unit normal stiffness, Xn is the normal relative displacement (Fig. 101.1b), Xb is the fracture displacement. When the units are connected to each other (Xn ≤ Xb ), they are subjected to the tensile force (Xn ≥ 0) or pressure force (Xn ≤ 0). When Xn between the two units exceeds the fracture displacement Xb , the spring breaks and the inter-unit tensile force disappears (Eq. 101.1). When the two units return to the compression contact state, the pressure between the units still exists. Shear deformation and tangential forces are simulated by tangential springs between the units and the tangential force of the shear spring is determined by Eq. (101.2): Fs = Ks Xs

(101.2)

where Fs is the tangential force, Ks is the tangential stiffness and Xs is the tangential relative displacement. For a complete unit connection, the maximum tangential force is determined by the Mohr-Coulomb criterion: Fsmax = Fs0 − μp Fn

(101.3)

where Fs0 is the shear resistance inside the unit, μp is the coefficient of friction and Fn is the normal force. When the unit is subjected to the shearing force and the force exceeds Fsmax in the Eq. (101.3), the joint between the units is broken. At this time, Fs should be less than or equal to the maximum friction at break:  = −μp Fn Fsmax

(101.4)

 , two adjacent units When the external force exceeds the maximum friction Fsmax  begin to slip and the sliding friction equals Fsmax . When the two units are separated from each other (Xn >Xb ), the normal force and the tangential force between the units are both 0.

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101.2.2 Deformation of Tetrahedral Elements The tetrahedral unit is composed of four identical particle stacking connections [14], as shown in Fig. 101.2a. In Fig. 101.2b, c, the Z-coordinates of the particles 2, 3, 4 is fixed to simulate a smooth rigid boundary. A slight external force Fz in the Z direction is applied to the particles 1, which produces a slight upward displacement dz . Owing to the pulling force of the particles 1, the particles 2, 3, 4 will move toward the centre of the equilateral triangle BCD. Since the unit body is centrally symmetrical, the normal displacement Xn1 and the tangential displacement Xx1 between particle 1 and the bottom three particles are identical, and the relative displacement and force between the bottom particles are also the same. Therefore, although the force and displacement are small, the deformation of the tetrahedron has an analytical solution. According to formula (101.1) and formula (101.2), the normal and tangential forces between the unit particles are: ⎧ ⎨ Fn1 = Kn · Xn1 F = Ks · Xs1 ⎩ s1 Fn2 = Kn · Xn2

(101.5)

where Fn1 and Fs1 are the normal and tangential force between particle 1 and particle 2 respectively, and Fn2 is the normal force between particle 2 and particle 3. The equilibrium equation of particle 1 in the Z-direction and the equilibrium equation of particle 2 in the X-direction are: 

Fz = 3 · (Fn1 · cos α + Fs1 · sin α) −FBO = Fn1 · sin α − Fs1 · cos α √ ⎧ ⎨ FBO = √3 · Fn2 cos α = √ 6/3 ⎩ sin α = 3/3

(101.6)

(101.7)

The normal displacement and tangential displacement between particle 1 and particle 2 are related to dz and Xn2 , as shown in Eq. (101.8):

Fig. 101.2 Deformation and destruction of tetrahedral elements under vertical tensile and compressive force

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Xn1 = dZ · cos α + XBO · sin α Xs1 = dZ · sin α − XBO · cos α √ XBO = 3/3 · Xn2

(101.8) (101.9)

In Eqs. (101.5), (101.6) and (101.8), there are seven equations including seven independent variables. By substituting the elimination, dz , Xn1 , Xs1 and Xn2 are obtained as follows: ⎧ n +Ks d = 9K5K · Fz ⎪ ⎪ n (Kn +Ks ) √ ⎪ z ⎪ ⎨ X = 6(3Kn +Ks ) · F n1 z 18K√ n (Kn +Ks ) (101.10) 2 3K n ⎪ Xs1 = 9K (K +K ) · Fz ⎪ n√ n s ⎪ ⎪ ⎩ X = − 6(Kn −Ks ) · F n2

18Kn (Kn +Ks )

z

101.3 Rock Fragmentation Model of TBM Disc Cutter The disc cutters assembled on TBM cutterhead are the most important tool to break the rock. According to the different cross-sections of the cutter ring, the disc cutters are mainly divided into the V-type disc cutter and the constant cross-section disc cutter (CCS). Currently, the wedge-shaped disc cutter is used less and has been gradually replaced by the normal-section cutter due to the contact area with the rock continuously changing because of the cutter wear, which causes the contact stress to change [17]. The disc cutter is subjected to the three directions of forces in the process of rock fragmentation, which are the normal force perpendicular to the rock surface, the rolling force along the cutting direction and the side force perpendicular to the plane of the disc cutter, as shown in Fig. 101.3a. As shown in Fig. 101.3b, when the disc cutter contacts the rock, the initial microcracks inside the rock are tightly closed, and the rock below the contact surface undergoes local compression deformation. With the increase of the normal force of the cutter, the stress concentration and outward expansion occur at both ends of Fig. 101.3 Disc cutter force and intrusion model: a the disc cutter force diagram; b cutter intrusion model

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the microcrack, and the internal damage of the rock is intensified and the locally damaged area appears subsequently. As the force further increases, the damaged area is continuously expanded, and a dense core of the rock will appear below the cutter at this time. The crack inside the rock gradually expands and forms an obvious macroscopic crack. Finally, the rock is broken and stripped [18–20].

101.4 Discrete Element Simulation of the Rock Fragmentation Process Based on the cutter-rock interaction principle, the discrete element software MatDEM was used to establish a large-scale three-dimensional model. The dynamic process of cutting rock of single cutter was simulated under various working conditions, which provides a new method for studying the rock fragmentation mechanism of TBM disc cutter.

101.4.1 Establishment of the Cutter and Rock Model The disc cutter model was built using the cluster elements in MatDEM according to the shape of a 19-in. disc cutter (CCS). Considering the calculation speed and efficiency, only the cutter ring part is constructed in the modelling process. The diameter of the disc cutter model is 483 mm and the width of the cutter blade is 20 mm, as shown in Fig. 101.4a. In addition, in order to accurately study the changing process of the disc cutter force, the rolling distance is supposed to ensure that the cutter can rotate in one circle at least. Therefore, the rock specimen model is set to be 1.5 m long, 0.6 m wide and 0.3 m high. In order to simulate the actual situation, the size of the three-dimensional discrete element model of the single cutter rock fragmentation should be as larger as possible. In this simulation, the total number of particles in the rock specimen unit is 275,845, and the total number of particles in the disc cutter unit is 20,946. The single cutter rock fragmentation model is shown in Fig. 101.4b. Fig. 101.4 Cutter and rock discrete element model: a the cutter model; b the rock specimen model

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Table 101.1 Raw and trained material mechanics parameters Material

Young’s modulus E(GPa)

Poisson’s ratio μ

Tensile strength τu (MPa)

Compressive strength cu (MPa)

Internal friction angle (°)

Rock 1

42.3

0.180

9.80

183.0

37

Mat.1

39.5

0.146

8.52

168.0

35

Rock 2

23.0

0.188

6.40

105.6

32

Mat.2

22.3

0.189

7.67

96.9

35

Rock 3

75.3

0.150

1.52

92.0

41

Mat.3

85.1

0.134

1.45

87.4

35

Rock 4

77.3

0.280

0.89

64.0

36

Mat.4

51.8

0.172

1.27

59.3

35

101.4.2 Simulation Material of Models In this study, the disc cutter was set as a rigid material, for that the wear of disc cutter was not considered. According to the hardness characteristic, four typical types of rock materials are selected for this simulation: extremely-hard rock (rock 1), hard rock (rock 2 and rock 3) and medium-hard rock (rock 4), of which the rock stiffness is also considered (rock 2 stiffness is less than rock 3). And the properties of each rock material are assumed to be isotropic. To obtain stable material properties, a training procedure of the simulated materials is required when using MatDEM. For this, the “material” program command had been written to train these four kinds of rock materials and the trained materials (Mat) were subsequently assigned to the rock specimen, which has solved the problem that the DEM is difficult to be quantitatively modelled for a long time [21]. The raw and trained rock material parameters are shown in Table 101.1.

101.4.3 Simulation Scheme While simulating the cutting process, the cutter firstly penetrated downward into a specific depth perpendicular to the rock surface, then did a linear cutting motion along the rock surface strike. The influence of cutting speed on the rock fragmentation process is weaker than other parameters [1] and it was not considered in this paper. The line speed of cutting was set to 0.5 m/s. During the cutting process, the normal force and rolling force and the amount of rock debris were monitored in real-time. Based on engineering practice experience and existing research results, the scheme for this simulation was determined as follows:

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Fig. 101.5 Rock combinations: a hard-soft combination; b soft-hard combination

(1) When simulating homogeneous rock materials, a total of 20 sets of single cutter rock fragmentation process were performed under different working conditions, In the conditions that for each of the four rock materials with different strength characteristics, the penetration P was simulated to be 2 mm, 4 mm, 6 mm, 8 mm, and 10 mm separately. (2) When simulating dissimilar rock combinations, the rock model was divided into two layers, the thickness of which was 0.1 m and 0.2 m respectively. Two rock materials with typical strength characteristics (rock 1 and rock 4) were assigned to the rock specimen model, using the function of layered material in MatDEM. The rock specimen model was divided into two combinations according to the hardness difference and the penetration was set to 2 mm considering the simulation efficiency, as shown in Fig. 101.5. During the simulation, the disc cutter motion and the rock force were limited by the boundary elements. To solve this problem, the connections between the boundary elements and the motion units need to be removed before the simulation. It can be implemented by the specialized commands (d.deleteConnection(‘boundary’)).

101.5 Analysis of Simulation Results 101.5.1 Influences of Penetrations on Disc Cutter Force It can be found through the analysis, the curves of the disc cutter force under different types of rock and different penetrations are similar. Due to the paper space limit, only the variation curves of the disc cutter rolling force and normal force monitored during the fragmentation process, under the condition that the rock material is rock 1 and the penetration is 2 mm, is shown in this paper, as shown in Fig. 101.6. It can be seen from Fig. 101.6 that during the process of cutting rock, the disc cutter force fluctuates constantly. So it is necessary to process and calculate the wave force to obtain the average values. According to the existing researches, the cutter force can be statistically determined using an arithmetic mean value. Table 101.2 shows the statistical results of the cutter force test data for the four rock materials.

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Fig. 101.6 Disc cutter force versus rolling distance: a rolling force; b normal force Table 101.2 Statistics of the disc cutter force test data Rock types

Rock.1

Rock.2

Rock.3

Rock.4

Penetration/mm

Average rolling force/kN

Average normal force/kN

Cutting coefficient

SE/(MJ/m3 )

2

47.6

375.6

0.12

101.6

4

90.6

490.9

0.18

156.5

6

111.9

547.2

0.20

160.5

8

118.8

828.7

0.14

142.8

10

120.1

906.7

0.13

117.0

2

9.9

66.2

0.15

42.0

4

49.4

320.5

0.15

88.8

6

63.1

324.2

0.19

73.1 52.8

8

71.8

370.7

0.19

10

74.8

387.7

0.19

49.6

2

13.4

119.4

0.11

21.1

4

26.0

133.2

0.19

43.4

6

32.1

272.6

0.11

35.6

8

38.0

354.6

0.10

37.2

10

49.0

365.5

0.13

46.6

2

11.4

107.7

0.10

20.7

4

22.0

112.7

0.19

38.6

6

25.4

247.1

0.10

34.0

8

31.9

336.0

0.09

34.3

10

36.8

362.1

0.10

36.8

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(a)

1199

(b)

Fig. 101.7 The relationship between average disc cutter force and penetrations: a average rolling force; b average normal force

Figure 101.7 is a graph showing the relationship between the average rolling force, normal force and penetrations of the cutter under different rock materials conditions. As can be seen from Fig. 101.7, the penetration and rock properties have significant effects on the cutting force: (1) For the same rock material, the average rolling and normal force increase with the increase of penetrations. The rolling force presents a minor value when the penetration is small, but the normal force generated at this time is relatively large. For instance, when the rock material is rock 1 and the penetration is 2 mm, the average rolling force of the cutter is 47.6 kN but the average normal force has reached 375.6 kN. This is because that the rock fragmentation is a dynamic load process that continuously changes along the cutting plane [1]. The rock fragmentation effect mainly depends on the normal intrusion of the cutter, so the cutter receives a normal force far greater than the rolling force. In addition, when the rock hardness is much large, such as rock 1 (extremely-hard rock), as the penetration increasing from 2 mm to 4 mm, the average rolling force is sharply increased from 47.6 to 90.6 kN. Therefore, in the case of that, the ground is very hard and the penetrations could change suddenly, it should be ensured that the TBM cutterhead can provide sufficient torques. (2) For different rock materials, the rock properties have a significant influence on the cutter force. When the rock gets harder, the uniaxial compressive strength will increase causing the critical stress of rock fracture to increase. Therefore, the internal crack propagation in rock requires a larger load and the cutter will be subjected to a greater force while forming harder rock pieces. For example, when the penetration is 2 mm, the average normal force of the extremely-hard rock (rock 1) is 267.9 kN larger than that of the medium-hard rock (rock 4). In TBM projects, the penetrations may change erratically but the rated thrust and torque of the machine are fixed. In order to ensure safe and effective constructions, the smaller penetrations should be adopted in a hard rock formation, and the penetrations can be appropriately increased when the rock formation is weak.

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101.5.2 Influences of Penetration on the Specific Energy of Rock Fragmentation Specific energy (SE) is an important index of the efficiency of rock fragmentation, and its specific meaning refers to the energy consumed by cutting a unit volume of rock. The smaller the SE, the less energy is consumed and the rock breaking efficiency is higher. The formula for calculating SE is: SE =

FR · L V

(101.11)

where SE is the specific energy (MJ/m3 ); FR is the average rolling force (kN); L is the cutting length (mm); V is the volume of rock debris (cm3 ). Figure 101.8 shows the relationship between penetrations and SE of rock fragmentation. It can be seen from Fig. 101.8 that for the same type of rock, the SE shows an irregular trend with penetration increasing. The SE value reaches smallest in 2 mm penetration, and the rock fragmentation efficiency gets highest at this time. For different types of rock material, the minimum SE increases with the uniaxial compressive strength of the rock. This indicates that more energy will be consumed when cutting harder rock. When the rock hardness is very large (such as rock 1 and rock 2), the penetration rate exceeds 4–6 mm and continues to increase, whereas the SE decreases. This is because that, when the disc cutter invades the rock, the rock under the cutter generates cracks due to the load and penetrates each other to form rock pieces, as shown in Fig. 101.9. Due to the small depth of cut, the volume of rock debris is small. When the penetration increases (represented by the dotted line in Fig. 101.9), the volume of broken rock pieces generated by the inter-fracture cracks increases, resulting in a decrease in SE. Fig. 101.8 The relationship between penetration and SE

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Fig. 101.9 Schematic diagram of rock pieces formation (the dotted line shows the case where the penetration becomes large)

101.5.3 Influences of Rock Combinations on the Disc Cutter Force When analyzing the influence of rock combinations on the disc cutter force, the layered rock simulation results were compared with that under the uniform rock conditions carried out in 5.1. It can be seen from Table 101.3 that, when the homogeneity of the rock changes, the force will be affected by the rock discontinuity. When the underlying rock becomes harder or softer, the average normal force decreases or increases accordingly. For example, in the hard-soft combination, when the underlying gets softer, the overall strength of the rock is reduced, the interior rock is easier to break during the invasion. And the normal force in this change is reduced consequently. But the average rolling force almost has no change. This is because that the cutter is always in contact with the upper layer rock during the cutting process, and the cutting direction is consistent with the invariable rock strike, so the rolling force changes a little. It is also noted that when the rock mass becomes weaker, the broken rock pieces volume increases and the SE decreases, which causing the torques of the disc cutter increasing. Table 101.3 Comparison of cutter force data under two rock models Combination scheme

Rock homogeneity

Penetration/mm

Average rolling force/kN

Average normal force/kN

Cutting coefficient

SE/(MJ/m3 )

Hard-soft combination

Uniform

2

47.6

375.6

0.12

101.6

Layered

2

47.3

308.3

0.15

79.6

Soft-hard combination

Uniform

2

11.4

107.7

0.10

20.7

Layered

2

12.1

135.6

0.09

22.0

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101.6 LCM Tests In order to verify the numerical simulation results of the rock fragmentation process, the TJ-TS500 linear cutting test platform independently designed by Tongji University was used. The platform pattern and test specimen (C40 fine aggregate concrete) are shown in Fig. 101.10. The disc cutter (19 in. CCS) used in the LCM test was consistent with that of the numerical simulation. Due to the rated cutting speed of the platform and excluding its influence, it was uniformly selected to be 400 mm/min. According to the experience of the project, the penetration of TBM in hard rock is generally 4–6 mm. To consistent with the numerical simulation test, the penetrations of the test was 2, 4, 6, and 8 mm respectively. The LCM tests under different penetrations were performed and the changes of cutter forces were recorded to analysis and compare in real-time by data acquisition instrument.

101.6.1 Analysis of Test Results The data monitored during the test were processed to obtain the relationships between the penetrations and the disc cutter force, as shown in Fig. 101.11. It is noted that the experimental data shows similar laws with respect to the simulation results. The normal force increases obviously with the increase of penetration as well as the rolling force. But they have different increase trends and the rolling force presents a smaller value under small penetrations. This is because that the normal force and the penetration are consistent in the direction, and the normal force changes more obviously with the penetrations. Providing that the cutter is in contact with the rock and a small penetration exists, a larger normal force can be generated. The disc cutter can only bear a significant resistance moment when the stress of the cutterrock interaction area is large with a certain cutting depth. Comparing the results of the LCM tests with the numerical simulation, it is found that the forces are smaller in physical tests due to the differences in rock materials and cutting parameters. But the

Fig. 101.10 Linear cutting test platform and concrete sample: a TJ-TS500 linear cutting test platform; b C40 fine aggregate concrete specimen

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Fig. 101.11 The relationship between penetrations and disc cutter force: a rolling force; b normal force

influences law of penetration on the disc cutter force obtained by the two methods are basically the same.

101.7 Conclusion In this paper, the discrete element software MatDEM are used to carry out a fine simulation study on the rock fragmentation mechanism of the TBM disc cutter, and the main conclusions are as follows: (1) The penetration has a significant effect on the disc cutter force. As the penetration increases, the average rolling force and normal force both increase correspondingly. The normal force is much greater than the rolling force in the same working condition. When cutting extremely hard rock, the deepening of penetration will cause the cutterhead torque to increase sharply. During construction, the TBM cutterhead should be able to provide sufficient thrust and torque. (2) The change of penetration has a direct influence on rock fragmentation efficiency. For the same rock adopted in this paper, the SE reaches smallest when the penetration is 2 mm. For different types of rock, the minimum SE value increases with the increase of rock hardness. For extreme-hard and hard rock, when the penetration exceeds 4–6 mm, the rock-fragmentation energy will be reduced due to the excessive volume of the rock debris. (3) The rock strength properties also have a certain influence on the disc cutter force. In this paper, the average cutter normal force of hard rock is 267.9 kN larger than that of the medium-hard rock under the same penetration (2 mm). Therefore, the penetrations of TBM should be adjusted according to the change of rock properties in time. The rock combinations can cause the cutter force to change. The increase of rock hardness causes the normal force to become larger, and vice versa, but the rolling force does not change much.

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(4) The numerical simulation results are verified by the LCM tests, which shows that the MatDEM software can well simulate the rock fragmentation process. By further improving the research, the TBM disc cutter rock fragmentation mechanism can be explored deeply and comprehensively. Acknowledgements The authors acknowledge the support of National Natural-Science Foundation of China (Grant No. 41072206), Science and Technology Commission of Shanghai Municipality (Grant No. 18DZ1205902).

References 1. Gong, Q.M.: Introduction to Tunnel Boring Machine Tunnelling. Science Press, Beijing (2014) 2. Yang, S.Q., Huang, Y.H.: TBM disc cutter rock breaking process and mesoscopic mechanism particle flow simulation. J. Coal 40(6), 1235–1244 (2014) 3. Gong, Q.M., He, G.W., Zhao, X.B., et al.: Influence of different penetration degree on rock breaking efficiency of TBM disc cutter. Mod. Tunn. Technol. 53(1), 62–68 (2016) 4. Rostami, J.: A new model for performance prediction of hard rock TBMs. In: Proceedings of the Rapid Excavation and Tunneling Conference, pp. 793–810 (1993) 5. Bruland, A.: Hard rock tunnel boring, Ph.D. thesis. Norwegian University of Science and Technology, Trondheim, Norway (1998) 6. Barton, N.: TBM tunnelling in jointed and faulted rock. Ph.D. thesis. Balkema (2000) 7. Cho, J.W., et al.: Optimum spacing of TBM disc cutters: a numerical simulation using the three-dimensional dynamic fracturing method. Tunn. Undergr. Space Technol. 25(3), 230–244 (2010) 8. Zhang, K., Li, L., Wang, H., et al.: Influence of rock breaking parameters of rolling knives on rolling force and specific energy based on ABAQUS. J. Shenyang Jianzhu Univ. (Nat. Sci. Ed.) 33(5), 914–922 (2017) 9. Tan, Q., Yi, N.E., Xia, Y.M., et al.: Research on rock breaking dynamics and optimal tool spacing of TBM disc cutter. J. Rock Mech. Eng. 31(12), 2453–2464 (2012) 10. Oñate, E., Rojek, J.: Combination of discrete element and finite element methods for dynamic analysis of geomechanics problems. Comput. Methods Appl. Mech. Eng. 193(27), 3087–3128 (2004) 11. Gong, Q.M., Zhao J., Hefny A.M.: Numerical simulation of rock fragmentation process induced by two TBM cutters and cutter spacing optimization. Tunn. Undergr. Space Technol. Inc. Trenchless Technol. Res. 21(3), 263 (2006) 12. Xu, L.Y., Xue, Y.D.: Numerical simulation of full particle flow in the interaction between disc cutter and rock in hard rock area. In: Proceedings of the 8th China-Japan Shield Tunnel Technical Exchange Conference. Hehai University Press, Nanjing, pp. 67–73 (2015) 13. Tan, Q., Li, J.F., Xia, Y.M., et al.: Numerical study on rock breaking process of disc cutter. Geotechnical 34(9), 2707–2714 (2013) 14. Liu, C., Xu, Q., Shi, B., Deng, S., et al.: Mechanical properties and energy conversion of 3D close-packed lattice model for brittle rocks. Comput. Geosci. 103, 12–20 (2017) 15. Hardy, S., Finch, E.: Discrete element modelling of the influence of cover strength on basementinvolved fault-propagation folding. Tectonophysics 415(1), 225–238 (2006) 16. Yin, H., et al.: Discrete element modeling of the faulting in the sedimentary cover above an active salt diapir. J. Struct. Geol. 31(9), 989–995 (2009) 17. Balci, C., Tumac, D.: Investigation into the effects of different rocks on rock cuttability by a Vtype disc cutter. Tunn. Undergr. Space Technol. Inc. Trenchless Technol. Res. 30(4), 183–193 (2012)

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18. Innaurato, N., et al.: Experimental and numerical studies on rock breaking with TBM tools under high stress confinement. Rock Mech. Rock Eng. 40(5), 429 (2007) 19. Gong, Q.M., Zhao, J.: Development of a rock mass characteristics model for TBM penetration rate prediction. Int. J. Rock Mech. Min. Sci. 46(1), 8–18 (2009) 20. Espallargas, N., et al.: Influence of corrosion on the abrasion of cutter steels used in TBM tunnelling. Rock Mech. Rock Eng. 48(1), 261–275 (2015) 21. Liu, C., Pollard, D.D., Shi, B.: Analytical solutions and numerical tests of elastic and failure behaviors of close-packed lattice for brittle rocks and crystals: analytical solutions of dem properties. J. Geophys. Res. Solid Earth 118(1), 71–82 (2013)

Chapter 102

Research on Pollution Accumulation Characteristics of Insulators in Natural Pollution Accumulation Experiments Mao Dong and Gao Qiang

Abstract Studying the characteristics of insulator surface contamination is the premise of studying the mechanism of insulator contamination flashover accident, and it is the inevitable need of anti-pollution work. In order to study the distribution of contaminated particles on the surface of insulators, natural contamination experiments of different insulators are carried out in this paper. The contaminated particles on the insulator surface were sampled and observed by scanning electron microscopy (SEM) through the establishment of experimental observation station. The image of the sample obtained after scanning is used to calculate the particle size of the contaminated particles by image processing method. The experimental results show: More than 90% of the contaminated particles on the surface of the insulator are less than 10 µm. Due to the anti-pollution double umbrella insulator, having two upper surfaces increases the probability of large particle deposition. Therefore, the proportion of large particle size particles is higher than that of ceramic and tempered glass insulators. The particle size distribution of the insulator surface obeys the lognormal distribution law, and its cumulative probability obeys the Boltzmann distribution law. The material and shape of the insulator do not affect this law. Keywords Insulators · Scanning electron microscopy · Particle size · Image processing method

102.1 Introduction Insulators are an important part of overhead transmission lines [1]. Contamination will increase the risk of insulator flashover and threaten the safe and stable operation of power system. The adsorption of contaminated particles on the surface of insulators results in electric field distortion of insulators, which reduces the insulation performance of insulators. It is easy to cause pollution flashover accidents in wet weather such as rain and fog [2, 3]. Statistics show that the number of pollution flash M. Dong · G. Qiang (B) School of Mechanical and Electric Engineering, Soochow University, Suzhou 215021, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_102

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accidents ranks second in the number of grid accidents, only after lightning accidents, but the losses caused by pollution flashover accidents are 10 times as much as those caused by lightning accidents [4]. Due to its occurrence mechanism, pollution flashover accidents have the characteristics of large area and long time, and often cause adverse social impact. It has seriously restricted the economic development, seriously hindered the safe and reliable operation of the power system, and seriously threatened the national life security [1, 4, 5]. Research on the deposition of contaminated particles on insulator surface is the premise of studying the mechanism of insulator contamination flashover accident, and it is the inevitable need of anti-contamination work. By studying the deposition law of the contaminated particles on the surface of the insulator, the grade of the contaminated area is determined to guide the design and cleaning of the outer insulation [5]. At present, domestic and foreign scholars have done a lot of research work on insulator fouling in the natural environment. Wang Shenghui of North China Electric Power University Co., Ltd. focused on the pollution distribution of glass, porcelain and composite insulators under the condition of natural contamination, and measured the distribution of salt and ash density on the surface of insulators [6]. Liang Xidong team of Tsinghua University has done a lot of research work on the natural contamination test of insulators under AC and DC voltage. The characteristics of the contaminated layer on the insulator surface are revealed through the surface ash-salt ratio, surface salt density and surface ash density [7–9]. S. M. Gubanski et al. studied the distribution of industrial pollutants on the surface of silica insulators [10, 11]. In order to further study the distribution characteristics of the contaminated particles on the insulator surface in the natural environment. In this paper, the natural contamination experiments was carried out, and the distribution characteristics of the particle size of the contaminated particles on the surface of the insulator were observed and analyzed by scanning electron microscopy (SEM). It reveals the influence of dirty particles on insulator fouling, provides reference for anti-pollution and cleaning of insulators in natural environment, and provides basis for simulation experiment of insulator accumulation.

102.2 Experimental Methods and Samples 102.2.1 Pollution Accumulation Experiment In this paper, a natural contamination experimental platform was built on the balcony of the 4th floor of the mechanical experiment building of Suzhou University in Xiangcheng District of Suzhou City (Fig. 102.1). Before setting up the test-bed, the insulators were cleaned and dried. There are teaching laboratory building, processing workshops, mechanical training buildings and off-campus farms around the experimental platform. The main pollution sources that have great influence on insulator fouling are industrial pollutants discharged from mechanical training buildings and

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Fig. 102.1 Natural fouling experimental platform

processing workshops, as well as dust from surrounding farms. To avoid the impact of trees and buildings on the natural wind field. The insulator is suspended from the 2 m high pollution test frame. The total distance from the ground is 13.2 m, which better simulates the actual environment where the insulator is located. Insulators are suspended on the contaminated test stand, which is 2 m high. The total distance from the ground is 13.2 m. The actual environment in which the insulator is located is better simulated [12, 13]. The natural sedimentation experimental observation platform mainly carried out a sediment observation experiment for three different types of insulators for one month (April 2018). The three kinds of insulators are ceramic insulators, tempered glass insulators and anti-pollution insulators with double umbrellas (Fig. 102.2). Specific parameters of sample insulators are shown in Table 102.1.

XP-70 Fig. 102.2 Three types of insulators

LXP-70

XWP-70

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Nominal diameter (mm)

Structural height (mm)

Creepage distance (mm)

XP-70 ceramics

255

146

295

LXP-70 tempered glass

255

146

320

XWP-70 ceramics

255

160

400

102.2.2 Particle Size Distribution Experiment Scanning electron microscopy (SEM) (Fig. 102.3) was used to obtain the particle size distribution of contaminated particles. From the scanning electron microscope image of Fig. 102.4, it can be seen that the microscopic morphology of the contaminated particles on the surface of the insulator is various. The shape of the contaminated Fig. 102.3 Scanning electron microscopy

Fig. 102.4 Scanning image of contaminated particles

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particles is complex and variable, with bands, spheres, flakes, and ellipsoids, and more are some extremely irregular shapes. In the statistics of the particle size of the contaminated particles, there are many scanned images obtained in the test, and there are many selected sample particles. It is extremely cumbersome to work by manually marking the particle size and performing mathematical statistics. At the same time, the dimension labeled is only a certain size of contaminated particles. Under the irregular shape of particles, this specific size is difficult to characterize the size of the particles. Therefore, this size may not be effective in the statistics of the size of contaminated particles. Therefore, when counting the particle size of the contaminated particles, it is necessary to treat the particle size of the contaminated particles in an equivalent manner. In the test measurement, the method of selecting the equivalent diameter of the particles is different according to different measurement methods. Since the SEM image is used to measure the particle size of the contaminated particles, the equivalent diameter of the contaminated particles is equivalent to the projected area diameter [12, 14]. In order to avoid cumbersome marking and statistical work, this paper uses image processing methods to calculate the particle size of particles. The image processing program mainly includes four modules, namely median filtering, binarization, hole filling and edge detection. The purpose of median filtering is to process salt and pepper noise in the image, so as to facilitate the subsequent high-quality binary images [15]. Then the adaptive threshold segmentation method is used to get the binary image, which distinguishes the particles from the background (Fig. 102.5b). From the binary image, it can be found that there are holes inside the particles. In this way, there will be some errors in the statistics of particle size. So the code for hole filling is added to the program to repair the binary image (Fig. 102.5c). Then, edge detection is performed on the particles in the image and the edge of the particles is obtained. In this way, the projected area of the particles is calculated and the equivalent projected diameter of the particles is calculated (Fig. 102.5d). In this experiment, the particle size is calculated by using the image of particle obtained by SEM, so the equivalent projection area diameter is chosen as the particle size.

102.3 Results and Analysis 102.3.1 Ceramic Insulators Figure 102.6 shows the data for the ceramic insulator, black for the upper surface and red for the lower surface. The overall distribution trend of the upper and lower surfaces is consistent, and there are some differences in the specific particle size ratio. The maximum particle size on the upper surface of the ceramic insulator is 30 µm, and the probability distribution of particles around 2.5 µm exceeds 50%. It can be known from the cumulative probability Fig. 102.6b that the proportion of particles below 10 µm is 97.7%, that is, most of the particles on the upper surface are below

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Fig. 102.5 Image processing (a Original image, b binary image, c hole filled image, d particle contour image)

(a)

(b) XP-70Upper surface Xp-70Lower surface

0.4

0.2

0.0

-5

0

5

10

15

20

Particle size(μm)

25

30

35

XP-70Upper surface XP-70lower surface Cumulative probability distribution

Probability distributions

0.6

1.0

0.5

0.0

-5

0

5

10

15

20

25

30

35

Particle size(μm)

Fig. 102.6 Particle distribution characteristics of ceramic insulator surface (a Probability distributions, b cumulative probability distribution)

10 µm in size. The maximum particle size of the contaminated particles on the lower surface is 34 µm. The probability of 2.5 µ particles is still the largest, but it is not half of the total. In the cumulative probability plot, the overall trend is consistent with the upper surface. The percentage of particles below 10 µm is 92.3%, which is less than that of the upper surface. The mean particle size on the upper surface is 3 µm, while that on the lower surface is 4 µm. In the cumulative probability plot, the cumulative probability of the upper surface increases faster than that of the lower

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surface. This directly shows that the proportion of fine particles on the upper surface is larger than that on the lower surface. Figure 102.7 is a comprehensive analysis of the particle data of the upper and lower surfaces of ceramic insulators. Here, the probability distribution and cumulative probability distribution of the integrated particle size analysis are fitted and analyzed. Figure 102.7a, b are the analysis diagrams of probability distribution. The probability of particle size distribution satisfies the law of lognormal distribution. Figure 102.7a is the fitting diagram, and Fig. 102.7b is the residual diagram. The square of the residual fit here is 0.998, which is close to 1. The weighted chi-square test coefficient is 3.06 × 10−5 . Figure 102.7c, d are analytical plots of the cumulative probability distribution of the particles, which are distributed to satisfy the Boltzmann distribution. The square of the residual fit here is 0.996, which is close to 1. The weighted chi-square test coefficient is 2.01 × 10−4 . Therefore, under the condition of natural contamination, the size distribution of insulator surface contamination obeys lognormal distribution, and its cumulative probability satisfies Boltzmann distribution.

(a)

(b)

Probability distributions

0.4

0.2

0.0 -2 0

2

4

6

Probability distribution regular residual

0.02

XP-70 Lognormal distribution curve

Regular residual 0.01

0.00

-0.01

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

0

5

10

1.0

Cumulative probability distribution

20

25

30

35

(d) XP-70 Boltzmann distribution curve

0.5

0.0 -2 0

2

4

6

8 10 12 14 16 18 20 22 24 26 28 30 32 34 36

Particle size(μm)

Cumulative probability distribution regular residual

(c)

15

Particle size(μm)

Particle size(μm)

0.04

Regular residual

0.02

0.00

-0.02

-5

0

5

10

15

20

25

30

35

Particle size(μm)

Fig. 102.7 Particle distribution characteristics of ceramic insulator surface (a Probability distribution fitting, b probability distribution fitting residual, c cumulative probability distribution fitting, d cumulative probability distribution fitting residual)

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102.3.2 Different Types of Insulators Figure 102.8 provides a statistical analysis of the particle size data of the contaminated particles, which is derived from the surface of different types of insulators. It can be seen that the three particle size distributions follow the lognormal distribution law. And their respective cumulative probabilities are also subject to the Boltzmann distribution law. It can be seen from Fig. 102.8a that the particle size data of the three different insulators are different. XP-70 ceramic insulators and LXP-70 tempered glass insulators have the largest proportion of particles with a diameter of about 2.5 µm, while XWP-70 anti-pollution double umbrella insulator has the largest proportion of particles about 4 µm. Considering the different size of the largest proportion particle, the main influencing factor is the shape of insulator. XP-70 ceramic insulator and LXP-70 tempered glass insulator have the same shape, while XWP-70 anti-pollution double umbrella insulator has two umbrella surfaces. For the antipollution double umbrella insulator, the area of the upper surface is increased, and the sedimentation probability of the large-sized particles is increased. Comparing XP-70 ceramic insulators with LXP-70 tempered glass insulators, they are identical in shape. However, the materials are different, which results in different roughness of the two insulating surfaces. From the probability diagram, the proportion of particles about 2.5 µm in diameter is higher on ceramic insulators than on toughened glass insulators. Moreover, the maximum particle size on ceramic insulators is much larger than that on toughened glass insulators. From the cumulative probability distribution, the cumulative probability growth rate of ceramic insulators is the fastest, and that of anti-pollution double umbrella insulators is the slowest. The proportion of particle size below 5 µm is 80% for ceramic insulator, 70% for toughened glass insulator and 62% for anti-pollution double umbrella insulator (as shown in Table 102.2). It can be seen that the proportion of large particles on the surface of anti-pollution double umbrella insulators is higher than that on ceramic and tempered glass insulators. It can be concluded that too large area of contamination on the upper surface will increase the sedimentation of large particle size. (a)

XWP-70 LXP-70 XP-70

Cumulative probability distribution

Probability distributions

0.5

(b)

0.4

0.3

0.2

0.1

XWP-70 LXP-70 XP-70

1.0

0.5

0.0

0.0 -5

0

5

10

15

20

25

30

35

40

Particle size(μm)

45

50

55

60

65

-5

0

5

10

15

20

25

30

35

40

45

50

55

60

65

Particle size(μm)

Fig. 102.8 Particle distribution characteristics of different types of insulators (a Probability distributions, b cumulative probability distribution)

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Table 102.2 Data of contamination particle size distribution for different types of insulators Insulator type

Particle size less than 10 µm (%)

Particle size less than 5 µm (%)

XP-70

95

80

3

LXP-70

92

70

4

XWP-70

90

62

5

Mean value (µm)

102.4 Conclusion The contamination characteristic of pollution particles on the surface of the insulator, this paper carried out the natural contamination experiments on different types of insulators. The transport and deposition of contaminated particles to the surface of insulators under natural forces (traction and gravity) are analyzed. Scanning electron microscopy (SEM) and image processing were used to analyze the particle size distribution characteristics of contaminated particles on insulator surface. The results are as follows: 1. XP-70 ceramic insulators and LXP-70 tempered glass insulators have the largest proportion of particles with a diameter of about 2.5 µm, while XWP-70 antipollution double umbrella insulator has the largest proportion of particles about 4 µm. 2. The proportion of particle size below 5 µm is 80% for ceramic insulator, 70% for toughened glass insulator and 62% for anti-pollution double umbrella insulator. Because the anti-pollution double umbrella insulator has two upper surfaces, which increases the probability of large particles deposition, the proportion of large particles is higher than that of ceramic and tempered glass insulators. 3. Statistical analysis of the experimental particle size data shows that the particle size distribution on the insulator surface obeys the lognormal distribution law, and the cumulative probability obeys the Boltzmann distribution law. The material and shape of insulators do not change this law.

References 1. Jiang, Z., et al.: Pollution accumulation characteristics of insulators under natural rainfall. IET Gener. Transm. Distrib. 11(6), 1479–1485 (2017) 2. Zhang, Z., et al.: Effects of pollution materials on the AC flashover performance of suspension insulators. IEEE Trans. Dielectr. Electr. Insul. 22(2), 1000–1008 (2015) 3. Xingliang, J., et al.: Study on fog flashover performance and fog-water conductivity correction coefficient for polluted insulators. IET Gener. Transm. Distrib. 7(2), 145–153 (2013) 4. Ghosh, P.S., Chatterjee, N.: Polluted insulator flashover model for AC voltage. IEEE Trans. Dielectr. Electr. Insul. 2(1), 128–136 (1995)

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5. Li, X., et al.: Statistics and analysis of lightning flashovers of transmission lines during 2000–2007. In: International Conference on High Voltage Engineering and Application, 2008, ICHVE 2008. IEEE (2008) 6. Wang, S., et al.: The pollution distribution characteristics of natural fouling insulators and the simulation study of surface current and electric field distribution. J. North China Electr. Power Univ. Nat. Sci. Edit. 43(6), 47–53 (2016) 7. Liang, X., et al.: Soil fixation on composite insulator surface and its effect on insulation properties. High Volt. Technol. 39(6), 1296–1303 (2013) 8. Wang, B., et al.: The natural fouling test of composite insulators and porcelain insulators under AC and DC voltages. High Volt. Technol. 35(9), 2322–2328 (2009) 9. Tang, C., Liang, X.: Foreign DC synthetic insulator operation and natural contamination test. Power Grid Technol. 23(9), 50–53 (1999) 10. Gubanski, S.M., Wankowicz, J.G.: Distribution of natural pollution surface layers on silicone rubber insulators and their UV absorption. IEEE Trans. Electr. Insul. 24(4), 689–697 (1989) 11. Fernando, M.A.R.M., Gubanski, S.M.: Leakage current patterns on contaminated polymeric surfaces. IEEE Trans. Dielectr. Electr. Insul. 6(5), 688–694 (1999) 12. Tu, Y., et al.: Soil particle size distribution characteristics of natural fouling insulators in a haze environment. High Volt. Technol. 40(11), 3318–3326 (2014) 13. Su, Z., Liu, Y.: Comparison of direct and AC natural pollution test results for insulators used in lines and substations in northern China. Power Grid Technol. 28(10), 13–17 (2004) 14. Li, Z., et al.: The natural fouling test of composite insulators under DC voltage. Power Grid Technol. 31(14), 10–14 (2007) 15. Zhang, X., Xu, B., Dong, S.: Adaptive median filtering for image processing. J. Comput. Aided Des. Comput. Graph. 17(2) (2005)

Chapter 103

Fast Reconstruction of Transient Heat-Flux Distributions in a Laser Heating Process with Time-Space Adaptive Mesh Refinement Qing-Qing Yang , Jiu Luo, Dong-Chuan Mo, Shu-Shen Lyu and Yi Heng Abstract The study of inverse heat transfer problems (IHTPs) have obtained extensive attention over the recent decades. However, IHTPs are often mathematically ill-posed, namely the solution stability suffers from measurement errors. Although there already exist established methods for IHTPs, their applications are mostly limited to simple geometric configurations in one- or multi-dimensions due to the high complexity. It is still a crucial task to develop special solution techniques that can ensure fast and accurate estimation. Recently, we developed a forward modeling approach and a time-space adaptive mesh refinement (TSAMR) strategy. In this work, we consider a realistic simulation study and use these methods to reconstruct laser heating boundary condition on the front surface of a 3D object subjected to laser beam heating and a combined radiation. By conducting groups of parameters tests, the solution quality is expected to be improved. The estimated heat-flux distributions and temperature distributions are presented to validate the proposed methods. Based on these results, a very large number of tests are supposed to be performed in the supercomputing plat-form in future work to obtain optimal parameters for the TSAMR strategy, so as to solve complicated IHTPs in a more efficient and accurate manner. Keywords Heat flux estimation · Inverse heat transfer problems · Time-space adaptive mesh refinement · Laser heating

Q.-Q. Yang · J. Luo School of Materials Science and Engineering, Sun Yat-Sen University, Guangzhou 510275, China D.-C. Mo · S.-S. Lyu School of Materials, Sun Yat-Sen University, Guangzhou 510006, China Y. Heng (B) School of Data and Computer Science, Sun Yat-Sen University, Guangzhou 510006, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_103

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103.1 Introduction Over the last several decades, inverse heat transfer problems (IHTPs) are of great importance for both industry and academic fields. They are variously known as boundary IHTPs, retrospective IHTPs, coefficient IHTPs, etc., which have already been extensively studied in many different fields. Example applications can be found in falling film [1], pool boiling [2, 3], microwave heating [4], material reactor [5] and steel continuous casting [6, 7]. Due to the high complexity, however, most of the numerical and application-oriented works on IHTPs focused on problems in one or two space dimensions. In our previous work, a time-space adaptive mesh refinement (TSAMR) strategy was proposed for the efficient solution of a three-dimensional (3D) IHTP in pool boiling applications [8]. From the mathematical point of view, the method still has great research potential in the sense that large-scale parameters estimation problems need to be solved with advanced parallel computing techniques, which is beyond the scope of this work. In this work, a nonlinear 3D IHTP arising in laser heating is to be studied. Such kind of IHTP are usually computationally expensive to obtain estimation results with satisfactory quality when nonlinearity, 3D geometric modeling and high-density mesh refinement are considered simultaneously. Our goal is to provide effective and accurate reconstruction of a radiation boundary condition in a 3D IHTP configuration, which is considered as the key step to accurately assess the resulting thermomechanical response of the target. For the implementation, it can be achieved by using our forward modeling approach [9] and TSAMR strategy [8] together with a heuristic parallel computing strategy.

103.2 The Inverse Problem The benchmark nonlinear 3D transient IHTP considered in this work that is related to laser heating on a plate is mathematically described in Eq. (103.1), where h, ε and T ∞ are assumed to be constant. The heat flux q on the front surface Γ S is unknown and needs to be reconstructed from temperature data on the back side of the plate Γ I , where a nonlinear radiation boundary condition is employed. Such an inverse problem can be solved by the forward modeling approach proposed in our previous work [9]. ⎧ ⎪ ρ(T )c P (T ) ∂∂tT = ∇ · (λ(T )∇T ), in Ω × (0, t f ), ⎪ ⎪ ⎪ ⎪ on Ω, ⎨ T (·, 0) = T0 (·), 4 (103.1) ) on Γ I × (0, t f ), −λ(T ) ∂∂nT = qi = h(T − T∞ ) + εσ (T 4 − T∞ ⎪ ∂T ⎪ ⎪ on Γ R × (0, t f ), −λ(T ) = 0 ⎪ ⎪ ⎩ −λ(T ) ∂∂nT = q =? on Γ S × (0, t f ), ∂n

103 Fast Reconstruction of Transient Heat-Flux Distributions …

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Fig. 103.1 The TSAMR computational strategy

103.3 Time-Space Adaptive Mesh Refinement A new TSAMR computational strategy proposed in our recent work [8] is given below to obtain accurate solutions at an acceptable computational cost in Fig. 103.1. Equation (103.2) is used to intensify regions with large heat-flux changes and guide the grid refinement at a particular time instant ti : ∀x, y, t i , S(x, y; q) ˆ =

  ∇ qˆ  0,

      ∇ qˆ   − D ·  ∇ qˆ  , Z =d Z =d

Z =d

          ≥ 0 and  qˆ  Z =d  − G ·  qˆ 

Z =d

   ≥ 0,

otherwise,

(103.2) where qˆ represents the estimated heat flux function at a certain mesh level. D, G are parameters used to trade-off peak and average values in the estimates. Based on parallel computing, towards the development of an automatic selection strategy to obtain optimal parameters, we make a first attempt to consider multi-groups of parameters tests in this work to perform error analysis for the heat-flux estimation task.

103.4 A Simulation Study A simulation study is presented in this subsection to validate the proposed method. We consider the simulation of a high-energy laser heat flux on the front surface of a target object. The temperature-dependent thermal properties of the target object material can be found in [10] and the object size is chosen as 68.4 × 68.4 × 1 mm3 . The total simulation time is 3 s with an equidistant time step of 1/12 s. The geometric model is shown in Fig. 103.2. The periodic laser heat flux qlaser imposed on the front surface is assumed to be:

  qlaser (x, y, t) = qmax · γ · exp − (x − 0.5M)2 + (y − 0.5N )2 /w 2 · (1 + sin(2π f t)),

(103.3)

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Fig. 103.2 The physical model of laser beams

where the simulation parameters are: qmax = 4000 W/m2 , γ = 0.05, M = 68.4 mm, N = 68.4 mm, w = 7.6 mm, f = 1 Hz. The simulated unknown heat flux q on the front surface is mathematically defined as follows: q = −λ(T )

∂T 4 = qlaser − h(T − T∞ ) + εσ (T 4 − T∞ ) on Γ S × (0, t f ) ∂n (103.4)

For the multi-group parameters tests, we focus on the analysis of the results at time instant t = 0.25 s, where heat-flux peaks occur. First, seven groups with different parameters D corresponding to a constant parameters G are tested in parallel; Then, after the optimal D value is fixed, the other seven groups of tests with different parameters values G are conducted in parallel. Furthermore, an error analysis of these fourteen groups of tests are given in Table 103.1, where the used error estimator E is defined by E = ω · E ave + (1 − ω) · E peak =ω·

n   

 qh (x ,y ) − q(x ,y ) /q(x ,y )  + (1 − ω) · |qh max − qmax |/|qmax |, i j i j i j i, j=1

(103.5) where qh (xi ,y j ) , q(xi ,y j ) are the reconstructed heat flux using the TSAMR strategy and the exact heat flux at the same position, respectively. n is the total number of spatial observation points. The weighting factor is a-priori chosen to be 0.7 in this study. Table 103.1 An error analysis of estimated heat flux at t = 0.25 s for different parameters (1) G = 3

D E (%)

(2) D = 0

G E (%)

22.04

10

5

3

1

0.1

0.05

0

30.59

23.24

23.09

22.44

21.84

21.68

21.63

10

5

3

1

0.7

0.5

0

21.80

21.63

20.61

20.51

21.19

26.18

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Fig. 103.3 From left to right: gradually refined meshes used for the inverse solution procedure at time instant t = 0.25 s

It is found that the error of estimated heat flux with D = 0 is the lowest when G = 3 is selected. Based on the result of the second group of tests, the choice of D = 0 and G = 0.7 leads to the most accurate estimation results in the sense of minimal errors, which will be shown below in detail. Figure 103.3 shows the spatial discretization information of gradually refined meshes used at the time instant t = 0.25 s. Explicitly speaking, the TSAMR computational strategy starts on a coarse initial mesh as shown in Fig. 103.3a. After two successive adaptive refinements, the inverse solution procedure will terminate when the error E * is less than 5% according to the pre-defined stopping criterion. This error estimator for the adaptive mesh refinement procedure is defined by ∗ + (1 − ω) · E ∗ E ∗ = ω · E ave peak n           =ω· qh+1 (xi ,y j ) − qh (xi ,y j ) /qh (xi ,y j )  + (1 − ω) · qh+1 max − qh max /|qh max |, i, j,h=1

(103.6) where h represents the level of refined meshes. The reconstructed heat-flux distributions and the simulated exact heat flux on the front surface at three representative time instants are depicted in Fig. 103.4. As we can see, the reconstructed heat flux is very close to the exact heat flux and captures the periodic dynamics. Figure 103.5 shows that the estimated temperature and the simulated temperature are similar both in shape and scale. All estimation results are obtained by using D = 0 and G = 0.7 (for the TSAMR strategy), which leads to the most accurate estimation results in the sense of minimal estimation errors.

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Fig. 103.4 a The exact and b the reconstructed heat flux distributions with the forward modeling approach [9] and the TSAMR strategy [8] at three representative time instants (from left to right: t 1 = 0.25 s, t 2 = 0.75 s, t 3 = 1 s)

Fig. 103.5 A comparison of the simulated exact temperature measurements (the left one) and the estimated temperature distribution with the forward modeling approach [9] and the TSAMR strategy (the right one) [8] at the time instant t 3 = 1 s

103.5 Conclusion and Outlook In summary, fourteen groups of tests with different D and G parameters are conducted in this work with the application of the developed forward modeling approach and the TSAMR strategy, which have been validated by a simulation case study here. It corresponds to a special type of IHTP with radiation boundary condition and temperaturedependent thermophysical properties. The proposed computational strategy can efficiently estimate the transient heat-flux distributions of the high-energy laser beams with an acceptable accuracy. In the future work, the proposed method will be targeted at various applications in other research fields. Besides, in order to find optimal D and G parameters pairs in a high-resolution solution domain, we will perform a large number of tests in parallel by using the supercomputing platform. This way, we can apply the timespace adaptive mesh refinement in an optimal sense by automatically adjusting the D and G parameters so that the solution error is minimized. Furthermore, in the context of high performance computing, a multi-nodes scalability analysis will also

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be conducted and used to optimize the IHTP solution codes so as to further improve the computational efficiency. Acknowledgements The corresponding author acknowledges support provided by the Thousand Talents Program for Young Scholars of China and the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) under Grant No. U1501501. The support from the Research Centre for Advanced Thermal Control Material and System Integration (ATCMSI) are acknowledged.

References 1. Gross, S., Soemers, M., Mhamdi, A., et al.: Identification of boundary heat fluxes in a falling film experiment using high resolution temperature measurements. Int. J. Heat Mass Transf. 48(25–26), 5549–5562 (2005) 2. Heng, Y., Mhamdi, A., Marquardt, W.: Efficient reconstruction of local heat fluxes in pool boiling experiments by goal-oriented adaptive mesh refinement. Heat Mass Transf. 46(10), 1121–1135 (2010) 3. Heng, Y., Mhamdi, A., Groß, S., et al.: Reconstruction of local heat fluxes in pool boiling experiments along the entire boiling curve from high resolution transient temperature measurements. Int. J. Heat Mass Transf. 51(21–22), 5072–5087 (2008) 4. García, E., Amaya, I., Correa, R.: Estimation of thermal properties of a solid sample during a microwave heating process. Appl. Therm. Eng. 129, 587–595 (2018) 5. Hafid, M., Lacroix, M.: An inverse heat transfer method for predicting the thermal characteristics of a molten material reactor. Appl. Therm. Eng. 108, 140–149 (2016) 6. Wang, Z., Yao, M., Wang, X., et al.: Inverse problem-coupled heat transfer model for steel continuous casting. J. Mater. Process. Technol. 214, 44–49 (2014) 7. Abbas Nejad, A., Maghrebi, M. Basirat Tabrizi, J., et al.: Optimal operation of alloy material in solidification processes with inverse heat transfer. Int. Commun. Heat Mass Transf. 37, 711–716 (2010) 8. Yang, Q.Q., Luo, J., Heng, Y., et al.: A time-space adaptive mesh refinement strategy for the inverse estimation of transient local heat flux. In: 2018 AIChE Annual Meeting, Pittsburgh, PA, USA (2018) 9. Luo, J., Yang, Q.Q., Lu, S., et al.: A novel formulation and sequential solution strategy with time-space adaptive mesh refinement for efficient reconstruction of local boundary heat flux. Int. J. Heat Mass Transf. 141, 1288–1300 (2019) 10. Cui, M., Gao, X.W., Zhang, J.B.: A new approach for the estimation of temperature-dependent thermal properties by solving transient inverse heat conduction problems. Int. J. Therm. Sci. 58, 113–119 (2012)

Chapter 104

Transfer Learning Approach in Automatic Tropical Wood Recognition System Rubiyah Yusof, Azlin Ahmad, Anis Salwa Mohd Khairuddin, Uswah Khairuddin, Nik Mohamad Aizuddin Nik Azmi and Nenny Ruthfalydia Rosli Abstract Automatic recognition of tropical wood species is a very challenging task due to the lack of discriminative features among intra wood species and very discriminative features among inter class species. While many conventional pattern recognition algorithms have been implemented and proven to solve wood image classification with 100% accuracy, when using deep learning however, the classification accuracy drops tremendously to only 36.3% due to small number of training samples. Deep learning requires large number of samples in order to work well, unfortunately, wood samples provided by the national forest institute are limited. In this paper, we explore the use of transfer learning in deep neural network for the classification of tropical wood species based on image analysis. Several model of deep learning techniques are tested and results have shown that the classification performance after transfer learning was added reaches 100% accuracy. Keywords Wood recognition system · Wood classification · Deep learning · Transfer learning

104.1 Introduction Illegal logging negatively impacts the economic, environment and ecological systems of optimal forest management. Environmental impacts include the loss or degradation of forests, as illegal logging tends to be associated with poor forest management. Any R. Yusof (B) · U. Khairuddin · N. M. A. N. Azmi · N. R. Rosli Centre of Artificial Intelligence and Robotic (CAIRO), Malaysia-Japan International Institute of Technology, Universiti Teknologi Malaysia, 50300 Kuala Lumpur, Malaysia e-mail: [email protected] A. Ahmad Faculty of Computer and Mathematical Sciences, Advanced Analytics Engineering Centre (AAEC), Universiti Teknologi MARA, 40450 Shah Alam, Selangor, Malaysia A. S. M. Khairuddin Department of Electrical Engineering, Faculty of Engineering, Universiti Malaya, Kuala Lumpur, Malaysia © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_104

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acts related to logging not with accordance to the national law are considered illegal logging including harvesting, transporting, buying, selling and processing illegally logged timber. These acts are made legal by fraudulent labelling of timber. In addition to the environmental damage, fraudulent labelling practiced by some timber exporters will reduce the country’s incomes generated by tax. This is because lower tax will be imposed when high quality timber is labelled as low quality timber. Illegal logging is one of the concerns encountered by timber exporting countries for instance Malaysia hence an intelligent wood species classification system that can automatically classify the species is essential. Besides that, timber importing countries such as European countries may also benefit from the proposed intelligent wood species recognition system where they can verify the authenticity of the wood species they received. Normally, wood species are identified manually by wood experts based on the pattern of the wood surface texture. Experts who examine wood surface texture scrutinize several important characteristics of the wood such as the arrangement of vessel or pores, wood parenchyma or soft tissue, rays parenchyma, fibers, phloem, latex traces and intercellular canals. This process of manual inspection is tedious and time consuming. Besides that, it is impractical and cost ineffective for a human to analyze and identify large number of timber species. Therefore, a reliable automatic wood recognition device is needed in order to classify the wood species efficiently to monitor every operation and export transaction to prevent illegal logging. Over the years, several automatic wood recognition systems have been developed. The system by Khalid et al. [1] is able to classify 20 tropical wood species accordingly based on macroscopic wood anatomy by using grey level co-occurrence matrix (GLCM) to extract the wood features, and Back Propagation Neural Network (BPNN) as classifier. Nasirzadeh et al. [2] used LBP histogram to extract features from the enhanced wood images and nearest neighbor classifier is employed in the computed space with Chi-square as a dissimilarity measure. Yusof and Khairuddin [3] proposed a new mutation operation for faster feature selection by Genetic Algorithm (GA) based on the exclusiveness of an allele to select only discriminative features. This method has increased the optimal convergence rate for feature selection while maintaining the classification accuracy. One of the main issues relating to the designing of wood species recognition system based on texture analysis is the lack of discriminative features among some species of the wood, and also some very discriminative features among inter class species, as well as noises due to illuminations, or uncontrolled environment. The wood features such as the size of pores, the density of pores, etc. depends very much on the age, weather and other factors, contributing to the irregularities of the features. Besides that, the other main problem is the overlapped issues among the wood species due to some of the wood species have different features even though they are of the same species. Moreover, the task has become harder and tougher caused by some influential factors; including (1) wood reaction, (2) fungus attack, (3) site condition and (4) weather and light condition. These factors have made the wood species classification to become more complicated and difficult.

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In [4], the performance of several classic methods of wood classification and deep learning were compared. The deep learning approach in particular the Res-Net— residual network based Convolutional Neural Network (CNN); has become a wellknown technique for image recognition due to its advantages; ruggedness to shifts and distortion in the image, fewer memory requirements and easier and better training. However, one of the main challenge in the use of ResNet is that its performance is highly dependent on the size of the training dataset. In eye recognition system by [5] using mixed CNN and ResNet, the accuracy of the system is only improved when the sample size is increased from 1000 to 940,000 by making several rotations and cropping procedures on the original images in the training set. In this paper, we investigate the effectiveness of employing transfer learning approach in classifying the wood species based on image analysis. Transfer learning is a popular approach in deep learning where pre-trained models of other image classification jobs are used as the starting point on another image recognition tasks. Presently, transfer learning has shown its worth in a wide variety of contexts including such as in analyzing different types of coral species based on texture images [6], detection of skin lesions [7], steganalysis system [8], person re-identification [9], liver steatosis assessment in ultrasound images [10], speech recognition [11], pedestrian recognition in multi-camera networks [12], and hologram classification for molecular diagnostics [13].

104.2 Methodology Wood Database. For this research, wood samples were obtained from the Forest Research Institute of Malaysia (FRIM) where it comes in a 1-inch by 1-inch cubic form (Fig. 104.1a). The wood surface images were captured using a built-in camera specially designed for the system which provides similar 10-times magnification (Fig. 104.1b) as per manual wood identification requires [1]. When building wood image database, from each wood sample, 10 images with different orientation are captured from the wood cube surfaces, and 9 images were put in the training set while

Fig. 104.1 a Wood samples in cubic form and b wood surface image in 10-times magnification

1228 Table 104.1 20 wood species dataset

R. Yusof et al. 20 wood species dataset Bintangor

Merbau

Chengal

Mersawa

Durian

Nyatoh

Gerutu

Penarahan

Jelutong

Perupok

Keledang

Punah

Kempas

Ramin

Kungkur

Sepetir

Mataulat

Sesendok

Melunak

Terentang

1 image was set aside to be in the training set. For each wood species, there are 10 wood cubes. Therefore, for each species, a total of 90 images are for training, and 10 images in testing set. Wood species included in the dataset are listed in Table 104.1. Each images are labelled according to their species name. Residual Neural Network (Res-Net-8). The Res-Net-8 was used in the preliminary stage of deep-learning exploration in the automatic wood species recognition due to its ability to train thousands of layers and can avoid the exploding gradient problem due to the incremental number of network depth. The architecture of ResNet is shown in Fig. 104.2. In order to improve the model capabilities, it used the identity skip-connection, which can help the gradient to flow back into layers without vanishing [14, 15]. In this application, two image-processing operations are used to pre-process the images, which are (1) image sharpening using high-pass filter and (2) contrast enhancement using HE algorithm. The high-pass filter is used to sharpen and emphasize the fine details of the image, while contrast enhancement process using HE algorithm is to equalize the input image intensity, in order to improve the visual quality of the images. Figure 104.3 shows the effect of the pre-processing techniques used on a wood image. The trained ResNet-8 models in both experiments are tested with testing dataset. The classification accuracy is calculated as follows:

Fig. 104.2 Res-Net architecture

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Fig. 104.3 a Original image, b highpass filter image, c highpass applied image, d HE applied image

Accuracy =

no. o f corr ectly classi f ied images no. o f test images

(104.1)

ResNet-50 Model. The ResNet-50 model used is proposed by Cho et al. [16] as shown in Fig. 104.4. The ResNet-50 architecture is the same as ResNet-8 model but with more hidden layers for learning and feature extractions added in the system. Softmax layer was added in the final step to perform classification. Another model tested was Bi-directional Gated Recurrent Unit (GRU) layers, which is proposed by He et al. [17], where the GRU layers are built on top of the ResNet-50 model (Fig. 104.5).

Fig. 104.4 ResNet-50 model

Fig. 104.5 ResNet-50-BiGRU model

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Transfer learning for ResNet-50. Transfer learning in general is using knowledge obtained by solving other image classification problem to solve another image classification task. It is useful when the latter has inadequate size of dataset. The 20 species wood image database contains only 1800 images for training and 200 images for testing and this database size was proven inadequate for deep learning in [4]. Therefore, the ResNet-50 model implemented here was pre-trained using transfer learning knowledge from MNIST dataset. MNIST dataset is a public database contains 60,000 training images and 10,000 images of handwritten digits by LeCun et al. [18]. The pre-training in ResNet-50 only uses 5 epochs and this transfer learning knowledge was also used by the ResNet-50-BiGRU model. Dataset splitting for ResNet-50 training. After ResNet-50 and ResNet-50BiGRU have been pre-trained with MNIST dataset, both models were used for training. There were two-types of training; with dataset splitting for validation and no validation set. For training with dataset splitting, some image samples from the training set were set aside for validation of losses and accuracy calculation after each epoch in order to measure the ResNet model performance. Therefore, the training set is randomly split into 95:5 where 5% of the images were put aside as the validation set. In order to ensure the random splitting process is fair, the process was repeated three times with different random seed number of 0, 1 and 2. At different seed number, images chosen for validation set will be different every time.

104.3 Results and Discussions ResNet-8 accuracy. In [4], ResNet-8 classification accuracy was only 36.3%. In using ResNet-8, no feature extraction is needed. Images are cropped and preprocessed and directly being feed as an input to the network. The classification accuracy was very low due to the inability of the Res-Net to generalize the features. The problem of overfitting is so prevalent in this method due to very small number of training images. ResNet-50 without training set split accuracy. The ResNet-8 model was upgraded with more additional hidden layers which made up ResNet-50 model and a Softmax layer was added in the end to solve classification task. Another model that was tested was ResNet-50-BiGRU and both models were tested with and without transfer learning, where all of them are without training set splitting. The results are as follows. Results in Table 104.2 shows that an upgraded ResNet-50 model have worse performance than it’s ResNet-8 predecessor where for training set, the accuracy was less than 1% and for testing set, only 15.56%. However, when the model was pretrained with transfer learning knowledge from MNIST database, the performance increases to 81.43% and 16.98% accuracy for training and testing set respectively. However, the performance was still inadequate if compared to other traditional image recognition methods in [4].

104 Transfer Learning Approach in Automatic Tropical Wood … Table 104.2 ResNet-50 and ResNet-50-BiGRU model performance without training set split

1231

Model

Transfer learning

Accuracy (%) Training set

Testing set

ResNet-50

Yes

81.43

16.98

ResNet-50

No

0.87

15.56

ResNet-50BiGRU

Yes

3.99

8.96

ResNet-50BiGRU

No

4.21

5.18

When Bi-directional Gated Recurrent Unit (GRU) layers were added, without transfer learning in place, the performance only increased to 4.2% for training set and 5.1% for testing set. Surprisingly, after it was pre-trained with transfer learning, the accuracy for training set dropped to 3.99% but increased for testing set to 8.96% accuracy. ResNet-50 with training set split accuracy. Training set splitting process were added where 5% of images in training set were put aside as validation set. After each epoch run during ResNet training, the validation losses and accuracy were calculated using this validation set. The process of taking out 5% of the images is random. Three random seed numbers; 0, 1 and 2; were used to ensure the model will work with any random validation set. The results after training set split are as follows (Table 104.3). With training set split, the performance of ResNet-50 increases and achieved 100% accuracy in most cases. When seed number was 0, without transfer learning in place, the training accuracy of ResNet-50 was 98.90%, and with transfer learning, both training and testing set achieved 100% accuracy. Similar performance was observed when seed number is 1. When seed number is 2, the performance for ResNet-50 were perfect for all cases; with or without transfer learning. For ResNet-50-BiGRU model, without transfer learning, it never achieves 100% accuracy for testing set. However, with transfer learning, for seed number equal 0 and 1, the performance may achieve 100%. It can be concluded that both ResNet-50 models will work well with transfer learning when training set splitting were in place. ResNet-50 model can also work well even without Bi-directional GRU layers added in the model.

104.4 Conclusions In this paper, we explore the execution of deep learning technique in automatic wood recognition system. The techniques tested was ResNet-50 model with and without additional Bi-directional GRU layers, with and without transfer learning, and with and without training set splitting. The transfer learning is essential to run deep learning technique on wood recognition where the sample size is small.

1232 Table 104.3 ResNet-50 and ResNet-50-BiGRU model performance with training set split

R. Yusof et al. Model

Transfer learning

Accuracy (%) Training set

Testing set

ResNet-50

Yes

100.00

100.00

ResNet-50

No

98.90

100.00

ResNet-50BiGRU

Yes

99.94

100.00

ResNet-50BiGRU

No

100.00

98.11

ResNet-50

Yes

100.00

100.00

ResNet-50

No

99.94

100.00

ResNet-50BiGRU

Yes

100.00

99.06

ResNet-50BiGRU

No

100.00

99.53

ResNet-50

Yes

100.00

100.00

ResNet-50

No

100.00

100.00

ResNet-50BiGRU

Yes

99.88

99.53

ResNet-50BiGRU

No

100.00

99.06

Seed number = 0

Seed number = 1

Seed number = 2

However, even with pre-trained model in transfer learning, the performance of ResNet-50 models was not satisfactory. Additional process of training set split was required in order for the model to work well. The training set split was used for validation losses and accuracy calculation during model training. Therefore, model’s performance increases and achieved 100% accuracy when combined with transfer learning. ResNet-50 model with both transfer learning and training set split achieved 100% accuracy and does not need additional Bi-directional GRU layers to achieve the high accuracy. Any seed number during training set split seems to work well for the model. The deep learning model used in this paper has been proven to work with 20 tropical wood species. For future work, the model will be further explored on more wood species and even on wood species with very small sample size.

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References 1. Khalid, M., Lew, E., Lee, Y.I., Yusof, R.: Design of an intelligent wood species recognition system. Most 9(3), 9–17 (2008) 2. Nasirzadeh, M., Khazael, A.A., Bin Khalid, M.: Woods recognition system based on local binary pattern. In: 2010 2nd International Conference on Computational Intelligence, Communication Systems and Networks, (2), pp. 308–313 (2010) 3. Yusof, R., Khairuddin, U.: A new mutation operation for faster convergence in genetic algorithm feature selection. Int. J. Innov. Comput., Inf. Control. (IJICIC), 8(10(B)), 7363–7379 (2012) 4. Yusof, R., Khairuddin, U., Rosli, N.R., Abdul Ghafar, H., Nik Azmi, N.A., Ahmad, A. Mohd Khairuddin, A.S.: A study of feature extraction and classifier methods for tropical wood recognition system. IEEE TENCON2018, pp 1–6 (2018) 5. Wang, Z., Li, C., Shao, H., Sun, J.: Eye recognition with mixed convolutional and residual network (MiCoRe-Net). IEEE Access 6, 17905–17912 (2018) 6. Gomez-Rios, A., Tabik, S., Luengo, J., Shihavuddin, A.S.M., Krawczyk, B., Herrera, F.: Towards highly accurate coral texture images classification using deep convolutional neural networks and data augmentation. Expert Syst. Appl. 118, 315–328 (2019) 7. Soudani, A., Barhoumi, W.: An image-based segmentation recommender using crowd sourcing and transfer learning for skin lesion extraction. Expert Syst. Appl. 118, 400–410 (2019) 8. Tian, J., Li, Y.X.: Convolutional neural networks for steganalysis via transfer learning. Int. J. Pattern Recognit. Artif. Intell. 33(2) (2019) 9. Xian, Y., Hu, H.: Enhanced multi-dataset transfer learning method for unsupervised person re-identification using co-training strategy. IET Comput. Vision 12(8), 1219–1227 (2018) 10. Byra, M., Styczynski, G., Szmigielski, C., Kalinowski, P., Michalowski, L., Paluszkiewicz, R., Bogna, Z.W., Krzysztof, Z., Piotr, S., Andrzej, N.: Transfer learning with deep convolutional neural network for liver steatosis assessment in ultrasound images. Comput. Assist. Radiol. Surg. 13, 1895–1903 (2018) 11. Qin, C.X., Qu, D., Zhang, L.H.: Towards end-to-end speech recognition with transfer learning. EURASIP J. Audio Speech Music. Process. (2018). https://doi.org/10.1186/s13636-0180141-9 12. Wang, J.T., Yan, G.L., Wang, H.Y., Hua, J.: Pedestrian recognition in multi-camera networks based on deep transfer learning and feature visualization. Neurocomputing 316, 166–177 (2018) 13. Kim, S.J., Wang, C., Zhao, B., Im, H., Min, J., Choi, H.J., Tadros, J., Choi, N.R., Castro, C.M., Weissleder, R., Lee, H., Lee, K.: Deep transfer learning-based hologram classification for molecular diagnostics. Sci. Rep. 8(17003) (2018) 14. He, K., Zhang, X., Ren, S., Sun, J.: Identity mappings in deep residual networks. arXiv:1603. 05027 (2016) 15. Abdi, M., Nahavandi, S.: Multi-residual networks. arXiv:1609.05672 [cs], September (2016) 16. Cho, K., Van Merriënboer, B., Gulcehre, C., Bahdanau, D., Bougares, F., Schwenk, H., Bengio, Y.: Learning phrase representations using rnn encoder-decoder for statistical machine translation. arXiv:1406.1078 (2014) 17. He, K., Zhang, X., Ren, S., Sun, J.: Deep residual learning for image recognition. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (pp. 770–778) (2016) 18. LeCun, Y., Cortes, C., Burges, C.: Mnist handwritten digit database. AT&T Labs. http://yann. lecun.com/exdb/mnist (2010)

Chapter 105

Moving Least Squares (MLS) Interpolation Based Post-processing Parametric Study in Finite Element Elastic Problems Mohd. Ahmed, Mohamed Hechmi El Ouni, Devender Singh and Nabil Ben Kahla Abstract The parametric effect on performance of Moving Least Squares (MLS) interpolation based recovery technique is evaluated in this paper. The Moving Least Squares (MLS) fitting recovers the field variable derivatives over nodes patch using background element meshes. The recovered errors at element and global levels in the finite element solution are presented in energy norm. The study considers three basic recovery parameters (i.e. parameters affecting the post-processed results) namely shape of influence (support) domain, dilation parameter and order of poly nominal basis function. Numerical experiments on elastic plate problems are carried out for parametric effect of interpolation based post processing technique on effectivity of error estimation and rate of convergence of the recovered solution with fineness of the meshing scheme. The linear and quadratic triangular elements have been used for the discretization of the problem domain. The circular and rectangular shape domain of influence is used to form the node patch. Six different dilation parameters and three different number of basis function terms are selected in the moving least squared interpolation formulation. The study shows that recovery parameters of MLS interpolation method have pronounced effect on the post-processing recovery of finite element solution and optimal alternatives are to be adopted for better performance of the recovery procedures. Keywords Effectivity · Error estimation · Error norm · Moving least square interpolation · Post-processing techniques

Mohd. Ahmed (B) · M. H. El Ouni · N. B. Kahla College of Engineering, Main Campus, Gregar, K. K. University, Abha 61411, Saudi Arabia e-mail: [email protected] D. Singh Ministry of Information, Soochan Bhavan, CGO Complex, Delhi 110003, India © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_105

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105.1 Introduction The finite element method yields approximate solutions for the problems, it become necessary to control the errors inherent in the method. The development of recovery of solution errors, priori and posteriori-error estimation techniques are underway for last several years. An extensive survey of a posteriori error estimation is due to Ainsworth and Oden [1] and Gratsch and Bathe [2]. Mirzaei [3] provides a comprehensive analysis of error estimation considering moving least square approximating. An error estimation method based on side curvature has been presented by Xing et al. [4]. It was used to estimate the accuracy of finite element discretization of a deformed sheet, especially one with sharp or quasi-sharp corners. Liu and Elmaraghy [5] discussed discretized errors, and their estimation in conjunction with quadrilateral finite element meshes which are generated by an intelligent mesh generator. Zienkiewicz and Zhu [6] proposed an error estimator which is not only accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes. The estimator computes energy norm of error on global as well as local levels. Liu et al. [7] has proposed technique to obtain the exact finite solution to mechanics problem using the alpha finite element method (αFEM). Kim and Lee [8] applied the equilibrated residual method with set of Neumann data for nonconforming elements to improve the finite element linear elasticity solution. Ullah et al. [9] proposed a coupled finite element method (FEM) and the element-free Galerkin method (EFGM) error estimation based on local maximum entropy shape functions, for linear and nonlinear problems. Many different post-processing techniques can be employed to improve the quality of the derivatives of the finite element approximations such as averaging [10], local or global projections [11], and those exploring the super-convergence phenomenon. The nodal average method is a simple technique that leads to optimal convergence rates when only linear elements are considered. A general recovery technique has been developed by Zienkiewicz and Zhu [12, 13] for determining the derivatives. The implementation of the recovery technique is simple and cost effective. The technique has been tested for a group of linear, quadratic and cubic elements for both one and two dimensional problems. Numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are super-convergent. One order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for derivatives with quadratic elements. In particular, an O(h4 ) convergence of the nodal values of the derivatives for a quadratic triangular element is reported. Zienkiewicz and Zhu [12, 13] derived a theorem on dependence of the effectivity index for the Zienkiewicz-Zhu error estimator and the convergence rate of a recovered solution. They found that with super-convergent recovery, the effectivity index tends asymptotically to unity. Numerical tests on various element types illustrate the effectivity of their error estimator in the energy norm and pointwise gradient (stress) error esti-

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mation. Li and Wiberg [14] presented a post-processing technique for obtaining a posteriori error estimators both in the energy and L2 norms, An element patch that represents the union of the considered element and its neighbours is introduced. The post-processing for determining more accurate solutions is made by fitting a higher order polynomial expansion to the computed solution at super-convergent points in the patch. The element error estimate norms are calculated directly from the improved solutions. The recovery by equilibrium in patches (REP) and the recovery by compatibility in patches (RCP) to obtain the improved stresses to the computed stress are due to Ubertini [15]. Parret et al. [16] propose a moving least squares (MLS) recovery-based procedure to obtain post- processed smoothed stresses field in which the continuity of the recovered field is provided by the shape functions of the underlying mesh. Sharma et al. [17] propose a stress recovery procedure for low-order finite elements in 3D in which the recovered stress field is obtained by satisfying equilibrium in an average sense and by projecting the directly calculated stress field onto a conveniently chosen space. More recently, Ahmed et al. [18] has effectively implemented the element free Galerkin based recovery technique in finite element analysis of elastic problems with super-convergent properties. The finite element analysis coupled with post-processing technique is influenced by several recovery parameters (i.e. parameters affecting the post-processed results) including meshing scheme, post-processed field variable, error norm, interpolation type for post processing, shape and size of influence domain, dilation parameter, order of polynomial expansion in basis function, weighting function etc. There are only a few published works on the role of recovery parameters in interpolation based post-processed finite element procedures. Rajendran and Liew [19] propose bestfit approach for predicting optimal stress sampling points for the patch recovery of nodal stresses. Wang and Liu [20] and Kanber et al. [21] have propose optimal shape parameters in uniform grids while Perko and Sarler [22] proposes shape parameters optimization in non-uniform grids. Neil et al. [23] proposes a practical mathematics model for obtaining the optimal radius of support of radial weights used in MLS methods. Onate et al. [24] has discussed the completeness, robustness, continuity and accuracy aspect of moving least squares (MLS) interpolation procedure. The present study deals with the influence of recovery parameters on MLS interpolation recovery based finite element analysis on elastic two-dimensional plates problems. The error at element and global levels in the finite element solution are presented in energy norm. The plate domain is discretization using linear and quadratic triangular elements. The three recovery parameter namely, shape of influence domain, dilation parameter and order of polynomial expansion in basis function are selected for the parametric investigation.

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105.2 Finite Element Elastic Formulation Two-dimensional linear elastic problems are governed by the following differential equations, L T σ + f = 0 in domain Ω

(105.1)

with boundary conditions as σ · n = t¯ and u = u¯ in which σ is the stress field, f is the force vector, LT is the derivative operator, t¯ and u¯ are prescribed tractions and displacements on  t and  u , respectively, and n is the unit outward normal on the boundary  =  t ∪ u. By finite element discretization, the displacements (u) of any point within an element are calculated based on the following equation, u=Nd

(105.2)

where, N is the matrix of the interpolation functions, also known as shape functions and d is the nodal displacement matrix,. The strains can be related to the nodal displacements by the following formula, ε= Lu= LNd = Bd

(105.3)

In which, B is the strain interpolation matrix. The relationship between nodal forces (F) and nodal displacements can be written as F = K d in which K is the stiffness matrix.

105.3 Moving Least Squares (MLS) Interpolation Technique The interpolation based recovery technique derived the approximate functions using a set of nodes distributed over a domain, without depending on the meshing scheme, in least square sense. The Moving Least Square (MLS) approximation, uh (x), can be defined as, uh (x) =

m 

φs (xi ) · u i = φ(x)u

(105.4)

i=1

where φ(x) is shape function and m is the total number of terms in the basis function. In the present study, m is taken as six for linear and nine for quadratic elements.   P T (x) = 1, x, y, x 2 , x y, y 2 for linear element

(105.5)

105 Moving Least Squares (MLS) Interpolation …

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  P T (x) = 1, x, y, x 2 , x y, y 2 , x 2 y, x y 2 , x 2 y 2 for quadratic element

(105.6)

The φ 1 (x) is given as, φ1 (x) = P T (x) · a −1 (x)P(x1 )w(x − x1 )

(105.7)

The vector of coefficients a(x) can be obtained by minimizing a weighted residual as follows, J=

n 

2  wi (x − xi ) P T (xi )a(x) − u ih

(105.8)

i=1

where n is the number of nodes i and w(x − xi ) is a weight function in 2-D associated to each node (domain of influence of that node) which is usually built in such a way that it takes a unit value in the vicinity of the point where the function and its derivatives are to be computed and vanishes outside a region i surrounding the point xi . The support of the shape function φ1 (x) is equivalent to the support of the weight function. The following cubic spline weight function with circular or rectangular domain of influence is considered in the present study. ⎧ 2 ¯2 ¯3 ¯ 1   ⎨ 4 3 − 4d +2 4d 4 3 f or d ≤ 2 w(x − xi) = w d¯ = 3 − 4d¯ + 4d¯ − 3 d¯ f or 1 ≤ d¯ ≤ ⎩ 0 f or d¯ > 1

1 2

⎫ ⎬ ⎭

(105.9)

where d¯ = x − xi /dw and dw (= dmax · dxi ) is the size of influence domain of the point x i where dmax is the dilation parameter. Minimization of weighted residual leads to φ(x) = P T (x) · a −1 (x)B(x)φ h

(105.10)

n T = [w1 (x − where a(x) = i=1 wi (x − x i )P(x i ) · P (x i ) and B(x) x 1 )P(x1 ), . . . , wn (x − x n )P(xn )] By putting MLS shape function into the well-known form of shape function equation, we get, φ(x) = N T (x) · φ h in which N T (x) = P T (x) · a −1 (x)B(x).

105.4 Error Evaluation The error in computed stress (σ), e*σ is defined as the difference between the exact or (recovered) values of σ and respective finite element solution value, σh i.e. eσ∗ = σ − σh ,

(105.11)

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The errors can be quantified in appropriate norms. The error in energy norm (E) may be defined as follows.  1/2 e E = eσ∗T Deσ∗ dΩ

(105.12)

An estimator is asymptotically exact for a particular problem if the problem global and local (element) effectivity index (θ ) i.e. ratio of estimated error and true error, converges to one when the mesh size approaches to zero. θ=

eex  e

(105.13)

where eex  and e denote the exact error and the evaluated error estimate in energy norm respectively.

105.5 Parametric Study of MLS Interpolation Based Recovery Technique Numerical experiments on elastic plate problems were carried out to study the effect of the recovery parameters (i.e. parameters affecting the post-processed results) in terms of effectivity and rate of convergence. The plate domain is discretization using linear and quadratic triangular elements. The error at element and global levels in the finite element solution are computed in energy norm. The effect on the performance of recovery technique is studied by varying three parameter of MLS interpolation technique namely, shape of influence domain, dilation parameter and number of terms in basis function. The circular and rectangular shape domain of influence is used to form the node patch from the background mesh. Six different dilation parameters namely, 2.4, 3.0, 4.0, 4.5, 5.5 and 6.5, are selected for the moving least squared interpolation formulation with consideration to order of element. Two set of number of basis function terms namely, 3, 6, 9 and 6, 9, 10 are selected for fitting the derivatives field variable over the node patches in influence domain to recover the solution error in linear and quadratic element respectively.

105.5.1 Elastic Plate Problem The square elastic plate having body loads (bx , by ) is tested to study the effect of recovery parameters. The body force of the plate in the form of polynomials and close form displacements solution (u, v) due to body load are given in Eqs. 105.11 to 105.13 [12, 13]. The linear and quadratic triangular elements meshing in structured as well as unstructured manner (Fig. 105.1) are employed to discretize the plate domain.

105 Moving Least Squares (MLS) Interpolation …

(a): Structured mesh

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(b): Unstructuredd mesh

Fig. 105.1 Elastic plate problem discretized with triangular elements

bx = (α + β) · (1 − 2x) · (1 − 2 · y) b y = −2 · β · y · (1 − y) − (α + 2 · β) · 2 · x · (1 − x) u = 0, and v = −x · y · (1 − x) · (1 − y)

(105.14) (105.15) (105.16)

where constants α and β are α = [E · ν/((1 – 2 · ν) · (1 + ν))], β = [E/(2 · (1 + ν))], and E and ν are Modulus of elasticity and Poisson’s Ratio respectively with a value of 1.0 N/mm2 and 0.3.

105.5.1.1

Recovery Parameter: Influence Domain Shape

The domain of influence shape construction for circular and rectangular shape used in moving least square based recovery finite element analysis is shown in Fig. 105.2. The dilation parameter for shape effect investigation is considered as 3.0 for linear elements and 4.5 for quadratic elements. The number of terms in polynomial basis is taken as 6 and 9 respectively for linear and quadratic elements. The computed error convergence and effectivity in finite element solution and, recovered solution with linear and quadratic meshing and with different order of fineness for circular and rectangular influence domain are given in Tables 105.1, 105.2, 105.3 and 105.4. The tables show that the performance of the linear element discretization is better than the quadratic element discretization of plate. The overall rate of convergence is higher

Field nodes Influence domains shape Fig. 105.2 Influence domains shape for MLS interpolation based recovery technique

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Table 105.1 Error convergence and global effectivity for square plate problem with influence domain shape in post-processing techniques (linear triangular structured meshing) Mesh size (1/h)

Error in FEM (energy norm)

Circular Error in stress recovery

Effectivity, θ

Error in stress recovery

Rectangular Effectivity, θ

1/4

0.09375

0.03123

0.918019

0.07077

0.995808

1/8

0.04845

0.00694

0.977688

0.01730

0.985682

1/12

0.03251

0.00285

0.989683

0.00717

0.997667

1/16

0.02444

0.00152

0.994092

0.00327

0.997166

1/24

0.01632

0.00063

0.997332

0.00122

0.998329

1/32

0.01225

0.00034

0.998489

0.00062

0.999086

Rate of convergence

0.97875

2.17052

2.27789

Table 105.2 Error convergence and global effectivity for square plate problem with influence domain shape in post-processing techniques (linear triangular unstructured meshing) Mesh Elem. no.

DoF

Error in FEM (energy norm)

Error in stress recovery

Circular

Rectangular Effectivity, θ

Error in stress recovery

Effectivity, θ

45

66

0.06789

0.01672

0.958672

0.04963

1.032141

88

118

0.05018

0.00918

0.970474

0.02505

0.999938

223

270

0.03064

0.00428

0.983250

0.00857

0.989058

925

1014

0.01387

0.00114

0.995079

0.00169

0.996055

1978

2106

0.00929

0.00057

0.995949

0.00073

0.997592

Table 105.3 Error convergence and global effectivity for square plate problem with influence domain shape in post-processing techniques (quadratic triangular structured meshing) Mesh size (1/h)

Error in FEM (energy norm)

Circular Error in stress recovery

Effectivity, θ

Error in stress recovery

Rectangular Effectivity, θ

1/4

0.013168

0.000632

0.997272

0.000186

0.999555

1/8

0.003388

0.000831

0.999601

0.000020

0.999924

1/12

0.001515

0.000241

0.999889

0.000007

1.000053

1/16

0.000854

0.000010

0.999960

0.000002

0.999993

1/24

0.000380

0.000006

1.000033

0.0000008

1.000004

Rate of convergence

1.978481

2.615516

3.038872

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Table 105.4 Error convergence and global effectivity for square plate problem with influence domain shape in post-processing techniques (quadratic triangular unstructured meshing) Mesh No. of elem.

DoF

Error in FEM (energy norm)

Circular Error in stress recovery

Rectangular Effectivity, θ

Error in stress recovery

Effectivity, θ

45

110

0.006953

0.000507

0.994336

0.000153

0.999081

88

205

0.003453

0.000233

0.993854

0.000071

1.000328

223

492

0.001328

0.000078

0.995469

0.000022

0.999859

925

1938

0.000319

0.000016

0.997628

0.000005

1.000142

for lower order elements. When comparing the influence domain shape effect on the post-processed analysis, the rectangular domain construction has better performance as compared to circular shape of domain for both lower and higher order element background mesh. The convergence rate is higher and the effectivity is closer to one in case of rectangular influence domain. Though, the order of error in energy norm obtained is higher with linear element mesh in rectangular domain used for recovery of the solution error. The element (local) effectivity frequency in plate problem domain is also plotted for different recovery parameters (Fig. 105.3). The local (element) effectivity frequency for linear and quadratic elements for influence domain shape is plotted in Figs. 105.4 and 105.5 at last refinement level considered. The stress error distribution in plate domain for circular and rectangular influence domain shape derivatives are depicted in Figs. 105.6 and 105.7. From Tables 105.1, 105.2, 105.3 and 105.4 and Figs. 105.4, 105.5, 105.6 and 105.7, it is clear that recovery parameter, influence domain shape significantly affect the performance of post-processing recovery based finite element analysis. The Figs. 105.4 and 105.5 shows that unstructured discretization and higher order element is more sensitive to influence domain shape in comparison to low order elements as the effectivity for the most of the element is nearly one with structured low order element meshing. From exact and recovered

(a) Linear Triangular Element

(b) Quadratic Triangular Element

Fig. 105.3 Histogram-element effectivity versus number of elements for plate problem using different shape of influence domain in recovery based finite element analysis

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(a) Exact FEM Error (Structured)

(b) Exact FEM Error (Unstructured)

(c) Recovered Error- Circular (Structured) (d) Recovered Error- Rectangular (Structured)

(e) Recovered Error- Circular (Unstruct.)

(f) Recovered Error- Rectangular (Unstruct.)

Fig. 105.4 Error distribution in plate problem using different influence domain shape in recovery based analysis (linear elements, structured mesh at size =1/32 and unstructured mesh at 1978 elements)

error distributions plots, for recovery parameter influence domain shape, confirms that structured linear element perform better than quadratic elements as the error distribution patterns for exact and recovered error distributions for linear element are in good agreement.

105.5.1.2

Recovery Parameter: Dilation Parameter

The density of the particle distribution in influence domain is measured by dilation parameter. The suitable dilation parameter supports the MLS interpolation function to automatically satisfy its completeness condition. Three dilation parameters namely, 2.5, 3.0 and 4.0, are considered in the moving least squared interpolation based post processing finite element analysis with linear element meshing. For quadratic element meshing, the dilation parameters are 4.5, 5.5 and 6.5. The effect of various dilation parameter values is observed six numbers of terms and nine numbers of terms in polynomial basis function respectively for linear and quadratic elements. The convergence rate of error and effectivity of error estimation with different dilation

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(a) Exact FEM Error (Structured)

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(b) Exact FEM Error (Unstructured)

(c) Recovered Error -Circular (Structured) (d) Recovered Error- Rectangular (Structured)

(e) Recovered Error–Circular (Unstruct.) (f) Recovered Error-Rectangular (Unstruct.) Fig. 105.5 Error distribution in plate problem using different influence domain shape in recovery based analysis (quadratic elements, structured mesh at size =1/32 and unstructured mesh at 1978 elements)

(a) Linear Triangular Element

(b) Quadratic Triangular Element

Fig. 105.6 Histogram-element effectivity versus number of elements for plate problem using different dilation parameters in recovery based finite element analysis

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(a) Linear Triangular Element

(b) Quadratic Triangular Element

Fig. 105.7 Histogram-element effectivity versus number of elements for plate problem using different order of basis function in recovery based finite element analysis

parameter with increasing fineness order of mesh in MLS interpolation based postprocessed finite element analysis are given in Tables 105.5 and 105.6. The tables Table 105.5 Error convergence and global effectivity for square plate problem with dilation parameter (dmax ) in post-processing techniques (linear elements) Mesh size (1/h)

dmax = 2.5 Error in stress recovery

dmax = 3.0 Effectivity, θ

Error in stress recovery

dmax = 4.0 Effectivity, θ

Error in stress recovery

Effectivity, θ

1/4

0.02601

0.909122

0.03123

0.918019

0.047584

0.933420

1/8

0.00606

0.977050

0.00694

0.977688

0.00942

0.973763

1/12

0.00606

0.989929

0.00285

0.989683

0.00374

0.987061

1/16

0.00138

0.994401

0.00152

0.994092

0.00194

0.992386

1/24

0.00058

0.997556

0.00063

0.997332

0.00078

0.996476

1/32

0.00032

0.998641

0.00034

0.998489

0.00041

0.997982

Conv. rate

2.11135

2.17052

2.28303

Table 105.6 Error convergence and global effectivity for square plate problem with dilation parameter (dmax ) in post-processing techniques (quadratic elements) Mesh size (1/h)

dmax = 4.5

dmax = 5.5

dmax = 6.5

Error in stress recovery

Effectivity, θ

Error in stress recovery

Effectivity, θ

Error in stress recovery

Effectivity, θ

1/4

0.000632

0.997272

0.000430

1.001457

0.000263

0.999218

1/8

0.000831

0.999601

0.000053

1.001100

0.000034

0.999982

1/12

0.000241

0.999889

0.000014

1.000633

0.000009

1.000045

1/16

0.000010

0.999960

0.000005

1.000400

0.000004

1.000044

1/24

0.000006

1.000033

0.000002

1.000199

0.000001

1.000030

Conv. rate

2.615516

3.157168

3.162190

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show that the order of error in energy norm increases with the increase of dilation parameter and effectivity of error estimation decreases with increase of the dilation parameter and, the rate of convergence is not significantly affected by the high value of dilation parameter in case linear element meshing. For higher order element, the order of error decreases and convergence rates increase with increase of dilation parameter and the finite element analysis with quadratic element shows improved performance under different dilation parameters. However, it is observed that there will be an optimal value of dilation parameter to provide an improved performance. The local (element) effectivity frequency using linear and quadratic elements for dilation parameter is plotted in Fig. 105.6 at last refinement level considered, to study the effect on recovery based finite element analysis. The figure depicts that finite element analysis using linear elements is not much affected by increasing the dilation parameter as the number of element effectivity, with value one, is more or less same with different dilation parameter. However, number of elements of effectivity with value one are shown to increase in analysis using quadratic elements mesh.

105.5.1.3

Recovery Parameter: Number of Basis Function Terms

In the post-processing technique, for the recovery, field variable or its derivative is assumed to belong to polynomial expansion of the same complete order or higher order polynomial as that present in the basis function. According to so-called ZZ super-convergent patch recovery technique [12, 13], the nodal values of stress belongs to a polynomial expansion of the same complete order as that present in the basis function. According to Ahmed and Singh [25] for element patch, the nodal values of field variables belong to a higher order polynomial expansion. The effect of order of polynomial expansion in the basis function i.e., number of basis function terms for MLS interpolation is investigated. Two set of number of basis function terms namely, 3, 6, 9 and 6, 9, 10 are used, for fitting the derivatives of field variable over the mesh independent node patches, considering respectively the background mesh of linear and quadratic elements. The convergence rate of error and effectivity of error estimation with set of number of basis function terms at different element fineness in MLS interpolation based post-processed finite element analysis are given in Tables 105.7 and 105.8. From the tables, it can be conclude that the best performance of the recovery based analysis will be obtained with one order higher polynomial expansion than the order of element of background meshing or in other words, in moving least square based post-processed analysis, the field variable derivative is considered to belong to polynomial expansion of the one order higher as that present in the basis function. The local (element) effectivity frequency using linear and quadratic elements for number of basis function terms is shown in Fig. 105.7 at last refinement level considered. The figure depicts that performance of recovery based finite element analysis will not improve by taking higher number of basis function terms as the number of element of effectivity with value one are not significantly increased by using more number of basis function terms.

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Table 105.7 Error convergence and global effectivity for square plate problem with number of basis function terms (m) in post-processing techniques (linear triangular meshing) Mesh size (1/h)

m=3 Error in stress recovery

Effectivity, θ

m=6 Error in stress recovery

Effectivity, θ

m=9 Error in stress recovery

Effectivity, θ

1/4

0.08186

1.012543

0.03123

0.918019

0.01600

0.960044

1/8

0.02494

1.003285

0.00694

0.977688

0.00453

0.987426

1/12

0.01232

0.999404

0.00285

0.989683

0.00206

0.994020

1/16

0.00751

0.997657

0.00152

0.994092

0.00117

0.996542

1/24

0.00377

0.996097

0.00063

0.997332

0.00053

0.998426

1/32

0.00233

0.995409

0.00034

0.998489

0.00030

0.999134

Conv. Rate

1.71037

2.17052

1.919061

Table 105.8 Error convergence and global effectivity for square plate problem with number of basis function terms (m) in post-processing techniques (quadratic triangular meshing) Mesh size (1/h)

m=6

m=9

Error in stress recovery

Effectivity, θ

Error in stress recovery

Effectivity, θ

Error in stress recovery

Effectivity, θ

1/4

0.01076

1.243404

0.000632

0.997272

0.003728

1.019333

1/8

0.00189

1.134483

0.000831

0.999601

0.000361

1.005534

1/12

0.00068

1.090981

0.000241

0.999889

0.000091

1.002630

1/16

0.00033

1.067465

0.000010

0.999960

0.000034

1.001536

1/24

0.00012

1.042609

0.000006

1.000033

0.000009

1.000710

Conv. rate

2.513851

2.615516

m = 10

3.387225

105.5.2 Elastic Plate Problem with Central Hole The elastic square plate with central circular opening, a typical stress concentration problem, is also selected for recovery parametric study of interpolation based recovery method. The side of the square plate is 5 unit with hole radius of one unit. The normal and vertical displacement components are zero along the circular arc boundary. The normal displacement component and shear stress are zero over the line of symmetry. Static boundary conditions are imposed from the given traction equations. The plate stress field are given in Eqs. 105.14 to 105.16 [12, 13]. The discretized domain for plate problem with circular opening is shown in Fig. 105.8 (one quarter due to symmetry).      σx = σ∞ 1 − a 2 /r 2 (1.5 cos 2θ + cos 4θ ) + (1.5a 4 /r 4 cos 4θ )

(105.17)

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Fig. 105.8 Central hole plate discretized with triangular elements

     σ y = σ∞ 0 − a 2 /r 2 (1.5 cos 2θ − cos 4θ ) + (1.5a 4 /r 4 cos 4θ )

(105.18)

     σx y = σ∞ 0 − a 2 /r 2 (1.5 sin 2θ + sin 4θ ) + (1.5a 4 /r 4 sin 4θ )

(105.19)

where r = y2 + x 2 and σ∞ is the uniaxial traction applied at infinity.

105.5.2.1

Recovery Parameter: Influence Domain Shape

The effect of shape effect is studied by taking dilation parameter as 3.5 for linear elements and 5.0 for quadratic elements. The number of terms in polynomial basis is taken as 6 and 9 respectively for linear and quadratic elements. The performance of the recovery analysis using linear and quadratic element meshing to form circular and rectangular influence domain error are presented in terms of convergence and effectivity in Tables 105.9 and 105.10. The tables show that the lower order element perform better as compared to higher order element for stress concentration type problem analysed with post-processed based finite element analysis. The rectangular influence domain construction has better performance as compared to circular shape of influence domain. The order of error in energy norm obtained is higher in rectangular domain used for recovery of the solution error. Table 105.9 Error convergence and global effectivity for central hole plate problem with influence domain shape in post-processing techniques (linear element meshing) Mesh No. of elem.

DoF

Error in FEM (energy norm)

Circular

Rectangular

166

202

0.010227

0.006150

0.866058

0.012778

1.237776

355

404

0.007326

0.003647

0.877953

0.007618

1.113363

567

630

0.005671

0.002364

0.903076

0.004810

1.061533

Error in stress recovery

Effectivity, θ

Error in stress recovery

Effectivity, θ

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Table 105.10 Error convergence and global effectivity for central hole plate problem with influence domain shape in post-processing techniques (quadratic element meshing) Mesh No. of Elem.

DoF

Error in FEM (energy norm)

Circular

Rectangular

Error in stress recovery

Effectivity, θ

Error in stress recovery

Effectivity, θ

166

734

0.001862

0.002613

1.328514

0.004672

2.388526

355

1516

0.001495

0.001529

0.813793

0.002062

1.176680

567

2392

0.001414

0.001321

0.501293

0.001568

0.815105

105.6 Conclusions In the present study, the post-processing based finite element analysis is carried out by varying the parameters (i.e. parameters affecting the post-processed results) of moving least square (MLS) based interpolation technique. The performance of finite element numerical experiments on elastic plate problems is investigated with different recovery parameters in terms of effectivity and rate of convergence. The linear and quadratic triangular elements have been used for the discretization of the numerical problem domain. The effect on the performance of recovery technique is studied by considering three parameter of MLS interpolation technique namely, shape of influence domain, dilation parameter and number of terms in basis function. The main conclusions of the study are summarized below. 1. The basic recovery parameters of MLS interpolation technique have pronounced effect on the post-processed finite element analysis results and order of effect depends on numerical problem type. 2. The higher order elements appear to be more sensitive to variation of basic recovery parameters. 3. In finite element analysis with different shape of influence domain construction, the rectangular domain construction has better performance as compared to circular shape of domain for both lower and higher element mesh. The convergence rate achieved is about two orders higher in case of rectangular influence domain. 4. The computational results oscillate with different dilation parameters and it appears to be an optimal value of dilation parameter for an improved performance. The optimal value of dilation parameters may be found in the range of 3.0–5.0. 5. The best performance of the moving least square based post-processed analysis will be obtained when the field variable derivative is considered to belong to higher order polynomial expansion as that present in the basis function. Acknowledgements Authors thank Deanship of Research, Ministry of Higher Education, KSA, for financial support to carry out the research work. The authors also acknowledge to the Dean, College of Engineering for his valuable support and help.

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References 1. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Comp. Methods Appl. Mech. Eng. 142, 1–88 (1997) 2. Gratsch, T., Bathe, K.: A posteriori error estimation techniques in practical finite element analysis. Comput. Struct. 83, 235–265 (2005) 3. Mirzaei, D.: Analysis of moving least squares approximation. Revisit. J. Comput. Appl. Math. 282, 237–250 (2015) 4. Xing, H.L., Wang, S., Makinouchi, A.: An adaptive mesh h-refinement algorithm for finiteelement modelling of sheet forming. J. Mater. Process. Technol. 91, 183–190 (1999) 5. Liu, Y.C., Elmaraghy, H.A.: Assessment of discretized errors and adaptive refinement with quadrilateral finite element. Int. J. Num. Methods Eng. 33, 781–798 (1992) 6. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Int. J. Num. Methods Eng. 24, 333–357 (1987) 7. Liu, G.R., Nguyen-Thoia, T., Lam, K.Y.: A novel alpha finite element method (αFEM) for exact solution to mechanics problems using triangular and tetrahedral elements. Comput. Methods Appl. Mech. Eng. 197, 3883–3897 (2008) 8. Kim, K.-Y., Lee, H.-C.: A posteriori error estimators for nonconforming finite element methods of the linear elasticity problem. J. Comput. Appl. Math. 235, 186–202 (2010) 9. Ullah, Z., Coombs, W.M., Augarde, C.E.: An adaptive finite element/meshless coupled method based on local maximum entropy shape functions for linear and nonlinear problems. Comput. Methods Appl. Mech. Eng. 267, 111–132 (2013) 10. Bramble, J.H., Schatz, A.H.: Higher order local accuracy by averaging in finite element method. 31, 94–111 (1977) 11. Hinton, E., Campbell, J.S.: Local and global smoothing of discontinuous finite element functions using a least square methods. Int. J. Num. Methods Eng. 8, 61–80 (1974) 12. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates, part I, the error recovery technique. Int. J. Num. Methods Eng. 33, 1331–1364 (1992) 13. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates. Part II: error estimates and adaptivity. Int. J. Num. Methods Eng. 33, 1365–1382 (1992) 14. Li, X.D., Wiberg, N.E.: An posteriori error estimate by element patch post-processing, adaptive analysis in energy and L2 norms. Comput. Struct. 53, 907–919 (1994) 15. Ubertini, F.: Patch recovery based on complementary energy. Int. J. Numer. Methods Eng. 59(11), 1501–1538 (2004) 16. Parret-Fréaud, A., Rey, V., Gosselet, P., Rey, C.: Improved recovery of admissible stress in domain decomposition methods—application to heterogeneous structures and new error bounds for FETI-DP. Int. J. Numer. Methods Eng. 111(1), 69–87 (2016) 17. Sharma, R., Zhang, J., Langelaar, M., van Keulen, F., Aragón, A.M.: An improved stress recovery technique for low-order 3D finite elements. Int. J. Numer. Methods Eng. 114, 88–103 (2018) 18. Ahmed, M., Singh, D., Desmukh, M.N.: Interpolation type stress recovery technique based error estimator for elasticity problems. Mechanika 24(5), 672–679 (2018) 19. Rajendran, S., Liew, K.M.: Optimal stress sampling points of plane triangular elements for patch recovery of nodal stresses. Int. J. Numer. Methods Eng. 58, 579–607 (2003) 20. Wang, J.G., Liu, G.R.: On the optimal shape parameters of radial basis functions used for 2-D meshless methods. Comput. Methods Appl. Mech. Eng., 191(23–24), 2611–2630 (2002) 21. Kanber, B., Bozkurt, O.Y., Erklig, A.: Investigation of RPIM shape parameter effects on the solution accuracy of 2D elastoplastic problems. Int. J. Comput. Methods Eng. Sci. Mech. 14, 354–366 (2013) 22. Perko, J., Šarler, B.: Weight function shape parameter optimization in meshless methods for non-uniform grids. CMES 19(1), 55–68 (2007) 23. Nie, Y.F., Atluri, S.N., Zuo, C.W.: The optimal radius of the support of radial weights used in moving least squares approximation. CMES 12(2), 137–147 (2006)

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Chapter 106

Genetically Aerodynamic Optimization of High-Speed Train Based on the Artificial Neural Network Method Fu Tao, Chen Zhaobo and Wang Zhonglong

Abstract The optimization of the car body cross-section shape of high-speed train under cross winds has been performed using genetic algorithms (GA), wherein the computational fluid dynamics (CFD) simulation model was established based on Design of Experiments (DOE), and then the aerodynamic performance of a train subjected to crosswind was calculated by CFD. However, directly using the CFD to compute the high-speed trains performances in an optimization scheme is not suitable for the optimization process. This is because the GA requires a large number of CFD solver calls, which increase the computational cost and time-consuming of the optimization process. To decrease the CFD simulation computational cost, an approximate substituted Artificial Neural Network (ANN) model was proposed. The ANN models were trained and tested using the data obtained from the CFD simulation model, and its correlation coefficient values are considerably close to 1, which indicates fine accuracy and prediction capability. Meanwhile, the impact of the car body cross-section design parameters on the aerodynamic performance has been also investigated, and the result show that the design parameters interrelate each other and jointly impact the aerodynamic performance of high-speed trains. After the optimization, the overturn moment coefficient has been reduced by 17.5%, and the lift force and slid force coefficient have been reduced by 11.5% and 8.05%, respectively. Compared to the original shape, the pressure coefficient of optimal shape has great improvement. The analysis method can provide a frame of reference for the high-speed train safe operation running on the track under crosswind conditions. Keywords Optimization · ANN · High-speed train · Genetic algorithm · Aerodynamic performance

F. Tao · C. Zhaobo (B) · W. Zhonglong School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_106

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106.1 Introduction For recent decades, high-speed trains have been developed rapidly from ICE in German and TGV in French to the star of China [1, 2], due to the rapid development of high-speed trains around the world, the performances of trains have been given more and more attention nowadays. In particular, when high-speed trains are running under strong crosswind conditions, the aerodynamic forces and moments on a train may be sufficiently large to overturn the train. These aerodynamic forces and moments are even intense when the train travels at high speed through bridges or narrow tunnel, and thus the crosswinds effect on the safe operation of trains is more prominent. Many trains blown over by crosswinds have been observed in the word. In Japan, there have been about 30 wind-induced accidents [3]. To guarantee the operation safety, numerous studies have been conducted across the world [4–9]. Xiang et al. [4, 5] proposed a dynamic analysis method of the crosswind and high-speed train and slab system and calculated the dynamic responses of the high-speed train and slab track under crosswind action. It is found that the crosswind has a significant influence on the lateral and vertical displacement responses of the car body, wheel load reduction factor and overturning factor. The lateral and vertical displacement responses of rail with the crosswind are almost the same as those without the crosswind. Hemida [6] investigated the influence of two different fronts of the train and yaw angles on the surrounding flow and the train aerodynamics. It is found that the lower vortex are responsible for the vortex shedding in the wake if it exists and there is a weak vortex shedding from the upper vortex. Carrarini [7] and Suzuki et al. [8] studied the train reliability based on crosswind stability. Cheli et al. [9] analyzed the performance of the new train AnsaldoBreda EMUV250 in terms of behavior to cross wind. The result shows that the characteristic wind curves can be defined to evaluate the aerodynamic improvements in terms of reduction of overturning risk. These studies mentioned above are mainly focused on the aerodynamic performance of high-speed trains under crosswinds conditions. However, for the new types of high-speed trains, optimal car body designs of high-speed trains, which consist in determining the optimal cross-section parameter and design the optimal head shape of the trains, also have significant influences on their operational safety when subjected to crosswinds [9, 10]. For the car body optimization of high-speed train, a large number of design and operating parameters have need to be optimized. Usually, the traditional optimization method of shape of high-speed train has been handled as a trial-and-error procedure. This method is not appropriate for the multiple parameter design process. This is because it leads to significant requirement of resources and time. However, with the gradual development of modern high-performance computer technology, computational fluid dynamics (CFD) has been effectively applied to the study of high-speed trains [11]. For example, Chen et al. [2] used the CFD method to compute and analyze aerodynamic performance of train at different wind speeds and yaw angles, the resulting flow showed significant complexities. Khier et al. [12] studied the application of CFD to the flow field structure around the German ICE high-speed train. The CFD

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method success in numerous areas, but the large computational demands required by optimization algorithm (such as Genetic algorithm and Particle Swarm optimization) made its use rather be limited. Therefore, we use a computation efficient surrogate model of Artificial Neural Network (ANN) model to conduct the optimization. The ANN [13, 14] is essentially a calculation mathematical model aimed at imitating the learning and memory instinct of the human brain, which can learn the complex relationship between the input and output data of nonlinear systems and construct an adaptive network model. Despite ANN are not exactly novel, the application of them in the aerodynamic optimization of high-speed trains is relatively new. In this study, the CFD simulation model of the cross-section optimization of highspeed trains is firstly established based on DOE (Design of Experiments), and then the aerodynamic forces of a train subjected to crosswind are calculated by CFD. The ANN models are trained and tested using the data obtained from the CFD simulation model. Based on the built ANN models, the GA (generic algorithm) has been proposed to optimize the design parameters of the cross-section of high-speed trains. According to this optimization method, optimal solutions is obtained. After that, aerodynamic performances of the optimized shape and the original shape are comparatively analyzed.

106.2 Methodology Optimal car body design of high-speed trains mainly focuses on the cross-sectional area in this paper, of which the key design parameters are: overturn moment acting on the mass center of the car body, aerodynamic lift force and aerodynamic side force, as is shown in Fig. 106.1. There are many structure parameters that can affect the overturn moment, lift force and side force, such as car body length and height; top radius arc structure; reinforced plate width and height, etc. Various combinations of these structure parameters can lead to different results. If calculations are performed for every parameter combination, the costs will be unacceptable. Therefore, the orthogonal test method is introduced, which would improve the efficiency. The orthogonal table L25 (65 ) is adopted, which includes the complete set 6 factors and each factor has 5 levels, as is shown in Table 106.2 (Appendix). The computational fluid dynamics (CFD) software performs the computation and gets required results through orthogonal table. The method selected to solve this optimization problem is the genetic algorithms (GA). GA is a population-based swarm intelligence algorithm inspired by the behavior of biological evolution, which conducts the selection, crossover and mutation operation after a population of potential solution is defined. Iteratively, objective function value is generated until the six optimization variables contained in population computed by CFD are reached. However, directly using the CFD to compute the high-speed trains performances in an optimization scheme is not suitable for the optimization process because the GA requires a large number of CFD solver calls and the increase of the computational power and time-consuming will be unacceptable.

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Fig. 106.1 Optimization scheme schematic of the car body cross-section of high-speed train

To decrease the CFD simulation computational cost, an approximate substituted artificial neural network (ANN) model is proposed. The ANN is essentially a calculation mathematical model aimed at imitating the learning and memory instinct of the human brain, which can learn complex relationship between the input and output data of nonlinear systems and construct adaptive network model. This is an advantage compared to CFD methods that need the repetitive calculation of objective function by CFD solver call. In Fig. 106.1, the coupling scheme among the CFD simulation model, GA model and ANN model is represented. This scheme refers to an optimization work flow, where after each generation of a new population is generated, the ANN model is called to compute the objective function value until all individuals are finished. Then the fitness function value is calculated for GA model.

106.3 Calculation Conditions and Relevant Definition Car body cross-section design of high-speed trains plays an important role in aerodynamic optimal design. An applicable cross-section geometry can not only improve lateral aerodynamic force but also can enhance operation stability. The optimal candidate is defined in terms of six design variables, {L, H, R1 , R2 , h, b}, as is shown in Fig. 106.2. The notations {L, H, R1 , R2 , h, b} refer to the car body length, car body height, top radius arc, bottom radius arc, cross-section reinforced plate thickness, reinforced plate height, respectively. The flow field around high-speed trains subjected to crosswind can be considered as a three-dimension incompressible viscous turbulent flow. To fully represent the flow around the train and to ensure the accuracy of results, a schematic of the computational domain was established, as is shown in Fig. 106.3. The computational fluid dynamics software was carried out using the

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Fig. 106.2 Schematic of the car body cross-section of high-speed train

Fig. 106.3 Schematic of computational domain and boundary condition

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commercial code FLUENT. The standard k-ε two-equation model was selected as the turbulence model and the equation was written as [15, 16]: ∂(ρφ) + div(ρuφ) = div(gradφ) + S ∂t

(106.1)

where t is time, ρ is the air density, u is the velocity vector,  is flow flux, S is the source term, and  is the diffuse coefficient. The computational range is 52 m in length, 4 m in width and 40 m in height. The distance from the inlet to the train is 12 m, the distance between the bottom of the train and the ground is 1 m, and the distance from the outlet to the train is 40 m. Under the crosswind conditions, the inlet is defined as velocity inlet, whose value is set as 20 m/s and as there is not much variation in the temperature. The outlet is defined as pressure outlet, whose value is set to constant pressure which is atmospheric and a gauge pressure of zero is applied. In order to obtain reasonable solutions, boundary conditions must be defined appropriately in numerical simulation. Therefore, the train surface and ground boundary are defined as no-slip condition. According to aerodynamic theory [17–20], the aerodynamic coefficients are defined as: Cs =

Fs FL M , CL = , CM = 0.5ρV 2 A 0.5ρV 2 A 0.5ρV 2 A3/ 2

(106.2)

where CS , CL and CM are the side force, lift force and overturn moment coefficients, respectively; FS , FL and M are the side force, lift force and overturn moment, respectively; ρ is the air density; V is the cross-wind speed; A is the area of train side face.

106.4 Artificial Neural Networks Models ANN accuracy that directly affects the construction of the regression surface and efficiency of optimization algorithm is the basis of the whole optimization process. The ANN model chosen in this paper is the generalized regression neural network (GRNN). GRNN, proposed by Donald F. S. pecht (1991), is a new type of neural network model based on the theory of the nonlinear regression [21]. Compared with radial basis function (RBF) and back propagation (BP) neural network, the weight and nodes number between each layer of GRNN is determined by training sample only. This is an advantage compared to other networks in the convergence speed and global convergence. The network structure of GRNN is as shown in Fig. 106.4, it is composed of input layer, pattern layer, summation layer and output layer. The input layer has six inputs, and each network has one output, which is lift force, side force and overturn moment coefficients, respectively.

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Fig. 106.4 The network structure of ANN model

The most important task of the GRNN application in a particular work is to obtain the optimum smooth factor. We have used the method of cross validation in the network to search for the optimum smooth factor. One data set that contains 200 smooth factor values is collected from interval [0.01, 2] by 0.01 increment to test the model. To better describe the generalization ability of GRNN model, the correlation coefficient (R) can be calculated as [22–24]: R=

( p − p)(y ¯ − y¯ )T  ( p − p)( ¯ p − p) ¯ T (y − y¯ )(y − y¯ )T

(106.3)

where p(y) is the target (network) output; p¯ and y¯ are the mean value of the p and y. Based on the R values, the optimum smooth factor is: 0.01, 0.02 and 0.01 for the side fore, lift force and overturn moment coefficients, respectively. Figure 106.5 presents the test data and the GRNN outputs. The correlation coefficient (R) values are considerably close to 1, which indicate fine generalization and excellent prediction accuracy of the GRNN model. Therefore, the GRNN model could be safely used for the optimization work of the high-speed trains.

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Fig. 106.5 ANN prediction values and target values: a CS ; b CL ; c CM

106.5 Effects of the Design Parameters Based on the optimum GRNN model, the effects of the design parameters on the aerodynamic performance of high-speed trains are analyzed. Some typical figures are plotted from the GRNN model and presented in Figs. 106.6, 106.7, 106.8, 106.9, 106.10 and 106.11. As is shown in Fig. 106.6, increasing the L can visibly improve the side force coefficients, and as the increase of L, the overturn moment coefficients will

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Fig. 106.6 Effect of L and H on the side force coefficients

Fig. 106.7 Effect of L and h on the overturn moment coefficients

decrease (Fig. 106.7), which is mainly due to the corresponding increase of the lateral projection area. Therefore, increasing the H will decrease the lift force coefficients (Fig. 106.8), and the side force coefficients can be preferably improved (Fig. 106.9). Furthermore, a larger h can effectively improve the side force and overturn moment coefficients, as is shown in Figs. 106.10 and 106.11. According to the analysis of the effects of the design parameters on the aerodynamic performance, one can find that the design parameters interrelate each other and jointly impact the aerodynamic performance of high-speed trains. In order to efficiently optimize the aerodynamic performance of high-speed trains, an optimization scheme is built in Sect. 106.6.

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Fig. 106.8 Effect of R1 and H on the lift coefficients

Fig. 106.9 Effect of L and H on the side force coefficients

Fig. 106.10 Effect of h and R1 on the overturn moment coefficients

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Fig. 106.11 Effect of R2 and h on the side force coefficients

106.6 Optimization Result and Discussion The performances of aerodynamic overturn moment have important influence on the stability of high-speed trains. As aerodynamic overturn moment increases, the aerodynamic performance of the train will deteriorate more severely, which may lead to train derailments and even overturning, thus impair the safety of the train. Therefore, in this paper, a single-objective optimization has been performed with the overturn moment as the objectives. Based on the GRNN model, the genetic algorithm is used as the optimization method, and the objective function is the minimization of the maximum overturn moment coefficients, which is denoted by f(x) for convenient description, and x = [L, H, R1 , R2 , h, b]. The optimization scheme is proposed as: Minimizing: f(x) Subject to: 3000 ≤ L ≤ 3400 3800 ≤ H ≤ 4200 675 ≤ R1 ≤ 900 400 ≤ R2 ≤ 800 50 ≤ h ≤ 70 2≤b≤6 Before dealing with the single-objective optimization problem, a parametric study is required to fit the GA parameters. As parameters for GA operation, the size of population is set to 50, while the roulette method is used as the selection operator. The probability of crossover and mutation are 0.6 and 0.01 respectively. After evolving 300 generations, the fitness converges to a constant value, as is shown in Fig. 106.12, indicating that the single-objective optimization problem could be solved by GA.

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Fig. 106.12 Convergence curve of fitness function

The original car body cross-section parameters are (3000, 3800, 600, 400, 50, 2), and the final optimal is optimal = (3757.1, 3889.8, 673.9, 498.2, 56.1, 3.1). The aerodynamic force coefficients of the optimal and original shapes are given in Table 106.1. After optimization, the overturn moment coefficient has been reduced by 17.5%, and the lift force and slid force coefficient have been reduced by 11.5% and 8.05%, respectively. The results from the GRNN model and CFD model are almost the same, the maximum error of the prediction is 0.42%, indicating that the GRNN model has excellent prediction accuracy. Figure 106.13 and Fig. 106.14 respectively shows comparison of the pressure contours of original shape and optimal shape. It is seen that the pressure of the train windward side increases significantly due to the air flow directly imposes on the windward side of the train, which generates a larger cross-wind forces, especially in the lateral direction where forms a stagnation point. In addition, the pressure of the train leeward side becomes small because the airflow is drawn out on leeward side and there is a rapid acceleration of the airflow above the train. In this way, the Table 106.1 Aerodynamic force coefficients of the optimal and original shape

Model type Original shape Optimal shape Reduction (%) ANN model

CS

CL

0.9866

0.5205

0.8731

0.4786

11.5 0.8730

8.05 0.4788

CM 0.4017 0.3315 17.5 0.3315

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Fig. 106.13 Pressure contours of original shape

Fig. 106.14 Pressure contours of optimal shape

pressure difference between the front and the rear part makes the car body subject to a large side force and overturn moment. On the other hand, due to the acceleration effect of the airflow above and under the train, a strong negative pressure region is formed respectively in both parts, which generates a large lift force upwards. After the optimization, the pressure zone between the top and bottom has been reduced. Meanwhile, the pressure of the train windward side also has been alleviated.

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Fig. 106.15 The pressure coefficient of the middle cross-section of high-speed trains

In order to better understand the aerodynamic performance of the shape due to the change of the structure parameters. The pressure coefficient (Cp ) of the middle cross-section of high-speed trains is presented in Fig. 106.15. It can be seen that the pressure coefficients get a lager difference between the original and optimal shape. The leeward side of the train almost remains the same pressure coefficient. However, compared to the original shape, the train bottom of optimal shape presents a great improvement. This is because the increase of the car body top and bottom radius arc, which slows the accelerating process of the passing airflow. Meanwhile, the windward side of optimal shape also has been improved due to the increase of the car body height. Therefore, the aerodynamic performance of the train has been improved.

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106.7 Conclusions In this work, the optimization of the car body cross-section shape of high-speed train under crosswind conditions has been performed. The CFD simulation model was established based on DOE, and then the aerodynamic performance of a train subjected to crosswind was calculated by CFD. The GA was selected to solve this optimization problem. However, directly using the CFD to compute the high-speed trains performances in an optimization scheme is not suitable for the optimization process. This is because the GA requires a large number of CFD solver calls, which increase the computational cost and time-consuming of the optimization process. To decrease the CFD simulation computational cost, an approximate substituted ANN model was proposed. The ANN models were trained and tested using the data obtained from the CFD simulation model, and its correlation coefficient values are considerably close to 1, which indicates fine accuracy and prediction capability. Meanwhile, the impact of the car body cross-section design parameters on the aerodynamic performance has been also investigated, and the result show that the design parameters interrelate each other and jointly impact the aerodynamic performance of high-speed trains. After the optimization, the overturn moment coefficient has been reduced by 17.5%, and the lift force and slid force coefficient have been reduced by 11.5% and 8.05%, respectively. Compared to the original shape, the pressure coefficient of optimal shape has great improvement. The analysis method can provide a frame of reference for the high-speed train safe operation running on the track under crosswind conditions. Acknowledgements This work presented here was supported by National Key R&D Program of China under the contract number 2017YFB1300600, and by the National Natural Science Foundation of China under the contract numbers 11772103 and 61304037.

Appendix: Orthogonal Table L25 (65 ) and the Numerical Results See Table 106.2.

Factor 1

3000

3400

3000

3100

3000

3300

3400

3300

3100

3200

3200

3000

3300

3400

3400

3300

3100

3400

3200

3000

3300

Case

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

4100

3900

4100

3800

3900

3900

3900

4200

4200

4200

4000

4200

4000

3800

4100

4000

4100

4200

3800

4000

4000

Factor 2

600

825

825

825

600

900

675

600

825

750

600

675

825

675

750

750

900

900

600

900

675

Factor 3

500

800

700

600

600

700

500

700

400

500

800

600

500

800

800

600

600

800

400

400

700

Factor 4

60

55

70

60

70

50

65

55

65

70

65

50

50

70

50

55

65

60

50

70

60

Factor 5

5

5

2

4

3

4

2

6

3

4

4

5

6

6

3

2

6

2

2

5

3

Factor 6

1.1328

1.062

0.8431

0.9819

1.0038

1.1061

0.9321

1.2155

1.0394

1.1076

1.012

1.4687

1.303

1.1041

0.9714

1.0032

1.2427

0.9368

0.9866

1.08

1.082

CS

Table 106.2 Orthogonal table L25 (65 ) and the numerical results for the aerodynamic performance of high-speed train CL

0.1982

0.5732

0.0842

0.2477

0.3481

0.8608

0.1768

0.3457

0.3848

0.1723

0.4559

0.4887

0.2603

0.1197

0.4068

0.2812

0.0332

0.0487

0.5205

0.1979

0.3813

CM

(continued)

0.5161

0.4904

0.3565

0.4445

0.4652

0.5141

0.4181

0.5848

0.3969

0.4466

0.4972

0.6589

0.5547

0.5541

0.4568

0.4358

0.5176

0.3867

0.4062

0.4398

0.4964

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Factor 1

3200

3200

3100

3100

Case

22

23

24

25

Table 106.2 (continued)

4100

3800

3800

3900

Factor 2

675

750

900

750

Factor 3

400

700

500

400

Factor 4

55

65

55

60

Factor 5

4

5

3

6

Factor 6

1.1293

1.0714

0.8629

1.152

CS

CL

0.2942

0.4547

0.5162

0.4363

CM

0.4585

0.5006

0.3365

0.4755

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References 1. Chen, R.L., Zeng, Q.Y., Li, D.J.: Advances in Rheology and Its Application. Science Press USA Inc., New York (2005) 2. Chen, R.L., Zeng, Q.Y., Zhong, X.G., et al.: Numerical study on restriction speed of train passing curved rail in cross wind. Sci. China Ser. E-Tech. Sci. 52(7), 2037–2047 (2007) 3. Fujii, T., Maeda, T., Ishida, H., et al.: Wind induced accidents of train vehicles and their measures in Japan. Q. Rep. RTRI 40(1), 50–55 (1999) 4. Xiang, J., Li, D.J., Zeng, Q.Y.: Simulation of spatially coupling dynamic response of train track time variant system. J. Cent. South Univ. Technol. 10(3), 226–230 (2003) 5. Xiang, J., He, D., Zeng, Q.Y.: Effect of crosswind on spatial vibration responses of train and track system. J. Cent. South Univ. Technol. 16, 520–524 (2009) 6. Hemida, H.: LES study of the influence of the nose shape and yaw angles on flow structures around trains. J. Wind. Eng. Ind. Aerodyn. 98, 34–46 (2010) 7. Carrarini, A.: Reliability based analysis of the crosswind stability of railway vehicles. J. Wind Eng. Ind. Aerodyn. 95(7), 493–509 (2007) 8. Suzuki, M., Tanemoto, K., Tatsuo, M.: Aerodynamics characteristics of train/vehicles under cross winds. J. Wind Eng. Ind. Aerodyn. 91(1), 209–218 (2003) 9. Cheli, F., Ripamonti, F., Rocchi, D., Tomasini, G.: Aerodynamic behaviour investigation of the new EMUV250 train to cross wind. J. Wind Eng. Ind. Aerodyn. 98, 189–201 (2010) 10. Fujii, T., Maeda, T., Ishida, H., Imai, T., Tanemoto, K., Suzuki, M.: Wind-induced accidents of train/vehicles and their measures in Japan. Q. Rep. Railw. Tech. Res. Inst. 40(1), 50–55 (1990) 11. Baker, C., Calleja, F., Jones, J., et al.: Measurements of the cross wind forces on trains. J. Wind Eng. Ind. Aerodyn. 92(7–8), 547–563 (2004) 12. Khier, W., Breuer, M., Durst, F.: Flow structure around trains under side wind conditions: a numerical study. Comput. Fluids 29(2), 179–195 (2000) 13. Najafi, G., Ghobadian, B., Tavakoil, T., et al.: Performance and exhaust emissions of a gasoline engine with ethanol blended gasoline fuels using artificial neural network. Appl. Energy 86, 630–639 (2009) 14. Togun, N., Baysec, S.: Prediction of torque and specific fuel consumption of a gasoline engine by using artificial neural networks. Appl. Energy 87, 349–355 (2010) 15. Bell, J.R., Burton, D., Thompson, M.C., et al.: Wind tunnel analysis of the slipstream and wake of a high-speed train. J. Wind Eng. Ind. Aerodyn. 134, 122–138 (2014) 16. Baker, C.: The flow around high speed trains. J. Wind Eng. Ind. Aerodyn. 98(6–7), 277–298 (2010) 17. Diedrichs, B.: Aerodynamic crosswind stability of a regional train model. J. Rail Rapid Transit 224(6), 580–591 (2010) 18. Sanquer, S., Barre, C., Virel, M.D., et al.: Effect of cross winds on high-speed trains: development of a new experimental methodology. J. Wind. Eng. Ind. Aerodyn. 92(7–8), 535–545 (2004) 19. Bell, J.R., Burton, D., Thompson, M.C., et al.: Moving model analysis of the slipstream and wake of a high-speed train. J. Wind Eng. Ind. Aerodyn. 136, 127–137 (2015) 20. Thomas, D., Diedrichs, B., Berg, M., et al.: Dynamics of a high-speed rail vehicle negotiating curves at unsteady crosswind. J. Rail Rapid Transit 224(6), 567–579 (2010) 21. Specht, D.F.: A general regression neural work. IEEE Trans. Neural Netw. (1991) 22. Lolas, S., Olatunbosum, O.A.: Prediction of vehicle reliability performance using artificial neural networks. Expert Syst. Appl. 4, 2360–2369 (2008) 23. Vapnik, V.: Statistical Learning Theory. Wiley, New York, NY (1998) 24. Lin, C.J.: Formulations of support vector machines: a note from an optimization point of view. Neural Comput. 13(2), 307–317 (2001)

Part XXIX

Theme: Sound and Vibration

Chapter 107

Predicting Structure Dynamic Acceleration Based on Measured Strain Wang Yuansheng, Lan Chunbo, Qin Weiyang and Yue ZhuFeng

Abstract For complex structures, to understand its dynamical characteristics, it is desired that the dynamical responses under excitation, e.g. displacement or acceleration, could be measured directly. But at present, the measuring method for acceleration is still limited and constrained by many factors. Especially for complex structures, the direct measurement for acceleration is quite difficult. In contrast, strain measurement is relatively easy, since the strain sensor is simple and can be easily pasted on the surface of structure. Thus if the measured strain data can be used to predict the dynamical response of system, it will result in great benefit. In this paper, a method is proposed to predict the acceleration from the measured strain response, which needs only a few strain sensors. The validation experiments were carried out on a cantilever beam and a rectangle cylinder. The stochastic motion was chosen as the excitation source. The results prove that the presented method is effective and could reach a high precision. Keywords Strain mode · Displacement mode · Prediction · Stochastic excitation

107.1 Introduction For complex structures and machinery, it is very important to monitor their dynamic states through measuring the displacements and accelerations with time. But in practice, directly measuring the displacements and accelerations is confined by many factors, sometimes it is even impossible. In contrast, strain sensor is simple in structure and small in size, especially it is easily integrated with the structures. So people hope to get the dynamical response at a point from the measured strain. Many research works focus on this subject. Li and Ulsoy[1] presents a strain-gauge-based method for the high-precision vibration measurement of a beam. It is based on the fact that

W. Yuansheng (B) · L. Chunbo · Q. Weiyang · Y. ZhuFeng School of Mechanics, Civil Engineering and Architecture, Northwestern Polytechnical University, Xi’an, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_107

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the vibration displacement can be expressed in terms of an infinite number of vibration modes. By placing multiple sets of strain gauges, multiple modes can be taken into account to achieve high-precision measurement. Experimental results proved that this method is applicable to high-precision flexible line boring. Kang et al. [2] investigates dynamic structural displacements estimation using displacement–strain relationship and measured strain data. Vibration experiments were performed. The beam structure’s deformed shapes were reconstructed by using strain signals. The estimated displacements show good agreements with those measured directly. Rapp et al. [3] investigates whole displacement field estimation using strain measurement and a displacement-strain-transformation approach. He discusses many issues related to the displacement field estimation of a dynamically excited plate using a transformation matrix based on a modal approach. A parametric study was performed and the sensor locations were optimized. In experiment the estimated displacements showed good agreements with those measured directly from laser displacement sensors. Kim et al. [4] investigates shape estimation of rotating structures using displacement-strain relationships and strain data measured with distributed fiber Bragg grating (FBG) sensors. The locations of strain sensors are optimized. Experimental validations for transient and steady rotations of the beam are performed. The estimated shapes of the rotating beam from FBG strain signals are compared with the directly captured shapes, and the results agree very well. Bang et al. [5] introduces a fiber Bragg grating (FBG) based arrayed sensor system for use in the measurement of strain of an wind turbine tower. Real-time shape estimation was accomplished using strain data gathered by surface mounted fiber Bragg grating sensors. The finite element model was created and the displacement-strain transformation (DST) matrix was obtained. The full deflection shapes of the tower were successfully estimated using arrayed FBG sensors. Glaser et al.[6] presents results of numerical simulations and an experimental investigation of a method to determine shape of a beam from curvature and/or strain measurements. A method based upon solving a set of continuity equations is proposed. The experiment was designed using an aluminum beam combined with a data acquisition system and a shape reconstruction algorithm. The real-time reconstruction of shape from curvature data was accomplished using strain gauges for the curvature estimates. Wang et al. [7] presents a new technique to estimate the dynamic displacement based on strain mode shapes of beam structures with strain sensors. Strain mode shapes are estimated from the cross-correlation function of the measured dynamic strain data. The displacement mode shapes were estimated from the strain mode shapes. After knowing the strain mode shapes and strain data, the dynamic displacement can be estimated. Iadicicco et al. [8] calculated the deflection of planar panels from the longitudinal surface strain measurements by means of FBG sensors arrays properly bonded to panels. The relationship between the longitudinal strain and the vertical deflection is derived by classical beam theory. Preliminary experiments have been carried out, showing that deflection measurements with resolution of few tens of microns can be successfully achieved. Zhang et al. [9] proposed a new scheme using long-gauge fiber optic sensors for estimating the deflection distribution of tied-arch bridges. The bending strain is separated from the measured long-gauge strain. The separated bending strain is utilized to estimate the deflection distribution

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of the tied-arch bridge through an improved conjugate beam method. Experimental examples illustrate the effectiveness of the proposed method. Hong et al. [10] proposed an indirect method for monitoring dynamic deflection of beam-like structures using strain responses measured. the influence of boundary conditions, load type and sensor gauge length on the method is investigated by numerical simulation. Experimental results show that both deflection at arbitrary points and deflection distribution along structures can be obtained with high-precision. It is desired that by measuring the strain of structure, the displacement and acceleration of structure could be recognized accurately. Many research works have been devoted to solving this problem. Up till now, it can be seen few works are developed on predicting the acceleration from strain. Especially for complex structures under stochastic excitation, the measuring points are limited and the response is stochastic. It is still difficult to predict the acceleration from strains in stochastic environment. In this study, we develop research on predicting accelerations from measured strains under stochastic base excitation. For a cantilever beam and a rectangular cylinder, we carried corresponding validation experiments under the stochastic excitation. The results proved that the proposed method could reach a high precision even under stochastic excitation.

107.2 Theoretical Analysis For a multi-degree of freedom system, we can establish its FEM model. Assuming that {δe }i is the displacement at ith element node, {δ}i represents the displacement vector at a point in ith element. Supposing that [N ] is the shape function describing the deformation of the element, then we have {δ}i = [N ]{δe }i

(107.1)

Then for any point in ith element, its strain vector {ε}i can be given by {ε}i = [D]{δ}i = [D][N ]{δ}i

(107.2)

where [D] is the differential operator. Defining [B] = [D][N ], Eq. 107.2 can be simplified as {ε} = [B]{δ}

(107.3)

Assuming that for a element, the transfer matrix between the local and global coordinates is [B], then we have   {ε} = [B][β]{δs } = B  {δs } where {δs } represents the node’s displacement in the global coordinate.

(107.4)

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Therefore for the multi-degree of freedom system, its dynamical equation can be given as     [Ms ] δ¨s + [Cs ] δ˙s + [K s ]{δs } = {Fs }

(107.5)

Denoting by ϕi the Multi-DOF system’s ith mode, then the mode matrix is [ϕi ]. From the concept of modal superposition, the solution for Eq. (107.5) can be given by {δs } = [ϕi ]{q}

(107.6)

where q is the modal coordinate. Substituting Eq. (107.6) into Eq. (107.5), the modal coordinate q could be solved and written as the following form {q} = [Yi ][ϕi ]T {Fs }

(107.7)

where [Yi ] is a diagonal matrix, its element is Yi = (ki − ω2 m i + jωci )−1

(107.8)

Substituting into the strain expression gives   {ε} = B  ][ϕi ][Yi [ϕi ]T {Fs } = [ψi ][Yi ][ϕi ]T {Fs } = [ψi ]{q}

(107.9)

  where ψi ] = [B  [ϕi ] is the strain modal matrix. From the expressions of displacement and strain in a MDOF structure, for both the displacement mode and the strain mode, their modal coordinate are identical. Suppose that for mth node its ith modal displacement is δim = qi ϕim

(107.10)

εim = qi ψim

(107.11)

ϕim δim = =α εim ψim

(107.12)

and its ith modal strain is

Then we have

α is the ratio of ith modal displacement to ith modal strain at the point. Following the similar procedure, we can obtain the ratios of other modal displacements to strains at the point. So if at a point the dynamic strain ε(t) under external excitation could be measured, since the modal coordinates are identical, then from the modal

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superposition the dynamic displacement δ(t) and the acceleration a(t) at the point could be obtained.

107.3 Validation Experiments and Analysis 107.3.1 Experiment on a Cantilever Beam Under Stochastic Excitation First, we chose a cantilever beam as the object of study. The cantilever beam was fixed on a shaker and subjected to a base stochastic excitation. We put two strain sensors on the beam, and a accelerometer at the middle of them. The measured strain signals were used to recognize the acceleration at the accelerometer’s position. The recognized results were verified with the measured acceleration. The schematic of experiment is as shown in Fig. 107.1. The steel beam had the dimension of 615 mm × 20 mm × 5 mm, which was fixed on a shaker (DongLing ES-15-150); two strain sensors were bonded at d = 60 mm (P1) and d = 276 mm (P2), and a accelerometer was at d = 200 mm (P3); the measured strain signals were imported to a dynamic strain instrument (DongHua, DH5920); The accelerometer used is PCB 333B30. The shaker produced a band-limited white noise as the excitation, whose mean is zero and standard deviation is 0.02 m/s2 . The excitation frequency band is 10–100 Hz. The measured strain responses are illustrated in Figs. 107.2 and 107.3. The recognized acceleration response is illustrated in Fig. 107.4, compared to the measured acceleration. From Fig. 107.4, it can be seen that the recognized acceleration reaches a high accuracy. The standard deviation of practical acceleration is σ p = 6.7811 m/s2 , and that of recognized acceleration is

Cantilever beam

Fixture

P1

P3 P2

Fig. 107.1 Experiment setup

Shaker

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Strain/με

200 100 0 -100 -200 -300 -400 -500 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.5

0.6

0.7

0.8

0.9

1

Fig. 107.2 Strain response at P1 400 300

Strain/με

200 100 0 -100 -200 -300 0.1

0.2

0.3

0.4

Time/s Fig. 107.3 Strain response at P2

σr = 6.0855 m/s2 . Their difference is δ = 8.9%. In view of that only two strain sensors were put and the first three modes were taken into account, the recognized result is satisfied.

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15 10

Acc./m/s2

5 0 -5 -10 -15 -20 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Time/s Fig. 107.4 Comparison between recognized acceleration and practical acceleration (red line: recognized response; blue line: practical response) (colour figure online)

107.3.2 Experiment on a Rectangular Cylinder Next we take a steel rectangular cylinder as the object of study. This rectangular cylinder has the dimension of 100 mm length, 200 mm width and 100 mm height. Its thickness is 2 mm. As Fig. 107.5 shows, the cylinder was fixed to the shaker by a slender fixture. There are two strain sensors and one accelerometer on the cylinder (Figs. 107.6, 107.7, and 107.8). In experiment, the excitation was set as a stochastic motion with frequency band of 50–120 Hz; its mean was zero and standard deviation was 0.02 m/s2 . For the recognized response, its standard deviation is σr = 17.31 m/s2 , while that of the practical response is σ p = 16.05 m/s2 , then the recognization error is

Rectangular cylinder

Strain sensor (P2)

Fixture

Fig. 107.5 Experiment setup of rectangular cylinder

Strain sensor (P1) Accelerometer (P3)

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100 50 0 -50 -100 -150 -200 35

35.5

36

36.5

37

37.5

38

38.5

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39

Time/s Fig. 107.6 Strain response at point P1 250 200 150

Strain/με

100 50 0 -50

-100 -150 -200 35

35.5

36

36.5

37

Time/s Fig. 107.7 Strain response at point P2

δ = 8.6%. Therefore the recognization precision is high and the recognized results are reliable. The recognization method was proved to be correct and effective in practice.

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60

40

Acc./m/s2

20

0

-20

-40

-60 35

35.5

36

36.5

37

37.5

38

38.5

39

Time/s

Fig. 107.8 Recognized acceleration and practical acceleration (blue line: practical response, black line: recognized response) (colour figure online)

107.4 Conclusions In this paper, we presented a method to recognize the displacement and acceleration of structure from the measured strain responses. First, the theoretical analysis was carried out. Then for a cantilever beam excited by stochastic motion, the corresponding experiments were performed. The results show that the recognization could reach a high precision. Finally for a rectangular cylinder, under stochastic excitation, the validation experiments were carried out. The results proved that the recognized acceleration was in good agreement with the measured one. So this method provide a new idea in obtaining the acceleration of structure. Acknowledgements The support of National Natural Foundation of China (Grant No. 11672237) is gratefully acknowledged.

References 1. Li, C.J., Ulsoy, A.G.: High-precision measurement of tool-tip displacement using strain gauges in precision flexible line boring. Mech. Syst. Signal Process. 13(4), 531–546 (1999) 2. Kang, L., Kim, D., Han, J.: Estimation of dynamic structural displacements using fiber Bragg grating strain sensors. J. Sound Vib. 305, 534–542 (2007) 3. Rapp, S., Kang, L., Han, J., Mueller, U., Baier, H.: Displacement field estimation for a twodimensional structure using fiber Bragg grating sensors. Smart Mater. Struct. 18, 025006 (2009) 4. Kim, H., Kang, L., Han, J.: Shape estimation with distributed fiber Bragg grating sensors for rotating structures. Smart Mater. Struct. 20, 035011 (2011) 5. Bang, H., Kim, H., Lee, K.: Measurement of strain and bending deflection of a wind turbine tower using arrayed FBG sensors. Int. J. Precis. Eng. Manuf. 13(12), 2121–2126 (2012) 6. Glaser, R., Caccese, V., Shahinpoor, M.: Shape monitoring of a beam structure from measured strain or curvature. Exp. Mech. 52, 591–606 (2012)

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7. Wang, Z., Geng, D., Ren, W., Liu, H.: Strain modes based dynamic displacement estimation of beam structures with strain sensors. Smart Mater. Struct. 23, 125045 (2014) 8. Iadicicco, A., Pietra, M.D., Gaudio, G., Campopiano, S.: Strain measurements of a multilayer panel via Fiber Bragg gratings as novel approach for deflection monitoring of tracking particle detectors. In: Proceedings of SPIE, 9506, Optical Sensors 95061L (2015) 9. Zhang, Q., Zhang, J., Duan, W., Wu, Z.: Deflection distribution estimation of tied-arch bridges using long-gauge strain measurements. Struct. Control Health Monit. 25(3), e2119 (2018) 10. Hong, W., Qin, Z., Lv, K., Fang, K.: An indirect method for monitoring dynamic deflection of beam-like structures based on strain responses. Appl. Sci. 8(811), 1–16 (2018)

Chapter 108

Experiment Investigation of Constrained Layer Damping Used for Vibration Suppression of Railway Wheel Wang Zhonglong, Jiao Yinghou and Chen Zhaobo

Abstract Noise is one of the major factors restricting the development of rail transit. Wheel/rail noise, especially wheel noise is one of the most important sources of rail traffic noise. Wheel noise is usually divided into rolling noise, curve squeal noise and impact noise. The rolling noise occurs on straight rail is considered to be generated by the web coupling vibration caused by the radial excitation of the wheel. In this paper, the axial and radial coupling frequencies of wheels are extracted by modal testing, and the dominant modes in each coupling frequencies are indicated. A constrained layer damping (CLD) ring mounted on the wheel tire is proposed. The principle of the damping ring is to reduce the vibration of the coupled web by reducing the radial vibration of the wheel tread so as to reduce the noise radiation. The experiments not only including the vibration suppression characteristics of the constrained damping ring, but also compares it with the friction damping ring which is widely used nowadays. The data of modal frequency and modal damping ratio are extracted, and the transfer functions of the wheels under the conditions of the constrained layer damping ring, the friction damping ring and the non-installed damping ring are tested. The results show that the constrained layer damping ring can effectively reduce the vibration of wheels in wide frequency domain. Keywords Wheel noise · Coupling vibration · Constrained layer damping ring · Vibration suppression

W. Zhonglong · J. Yinghou (B) · C. Zhaobo Harbin Institute of Technology, Harbin 150001, Hei Longjiang, China e-mail: [email protected] W. Zhonglong e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_108

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108.1 Introduction Rail transit has a deeply development as an important mode of transportation in recent years in many countries. But some problems also emerge and have to be solved, such as higher requirements for ride comfort and lower environmental noise. Wheel, as an important noise source of rail transit noise has attracted wide attention of researchers. Wheel noise is generally classified into three categories, rolling noise, squeal noise and impact noise which correspond to different theoretical models [1–3]. Subsequently, scholars have studied the wheel vibration from the modal point of view. The rolling noise of the wheel is assigned to 1 nodal circle and n (n ≥ 2) nodal diameter modes, and the squeal noise is attributed to a 0 nodal circle and n nodal diameter modes. In view of the vibration and noise characteristics of wheels, researchers have designed many vibration and noise reduction devices. For instance, tuned mass absorber [4], elastic wheel [5], constrained layer damping [6], wheel noise barrier [7], etc., and the most widely used, more research should be the friction damping ring. Friction damping rings are usually made of metal with circular or rectangular cross sections and mounted in the grooves on wheel tires. There are many factors affecting the effect of vibration and noise mitigation effect of friction damping rings, including mass and quantity. However, the most studied parameter is the positive pressure between the friction damping ring and the wheel. In the study of this parameter, scholars usually use the finite element method and theoretical analysis. The finite element method regards the positive pressure as the contact stiffness between the elements [8], while the theoretical analysis usually adopts a two-degree-of-freedom system [9]. A large number of researches show that the friction damping ring has the optimum preload to make the noise of the wheel lowest. However, this is exactly one of the drawbacks of the friction damping ring. The effect of the friction damping ring will be greatly changed by the slight difference in temperature, machining error and so on. At the same time, a large number of studies have shown that friction damping rings have a good suppression effect on curve squeal noise, but the effect on rolling noise is very limited. In this paper, a constrained damping ring structure is proposed to solve the problem of positive pressure sensitivity of friction damping ring through viscoelastic materials with high damping coefficient, and to achieve better vibration and noise reduction effect than friction damping ring, which can provide reference for engineering application.

108.2 Experiment Setup Viscoelastic materials have higher damping coefficients than other damping materials and have been extensively used in vibration and noise reduction of automotive and aircraft structures. Constrained layer damping is a commonly used damping form for

108 Experiment Investigation of Constrained Layer …

1285

τ

τ0 γ

ωt

α γ0

Fig. 108.1 Stress-strain curves of viscoelastic materials

viscoelastic materials. The shear, tension and compression deformation of viscoelastic materials are induced by the relative displacement between the main structure and the constrained layer, and the vibration energy is correspondingly consumed. The main energy dissipation mode of viscoelastic materials is shear deformation, which is characterized by the strain lagging behind stress, forming a hysteresis curve as shown in Fig. 108.1. The stress and strain of viscoelastic materials are assumed to be sinusoidal and the phase angle of strain behind stress is ϕ, the shear modulus of viscoelastic material is G, then

G∗ =

τ = τ0 e jωt , γ = γ0 e j (ωt−ϕ) , G = τ0 /γ0

(108.1)

τ0 τ = e jϕ = G(cos ϕ + j sin ϕ) = G  (1 + jβ) γ γ0

(108.2)

where G* is the complex shear modulus and G is the real part of it, β is the loss factor of the material and can be expressed as β = tan ϕ.

108.2.1 Structure of CLD Ring Structural and installation diagrams of circle ring and CLD ring are separately shown in Fig. 108.2a, b. Figure 108.2a shows two commonly used circle damping ring, which was installed on both sides of the wheel tire. Figure 108.2b is the structural and installation diagram of a CLD ring. The CLD ring consists of viscoelastic material and constrained ring of which the material is metal. The constrained layer is first pasted into the rectangular groove inside the wheel tire, and then the metal ring is installed. The size of the CLD ring is determined by the size of the wheel structure. The diameter of the wheel tested in this paper is 860 mm, the diameter of the CLD ring is 740 mm and the width is 30 mm, the thickness of the viscoelastic material is 1.5 mm while the thickness of the metal ring is 15 mm. In order to reduce the weight, the metal ring is

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Circle ring

Wheel

Viscoelastic layer

Metal ring

(b)

(a)

Fig. 108.2 Structural and installation diagrams of circle ring and CLD ring

Table 108.1 Parameters of viscoelastic material

Density (kg/m3 )

Shear modulus (Pa)

Poisson ratio

Loss factor

1990

6.8 × 105

0.47

0.35

a hollow structure with a wall thickness of 2 mm. The principle physical properties of viscoelastic materials are shown in Table 108.1.

108.2.2 Test Setup Data acquisition and analysis system of B&K was used in the experiments. The wheel was excited by modal hammer and high precision accelerometers were used for signal acquisition. In the vibration response test, sensors are attached to the wheel tread radial, the wheel tire axial and the web axial respectively. The hammer excited the wheel tread and the wheel tire from radial and axial directions separately. The datum is given in the form of acceleration admittance, that is, acceleration divided by force. In the modal test experiment, 80 measuring points were arranged in four circles on the axle of the wheel, while 20 measuring points were arranged in one circle on the radial direction. The modal hammer excited a fixed point on the wheel tread and tire respectively. The modal frequency and modal damping of the wheel are extracted by post-processing. The modal recognition algorithm uses Rational Fraction Polynomial-Z method. Test equipment and test objects are presented in Fig. 108.3.

108 Experiment Investigation of Constrained Layer …

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Sensor 1

Sensor 2

Sensor 3

(a)

(b)

Fig. 108.3 Accelerometers distribution diagram (a) of FRF test and test equipment and target wheel (b)

108.3 Results and Discussion 108.3.1 Coupling Analysis The axial and radial modal frequencies of standard wheels (without any vibration suppression devices) are extracted. The modes with the same axial and radial modal frequencies are considered as coupled modes. The coupling modal frequencies of 0–6000 Hz are presented in Table 108.2. As can be seen from Table 108.2, 18 coupled frequencies of axially and radially frequencies of the wheel within 6000 Hz are presented, which means that most modes of the wheel are coupled. However, each mode has the main frequency. To illustrate the dominant frequencies of these modes, the Frequency Response Function (FRF) data of tread and web of the wheel subjected to radial excitation are presented in Fig. 108.4. Table 108.2 Coupling frequencies of the standard wheel Mode

Axial frequency

Radial frequency

1

416

416

2

606

606

3

953

4 5 6

Mode

Axial frequency

Radial frequency

Mode

Axial frequency

Radial frequency

7

2102

2102

13

4049

4049

8

2248

2248

14

4560

4560

953

9

2867

2867

15

4683

4683

1111

1111

10

2969

2969

16

4784

4783

1967

1968

11

3651

3651

17

5130

5130

2001

2001

12

3738

3738

18

5665

5666

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Fig. 108.4 FRF of the web and tread of the wheel subjected to radial excitation

It can be seen from Fig. 108.4 that except for a few radial frequencies, the peak values of the web are larger than that of the tread. Besides, sound pressure level (SPL) of the wheel is relative to the acoustic radiation area. It is clear that the web area is much larger than the tread. However, because of the coupling of axial and radial modes of the wheel, CLD ring decrease web vibration of the wheel by decreasing the vibration of the wheel tire.

108.3.2 Contrast Between Circle Ring and CLD Ring The modal and vibration of standard wheel, wheels with circle ring and with CLD ring were separately tested in laboratory. Through Reflex post-processing software, the natural frequencies and modal damping ratios (%) of the wheels under the three states are separately extracted, as shown in Figs. 108.5 and 108.6. Some modifications have been made to the wheels for the installation of the damping rings. There is a certain difference in the frequency of installation of circle ring and CLD ring. Even for the same wheel, there are some changes of the natural frequencies of each mode before and after installing the damping ring. Compared with the change of frequencies, the change of modal damping ratios is more obvious. The FRF of the wheel subjected to axial and radial excitation are listed in Fig. 108.7 to illustrate the vibration suppression effect of the CLD ring. Figure 108.7a shows the data of sensor 2 and sensor 3 in Fig. 108.3a while Fig. 108.7b shows the data of sensor 1 and sensor 3. In fact, Fig. 108.7a, b represent squeal noise and rolling noise which are mainly noise of metro and high speed rail. These two figures show that CLD ring has better performance than circle ring in the frequency band between 1500 and 6000 Hz, which means that it is more suitable for the rolling noise suppression for high speed railway wheel.

108 Experiment Investigation of Constrained Layer …

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Fig. 108.5 Frequencies of the wheels under different states

Fig. 108.6 Modal damping ratios (%) of the wheel under different states

108.4 Conclusions A CLD ring for high speed railway wheel is presented in this paper. Coupling analysis was carried out and the coupling frequencies of the wheel are extracted. Result shows that most modes of the wheel are coupled and the CLD ring is suitable for the vibration suppression of the wheel when it was excited from the radial direction which corresponding to rolling noise. FRF test of the wheel under different states were conducted to verify the effect of the CLD ring. Results show that CLD ring is more effective in the frequency of 1500–6000 Hz which is the main frequency range of rolling noise.

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(a)

(b)

Fig. 108.7 FRF of the wheel under different states subjected to a axial excitation and b radial excitation

References 1. Rudd, M.J.: Wheel/rail noise–part II: wheel squeal. J. Sound Vib. 46(3), 381–394 (1976) 2. Vér, I.L., Ventres, C.S., Myles, M.M.: Wheel/rail noise–part III: impact noise generation by wheel and rail discontinuities. J. Sound Vib. 46(3), 395–417 (1976) 3. Remington, P.J.: Wheel/rail noise–part IV: rolling noise. J. Sound Vib. 46(3), 419–436 (1976) 4. Inaki Merideno, F., Javier Nieto, S., Nere Gil-Negrete, T.: Theoretical prediction of the damping of a railway wheel with sandwich-type dampings. J. Sound Vib. 333(20), 4897–4911 (2014) 5. Koo, D.H., Kim, J.C., Yoo, W.H.: An experimental study of the effect of low-noise wheels in reducing noise and vibration. Transp. Res. D 7(6), 429–439 (2002) 6. Jones, C.J.C., Thompson, D.J.: Rolling noise generated by railway wheels with visco-elastic layers. J. Sound Vib. 231(3), 779–790 (2000) 7. Thompson, D.J., Gautier, P.E.: A review of research into wheel/rail rolling noise reduction. Proc. Inst. Mech. Eng. F J. Rail Rapid Transit 220(4), 385–408 (2006) 8. Brunel, J.F., Dufrenoy, P., Demilly, F.: Modeling of squeal noise attenuation of ring damped wheels. Appl. Acoust. 65(5), 457–471 (2004) 9. Brunel, J.F., Dufrénoy, P., Charley, J.: Analysis of the attenuation of railway squeal noise by preloaded rings inserted in wheels. J. Acoust. Soc. Am. 127(3), 1300–1306 (2010)

Chapter 109

Free Transverse Vibration of Mindlin Annular and Circular Plate with General Boundary Conditions Qingjun Hao, Zhaobo Chen and Wenjie Zhai

Abstract In this paper, free vibration of Mindlin annular plate with arbitrary boundary conditions is investigated by an accurate solution method. The effects of the restrained stiffness of the springs on modal properties are incorporated in the current framework. The displacement components can be expressed as Fourier cosine series plus auxiliary polynomial functions, which aim to eliminate the discontinuities of displacement and its derivatives at both ends and to enhance the convergence of the results effectively. The current solution method can be used to general boundary conditions without changing the calculate method, which is different from the most existing studies. Numerical examples were presented and discussed for several different geometric and material properties as well as distinct boundary conditions. Keywords Mindlin annular plate · Fourier–Ritz method · General boundary conditions · Free vibration

109.1 Introduction For the high strength-to-weight and stiffness-to-weight ratios, the annular and circular plates are widely used in various fields, such as mechanical, architectural, aeronautical, and marine engineering due to their special geometric shapes. These important applications impel many researchers to investigate and predict the vibration behavior of annular and circular plates since they are prone to damage and fatigue for the highly applications in dynamic environment as well as abnormal and extensive vibrations. There are a comprehensive literature for transverse vibration of annular plates and circular plates with various kinds of boundary conditions mainly based on two-dimensional theories, such as classical plate theory (CPT), the first-order Q. Hao · Z. Chen (B) · W. Zhai School of Mechatronics Engineering, Harbin Institute of Technology, Harbin 150001, People’s Republic of China e-mail: [email protected] Q. Hao e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_109

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shear deformation theory (FSDT), and the higher-order shear deformation theory (HSDT). The CPT neglect the influence of shear deformation on thickness, thus it is suitable for thin plate structures. In order to consider the effect of shear deformation through the thickness, the FSDT [1, 2] introduces the shear correction factor as a constant to adjust the transverse shear stiffness. In the HSDT [3, 4], the effects of both shear and normal deformations are considered by an assumption of nonlinear stress variation through the thickness. The HSDT theory will result in mathematical and computational complexities and need more computational requirements. Under the appropriate shear correction factor, FSDT is sufficient to predict the vibration characteristics of moderately thick plates. In view of this, the FSDT is just applied to formulate the theoretical model in current work. Soni and Amba Rao [5] deal with the free axisymmetric vibrations of Mindlin circular plates with linear variation in thickness by using Chebyshev collocation methods. Based on Rayleigh–Ritz method, Chakraverty, et al. [6, 7] presented the transverse vibration of annular elliptic and circular plates by two-dimensional boundary characteristic orthogonal polynomials. Wu et al. [8, 9] presented the free vibration of circular plates with variable thickness or not by the generalized differential quadrature rule. By discrete singular convolution method, Civalek et al. [10] researched the free vibration of annular Mindlin plates with free inner edge. Chen [14] proposed a meshless method to solve the eigenfrequencies of clamped plates using the radial basis function (RBF). In his paper, singular value decomposition technique is used to obtain the eigenvalues and boundary modes at the same time. In this paper, a modified Fourier–Ritz method, which is proposed by Li et al. to solve vibrations of single beam [11, 12] or plate [13, 14], is used to solve the transverse vibration of Mindlin annular plate and circular plate with general boundary conditions. The transverse vibration can be expressed as combinations of Fourier cosine series and sine series functions. Compared to the existing solution methods, the transverse vibration of annular plate and circular plate under different boundary conditions, including classical and elastic boundary conditions or their combinations, can be easily obtained just by changing the stiffness of the three sets of springs at inner and outer edge without altering the solution procedures. In order to illustrate the convergence and accuracy of this method, the current results are compared with those results of other published papers.

109.2 Theoretical Formulations 109.2.1 Description for the Mindlin Annular Plate and Circular Plate The Mindlin annular plate with the dimension of a×b×h is considered here, as shown in Fig. 109.1. At inner and outer edge of the considered annular plate, the arbitrary boundary conditions can be expressed as kinds of linear springs and torsional springs.

109 Free Transverse Vibration of Mindlin Annular …

a co-ordinate system

1293

b partial view

Fig. 109.1 The theoretical model of Mindlin annular plate

Based on the first-order shear deformation, all kinds of boundary  a b conditions can be expressed by defining the stiffness of the two linear springs kw , kw as well as the four   torsional springs K ar , K aθ , K br , K bθ . In this way, all different boundary conditions, including the classical and elastic boundaries as well as their combinations, can be expressed by setting the values of liner and torsional spring stiffness.

109.2.2 Kinetic Energy Relations and Stress-Strain Relations The assumed displacement for the annular plate based on the first-order shear deformation plate theory can be shown as follows: u r (r, θ, z, t) = −zϕr (r, θ, t) u θ (r, θ, z, t) = −zϕθ (r, θ, t) u z (r, θ, z, t) = w(r, θ, t)

(109.1)

where u r , u θ and u z represent displacement functions in the r, θ and z directions respectively. ϕr and ϕθ are the rotations of the normal to the middle surface about r and θ directions. t is time variable. The relationship of strain and displacement are given as follows r εr = ∂u ∂r ∂w εz = ∂z γr θ = r∂u∂θr +

∂u θ ∂r

−

uθ r

εθ = r∂u∂θθ + urr r γr z = ∂w + ∂u ∂r ∂z ∂u θ ∂w γθ z = r ∂θ + ∂z

(109.2)

where εr , εθ , εz are normal strains and γr θ , γr z , γθz are shear stains on middle surface of Mindlin annular plate.

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According to Hook’s law and considering the Mindlin plate theory, the stressstrain relationship of the considered model by elastic theory, as shown in Fig. 109.1, can be expressed as ⎡

E/1 − υ 2 υ E/1 − υ 2 ⎢ υ E/1 − υ 2 E/1 − υ 2 ⎢ ⎢ 0 0 ⎢ ⎢ ⎣ 0 0 0 0

0 0 G 0 0

0 0 0 κG 0

⎤ ⎡ ⎤ ⎤⎡ σr εr 0 ⎢ ⎥ ⎢ ⎥ 0 ⎥⎢ εθ ⎥ ⎢ σθ ⎥ ⎥ ⎥ ⎢ ⎥ ⎥⎢ 0 ⎥⎢ γr θ ⎥ = ⎢ τr θ ⎥ ⎥ ⎢ ⎥ ⎥⎢ 0 ⎦⎣ γr z ⎦ ⎣ τr z ⎦ γθ z τθ z κG

(109.3)

where σr and σ θ are the normal stresses, τrθ , τrz and τθz are the shear stresses, E, υ, κ and G are the young’s modulus, poisson’s ratio, shear correction factor and shear modulus, respectively. Based on Eqs. (109.1) and (109.2), the normal and shear strains of the considered Mindlin annular plate can be written as follows ⎤ ∂ϕr ⎤ ⎡

−z ∂r  εr ⎥ ⎢ε ⎥ ⎢ −z r∂ϕ∂θθ + ϕrr ⎥ ⎢ θ ⎥ ⎢ 

⎢ ⎥ ⎥ ⎢ ⎢ ∂ϕ ∂ϕ ϕ ⎢ γrθ ⎥ = ⎢ −z r + θ − θ ⎥ ⎥ r ∂θ ∂r r ⎥ ⎢ ⎥ ⎣ γrz ⎦ ⎢ ∂w ⎣ ⎦ − ϕr ∂r γθz ∂w − ϕθ r ∂θ ⎡

(109.4)

109.2.3 Energy Expressions The linear strain potential energy during vibration is defined as 1 Uv = 2

˚ (σr εr + σθ εθ + τr θ γr θ + τr z γr z + τθ z γθ z ) r dr dθ dz.

(109.5)

V

Substituting Eqs. (109.3) and (109.4) into (109.5), the strain energy of the considered model can be written in terms of displacement and rotations of the middle surface.

¨ ∂ϕθ ϕr 2 ∂ϕr E h3 × + + r dr dθ Uv =  12 ∂r r ∂θ r 2 1 − υ2

¨ ∂ϕr ∂ϕθ ϕθ 2 1 h3 G + − + × r dr dθ 2 12 r ∂θ ∂r r 

2

2  ¨ ∂w ∂w 1 − ϕr + − ϕθ + κGh r dr dθ. 2 ∂r r ∂θ

(109.6)

109 Free Transverse Vibration of Mindlin Annular …

1295

The strain potential energy of boundary springs is defined as Usp

1 = 2



1 + 2

2π

0



  a kaw w2 + Kra ϕr2 + Kaθ ϕθ2 dθ

2π

0

  b kbw w2 + Krb ϕr2 + Kθb ϕθ2 dθ.

(109.7)

The kinetic energy of the considered annular plate is given by 1 T = 2



˚ ρ V

∂u r ∂t

2

+

∂u θ ∂t

2

+

∂u z ∂t

2  r dr dθ dz.

(109.8)

In terms of the transverse displacement and rotation angles, the bending and twisting moments and transverse shearing forces for annular plate and circular plate can be expressed as

  1 ∂ϕr ∂ϕθ + ϕr + Mθ = D υ ∂r r ∂θ  

1 ∂ϕr ∂ϕθ D Mθr = (1 − υ) − ϕθ + 2 r ∂θ ∂r

∂W Qθ = κGh − ϕθ . r ∂θ

(109.9) (109.10) (109.11)

To guarantee the continuities of the displacement and their derivatives at θ = 0 and 2π of the considered model being same, the modified potential energy are introduced to incorporate the continuity conditions, given as  1 b [Q θ (w|θ=2π − w|θ=0 ) + Mθ (ϕθ |θ=2π − ϕθ |θ=0 ) 2 a + Mθr (ϕr |θ=2π − ϕr |θ=0 )]dr. (109.12)

Ucp =

109.2.4 Admissible Displacement Functions Ritz method is widely used in the analysis of structures for its simplicity and high accuracy. The solutions can be easily acquired by minimizing the energy functions with regard to the unknown coefficients of admissible displacement functions. When choosing the admissible functions, there are two conditions must be satisfied when choosing the permissible functions of equations. Firstly, the permissible functions must be integrable, differential, continuous and linearly independent. Secondly,

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the permissible functions must satisfy the boundary conditions. Herein, a modified Fourier series are used here to obtain the displacement function of translation and rotation, which including a standard Fourier series and a couple of auxiliary functions. The introduction of auxiliary functions are purpose to eliminate the discontinuities of the original displacement functions as well as their derivatives for translation and rotation. The detailed expressions of the displacement w, ϕr and ϕθ can be given as ⎡

∞ ∞  

A1mn cos(λm s) cos(λn θ ) +

⎢ ⎢ w(r, θ, t) = ⎢ m=02n=0∞ ⎣   3 + Amn sin(λm s) cos(λn θ )

2 ∞  

⎤

A2mn cos(λm s) sin(λn θ) ⎥ ⎥ jωt m=0 n=1 ⎥e ⎦

m=1 n=0

⎡

(109.13.a) ∞ ∞  

1 cos(λ s) cos(λ θ ) + Bmn m n

2 ∞  

⎢ ⎢ m=0 n=1 ϕr (r, θ, t) = ⎢ m=02n=0∞ ⎣   3 + Bmn sin(λm s) cos(λn θ )

⎤

2 cos(λ s) sin(λ θ) Bmn m n ⎥

⎥ jωt ⎥e ⎦

m=1 n=0

⎡

(109.13.b) ∞ ∞  

1 cos(λ s) cos(λ θ ) + Cmn m n

⎢ ⎢ ϕθ (r, θ, t) = ⎢ m=02n=0∞ ⎣   3 + Cmn sin(λm s) cos(λn θ )

2 ∞  

⎤

2 cos(λ s) sin(λ θ) Cmn m n ⎥ ⎥ jωt m=0 n=1 ⎥e ⎦

m=1 n=0

(109.13.c) 1 2 3 1 2 , Bmn , Bmn , Cmn , Cmn and where λm = mπ/(b − a), λn = n/2, A1mn , A2mn , A3mn , Bmn 3 Cmn are unknown coefficients. Mathematically, the displacement functions given in Eq. (109.10) can be consistently converge and expand to any functions for F(r, θ ) ∈ C 2 over S:([(a, b)] × [(0, 2π )]). The obtained solutions can be set as arbitrary calculation precision for the truncation of the series expressions. So the acceptable precision can be obtained by setting truncation numbers M × N properly.

109.2.5 Solution Procedures In this subsection, the main purpose is to calculate the unknown coefficients of all displacement functions by Ritz method. The Lagrangian energy functions is expressed as: L = UV + Usp − Ucp − T.

(109.14)

Substituting Eqs. (109.6)–(109.9) into Eq. (109.11) and taking The Lagrangian expression’s derivatives with respect to these coefficients based on the Ritz method, it’s easy to obtain the follows

109 Free Transverse Vibration of Mindlin Annular …

∂L = 0, ϑ = Amn , Bmn , Cmn ∂ϑ

1297

(109.15)

where ϑ represents the unknown coefficients Amn , Bmn , Cmn . The acceptable precision is acquired by properly setting the truncated number m = M and n = N. Obviously, it’s easy to obtain the natural frequencies and corresponding eigenvectors of the considered annular and circular plate model by solving a standard matrix eigenvalue problem which can be obtained from Eq. (109.15).

109.3 Numerical Results and Discussions The analysis method in this paper can deal with all kinds of boundary conditions, including clamped, free, simply supported and elastic ones as well as their combinations. In order to verify the accuracy and convergence of the solution method, a numerical example of transverse vibration of considered annular plate is presented and compared with the theoretical results from other researchers.

109.3.1 Convergence and Accuracy Analysis For demonstration of the method’s effectiveness, the fully clamped annular plate both inner and outer boundaries is calculated and the calculation results are compared with other published research by different analysis methods. The material properties of the numerical modal are defined as follows ρ = 7850 kg/m3 E = 2.1 × 1011 Pa υ = 1/3 b = 1 m a/b = 0.4 h/b = 0.001. Table 109.1 shows the calculated first six non-dimensional frequencies and from published results by Zhou [15]. It’s easy to obtain that the maximum difference of frequency parameters between M = N = 20 and M = N = 30 is 0.055%, thus the natural frequencies convergence quickly at M = N = 20. Therefore, the truncation number is set as 20 in all the following numerical calculations. The worst differentia of natural frequencies is 0.023% between the numerical analysis at M = N = 30 and the results from Zhou [15], where the classical thin plate theory is employed in their research, which indicates the method’s high accuracy. According to the above analysis, we can conclude that the modified Fourier method based on the Mindlin plate theory is able to correctly deal with and predict the free vibration characteristics of the moderately thick circular or annular plate with various kinds of boundary conditions.

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√ Table 109.1 The first six natural frequencies  = ωb2 ρh/D with M and N increasing M and N

Modal sequence 1

2

3

4

5

6

10

13.611

19.457

31.461

48.109

67.401

72.351

12

13.558

19.423

31.385

47.337

67.181

68.401

16

13.520

19.401

31.349

46.953

66.538

67.019

20

13.501

19.399

31.342

46.883

66.013

66.940

30

13.502

19.399

31.338

46.857

65.999

66.934

Ansys

13.566

19.609

31.588

46.970

65.904

66.448

Zhou [15]

13.500

19.389

31.338

46.855

65.984

66.924

109.3.2 Free Vibration of Annular and Circular Plate This method can deal with various combinations of classical and elastic supported boundary conditions. This section analyses five edge constraints of circular and annular plates. Natural frequency parameters are listed in the tables, where existing solutions validate the accuracy of these results. Tables 109.2, 109.3, 109.4, 109.5, and 109.6 present the lowest five frequency parameters for the circular and annular plates with different internal and external diameter ratios a/b = 0(0.2)0.4. The boundary conditions can be obtained by setting √ Table 109.2 Dimensionless frequency parameters with  = ωb2 ρh/D for the annular and circular plate with clamped outer and inner edge (CC) (υ = 1/3) a/b

Modal number

Present

Vera [16]

Zhou et al. [15]

0

1

10.2160

–

10.216

2

21.2224

–

21.260

3

34.8838

–

34.877

4

39.7773

–

39.771

5

51.0703

–

51.031

1

34.6236

34.6092

34.609

2

36.1188

36.1032

36.103

3

41.8382

41.8196

41.820

4

53.4095

–

53.388

0.2

0.4

5

70.4430

–

70.501

1

61.8812

61.8722

61.872

2

63.0056

62.9959

62.996

3

66.6826

66.6716

66.672

4

73.6453

–

73.630

5

84.6330

–

84.594

109 Free Transverse Vibration of Mindlin Annular … Table 109.3 Dimensionless frequency parameters with √  = ωb2 ρh/D for the circular plate with free outer and inner edge (FF) (υ = 1/3)

a/b

Modal number

0

1

5.3231

–

2

9.0790

–

3

12.2393

–

4

20.5190

–

5

21.7467

–

1

5.0542

2

8.4560

8.4419

3

12.1984

12.187

4

19.6796

19.675

5

21.3360

21.209

1

4.5386

4.5325

0.2

0.4

Table 109.4 Dimensionless frequency parameters with √  = ωb2 ρh/D for the circular plate with simply supported outer and inner edge (SS) (υ = 1/3)

1299 Present

Zhou et al. [15]

5.0508

2

8.5798

8.551

3

11.7772

11.765

4

17.0573

17.043

5

21.3090

21.262

a/b

Modal number

Present

Vera [16]

0

1

13.9326

–

2

14.8818

–

3

25.6046

–

4

40.0051

–

5

48.5224

–

1

16.7303

16.7298

2

19.1966

19.1960

3

27.2550

27.2530

4

40.3622

–

5

57.0724

–

1

28.0867

28.0853

2

30.0790

30.0794

3

36.1424

36.1431

4

60.5316

–

5

78.5549

–

0.2

0.4

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√ Table 109.5 Dimensionless frequency parameters with  = ωb2 ρh/D for the circular plate with clamped outer and simply supported inner edge (CS) (υ = 1/3) a/b

Modal number

Present

Vera [16]

Zhou et al. [15]

0

1

21.2224

–

–

2

22.7710

–

–

3

34.8838

–

–

4

51.0703

–

–

0.2

0.4

5

60.9667

–

–

1

26.6304

26.6196

26.619

2

29.1682

29.1576

29.158

3

37.5911

37.5785

37.579

4

51.7208

–

51.685

5

69.9505

–

69.807

1

44.9453

44.9323

44.932

2

46.7474

46.7345

46.735

3

52.3656

52.3526

52.353

4

62.1769

–

62.148

5

76.3522

–

76.230

Table 109.6 Dimensionless frequency parameters with √  = ωb2 ρh/D for the circular plate with simply supported outer and free inner edge (SF) (υ = 1/3)

a/b

Modal number

0

1

4.9869

–

2

13.9320

–

3

25.5569

–

4

29.7581

–

0.2

0.4

Present

Zhou et al. [15]

5

39.9913

1

4.7482

4.7325

–

2

13.5855

13.583

3

24.937

24.935

4

31.2739

–

5

39.7337

39.710

1

4.7809

4.7436

2

11.9196

11.907

3

23.1045

23.098

4

37.2892

37.272

5

47.3130

–

109 Free Transverse Vibration of Mindlin Annular …

1301

the linear and torsional spring stiffness as 0 or 107 D according to the boundary conditions in all directions of the inner and outer edges. It’s easy to obtained that the fundamental five natural frequency tend to increases with the increase of radius ratio a/b except for Tables 109.3 and 109.6. For the free boundary conditions in both inner and outer edges in Tables 109.3 and 109.6, the frequency parameter varies kinds of irregular with a/b.

109.4 Conclusion In this paper, a unified solution has been presented to solve the free vibration of Mindlin circular and annular plate with different kinds of boundary conditions. Based on the Fourier–Ritz method, the displacements can be expressed as the combination of one standard Fourier method and two kinds of auxiliary functions. Then the modal parameters can be obtained by solving an eigenvalue matrix. Thus, the natural frequencies can be obtained for different kinds of boundary conditions. a numerical example of transverse vibration of considered circular and annular plate is presented and compared with the theoretical results from other researchers, which validate the accuracy of this method. This solution method can be extended to the static and dynamic problems of circular and annular plate and others. Acknowledgements The authors would like to thank the anonymous reviewers for their very valuable comments. Conflict of Interest None declared.

References 1. Reissner, E.: The effect of transverse shear deformation on the bending of elastic plates. J. Appl. Mech. 12(2), 69–72 (1945) 2. Mindlin, R.D.: Influence of rotatory inertia and shear on flexural motions of isotropic, elastic plates. J. Appl. Mech. 18(1), 31–38 (1951) 3. Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech. 51, 745–752 (1984) 4. Librescu, L.: On the theory of anisotropic elastic shells and plates. Int. J. Solids Struct. 3(1), 53–68 (1967) 5. Soni, S.R., Amba Rao, C.L.: On radially symmetric vibrations of orthotropic non uniform disks including shear deformation. J. Sound Vib. 42, 57–63 (1975) 6. Chakraverty, S., Petyt, M.: Natural frequencies for free vibration of nonhomogeneous elliptic and circular plates using two-dimensional orthogonal polynomials. Appl. Math. Model. 21(7), 399–417 (1997) 7. Chakraverty, S., Bhat, R.B., Stiharu, I.: Free vibration of annular elliptic plates using boundary characteristic orthogonal polynomials as shape functions in the Rayleigh–Ritz method. J. Sound Vib. 241(3), 524–539 (2001) 8. Wu, T.Y., Liu, G.R.: Free vibration analysis of circular plates with variable thickness by the generalized differential quadrature rule. Int. J. Solids Struct. 38, 7967–7980 (2001)

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9. Wu, T.Y., Wang, Y.Y., Liu, G.R.: Free vibration analysis of circular plates using generalized differential quadrature rule. Comput. Methods Appl. Mech. Eng. 191, 5365–5380 (2002) 10. Civalek, O., Gurses, M.: Free vibration of annular Mindlin plates with free inner edge via discrete singular convolution method. Arab. J. Sci. Eng. 34 (2009) 11. Shi, X., Shi, D., Qin, Z., Wang, Q.: In-plane vibration analysis of annular plates with arbitrary boundary conditions. Sci. World J. 2014 (2014) 12. Shi, X., Shi, D., Li, W.L., et al.: A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions. J. Vib. Control 22(2), 442–456 (2016) 13. Hao, Q., Zhai, W., Chen, Z.: Free vibration of connected double-beam system with general boundary conditions by a modified Fourier–Ritz method. Arch. Appl. Mech. 88(5), 741–754 (2018) 14. Chen, J.T., Chen, I.L., Chen, K.H., Lee, Y.T., Yeh, Y.T.: A meshless method for the free vibration analysis of circular and rectangular clamped plates using radial basis function. Eng. Anal. Boundary Elem. 28, 535–545 (2004) 15. Zhou, Z., et al.: Natural vibration of circular and annular thin plates by Hamiltonian approach. J. Sound Vib. 330(5), 1005–1017 (2011) 16. Vera, S.A., Sanchez, M.D., Laura, P.A.A., et al.: Transverse vibrations of circular, annular plates with several combinations of boundary conditions. J. Sound Vib. 213, 757–762 (1998)

Chapter 110

Underdetermined Blind Source Separation for Multi-fault Diagnosis of Planetary Gearbox H. Li, Q. Zhang, X. R. Qin and Y. T. Sun

Abstract Due to the advantage of strong load capacity and large transmission ratio, planetary gearboxes are widely used in heavy-duty machinery. The complex structure and tough working environment make the gears and bearings in planetary gearbox prone to failure, especially the occurrence of multi-faults. If these faults cannot be diagnosed in time, the shutdown of the entire equipment and even major accidents may happen. The analysis of vibration signal collected from acceleration transducer is regarded as an effective method for fault diagnosis of planetary gearbox to prevent accidents and save cost. In practical application, the collected measurements are always a mixture of signals from many unknown sources, which is considered challenging as an underdetermined blind source separation (UBSS) problem for fault location and recognition. A novel method is proposed in this study to tackle this challenge by three stages: vibration signal sparsity, mixing matrix estimation and source signal recovery. In the first stage, the vibration signals are transformed into the time-frequency (TF) domain by Wigner-Ville distribution (WVD) and then the K-singular value decomposition (K-SVD) is used to further improve the sparsity of time-frequency distribution (TFD) of vibration signals. In the second stage, according to the clustering distribution characteristics of TFDs, a clustering algorithm, namely the Variational Bayesian Gaussian Mixture Model (VBGMM), is used to estimate the number of sources and the values of mixing matrix at the same time. In the last stage, the source signal can be recovered from the source TFD estimate by a TF synthesis algorithm. In order to verify the practicability and effectiveness of the proposed method, a scale-down test rig for planetary gearbox of quay crane is built to simulate the occurrence of fault under variable work conditions. And several acceleration transducers are fixed on the gearbox to collect multi-channel vibration signals. The experimental results, compared with some existing methods, show that the source signals separated by using the proposed method have a clearer frequency spectrum characterization, which remarkably reduces the difficulty of fault feature extraction and improves the accuracy of multi-fault diagnosis.

H. Li · Q. Zhang (B) · X. R. Qin · Y. T. Sun College of Mechanical Engineering, Tongji University, Shanghai 201800, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_110

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Keywords Planetary gearbox · Fault diagnosis · Underdetermined blind source separation

110.1 Introduction Planetary gearbox is widely used as the main transmission device in the large engineering machinery: quayside container crane, wind turbine and heavy machine tool, to name a few. Non-stationary work conditions, for example, shock load and variable speed, make these equipment easy to failure. Furthermore, due to the large gear configuration and time-vary transmission path, using the single vibration sensor signal is difficult to pinpoint the fault feature, especially the multi-fault features that may overlap in frequency domain. Fault diagnosis algorithm based on multi-sensor is an effective method to address this problem. Suppose source vibration signals produced by different components of gearbox can be separated from multi-sensor signals, fault location will be detected accurately. Source Separation aims to different vibration sources from multiple observation vibration signals collected by a set of sensors. This procedure is always said to be “blind” because there is no a priori knowledge of the mixed mode of the multi-fault sources in the observations. This blind source separation (BSS) problem is fully detailed by Jean-Francois Cardoso in 1998 [1]. In the past two decades, two kinds of methods have been extensively studied. One assumes that the source signals are statistically independent, for example, independent components analysis (ICA) [2]. And the other relies on the assumption that source signals are sparse in some signal representation domain [3]. Independent-based methods work well in the condition that the number of sources N is less than or equal to the number of sensors M. But in the application of fault diagnosis for planetary gearbox, the number of vibration source signals is always more than the acceleration transducers, which is regarded as the Underdetermined Blind Source Separation (UBSS) problem. Therefore, many sparseness-based methods are proposed to address this problem [4]. A prerequisite for using these methods is to meet the requirements of sparseness. Time-frequency distribution (TFD) is selected as an ideal signal representation with the assumption that there is no more than one source present at any point in the TF domain [5]. Thus, the clustering approach can be used to separate TF-points that belong to different vibration sources. TFDs can be divided into two categories: linear TFDs and quadratic TFDs [6]. The Wigner-Ville distributed (WVD) is the most widely studied quadratic TFD. It gives ideal concentration for monocomponent frequency signals in comparison with the linear TFDs, but also produces the undesired “cross-terms” that reduce the resolution for multi-component signals. In order to enhance the resolution of the WVD, the K-singular value decomposition (K-SVD) [7] is used to minimize cross-terms. This is one main concentration of this paper and the other concentration is to use the Variational Bayesian Gaussian Mixture Model (VBGMM) [8] for mixing matrix estimation. Then, time series of vibration sources are recovered from corresponding TFDs by a TF synthesis algorithm [9, 10].

110 Underdetermined Blind Source Separation … Table 110.1 TF-UBSS algorithm description

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Step 1: Mixture STFD computation by WVD Step 2: STFD resolution enhancement via KSVD Step 3: Noise thresholding and auto-source points selection Step 4: Auto-source clustering by VBGMM Step 5: Source signals recovery by TF-synthesis algorithm

In this study we choose the sparseness-based assumption as a fundamental framework to solve the UBSS problem, which is formulated in Sect. 110.2. Detailed algorithm flow is also included in Sect. 110.2. Our method is first validated by a numerical simulation example in Sect. 110.3 and put into practice in an experimental platform of planetary gearbox in Sect. 110.4. Conclusions are drawn in Sect. 110.5.

110.2 Algorithm Description 110.2.1 Problem Formulation Suppose that the n × 1 underlying source vector s(t) = [s1 (t), . . . , sn (t)]T composes of the desired sources to be recovered from the m × 1 observation vector x(t) = [x1 (t), . . . , xm (t)]T . The blind source separation (BSS) problem can be formulated as follows: x(t) = As(t)

(110.1)

where A = [a1 , . . . , an ] is the mixing matrix with the size of m × n. The column ai represents the steering vector related to the source si (t). Because the acceleration transducers are often less than the vibration source in the application of fault diagnosis, this paper concentrates on the ill-conditioned case with m < n, which is called the underdetermined BSS (UBSS). Based on the assumption of source sparseness for this problem, the TF-UBSS algorithm is descripted in Table 110.1.

110.2.2 Time Frequency Distribution by Using WVD The discrete-time form of WVD for observation x(t), is given by [11] ρx x (t, f ) =

N −1 

Z [t + k]Z ∗ [t − k]e− j

4π N

kf

(110.2)

k=−N

where Z [t] = x[t] + H{x[t]} and H{·} represents the Hilbert transform (HT). Z ∗ [t] denotes the conjugate of Z [t]. For two different observation xi (t) and x j (t), the cross-WVD is given as

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ρxi x j (t, f ) =

N −1 

Z i [t + k]Z ∗j [t − k]e− j

4π N

kf

(110.3)

k=−N

WVD has a good resolution both in time and frequency domain, but suffers the problem of undesired cross-terms in TF domain. In order to obtain high-quality TFDs, K-singular value decomposition (K-SVD), which is initially proposed in [7] for image denoising, is used to remove the cross-terms in TF domain. The main idea of this algorithm is to represent TF patches of WVD sparsely over a trained dictionary, which can be descripted as min{yi − Dαi } s.t. αi 0 < T0 αi

(110.4)

where yi represents patches of WVD, D is the trained dictionary and •0 is the 0 norm, calculating the nonzero entries of the coefficient vector αi . T0 is a predefined constraint factor. yi − Dαi  denotes the reconstruction error. The KSVD algorithm continuously minimizes reconstruction error by iteration.

110.2.3 Spatial Time-Frequency Distribution (STFD) According to Eqs. (110.2) and (110.3), the STFD matrix [12] is defined as ⎤ ρx1 x1 (t, f ) · · · ρx1 xm (t, f ) ⎥ ⎢ .. .. .. Dxx (t, f ) = ⎣ ⎦ . . . ρxm x1 (t, f ) · · · ρxm xm (t, f ) ⎡

(110.5)

where Dxx (t, f ) is of dimension m × m and its diagonal elements are auto TF points while non-diagonal elements are cross TF points. Let the STFD matrix work on both side of the Eq. (110.1), the TF-transformed form of BSS problem is given by, Dxx (t, f ) = ADss (t, f )A H

(110.6)

where the superscript H represents the complex conjugate transpose of a matrix. Dss (t, f ) is the source STFD matrix with the dimension of n × n. The entries of the source STFD matrix can be divided into three parts, the auto-source TF points at which there is a true energy concentration, the cross-source TF points at which there is a false energy concentration, and the noise TF points without energy concentration. According to the TF-disjoint assumption, there is only one source activate at any auto-source TF points. Thus, the noise TF points and the cross-source TF points need to be first removed. Noise Thresholding. By setting a small threshold 1 (default, 1 = 0.05), the TF points that has insufficient energy will be removed by

110 Underdetermined Blind Source Separation …

Dxx (t p , f q )  < 1 , set (t p , f q ) to 0

If max f Dxx (t p , f )

1307

(110.7)

Auto-source TF points selection. A criterion proposed in [13] can be used to selection auto-source TF points, which is presented as λ {Dxx (ta , f b )} − 1 If max < 2 , Dxx (ta , f a ) then (ta , f a ) is atuo-source TF points

(110.8)

where 2 is a small threshold (default, 2 = 0.3) and λmax is the largest eigenvalue of TF points. After removing the noise and cross-source TF points in STFD matrix Dxx (t, f ), Eq. (110.5) is rewritten as Dxx (ta , f b ) = ρxi xi (ta , f b )ai aiT ∀(ta , f a ) ∈ i

(110.9)

According to the above equation, the auto-source TF points (ta , f b ) that belong to the same TF domain of si (t) will have the same principal eigenvector ai . Through clustering these auto-source TF points into the corresponding classes, the columns of mixing matrix A are estimated as the cluster centroid of each class.

110.2.4 Auto-source Clustering The spatial vector calculated from corresponding auto TF point is described as: 

diag Dauto xx (ta , f a )

 v(ta , f a ) = diag Dauto (ta , f a ) xx

(110.10)

By clustering these spatial vectors into N classes, the TF points set i of source si (t) can be formed by the auto-source TF points belong to the class Ci , ({Ci |i ∈ N }). However, because the number of sources (N) is usually unknowable, some commonly used clustering algorithm, for example k-means algorithm [14], have no effect in this case. To estimate the number of source signals automatically, the Variational Bayesian Gaussian Mixture Model (VBGMM) is selected to cluster the spatial vectors into different classes. The operation procedure of the VBGMM is detailed in [8]. Then, the WVD of source signal si (t) can be arranged as

ρˆsWi V D (ta ,

fb ) =



trace Dauto xx (ta , f a ) , (ta , f b ) ∈ i 0 (ta , f b ) ∈ / i

(110.11)

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WVD reflects the energy distribution of source signals in time-frequency domain. The time series of source signals can be recovered from corresponding WVDs by a TF synthesis algorithm [9, 10], based on the following inversion form of Eq. (110.2). N −1 1  wvd si (t) = ∗ ρ (t, f )e− j si (0) f =−N si

4π N

tf

(110.12)

110.3 Simulation Results 110.3.1 Description A simulation example is used to explain the proposed method in detail. Four original source signals are shown in Fig. 110.1a, including an amplitude modulation signal s1 (t), a sinusoidal signal s2 (t), an incremental frequency modulation signal s3 (t), and a decreasing frequency modulation signal s4 (t). Their mathematical expressions are given as: s1 (t) = sin(2π 150t)[cos(2π 10t) + 1] s2 (t) = sin(2π 100t) s3 (t) = sin[2π(200t2 + 200t)] s4 (t) = sin[2π(−200t2 + 400t)]

(110.13)

Then these source signals are mixed by a randomly generated mixing matrix A with the dimension of 3 × 4. According to Eq. (110.1), three observation signals are obtained as shown in Fig. 110.1b. In this simulation example, we need to estimate

Fig. 110.1 The simulation example a four source signals b three mixed signals

110 Underdetermined Blind Source Separation …

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the mixing matrix A and recover the source signals from the observation signals. More details are described below.

110.3.2 Detail Procedure The basic assumption of the proposed method is that the source signals are disjoint in time-frequency (TF) domain, in other words, at most, only one source present at any point in the TF domain. Therefore, the Wigner-Ville distribution (WVD) is chosen to transform the time series of source signals into TF domain. The biggest advantage of this algorithm is a good resolution both in time and frequency scale. However, it also suffers the problem of unexpected cross-terms when there are many frequency components in TF domain. Figure 110.2a presents the TFD of the simulation observation signal x3 . The points in red circle are the cross-terms, which introduce some false information to the TF domain. It is difficult to distinguish different frequency components from this figure. Hence, before going to the next step, cross-terms need to be removed at first. Figure 110.2b shows the enhanced TFD of x3 by using the KSVD to remove the cross-terms and noise, and the active points can be figure out easily. Four frequency components are clearly shown in the figure that from four different source signals. In the next step, the points in the TFD of observation signals are restructured into a STFD matrix according to Eq. (110.5). Its entries are TF points with the size of 3 × 3, which can be divided into 3 types: auto-source TF points (diagonal matrix), cross-source TF points (off-diagonal matrix) and noise TF points. The autosource TF points can be clustered into different categories because the corresponding Dxx (ta , f a ) will have the same principal eigenvector ai . The number of categories

(a)

(b)

Fig. 110.2 The TFD of third mixed signal a without KSVD b with KSVD

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Fig. 110.3 The clustering results visualization

is the same as the number of source signals and the TF points in each category belong to the same source signals. In order to acquire all the auto-source points for clustering, Eqs. (110.7) and (110.8) are used to remove the undesired noise points and cross-terms respectively. The set of vectors {v(ta , f a )|(ta , f a ) ∈ } are obtained according to the Eq. (110.11) and the symbol  denotes the set of auto-source TF points from the same source signals. The VGBMM algorithm is selected to cluster them into different categories because it let the model choose a suitable number of effective components automatically. As shown in Fig. 110.3, the auto-source are clearly divided into 4 categories that correspond to four source signals respectively. The column vector ai of mixing matrix A is estimated as the centroid of this set of vectors in each category and the TFD of each source signal can be obtained by calculating the trace of auto-source points according to Eq. (110.10). After the TFDs of source signals (Fig. 110.4) are obtained, the original time series can be recovered by a TF synthesis algorithm. According to the Fig. 110.5a, the recovered source signals are similar to real source signals. However, two obvious disadvantages are also shown in this figure: end distortion and amplitude reduction. For comparison, the correlation coefficient between column vectors of different mixing matrix are calculated, which is given as 



Cov(ai , ai )

R(ai , ai ) =  Var(ai )Var(ai ) 

(110.14)

Figure 110.5b gives the recovery source signals without using the KSVD and Table 110.2 gives the comparison results. We can point out that the application of KSVD can improve the quality of source separation.

110 Underdetermined Blind Source Separation …

Fig. 110.4 TFDs of recovery signals

Fig. 110.5 Recovery source signals a with KSVD, b without KSVD

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1312 Table 110.2 The comparison results between column vectors of different mixing matrix

H. Li et al. KSVD

Non-KSVD

a1

0.99

0.72

a2

0.84

0.91

a3

0.89

0.99

a4

0.99

0.71

110.4 Experimental Results The proposed methods are put into practice in an experimental platform, which is shown in Fig. 110.6. The scale ratio of planetary gearbox is 1:5 and multi-sensor system are fixed to this gearbox. The sampling frequency is 10 kHz. First, we focus on tooth broken of sun gear (Fig. 110.7a). Two-channel sensor signals are presented in Fig. 110.7b. Three recovered source signals and their frequency spectrums are presented in Fig. 110.7c, d. According to these figures, we can summary below that • The fundamental frequency associated with the speed can be separated. • The recovered source signal has clear impact phenomenon and distinct spectrum characteristics. Figure 110.8a shows the crack failure of rolling bearings, which occurs in both inner and outer rings. It’s a multi-fault diagnosis problem. Figure 110.8c represents recovered source signals and their spectrums (Fig. 110.8d). Shock source signals with different spectral characteristics are separated, which correspond to two kinds of faults respectively. Some fault indicators extracted from both time and frequency domain can be used for further diagnosis.

Fig. 110.6 The experiment platform fixed with multi-sensor system

110 Underdetermined Blind Source Separation …

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Fig. 110.7 The tooth broken of sun gear: a failure schematic, b the observation sensor signals, c the recovery source signals and d their frequency spectrums

Kurtosis and Shape Factor (SF) are two widely used fault indicators [15]. The former indicates impulsive signals by calculating the fourth order standardized moment of time series. And, the later represents the distribution of signals in time domain. These two indicators are used to distinguish different health states of planetary gearbox, such as normal, the broken of sun gear and the crack of bearing. According to Fig. 110.9, based on kurtosis and SF, different health states are easier to distinguish by using the recovery source signal of energy concentration than the observation sensor signals.

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Fig. 110.8 The crack of rolling bearing: a failure schematic, b the observation sensor signals, c the recovery source signals and d their frequency spectrums

Fig. 110.9 Distinguishability comparison of fault indicators. a Kurt; b shape factor

110.5 Conclusions In this paper, we proposed an effective WVD-enhancement approach to tackle the problem of undetermined blind source separation for fault diagnosis of planetary gearbox. According to the assumption that the sources are disjoint in TF domain, the fault source signal can be recovered from the multi-channel vibration signal. The main contributions over the proposed methods are that, first, the undesired cross-terms

110 Underdetermined Blind Source Separation …

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of WVD are removed by KSVD, which makes the transformed TF representation sparser; second, a clustering algorithm based on VBGMM is used to extract source TF points; and final, with a WVD synthesis algorithm, source signals are recovered from TF domain for further fault diagnosis. Simulation and experiment results validate the effectiveness of the proposed methods. Two potential research directions are worth studying in further work. One is the problem of assumption relaxation. The one-source activation assumption used in this study is too strict to apply to more situations. The other direction is a more high-performance WVD synthesis algorithm that can overcome the disadvantage of endpoint distortion. Acknowledgements This work was supported by the International Exchange Program for Graduate Students, Tongji University and the Project supported by the National Key Research and Development Program of China [Grant No. 2018YFC0808902].

References 1. Cardoso, J.-F.: Blind signal separation: statistical principles. In: Proceedings of the IEEE, vol. 86, No. 10, pp. 2009–2025 (1998). https://doi.org/10.1109/5.720250 2. Puntonet, C.G., Lang, E.W.: Blind source separation and independent component analysis. Neurocomputing 69(13), 1413–1413 (2005). https://doi.org/10.1016/j.neucom.2005.12.018 3. Xie, S., Yang, L., Yang, J.M., et al.: Time-frequency approach to underdetermined blind source separation. IEEE Trans. Neural Netw. Learn. Syst. 23(2), 306–316 (2012). https://doi.org/10. 1109/tnnls.2011.2177475 4. O’Grady, P.D., Pearlmutter, B.A., Rickard, S.T.: Survey of sparse and non-sparse methods in source separation. Int. J. Imaging Syst. Technol. 15(1), 18–33 (2010). https://doi.org/10.1002/ ima.20035 5. Aissa-El-Bey, A., Linh-Trung, N., Abed-Meraim, K., et al.: Underdetermined blind separation of nondisjoint sources in the time-frequency domain. IEEE Trans. Signal Process. 55(3), 897–907 (2007). https://doi.org/10.1109/tsp.2006.888877 6. Boashash, B., Khan, N., Ben-jabeur, T.: Time-frequency features for pattern recognition using high-resolution TFDs: a review. Digit. Signal Process. Rev. J. 40(1), 1–30 (2015). https://doi. org/10.1016/j.dsp.2014.12.015 7. Elad, M., Aharon, M.: Image denoising via sparse and redundant representations over learned dictionaries. IEEE Trans. Image Process. 15(12), 3736–3745 (2006). https://doi.org/10.1109/ TIP.2006.881969 8. Yupeng, L., Jianhua, Z., Zhanyu, M., et al.: Clustering analysis in the wireless propagation channel with a variational gaussian mixture model. IEEE Trans. Big Data 1–1 (2018). https:// doi.org/10.1109/tbdata.2018.2840696 9. Nelatury, S.R., Mobasseri, B.G.: Synthesis of discrete-time discrete-frequency wigner distribution. Signal Process. Lett. IEEE 10(8), 221–224 (2003). https://doi.org/10.1109/LSP.2003. 814391 10. Boudreaux-Bartels, G.: Time-varing filtering and signal estimation using the Wigner distribution synthesis techniques. IEEE Trans. Acoust. Speech Signal Process. 34(3), 442–451 (1986). https://doi.org/10.1109/tassp.1986.1164833 11. Climente-Alarcon, V., Antonino-Daviu, J.A., Riera-Guasp, M., Puche-Panadero, R., Escobar, L.: Application of the Wigner-Ville distribution for the detection of rotor asymmetries and eccentricity through high-order harmonics. Electr. Power Syst. Res. 91, 28–36 (2012). https:// doi.org/10.1016/j.epsr.2012.05.001

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12. Belouchrani, A., Amin, M.G.: Blind source separation based on time-frequency signal representations. IEEE Trans. Signal Process. 46(11), 2888–2897 (2002). https://doi.org/10.1109/78. 726803 13. Linh-Trung, N., Belouchrani, A., Abed-Meraim, K., et al.: Separating more sources than sensors using Time-Frequency distributions. EURASIP J. Adv. Signal Process. 2005(17), 2828–2847 (2005). https://doi.org/10.1155/ASP.2005.2828 14. Coates, A., Ng, A.Y.: Learning feature representations with K-means. Lect. Notes Comput. Sci. 7700, 561–580 (2012). https://doi.org/10.1007/978-3-642-35289-830 15. Goyal, D., Vanraj, D.G., Pabla B.S., et al.: Condition monitoring parameters for fault diagnosis of fixed axis gearbox: a review. Arch. Comput. Methods Eng. 24(3), 543–556 (2017). https:// doi.org/10.1007/s11831-016-9176-1

Chapter 111

Study on Sound Transmission Loss of Lightweight FGM Sandwich Plate C. Li, Z. Chen and Y. Jiao

Abstract Lightweight sandwich structure is widely used to prepare the body of moving vehicle, which requires its light weight, high mechanical strength, good sound transmission loss and other multi-functional characteristics. Functionally gradient materials can effectively reduce the stress mutation caused by mismatching of material properties, so the mechanical strength of the structure can be improved. The main objective of this research work is focused on sound transmission loss analysis of ribbed plate functionally graded material (FGM) sandwich plates filled with air, wherein two types of FGM sandwich structures are considered, which include one with FGM face sheets and homogeneous metal core, and the other with FGM core and homogeneous face sheets. Firstly, the acoustic projection loss of the sandwich panel filled with a cavity is analyzed by simulation software, and compared with the existing numerical results, the validity of the model is verified. Based on the simulation model, the effects of boundary conditions, the spacing between the stiffeners, the gradient index of functionally graded materials, and the thickness ratio of the panels to the stiffeners were analyzed. Keywords Sandwich structure · Stiffened plate · FGM

111.1 Introduction Sound transmission loss (STL) of panels is one of important acoustic issues for investigators in the past decades. The most appealing structures for sound transmission are sandwich structures made of multiple layer panels and cores [1–6]. Sandwich panels can be designed to have low density, high stiffness to mass ratio, excellent mechanical properties and acoustic properties, and is therefore widely used in the field of sound insulation. A variety of sandwich construction cores exist,such as air cavity, foams, honeycombs, pyramids, and corrugations, to mention a few. Extensive research is devoted to evaluating the STL of various sandwich panels that can C. Li · Z. Chen (B) · Y. Jiao Harbin Institute of Technology, 92 West Dazhi Street, Nan Gang District, Harbin 150001, China e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_111

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be classified by core type. Sandwich structures are widely applied in engineering applications. As a typical sandwich panel, stiffeners are widely used in high-speed trains. Stiffeners or structural links are commonly used to reinforce the laminated structure, adding complexity for exploring its dynamical and acoustical behaviors. Double wall partitions with air cavity, perhaps the simplest sandwich structure, received much attention [7–16]. Among them, the classical laminated plate theory (CLPT) and first-order shear deformation theory (FSDT) are the commonly used analysis of orthotropic laminated plate theory. However, the biggest limitation of FSDT is that it needs shear correction factor to correct the unreal change of transverse shear strain in thickness [8, 9]. In order to overcome the limitations of FSDT, Reddy [10] and Librescu [11] et al. proposed the HOSDT theory based on the assumption of nonlinear stress change in thickness, ignoring the shear correction factor to make the transverse shear strain more accurate and stable. Compared with monolayer composite plates, periodic ribbed composite plates are widely used in engineering structures. Therefore, enough attention should be paid to the function of reinforcing tendon. Functionally graded material (FGM) is a type of heterogeneous composite material that exhibits a continuous variation of mechanical properties from one point to another [17–21]. This material is produced by mixing two or more materials in a certain volume ratio. Material properties of FGM vary along the material size depending on a function. In many applications, the sandwich plate is a laminated construction, consists of two or more thin face sheets connected by one or more thick core in order to achieve superior properties such as light weight, high strength for noise, vibration, thermal isolation and long fatigue life, wear resistance. Through a brief review of the literature, it is noted that a great deal of work has been done in the past to study the vibrational acoustic behavior of various sandwich plate structures, and the influence of structural geometric parameters on sound insulation performance has been described very completely. However, these studies mainly focus on the traditional engineering materials such as aluminum and steel. The application of new composite materials in sound insulation, especially the application of functional gradient materials, is relatively rare [22].

111.2 FE Simulation for the Stiffened Sandwich Plates The stiffened plates shown in Fig. 111.1 are composed of orthotropic plies with identical material properties and different plies orientations. As an expression of an incident plane sound wave at an Angle (θ , ϕ): pi (x, y, t) = Pi ei (ωt−kx x−ky y )

(111.1)

where, ω is the circular frequency of sound source simple harmonic vibration, Pi is the amplitude of incident plane sound wave, k x = k sin(θ ) cos(ϕ) is the wave number

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Fig. 111.1 The model sandwich structure with stiffened plate

Ly Lx

of incident plane sound wave in the x direction, k y = k sin(θ ) sin(ϕ) is the wave number of incident plane sound wave in the y direction, k is the wave number. Under the excitation of incident harmonic plane acoustic wave, the FE motion equation without considering the effect of fluid loading (structure in air) is v = [G][A] pi [Z ]

(111.2)

where, [Z] = (−ω 2[M] + iω[C] + [K])/iω is the structure impedance matrix, [M], [C] and [K] are structure mass matrix, proportional damping matrix and stiffness matrix, v is the structural velocity column vector, [G] is convert matrix, [A] is structure surface area matrix, pi is the incident sound pressure column vector of the node on the structure surface. For the FE discretization of stiffened plate, the plate element based on Mindlin plate bending theory and the eccentricity beam element after the rigid arm transformation considering the shear deformation (the influence of shear deformation is introduced on the basis of the classical beam element) are adopted. v of (2) can be directly solved by modal analysis method, and then the transformation relation between vn and v is obtained from the method of structure surface: − → vn = [G]T v

(111.3)

After the vn is obtained, the radiated sound pressure P(r, ω) generated by the plate structure on the infinite rigid baffle under the action of a simple harmonic excitation force at the center point P (position vector r) in the air medium in the semi-infinite region on one side of the plate can be obtained by the Rayleigh integral

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p(r, ω) =

e−ik R iωρ Svn dS 2π R

(111.4)

where, ρ is the air density, S is the surface area of the stiffened board structure, R = Q−P, Q is any point on the structure surface S. The radiated acoustic power of the structure can be obtained from the following equation: 

=A

| p(r, ω)|2 dA 2ρc

(111.5)

where, A is the hemispherical sphere connected with the plane of infinite rigid baffle in the air medium in the semi-infinite domain, and c is the sound velocity in the air. In the air, the incident sound power of an incident plane sound wave at an Angle (θ , ϕ) is  i

=

Pi2 abcos(θ ) 2ρc

(111.6)

where a and b are the length and width of the board respectively. Thus, the sound transmission loss of the stiffened plate can be defined as     /

TL = 10log10 i

(111.7)

So far, a calculation model of sound transmission in a semi-infinite domain of air medium on the other side of the plate surface has been established for the stiffened plate structure with respect to plane sound waves incident at an Angle (θ , ϕ) in the air.

111.3 Verification of Simulation In this section, the finite element model is experimentally verified. B&K equipment as shown in Fig. 111.2 is used for experiment, which are standing wave tube, amplifier and signal collector respectively. The device uses four microphones to collect signals, analyzes and compares the signals of the two-load modes, and calculates the sound transmission loss of the sample by four-microphone in standing wave tube. The standing-wave tube diameter is 100 mm, so the measured sound transmission loss range is 50–1600 Hz. The right end of the standing-wave tube is a fully absorption, with a total depth of about 75 mm, which is composed of three standard foam sound absorption specimens.

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Fig. 111.2 Test equipment, a standing wave tube b amplifier c signal collector

Figure 111.3 shows the four test samples A, B, and C, corresponding to the three types of panel. The samples were manufactured using a 3D printer with a density of 1003 kg/m3 , Young’s modulus of 1290 MPa and Poisson ratio of 0.37. The thickness and spacing of stiffeners of each sample are defined in Table 111.1. During the measurement, the samples were free in the tube. FE models of finite size identical to the tested samples are set up (see Fig. 111.4) by applying the FE method presented in the previous section. Free constrains and sound hard wall boundary conditions are applied to the boundaries of the solid panel and pressure acoustic field of the FE models respectively. The meshes of the calculated FE models are shown in Fig. 111.4 with the convergences checked by mesh refinement. Physical parameters of the air are shown in Table 111.2.

Fig. 111.3 Pictures of stiffened plate for impedance tube test, a specimen A b specimen B c specimen C

Table 111.1 Geometrical parameters of sandwich structure with stiffened plate for experiment Parameters

Value A (mm)

B (mm)

C (mm)

Thickness of stiffened plate/t1

1.5

1.5

3

Thickness of face plate/t2

2

2

2

Separation distance of stiffener/Lx = Ly

10

20

20

Thickness of core plate/H

20

20

20

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Fig. 111.4 Representative FE model for test

Table 111.2 Physical parameters of the air

Parameters

Value

Density/ρ0

1.21 kg/m3

Sound speed/c0

340 m/s

The pressure/p0

101 kPa

Temperature/T0

293 K

Figure 111.5 compares the measured STLs with those obtained from simulations. The simulation results have good agreement with the experimental data for all three specimens, demonstrating that the simulation model established in this paper can effectively characterize the acoustic characteristics of the stiffened plate. These differences between simulations and experiments at low frequencies are introduced by nonideal experimental conditions, including measuring errors by microphones, air leaks at the interface between sample edges and impedance tube, and so on. All of the first structural resonance frequencies for these specimens are higher than 1600 Hz which exceeds the tested frequency range, so the STL rises with frequency as shown in Fig. 111.5.

Fig. 111.5 Comparison between the STLs obtained by FE simulation and experimental measurement, a specimen A b specimen B c specimen C

111 Study on Sound Transmission Loss … Table 111.3 Main parameters of the model

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Parameters

Value

Thickness of face plate/t2

2 mm

Thickness of core plate/H

20 mm

Nylon, Young’s modulus

1.290 GPa

Nylon, density

1003 kg/m3

Nylon, Poisson ratio

0.37

111.4 Parameter Analysis 111.4.1 Simulation Parameters In this section, based on the above FE model, the sound transmission loss of stiffened plate structure is simulated. The boundary conditions, the spacing between bars, the gradient distribution index of FGM and the thickness of bars were analyzed. The main parameters are shown in Table 111.3.

111.4.2 Influence of Boundary Conditions This section discusses the influence of different boundary conditions on sound transmission loss in different materials. Material and model parameters are shown in Table 111.4. In Fig. 111.6, the sound transmission loss of the stiffened plate structure with aluminum and nylon materials are compared under the free boundary and the constraint boundary respectively. It is observed that no matter in which boundary conditions, the sound transmission loss of the aluminum stiffened plate structure is higher than that of the nylon material, and the first natural frequency is higher than that of the nylon material. By comparing the effects of different boundary conditions, it is observed that under free boundary conditions, the sound transmission loss increases monotonically within a certain range until the first natural frequency. On the other hand, under the constrained boundary condition, the frequency decreases monotonically until the first natural frequency. Table 111.4 Material and model parameters of the model

Parameters

Value

Thickness of stiffened plate/t1

1.5 mm

Separation distance of stiffener/Lx = Ly

20 mm

Aluminum, Young’s modulus

70 GPa

Aluminum, density

2700 kg/m3

Aluminum, Poisson ratio

0.33

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Fig. 111.6 Comparison of different boundary conditions on sound transmission loss in different materials, a free boundary b constraint boundary

First, because the density of aluminum is greater than that of nylon, according to the law of sound insulation mass, the increase of sound transmission loss by 6 dB for every doubling of mass is consistent with the result of sound transmission loss of the two materials under the free boundary. On the other hand, the young’s modulus of aluminum is 70 times that of nylon. Under the constraint condition, due to the small sample size and the large impact of stiffness, the sound transmission loss of aluminum stiffened plate structure is nearly 40 dB higher than that of nylon.

111.4.3 Influence of the Spacing Between the Stiffeners This section discusses the effect of the spacing of the ribs on the sound transmission loss. The boundary condition is free, the material is nylon, the young’s modulus is 1290 MPa, the density is 1003 kg/m3 , and the poisson ratio is 0.37. Spacing Lx = Ly = 10 mm, 20 mm, 30 mm. Figure 111.7 compares the sound transmission losses of three types of stiffenedplate spacing structures. It is observed that the sound transmission loss is the highest when the spacing of the ribs is 10 mm. On the one hand, the shortening of the spacing of the ribs leads to the increase of the quality of the core layer per unit area, so the sound transmission loss decreases with the increase of the spacing of the ribs. It can also be seen from Fig. 111.7 that the first order frequency decreases with the increase of the spacing between the ribs, which is caused by the decrease of the structural stiffness and the shift of the first order frequency to the low frequency region.

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Fig. 111.7 Comparison of the spacing of the ribs on sound transmission loss

111.4.4 Influence of Different Gradient Indexes on Sound Transmission Loss This section discusses the effect of different gradient exponents on sound transmission loss of FGM panels. The boundary conditions are free, and to make it easier to understand, considering the functionally gradient material limitations, these panels assume functionally gradient materials made of aluminum and alumina. Its properties are shown in the Table 111.5. The gradient exponents are p = 1, 2, 5, 10. In Fig. 111.8, it is observed that there is no significant difference in sound transmission loss under different gradient exponentials. Since alumina has higher stiffness and density than aluminum, its sound transmission loss decreases with the increase of gradient index. Table 111.5 Material and model parameters of the model

Parameters

Value

Thickness of stiffened plate/t1

1.5 mm

Separation distance of stiffener/Lx = Ly

20 mm

Aluminum, Young’s modulus

70 GPa

Aluminum, density

2700 kg/m3

Aluminum, Poisson ratio

0.33

Alumina, Young’s modulus

380 GPa

Alumina, density

3800 kg/m3

Alumina, Poisson ratio

0.33

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Fig. 111.8 Comparison of different gradient exponents on sound transmission loss

111.4.5 Influence of the Thickness of the Stiffeners This section discusses the effect of the thickness of the ribs on the sound transmission loss. The boundary condition is free, the material is nylon. Thickness of steel plate t2 = 1.5 m, 3 mm, 4.5 mm. Figure 111.9 compares the sound transmission losses of three structures with different thickness of ribs. It is observed that the sound transmission loss is the largest when the thickness of the reinforcement plate is 4.5 mm. However, when the thickness of the ribs increases from 1.5 to 3 mm, the acoustic transmission loss increase is obviously about 3 dB, while the acoustic transmission loss increase from 3 to 4.5 mm is less about 1.2 db. Fig. 111.9 Comparison of the thickness of the ribs on sound transmission loss

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111.5 Conclusions In this paper, the numerical study of sandwich structural plates with stiffeners is carried out from the perspective of sound transmission loss. Comsol was applied to establish the finite element model. The correctness of the calculated results is verified by comparison with the measured results, and good consistency is obtained. The comparison of the sound transmission loss of the stiffened plate structure with different boundary conditions and different materials shows that the density of the small size specimen has a great influence on the sound transmission loss under the free boundary condition, while the material stiffness has a great influence on the sound transmission loss under the constrained boundary condition. By analyzing the spacing and thickness of the ribs, it is found that the sound transmission loss of the stiffened plate structure increases with the decrease of the spacing and the increase of the thickness. The comparison of different material gradient indices for the faceplate of the stiffened plate structure with functionally gradient materials shows that the influence of gradient index on the sound transmission loss is mainly due to the influence of material proportion. Combined with the research results of this paper, it is helpful for the researchers to design a sandwich structure that can improve the sound transmission loss while ensuring the high bearing capacity of the structure. Acknowledgements This work presented here was supported by National Key R&D Program of China under the contract number 2017YFB1300600, and by the National Natural Science Foundation of China under the contract numbers 11772103 and 61304037.

References 1. Wen, Z.-H., Wang, D.-W., Ma, L.: Sound transmission loss of sandwich panel with closed octahedral core. J. Sandw. Struct. Mater. (2019) 2. Hengchun, W., Zhaoxiang, D., Weidong, S.: Sound transmission loss characteristics of unbounded orthotropic sandwich panels in bending vibration considering transverse shear deformation. Compos. Struct. 92(12), 2885–2889 (2010) 3. Meng, H., Galland, M.A., Ichchou, M., et al.: Small perforations in corrugated sandwich panel significantly enhance low frequency sound absorption and transmission loss. Compos. Struct. 182, 1–11 (2017) 4. Sadri, M., Younesian, D.: Vibroacoustic analysis of a sandwich panel coupled with an enclosure cavity. Compos. Struct. 146, 159–75 (2016) 5. Liu, Y., Daudin, C.: Analytical modelling of sound transmission through finite clamped doublewall sandwich panels lined with poroelastic materials. Compos. Struct. 172, 359–73 (2017) 6. Wang, D.-W., Ma, L.: Sound transmission through composite sandwich plate with pyramidal truss cores. Compos. Struct. 164, 104–17 (2017) 7. Fu, T., Chen, Z.: Sound transmission from stiffened double laminated composite plates. Wave Motion 27, 331–341 (2017) 8. Thai, H.-T., Choi, D.-H.: A simple first-order shear deformation theory for laminated composite plates. Compos. Struct. 106, 754–763 (2013)

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9. Mantari, J.L.: Free vibration of single and sandwich laminated composite plates by using a simplified FSDT. Compos. Struct. 132, 952–959 (2015) 10. Reddy, J.N.: A simple higher-order theory for laminated composite plates. J. Appl. Mech 51, 745–752 (1984) 11. Librescu, L.: On the theory of anisotropic elastic shells and plates. Internat. J. Solids Struct. 3(1), 53–68 (1967) 12. Chazot, J.D., Guyader, J.L.: Prediction of transmission loss of double panels with a patch mobility method. J. Acoust. Soc. Am. 121(1), 267–78 (2007) 13. Craik, R.J.M.: Non-resonant sound transmission through double walls using statistical energy analysis. Appl. Acoust. 64(3), 325–41 (2003) 14. Xin, F.X., Lu, T.J.: Analytical and experimental investigation on transmission loss of clamped double panels: implication of boundary effects. J. Acoust. Soc. Am. 125(3), 1506–17 (2009) 15. Sun, W., Liu, Y.: Vibration analysis of hard-coated composite beam considering the strain dependent characteristic of coating material. Acta Mech. Sin. 32(4), 731–742 (2016) 16. Wang, T., Li, S., Rajaram, S., Nutt, S.R.: Predicting the sound transmission loss of sandwich panels by statistical energy analysis approach. J. Vib. Acoust. 132(1), 1–7 (2010) 17. Reddy, J.N.: Analysis of functionally graded plates. Int. J. Numer. Methods Eng. 47, 663–684 (2000) 18. Cheng, Z.Q.: Exact correspondence between eigenvalues of membranes and functionally graded simply supported polygonal plate. J. Sound Vib. 299, 879–895 (2000) 19. Loy, C.T., Lam, K.Y.: Vibration of functionally graded cylindrical shells. Int. J. Mech. Sci. 41, 309–324 (1999) 20. Jin, C.H., Wang, X.W.: Accurate free vibration analysis of Euler functionally graded beams by the weak form quadrature element method. Compos. Struct. 125, 41–50 (2015) 21. Dorduncu, M., Apalak, M.K.: Elastic wave propagation in functionally graded circular cylinders. Compos. Part B 73, 35–48 (2015) 22. Fu, T., Chen, Z.: An analytical study of sound transmission through corrugated core FGM sandwich plates filled with porous material. Compos. Part B 151, 161–172 (2018)

Chapter 112

Supervised Learning for Finite Element Analysis of Holes Under Tensile Load Wai Tuck Chow

Abstract As the use of machine learning becomes more common, there are many algorithms that are readily available to perform supervised learning. This paper is to evaluate the feasibility of supervised learning in simplifying the finite element analysis of holes under tensile load. The objective of this approach is to determine if the mesh size can be significantly reduced with supervised learning. The neural network training is performed with just a small set of 55 course mesh with 2-D linear element against the analytical solution of a hole under tensile load in an infinite width plate. The coarse mesh only has 2 elements along the quarter hole perimeter. The training would be done using the displacement nodal solution of the nearest 6 nodes to the hole’s edge. Three common back propagation network algorithms are evaluated; Conjugate Gradient, Bayesian and Levenberg-Marquart methods. These algorithms are used along with the tangent sigmoid and pure linear transfer functions. In the infinite width problem, the Bayesian algorithm with the tangent sigmoid function offers the highest accuracy in the testing of the network model. However, this model performs poorly when it is applied to the finite width problem. To reduce the prediction error, the training would be done solely based on the displacement u y component. The displacement u x component is removed from the network training since the displacement field in the x direction is quite different between the infinite hole solution and finite hole solution. Further testing shows the prediction can be improved by using the Levenberg-Marquart method with pure linear function. With these options, the prediction error is just 3% even though the mesh size is relatively coarse with only 2 elements along the perimeter. In contrast, the normal finite element method with this coarse mesh has an error of 26%. To achieve similar accuracy, the standard FEM would require 3 times the number elements along the perimeter to achieve similar accuracy. This initial result shows there is synergy between machine learning and finite element method in reducing the mesh size requirement and yet achieve good accuracy.

W. T. Chow (B) School of Mechanical and Aerospace Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore e-mail: [email protected] © Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7_112

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Keywords Finite element method · Machine learning · Supervised learning · Neural network

112.1 Introduction The use of finite element modeling using 3D element has grown significantly over the years due to the increasing power of computer hardware and the affordability of CAD solid modeling and finite element software. However, to keep the computational time manageable and be able to model a larger segment of the component, the mesh of the 3D model is often coarse and the element of choice is usually linear elements. As a result, the accuracy of the model is often limited and sub-modelling is usually required to obtain sufficient stress convergence. Unfortunately, as sub-modelling is often performed manually (or semi-manual), the amount of effort to setup the submodel is often quite expensive. As a result, to perform 3D design iterations with geometry change or topology optimization are often challenging and requires significant time resources. The aim of this paper is to evaluate if machine learning with neural network can be used to significantly reduce the mesh size and yet be able to accurately calculate the stress level at stress concentration areas. There are some early works of using machine learning with finite element methods. Hashash et al. [1] use neural network based constitutive models to capture the nonlinear material behavior and implement it into the finite element method. Manevitz et al. [2] use neural network time series methodology to predict the area for finite element mesh refinement at the appropriate times. Since then, the use of machine learning has become more wide spread and more engineers are trained to use machine learning algorithm. Hence, it would be beneficial to evaluate if machine learning can be used to support finite element method to calculate stress accurately with a coarser mesh size.

112.2 Method 112.2.1 Stress and Displacement Around a Hole For a hole in an infinite width plate [3], the stress field in polar coordinate system can be defined as follows:     a2 4a 2 1 1 3a 4 (112.1) σrr = σ 1 − 2 + σ 1 − 2 + 4 cos2θ 2 r 2 r r     1 1 a2 3a 4 σθθ = σ 1 + 2 − σ 1 + 4 cos2θ (112.2) 2 r 2 r

112 Supervised Learning for Finite Element Analysis …

τr θ

  2a 2 1 3a 4 = − σ 1 + 2 − 4 sin2θ 2 r r

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(112.3)

where σ is the applied stress, a is the radius of the hole, r and θ are the polar coordinates of a point in the element. Assuming Hooke’s law, the strain can be defined as follow: Plane Stress εrr =

1 [σrr − νσθθ ] E

(112.4)

εθθ =

1 [σθθ − νσrr ] E

(112.5)

1+ν σr θ E

(112.6)

εrr =

1+ν [(1 − ν)σrr − νσθθ ] E

(112.7)

εθθ =

1+ν [(1 − ν)σθθ − νσrr ] E

(112.8)

1+ν σr θ E

(112.9)

εr θ = Plane Strain

εr θ =

where E is the elastic Young’s modulus and ν is the Poisson ratio. The strain displacement relationship can be expressed as: ∂u r ∂r

(112.10)

ur 1 ∂u θ + r ∂θ r

(112.11)

εrr = εθθ = εr θ

  ∂u θ uθ 1 1 ∂u r + − = 2 r ∂θ ∂r r

(112.12)

Integrating the strain, the displacement for a hole under applied stress, σ , is derived as follows: Plane Stress σr ((1 + ν)cos2θ + (1 − ν)) 2E   σ a 2  2 a (1 + ν) − 4r 2 cos2θ − (1 + ν)r 2 − 3 2r E

ur =

(112.13)

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uθ = −

    2 σ sin2θ  2 2 2 2 2 a + r + ν a − r 2r 3 E

(112.14)

112.2.2 Neural Networks One of the important aspect of machine learning is the ability of automatically detect the patterns and fit the observation to a certain minimization goal. The goal is to train the neural network model with a set of coarse mesh with known solution to predict the solution for another set of coarse mesh. The artificial neural networks consist of simple calculation elements, called neurons, and weighted connections between them called weights. Typically, the network contains three types of processing units (neurons), input units, output units and hidden units, organized in a hierarchy of layers: input layer, hidden layers and output layer. The data from input layer or hidden layers are multiplied by the weights associated with a next layer unit and summed together before processing by the unit. Most of the recent supervised learning is based on back-propagation training algorithm. Here, the training data first propagates forward through the network and the output data are calculated. Using a minimization procedure, the error between the expected output and the calculated output is used to adjust the weights between two connection layers starting backwards from the output layer to input layer. Some of the minimization procedures that are based on different optimization methods such as Conjugate Gradient, Bayesian and Levenberg-Marquart methods [3–7].

112.2.3 Supervised Training In this paper, Matlab’s library [8], fitnet, is used for the neural network supervised training. Three network model are evaluated; ‘trainscg’ (scaled conjugate gradient backpropagation), ‘trainbr’ (Bayesian regularization backpropagation) and ‘trainlm’ (Levenberg-Marquart backpropagation). The neural network training is performed with a set of course mesh with linear mesh element against the analytical solution of a hole under tensile load in an infinite width plate. A simple quarter model of the hole is shown in Fig. 112.1. The infinite width effect is approximated using half width = 20 × hole radius. Ansys 2D linear element, Plane 182 [9], is used. The mesh size is relatively coarse with only 2 elements along the perimeter of the quarter hole. The training is performed using the nodal solution at the 6 nodes near the hole’s edge as shown in Fig. 112.1. The material constant used is based on Aluminum, where E = 10 × 106 psi and ν = 0.3. The mesh is subjected to unit stress load, σ = 1 psi. The training is done based on a total of 55 mesh variation. The variation is listed in Table 112.1. The training is based on the displacement nodal solution for the 6 nodes nearest to the hole’s edge as shown in Fig. 112.1. For the supervised training, both the input

112 Supervised Learning for Finite Element Analysis …

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Fig. 112.1 Coarse mesh of a quarter hole with 2 elements along the perimeter

Table 112.1 Variation in the coarse mesh to train the network model

Min

Max

Interval

b/a

1.5

3

0.15

Elementsize/a

0.6

1.4

0.2

and target data is normalized to a magnitude near 1. As a result, the displacement on the 6 nodes is normalized with the u y : y displacement component of node 3. (i) (3) u (i) j = u j /u y

where i is node = 1 to 6 and j = x or y.  ∗ , maximum stress at the hole r = a, θ = The target, σθθ follows: σθθ =

∗ u r∗ σθθ u r∗ = 3σ u (3) u (3) y y

(112.15) π 2



is normalized as

(112.16)

where u r∗ is the analytical radial displacement at r = a, θ = 0 given in Eq. 112.13 under unit stress, σ = 1. In order to train the data to be sensitive to the magnitude change in displacement, the training is done using a simple linear scale of 70 and 150%.  0.7 u (ι) j u train = (112.17) 1.5 u (ι) j

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112.3 Result 112.3.1 Infinite Hole: Neural Network Option The displacement solution data from the 55 mesh variation is randomly split with a ratio of 70% for training the neural network, 15% for validation of the neural network model and the remaining 15% for testing the accuracy of the model. Given the sample size of 110 (55 × 2) is relatively small, a single layer with 10 hidden neurons are used. Three common training algorithms are evaluated; Conjugate Gradient, Bayesian and Levenberg-Marquart methods. Two neural transfer functions are evaluated: tangent sigmoid and pure linear. The accuracy of the test result on the various neural training model for both the non-zero x and y displacement components for the 6 nodes (Fig. 112.1) is shown in Table 112.2. The result shows the ‘tangent sigmoid’ transfer function performs better than the ‘pure linear’ option. Using sigmoid transfer function, the Bayesian algorithm provides better accuracy. This is expected as Bayesian algorithm are better suited for small training data sets. However, given the variation of the displacement due to meshing is significantly more than 0.0012%, this is probably a case of ‘over-fitting’. The poor performance of the Bayesian algorithm with ‘pure linear’ function also indicates that the algorithm may not perform as well if the mesh variation is quite different from the training sets. From that perspective, Table 112.2 shows the Levenberg-Marquart method may perform better for diverse variation of mesh. Since the loading is in the y direction, it is expected that the y component of the displacement, u y , may play a bigger role in predicting stress concentration of the hole than u x . The result for training using just u y is shown in Table 112.3. The result shows the prediction error is slightly higher than using both the x and y components of the displacement. Table 112.2 Root Mean Squared Error (RMSE) of the test sample using different neural models and transfer function. Training done using both x and y components of the displacement; u x and u y

Table 112.3 Root Mean Squared Error (RMSE) of the test sample using different neural models and transfer function. Training done using only y component of the displacement; u y

Model

RMSE (%)*

Conjugate gradient

4.53

Tangent sigmoid Bayesian

0.0012

Levenberg-Marquart

0.15

Model

RMSE (%)*

Pure linear 5.22 12.11 0.22

Tangent sigmoid

Pure linear

Conjugate gradient

4.78

2.95

Bayesian

0.18

8.88

Levenberg-Marquart

0.19

0.39

*Based on maximum error with 20 trials

112 Supervised Learning for Finite Element Analysis … Table 112.4 Root Mean Squared Error (RMSE) of the test sample using different neural models and transfer function. Training done using only y component of the displacement of Node 2 and Node 5

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Model

RMSE (%)*

Conjugate gradient

4.76

0.93

Bayesian

0.23

15.15

Levenberg-Marquart

0.27

0.92

Tangent sigmoid

Pure linear

*Based on maximum error with 20 trials

Finally, this research also looks at the use of displacement on the element attached to the maximum stress location at Node 1. The non-zero u y on this element is just Node 2 and Node 5. The training result for this option is shown in Table 112.4. Again, the result shows the prediction error is slightly higher than Table 112.2.

112.3.2 Finite Hole To assess if supervised learning can be used to support finite element method to calculate stress accurately with a coarser mesh size, it is necessary to apply the neural network model on a separate set of problem. The neural network models trained using the mesh result from the hole in an infinite width plate is applied onto a finite width problem. Four different configurations are tested with w/a (width to hole radius) of 3, 4, 5 and 6. A coarse mesh with linear element is used with only 2 elements along the perimeter as shown in Fig. 112.2a. The result is compared against the Roark’s solution [10] given as follow: σmax = K t σnom

(112.18)

where P 2t(w − r )  r 2 r   r 3 + 3.66 K t = 3.00 − 3.13 − 1.53 w w w σnom =

(112.19) (112.20)

The result in Table 112.5 shows the prediction errors using the neural network model based on both u x and u y are high. This neural model trained using the infinite hole mesh cannot be applied to the finite hole mesh. The training based on the ‘tangent sigmoid’ transfer function, even though performs quite well on its own testing on the infinite hole mesh in Table 112.2, have much higher prediction error (Table 112.5) than the ‘pure linear’ function. The ‘linear’ function performs better could be attributed to the fact that only a small 55 sample mesh is used to train the model while many of the neural network models based on sigmoid function typically used for speech or image recognition are based on more than 10,000 samples. The

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a. Coarse mesh with 2 elements along perimeter

b. Fine mesh with 6 elements along perimeter

Fig. 112.2 Coarse and fine mesh of a quarter hole with finite width of w/a = 2

Table 112.5 Prediction error (%) using different neural models and transfer function. Training done using both x and y components of the displacement; u x and u y

Model

Absolute error (%)* Tangent sigmoid

Pure linear

Conjugate gradient

53.36

53.12

Bayesian

36.62

15.11

117.87

17.90

Levenberg-Marquart

*Based on maximum error with 20 trials

sample size is kept small as the intend of this paper is to reduce the overall computational cost by using a small set of coarse mesh to train the neural model rather than perform a fine mesh analysis. To reduce the prediction error, it may be better to remove the u x component from the neural training since the displacement field in the x direction is quite different between the infinite hole solution and finite hole solution. Table 112.6 shows that trained model using u y component only performs well with the ‘pure linear’ transfer function. The error using different algorithm model is about 3%, which is adequate for most practical engineering analysis. The Levenberg-Marquart algorithm with ‘pure

112 Supervised Learning for Finite Element Analysis …

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Table 112.6 Prediction error (%) using different neural models and transfer function. Training done using only y components of the displacement; u y Model

Absolute error (%)* Tangent sigmoid

Conjugate gradient

Pure linear

47.31

3.23

Bayesian

102.47

3.21

Levenberg-Marquart

127.60

3.18

*Based on maximum error with 20 trials

Table 112.7 Prediction error (%) using different neural models and transfer function. Training done using only y components of the displacement of Node 2 and Node 5 Model

Absolute error (%)* Tangent sigmoid

Conjugate gradient Bayesian Levenberg-Marquart

Pure linear

83.20

3.22

287.06

5.29

85.23

3.21

*Based on maximum error with 20 trials

Table 112.8 Sensitivity study on the mesh size where n is the number of elements on the hole perimeter n

w/a = 3 Kt

Roark

w/a = 4 Error (%)

3.460

Kt

w/a = 5 Error (%)

3.230

Kt

w/a = 6 Error (%)

3.135

Supervised learning*

2

3.348

−3.2

3.153

−2.4

FEM

2

2.552

−26.2

2.420

FEM

4

3.281

−5.2

3.072

FEM

6

3.494

1.0

3.263

Kt

Error (%)

3.088

3.148

0.4

3.121

1.1

−25.1

2.404

−4.9

2.984

−23.3

2.377

−23.0

−4.8

2.940

−4.8

1.0

3.167

1.0

3.118

1.0

*Based on Levenberg-Marquart algorithm, pure linear function and u y

linear’ function provide an overall better model with low error in both Tables 112.2 and Table 112.6. To evaluate if the trained model can be further improved by focusing the training on just the element with the highest stress, the training is based on u y of Node 2 and Node 5. Table 112.7 shows the prediction error with the ‘pure linear’ function is comparable with Table 112.6. Table 112.8 shows that for the same course mesh (with just 2 elements along the hole perimeter), using the standard finite element approach, the stress concentration, K t , obtained has an error of 26.2%. However, with neural network model, the error is reduced to just 3.2%. To obtain similar accuracy without the supervised learning, a much finer mesh is required. A fine mesh with 6 elements along the perimeter, as

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shown in Fig. 112.2b, would be necessary. Table 112.8 also shows that with higher w/a, the prediction error is much lower given the similarity of this problem with an infinite width problem that the neural model is being trained.

112.4 Conclusion In this paper, the neural network approach to reduce the finite element mesh size required to analyze the stress concentration of a hole under tensile load is presented. Supervised learning is based on a small set of 55 variation of coarse mesh using 2-D linear element of a hole in an infinite width plate. The coarse finite element displacement solution on the 6 nodes nearest to the hole’s edge is set as the network input while the analytical solution of a hole is set as the output. The training is performed with a single layer of 10 neurons. Using the test data from the infinite hole solution, the Bayesian method with ‘tangent sigmoid’ transfer function offers the highest accuracy with root mean squared error (RMSE) of 0.0012%. Unfortunately, when this trained model is used to predict the stress concentration of a hole in a finite width plate, the prediction error is high with absolute error of 36.6%. This is could be due to the fact that the initial training is based on both displacement components; u x and u y . Since the stress concentration of a hole is more dependent on the u y component than the u x component, the neural model can be improved by removing the u x displacement data from the training. With the removal of the u x component from the training set, the result significantly improved. Even with only 2 elements along the quarter hole perimeter, the prediction error is about 3%, well within the requirement for typical engineering analysis. To obtain similar accuracy with a fine finite element mesh, 6 elements would be necessary for the quarter hole perimeter. Hence, it is demonstrated that supervised learning can be used to significantly reduce the mesh size requirement of finite element method. This approach would provide large computational savings if it can be applied to 3-D finite model.

References 1. Hashash, Y.M.A., Jung, S., Ghaboussi, J.: Numerical implementation of a neural network based material model in finite element analysis. Int. J. Numer. Meth. Eng. 59(7), 989–1005 (2004) 2. Manevitz, L., Bitar, A., Givoli, D.: Neural network time series forecasting of finite-element mesh adaptation. Neurocomputing 63, 447–463 (2005) 3. Pilkey, W.D., Pilkey, D.F.: Peterson’s stress concentration factors. John Wiley (2008) 4. Bishop, M.C.: Neural networks for pattern recognition. Oxford University Press, Oxford (1995) 5. Haykin, S.S.: Neural networks: a comprehensive foundation. Prentice Hall (1999) 6. Levenberg, K.: A method for the solution of certain problems in least squares. Quart. Appl. Math. 2, 164–168 (1944) 7. Marquardt, D.: An algorithm for least-squares estimation of nonlinear parameters, SIAM. J. Appl. Math. 11, 431–441 (1963)

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8. Matlab: Statistics and Machine Learning Toolbox: User’s Guide. MathWorks (2018) 9. Ansys: ANSYS Mechanical APDL Theory Reference. Ansys Inc (2018) 10. Young, W.C., Budynas, R.G., Sadegh, A.M.: Roark’s Formulas for Stress and Strain. McGrawHill (2012)

Index

B Biodiesel fuel, 237–241 Biomedical applications, 1082

E Elastic-Plastic, 151, 155, 381, 382, 389, 390, 393, 394, 491, 585, 586 Electrocoagulation, 175–180, 183, 185

C Carbon fiber reinforced plastic, 27, 28, 31–34 Composites, 27, 58, 63, 587, 612, 621, 622, 679–683, 685, 687, 690, 691, 737, 738, 740, 741, 744, 763–765, 767, 771, 772, 774, 775, 778, 803, 804, 811, 868, 1025, 1038, 1103, 1208, 1318 Computer simulation, 223, 296–298, 414, 418, 824, 827, 828

F Flow field, 190, 233, 235, 249, 252, 253, 461, 465–468, 470, 472, 679, 885, 888, 916, 917, 920, 924, 925, 1171, 1173, 1175, 1176, 1178, 1180, 1183, 1184, 1254, 1256 Fluid simulation, 207, 208, 221, 989 Fracture, 83, 149, 151, 152, 154–159, 162, 491–500, 502–508, 657, 659, 662, 804, 808, 811, 1028, 1113, 1136, 1137, 1147–1154, 1190–1192, 1199, 1200 Fracture mechanics, 492, 497 Functional materials, 842

D Disaster prevention, 709 Dynamics, 11, 18, 22–24, 70, 77, 107, 108, 110, 117, 119, 120, 123, 135–138, 146, 147, 161, 162, 164, 165, 171, 201, 203, 223, 249, 250, 252, 253, 259, 260, 264, 279, 390, 391, 450, 472, 537–539, 545, 568, 577, 580, 621, 622, 657, 659, 661, 662, 665, 669, 719, 764, 772, 773, 819, 867, 868, 879, 880, 884–886, 888, 916, 917, 970, 973, 976, 981, 1009, 1125, 1126, 1171–1173, 1175, 1177–1179, 1182, 1183, 1190, 1191, 1195, 1199, 1221, 1253–1256, 1273–1277, 1291, 1301

G Geo-technical engineering, 154, 1190, 1191 H Heat transfer, 99–104, 189, 190, 201–204, 297, 302, 310, 313, 513, 514, 519, 647–650, 653, 820, 831, 834, 927, 929–931, 973, 1015, 1016, 1217, 1218 I Infinite element method, 259–161, 263, 269, 270, 272, 273, 275, 277, 279

© Springer Nature Switzerland AG 2020 H. Okada and S. N. Atluri (eds.), Computational and Experimental Simulations in Engineering, Mechanisms and Machine Science 75, https://doi.org/10.1007/978-3-030-27053-7

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1342

Index M Meshless methods, 260, 422, 423, 427, 816, 1292 Metamaterials, 1036 Modeling, 36, 75, 119, 229, 250, 283–286, 289, 291, 296–299, 319–322, 324–327, 331, 332, 422, 443, 466, 609, 616, 629, 634, 642, 679, 698, 752, 758, 764, 803, 804, 816, 868, 869, 886, 970, 971, 973, 1005, 1041, 1048, 1134–1136, 1138, 1141, 1142, 1148, 1157, 1163, 1173, 1217, 1218, 1222, 1330 Multi-scale & multi-physics, 764, 816, 1174

N Numerical approach, 302, 310, 421, 738, 853, 855–857 Numerical method, 108, 260, 439, 449, 450, 457, 466, 512, 737, 738, 740, 803, 816, 818, 917, 921

O Offshore, 201, 285, 512, 518, 585, 586, 666, 679, 867, 875, 876, 935

P Particle method, 207, 208, 213, 223, 301, 302 Polymeric materials, 11

S Simulation, 3–6, 8, 100, 101, 107, 114, 119–122, 136, 169, 208, 211, 213, 215, 219, 220, 223–226, 230,

249–255, 284, 295–299, 301, 302, 307, 309–311, 320, 322, 326, 330, 375, 376, 382–384, 413, 418, 422, 444, 445, 449, 461, 463, 465–467, 469, 470, 472, 476, 511, 512, 516–518, 522, 548, 551, 586, 590, 591, 612, 613, 616, 657, 663, 666, 679, 686, 691, 709–714, 718, 763, 767, 777, 778, 803, 804, 808–810, 815, 816, 818–821, 824, 827–834, 842, 861, 864, 879–881, 884–888, 915, 916, 933–935, 940, 942–945, 951, 955, 956, 967, 970, 971, 975, 978, 981, 988–990, 999, 1002, 1005, 1006, 1025, 1029, 1030, 1037–1039, 1042, 1047, 1053, 1057, 1081–1085, 1087, 1088, 1102, 1104, 1105, 1108–1112, 1114–1116, 1126, 1154, 1157, 1160, 1163, 1171, 1174, 1175, 1179, 1182, 1189–1191, 1195–1197, 1201–1204, 1208, 1217, 1219, 1220, 1222, 1253, 1255, 1256, 1258, 1267, 1274, 1275, 1305, 1308, 1309, 1315, 1317, 1318, 1320, 1322, 1323 Soft computing, 400 Soil, 127, 128, 133, 135–147, 150, 151, 153–156, 159, 869 Steel structure, 35, 491, 492, 495, 507, 508

T Thermal efficiency, 238, 439, 440, 443–446 Topology optimization, 1330

W Welding, 491, 492, 495–498, 984