Isogeometric Topology Optimization: Methods, Applications and Implementations (Engineering Applications of Computational Methods, 7) [1st ed. 2022] 9811917698, 9789811917691

Table of contents :
Preface
Contents
Acronyms
1 Introduction
1.1 Topology Optimization (Top-Opt)
1.2 Isogeometric Analysis (IGA)
1.3 Isogeometric Topology Optimization (ITO)
1.3.1 Density-Based ITO Methods
1.3.2 Boundary-Based ITO Methods
1.4 Applications of Topology Optimization
1.4.1 Multi-material Structures
1.4.2 Stress-Related Problems
1.4.3 Piezoelectric Structures
1.4.4 Architected Materials
1.4.5 Auxetic Meta-Materials/composites
1.5 Implementations of Topology Optimization
1.6 Main Focus of the Current Monograph
2 Density-Based ITO Method
2.1 NURBS-Based IGA
2.1.1 NURBS Basis Functions
2.1.2 Galerkin’s Formulation for Elastostatics
2.2 Density Distribution Function (DDF) for Material Description Model
2.2.1 NURBS for Structural Geometry
2.2.2 Density Distribution Function (DDF)
2.2.3 Material Interpolation Model
2.3 ITO Formulation for Stiffness-Maximization
2.4 Numerical Implementations
2.5 Numerical Examples
2.5.1 Several Numerical Examples in 2D
2.5.2 Several Numerical Examples in 3D
2.6 Discussions on the Indispensability of ITO
2.6.1 Extension of the DDF
2.6.2 Comparisons Between ITO and FEM-Based Three-Field SIMP
2.6.3 Numerical Examples
2.7 Appendix for Sensitivity Analysis
2.8 Summary
3 Density-Based Multi-material ITO (M-ITO) Method
3.1 NURBS-Based Multi-material Interpolation (N-MMI)
3.1.1 Field of Design Variables (DVF)
3.1.2 Field of Topology Variables (TVF)
3.1.3 Multi-material Interpolation Model
3.2 Multi-material ITO (M-ITO)
3.3 Design Sensitivity Analysis
3.4 Numerical Examples in 2D
3.4.1 Two-Material Design
3.4.2 Three-Material Design
3.4.3 Discussions on the Stiffness-To-Mass Ratio
3.4.4 Quarter Annulus
3.5 Numerical Examples in 3D
3.6 Summary
4 ITO for Structures with Stress-Minimization
4.1 Topology Description Model
4.2 NURBS-Based IGA for Stress Computation
4.3 Induced Aggregation Formulations of p-Norm and KS
4.4 ITO for Stress-Minimization Designs
4.4.1 Stress-Minimization Design Formulation
4.4.2 Design Sensitivity Analysis
4.4.3 Numerical Implementations
4.5 Numerical Examples
4.5.1 Discussions on Aggregation Formulations of p-Norm and the Induced p-Norm
4.5.2 Discussions on Aggregation Formulations of KS and the Induced KS
4.5.3 An Inverse L-Type Structure with the Curved Design Domain
4.5.4 A MBB Beam with One Preexisting Crack Notch
4.5.5 A Half Annulus with a Square Hole
4.6 Summary
5 ITO for Piezoelectric Structures with Manufacturability
5.1 NURBS-Based IGA for Piezoelectric Materials
5.1.1 Piezoelectric Constitutive Relations
5.1.2 IGA Formulation for Piezoelectric Materials
5.2 DDF with the Erode–Dilate Operator
5.3 Piezoelectric Materials Interpolation Schemes
5.4 ITO and RITO for Piezoelectric Actuators
5.4.1 ITO Formulation Without Uniform Manufacturability
5.4.2 RITO Formulation with Uniform Manufacturability
5.5 Design Sensitivity Analysis
5.6 Numerical Examples
5.6.1 Design of Piezoelectric Actuators Using ITO
5.6.2 Design of Piezoelectric Actuators Using RITO
5.7 Summary
6 ITO for Architected Materials
6.1 Numerical Implementations of the Homogenization Using IGA
6.2 ITO for Micro-Architected Materials
6.3 Design Sensitivity Analysis
6.4 Optimality Criteria (OC)
6.5 Numerical Examples
6.5.1 2D Micro-Architected Materials
6.5.2 3D Micro-Architected Materials
6.6 Summary
7 ITO for Auxetic Metamaterials
7.1 ITO Formulation for Auxetic Metamaterials
7.2 Design Sensitivity Analysis
7.3 A Relaxed OC Method
7.4 2D Auxetic Metamaterials
7.5 Discussions of the Weight Parameter
7.6 3D Auxetic Metamaterials
7.7 Summary
8 M-ITO for Auxetic Meta-Composites
8.1 Computational Design Framework
8.2 M-ITO Formulation for Auxetic Meta-Composites
8.3 Design Sensitivity Analysis
8.4 Numerical Implementations
8.5 Numerical Examples
8.5.1 2D Auxetic Meta-Composites with Two Materials
8.5.2 2D Auxetic Meta-Composites with Three Materials
8.5.3 3D Auxetic Meta-Composites with Two Materials
8.6 Summary
9 An In-House MATLAB Code of “IgaTop” for ITO
9.1 GeomMod: Construct Geometrical Model Using NURBS
9.2 PreIGA: Preparation for IGA
9.3 BounCond: Define Dirichlet and Neumann Boundary Conditions
9.4 Initializing Control Densities and DDF at Gauss Quadrature Points
9.5 ShepFun: Define the Smoothing Mechanism
9.6 IGA to Solve Structural Responses
9.6.1 StiffEle2D
9.6.2 StiffAss2D
9.6.3 Solving
9.7 Objective Function and Sensitivity Analysis
9.8 OC: Update Design Variables and DDF
9.9 PlotData and PlotTopy: Representation of Numerical Results
9.10 Demos for Several Examples
9.11 Summary
References

Citation preview

Engineering Applications of Computational Methods 7

Jie Gao Liang Gao Mi Xiao

Isogeometric Topology Optimization Methods, Applications and Implementations

Engineering Applications of Computational Methods Volume 7

Series Editors Liang Gao, State Key Laboratory of Digital Manufacturing Equipment and Technology, Huazhong University of Science and Technology, Wuhan, Hubei, China Akhil Garg, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, Hubei, China

The book series Engineering Applications of Computational Methods addresses the numerous applications of mathematical theory and latest computational or numerical methods in various fields of engineering. It emphasizes the practical application of these methods, with possible aspects in programming. New and developing computational methods using big data, machine learning and AI are discussed in this book series, and could be applied to engineering fields, such as manufacturing, industrial engineering, control engineering, civil engineering, energy engineering and material engineering. The book series Engineering Applications of Computational Methods aims to introduce important computational methods adopted in different engineering projects to researchers and engineers. The individual book volumes in the series are thematic. The goal of each volume is to give readers a comprehensive overview of how the computational methods in a certain engineering area can be used. As a collection, the series provides valuable resources to a wide audience in academia, the engineering research community, industry and anyone else who are looking to expand their knowledge of computational methods. This book series is indexed in SCOPUS database.

More information about this series at https://link.springer.com/bookseries/16380

Jie Gao · Liang Gao · Mi Xiao

Isogeometric Topology Optimization Methods, Applications and Implementations

Jie Gao Department of Engineering Mechanics School of Aerospace Engineering Huazhong University of Science and Technology Wuhan, Hubei, China

Liang Gao State Key Laboratory of Digital Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan, Hubei, China

Mi Xiao State Key Laboratory of Digital Manufacturing Equipment and Technology Huazhong University of Science and Technology Wuhan, Hubei, China

ISSN 2662-3366 ISSN 2662-3374 (electronic) Engineering Applications of Computational Methods ISBN 978-981-19-1769-1 ISBN 978-981-19-1770-7 (eBook) https://doi.org/10.1007/978-981-19-1770-7 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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 Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Recently, topology optimization has been regarded as a powerful tool to seek for the optimized distribution of materials in a design domain subject to several prescribed constraints, and the concerned structural performance can reach the expected value. In the topology optimization, the classic finite element method is broadly adopted to solve the unknown structural responses, and then which are employed to compute the objective function in the optimization. Hence, the finite element method extensively affects the effectiveness of topology optimization on several design problems. However, as we know, several deficiencies of the finite element method might damage the capabilities of topology optimization, as follows: (1) The finite element mesh cannot exactly capture the structural geometry, which strongly lower the numerical precision; (2) the lower-order (C0) continuity of structural responses between the neighboring finite elements, also in the higher-order elements, extensively hurt the effectiveness of the optimization; and (3) the lower efficiency in achieving a high quality of the finite element mesh. Isogeometric analysis (IGA) employs same basis functions in structural geometry and numerical analysis, which can effectively guarantee the consistence between the models of computer-aided design (CAD) and computer-aided engineering (CAE). Hence, a new framework of topology optimization using IGA, namely the isogeometric topology optimization (ITO), gradually attracts interests among many researchers in recent years. The current monograph attempts to provide a systematic and complete presentation of the ITO method with the density and its applications in many problems, consisting of the multi-material topology optimization, stress-minimization problems, the design of piezoelectric structures, the design of architected materials, the rational design of auxetic metamaterials/composites and a complete implementation of the ITO. It mainly contains nine chapters. In Chap. 1, a detailed introduction for topology optimization methods and implementations, IGA, ITO methods and applications is given. Chapter 2 presents the development of the ITO method with the detailed construction of the DDF to show structural topology within the optimization, and the NURBS-based topology description model and numerical analysis model are developed. Chapter 3 gives the development of a multi-material isogeometric topology optimization (M-ITO) method, which contains the construction of v

vi

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the N-MMI model to present the layout of multiple materials. In Chap. 4, the ITO method on several optimization problems of stress-minimization design is studied in detail, and the induced aggregation formulations to replace conventional p-norm and KS are employed to remove the influence of mesh dependency on the optimization. Chapter 5 presents the effectiveness of the ITO on the design of piezoelectric structures and also offers the development of the robust ITO (RITO) method for piezoelectric actuators to eliminate the one-node-connected pins and joints and maintain the flexible deformations to transmit the micrometer or nanometer displacements. Then, Chap. 6 develops the ITO formulation for the rational design of ultra-lightweight architected materials with the higher stiffness-mass ratio, which gives a series of new and interesting 2D and 3D microstructures. In Chaps. 7 and 8, the computational design of auxetic metamaterials/composites by the ITO method and the MITO method, respectively, is given in detail and also seeks for a family of novel and interesting auxetic microstructures in 2D and 3D with one, two and three materials. Finally, Chap. 9 presents a detailed implementation of the ITO in the platform of MATLAB and gives an in-house MATLAB code of the “IgaTop” for the ITO method. We wish to thank the support from the National Natural Science Foundation of China (Grant No. 52105255), the National Key R&D Program of China (Grant No. 2020YFB1708300), the Tencent Foundation or XPLORER PRIZE, the KTP program of HUST (Huazhong University of Science and Technology), UTS (University of Technology Sydney) and the Fundamental Research Funds for the Central Universities of HUST (Grant No. 5003123021). The first author would like to express his sincere thanks to Prof. Peigen Li, A/Prof. Jinhui Yan, A/Prof. Wentao Yan, A/Prof. Hao Li, A/Prof. Zhen Luo, Dr. Yan Zhang, Dr. Sheng Chu, Dr. Jie Xu, A/Prof. Liang Xia, A/Prof Yiqiang Wang, A/Prof Qinghai Zhao, Dr. Zhenpei Wang and Dr. Jing Zheng for their valuable suggestions in the field of topology optimization. As aforementioned, there are tremendous high-quality papers in the research fields of topology optimization and isogeometric topology optimization. Part of them has been cited at the end of the monograph. The readers can benefit from referring to the articles and references therein. The authors of these articles are also greatly appreciated. Wuhan, China February 2022

Jie Gao Liang Gao Mi Xiao

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Topology Optimization (Top-Opt) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Isogeometric Analysis (IGA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Isogeometric Topology Optimization (ITO) . . . . . . . . . . . . . . . . . . . 1.3.1 Density-Based ITO Methods . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Boundary-Based ITO Methods . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Applications of Topology Optimization . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Multi-material Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Stress-Related Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Piezoelectric Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 Architected Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Auxetic Meta-Materials/composites . . . . . . . . . . . . . . . . . . . 1.5 Implementations of Topology Optimization . . . . . . . . . . . . . . . . . . . 1.6 Main Focus of the Current Monograph . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4 5 7 9 9 9 11 12 12 13 14

2 Density-Based ITO Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 NURBS-Based IGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 NURBS Basis Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Galerkin’s Formulation for Elastostatics . . . . . . . . . . . . . . . . 2.2 Density Distribution Function (DDF) for Material Description Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 NURBS for Structural Geometry . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Density Distribution Function (DDF) . . . . . . . . . . . . . . . . . . 2.2.3 Material Interpolation Model . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 ITO Formulation for Stiffness-Maximization . . . . . . . . . . . . . . . . . . 2.4 Numerical Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Several Numerical Examples in 2D . . . . . . . . . . . . . . . . . . . . 2.5.2 Several Numerical Examples in 3D . . . . . . . . . . . . . . . . . . . . 2.6 Discussions on the Indispensability of ITO . . . . . . . . . . . . . . . . . . . . 2.6.1 Extension of the DDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 19 21 21 22 25 26 27 29 29 35 37 37

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2.6.2 Comparisons Between ITO and FEM-Based Three-Field SIMP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix for Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

38 40 47 49

3 Density-Based Multi-material ITO (M-ITO) Method . . . . . . . . . . . . . . 3.1 NURBS-Based Multi-material Interpolation (N-MMI) . . . . . . . . . . 3.1.1 Field of Design Variables (DVF) . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Field of Topology Variables (TVF) . . . . . . . . . . . . . . . . . . . . 3.1.3 Multi-material Interpolation Model . . . . . . . . . . . . . . . . . . . . 3.2 Multi-material ITO (M-ITO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Design Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Numerical Examples in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Two-Material Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Three-Material Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Discussions on the Stiffness-To-Mass Ratio . . . . . . . . . . . . . 3.4.4 Quarter Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Numerical Examples in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 51 51 52 53 54 56 58 60 63 66 67 68 70

2.7 2.8

4 ITO for Structures with Stress-Minimization . . . . . . . . . . . . . . . . . . . . . 71 4.1 Topology Description Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 NURBS-Based IGA for Stress Computation . . . . . . . . . . . . . . . . . . . 74 4.3 Induced Aggregation Formulations of p-Norm and KS . . . . . . . . . . 75 4.4 ITO for Stress-Minimization Designs . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.4.1 Stress-Minimization Design Formulation . . . . . . . . . . . . . . . 78 4.4.2 Design Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4.3 Numerical Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.5.1 Discussions on Aggregation Formulations of p-Norm and the Induced p-Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5.2 Discussions on Aggregation Formulations of KS and the Induced KS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.5.3 An Inverse L-Type Structure with the Curved Design Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5.4 A MBB Beam with One Preexisting Crack Notch . . . . . . . . 96 4.5.5 A Half Annulus with a Square Hole . . . . . . . . . . . . . . . . . . . 98 4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 5 ITO for Piezoelectric Structures with Manufacturability . . . . . . . . . . . 5.1 NURBS-Based IGA for Piezoelectric Materials . . . . . . . . . . . . . . . . 5.1.1 Piezoelectric Constitutive Relations . . . . . . . . . . . . . . . . . . . . 5.1.2 IGA Formulation for Piezoelectric Materials . . . . . . . . . . . . 5.2 DDF with the Erode–Dilate Operator . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Piezoelectric Materials Interpolation Schemes . . . . . . . . . . . . . . . . .

101 101 101 102 103 105

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5.4

ix

ITO and RITO for Piezoelectric Actuators . . . . . . . . . . . . . . . . . . . . 5.4.1 ITO Formulation Without Uniform Manufacturability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.2 RITO Formulation with Uniform Manufacturability . . . . . . Design Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Design of Piezoelectric Actuators Using ITO . . . . . . . . . . . 5.6.2 Design of Piezoelectric Actuators Using RITO . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

106

6 ITO for Architected Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Numerical Implementations of the Homogenization Using IGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 ITO for Micro-Architected Materials . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Design Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Optimality Criteria (OC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 2D Micro-Architected Materials . . . . . . . . . . . . . . . . . . . . . . 6.5.2 3D Micro-Architected Materials . . . . . . . . . . . . . . . . . . . . . . 6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

121 121 124 126 127 127 128 136 143

7 ITO for Auxetic Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 ITO Formulation for Auxetic Metamaterials . . . . . . . . . . . . . . . . . . . 7.2 Design Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 A Relaxed OC Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 2D Auxetic Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Discussions of the Weight Parameter . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 3D Auxetic Metamaterials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

145 145 147 148 149 153 155 162

8 M-ITO for Auxetic Meta-Composites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Computational Design Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 M-ITO Formulation for Auxetic Meta-Composites . . . . . . . . . . . . . 8.3 Design Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Numerical Implementations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 2D Auxetic Meta-Composites with Two Materials . . . . . . . 8.5.2 2D Auxetic Meta-Composites with Three Materials . . . . . . 8.5.3 3D Auxetic Meta-Composites with Two Materials . . . . . . . 8.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

165 165 166 167 169 170 170 173 178 185

9 An In-House MATLAB Code of “IgaTop” for ITO . . . . . . . . . . . . . . . . 9.1 Geom_Mod: Construct Geometrical Model Using NURBS . . . . . . 9.2 Pre_IGA: Preparation for IGA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Boun_Cond: Define Dirichlet and Neumann Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 190 193

5.5 5.6

5.7

106 107 108 109 110 114 119

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Contents

9.4

Initializing Control Densities and DDF at Gauss Quadrature Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5 Shep_Fun: Define the Smoothing Mechanism . . . . . . . . . . . . . . . . 9.6 IGA to Solve Structural Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Stiff_Ele2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.2 Stiff_Ass2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.3 Solving . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.7 Objective Function and Sensitivity Analysis . . . . . . . . . . . . . . . . . . . 9.8 OC: Update Design Variables and DDF . . . . . . . . . . . . . . . . . . . . . . . 9.9 Plot_Data and Plot_Topy: Representation of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.10 Demos for Several Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

199 200 201 202 204 205 206 207 208 210 212

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213

Acronyms

BDMs CAD CAE CAMD DDF DMO DVFs ESO FEM H-J PDEs IGA ITO LSM MDMs M-ITO MMA MMC/V N-MMI NPR NURBS OC PEMAP PEMAP-P PUCs RITO RMMI SIMP Top-opt TVFs UMMI

Boundary description models Computer-aided design Computer-aided engineering Continuous approximation of material distribution Density distribution function Discrete material optimization Fields of design variables Evolutionary structural optimization Finite element method Hamilton Jacobi partial differential equations Isogeometric analysis Isogeometric topology optimization Level-set method Material description models Multi-material isogeometric topology optimization Methods of moving asymptotes Moving morphable components/voids NURBS-based multi-material interpolation Negative Poisson’s ratio Non-uniform rational B-splines Optimality criteria Piezoelectric material with penalization Piezoelectric material with penalization and polarization Periodical unit cells Robust isogeometric topology optimization Recursive multi-phase materials interpolation Solid isotropic material with penalization Topology optimization Fields of topology variables Uniform multi-phase materials interpolation

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Chapter 1

Introduction

1.1 Topology Optimization (Top-Opt) Structural optimization is a discipline that focuses on dealing with the optimal design of load-carrying mechanical structures, which has been considered as a powerful tool to determine structural features, such as the connectivity of holes, the shapes of boundaries and the sizes, in the past decades. Overall speaking, in the case of the design stage, three subfields are mainly involved in the field of structural optimization, namely topology optimization, shape optimization, size optimization, shown in Fig. 1.1. In the size optimization, structural performance is optimized by changing structural features, such as the width, height and the cross-sectional scales, under the predefined structural topology and shapes. The main focus of shape optimization is to advance the shapes of structural boundaries, like changing from the straight line to a curve of geometry, and thereby improve structural performance. A critical characteristic is that structural topology keeps unchanged. Topology optimization lies in the stage of the conceptual design, which determines the numbers, connectivity and existence of holes in design domain and focuses on advancing structural topology to improve the concerned performance. Overall speaking, topology optimization can be identified as one of the most promising subfield of structural optimization owing to its superior advantages occurring in the conceptual design stage without prior knowledge of design domain. The highly expected requirements of topology optimization also pose more challenges during the solving of several different design problems compared to size and shape optimization. Since the seminar work of topology optimization where a homogenization approach is proposed was performed by Bendsøe MP and Kikuchi N [1], topology optimization has been obtained a myriad of applications ranging from the classic mechanical field to other physical disciplines, including fluids, acoustics, optics, electromagnetics, etc. [2–5]. As shown in Fig. 1.2, an engineering case using the Top-opt, namely the chassis of ford car, is provided. We can see that a suitable

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gao et al., Isogeometric Topology Optimization, Engineering Applications of Computational Methods 7, https://doi.org/10.1007/978-981-19-1770-7_1

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1 Introduction

Fig. 1.1 Structural optimization: topology optimization, shape optimization and size optimization

Fig. 1.2 An engineering case using topology optimization

design domain needs to be firstly predefined before the optimization, which determines the distribution space of materials. Then, topology optimization is applied to optimize structural performance by determine the existence or not of the related design elements in structural design domain, so that a basic material distribution, including the numbers, connectivity and locations of many holes, can be gained. We can also see that topology optimization can effectively reduce material consumptions and guarantee the improvement of the concerned performance, which can arrive at the requirement in the application. In recent years, topology optimization has

1.2 Isogeometric Analysis (IGA)

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been adopted to solve many design problems, like the multiscale topology [6–11], materials design or functional graded materials [11–16] and robust designs with uncertainties [17–20]. It is known that the field of topology optimization initially stems from the pioneering work in 1904 by Michell [21], in which a critical problem of the limits of economy of material in a frame structures was studied. In practice, the seminar research on seeking for the optimal design of solid elastic plates addressed by Cheng and Olhoff [22, 23] opens a new window of structural optimization for continuum structures, which has been regarded as the pioneering work in the field of topology optimization and also attracted a wide of discussions in the latter three decades. In 1988, the homogenization approach based on the theory of porous continuum media distribution was proposed, where structural topology was evolved by changing the sizes and orientations of micro holes in the design domain. Up to now, several topology optimization methods with their unique positive characteristics have been developed. In the viewpoint of the topology representation, the existing topology optimization methods can be roughly divided into two main branches, where one corresponds to material description models and the other refer to boundary description models. In the first branch, the homogenization approach [1], the solid isotropic material with penalization (SIMP) approach [24, 25], the Evolutionary Structural Optimization (ESO) [26] method, etc. are mainly included. In the second branch, the classic level-set method (LSM) [27–29], the phase field method [30, 31], the moving morphable components (MMC) method [32,33], etc. are mainly included. A similar feature of these methods in the second branch is that the isocontour of a high-dimensional function is applied to present structural boundaries. In terms of the first branch, these methods have several positive features, like the conceptual clarity and easily numerical implementations. The second branch has the rigorous theoretical derivations and smooth structural boundaries of the optimized solutions.

1.2 Isogeometric Analysis (IGA) As we know, a complete period from the initial design to the last manufacturing is required for each engineering structure. In the period, the computer-aided design (CAD) model should be translated into the analysis-suitable geometries, was then meshed and was later transformed to an analysis-suitable model, namely the computer-aided engineering (CAE) model in the engineering software. It is known that this period is trivial owing to a prohibitive computational cost, specifically for the complex engineering structures. The estimation of time costs of each component for the engineering structure is provided [34, 35]. We can see that the task for the development of the CAD takes over 80% of the overall analysis time, and engineering products are becoming more complex. A critical issue is that the analysis-suitable models cannot be automatically obtained from the meshes of the CAD model. The time is consumed to a great extent for the preparatory steps involved, and no mesh is no longer enough. Hence, the today’s bottleneck in CAD-CAE integration is not

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1 Introduction

only located at automated mesh generation, it also lies in how to efficient create an appropriate “simulation-specific geometry”. Mesh generation costs 20% of overall analysis time, and the creation of the “simulation-specific” geometry accounts for about 60% and only 20% of overall time is actually devoted to the analysis. The 80/20 modelling/analysis ratio is a very common industrial experience, and it is desirable to reverse it. Hence, it is compulsory to perform the integration of CAD and CAE to save the time cost for engineering structures from the CAD to CAE. Meanwhile, it also seems that fundamental changes should take place to fully integrate the engineering design and analysis processes, namely the way to construct CAD and CAE. On the other side, the conventional finite element method (FEM) is adopted to perform the numerical analysis to predict the unknown structural responses. The inherent deficiencies of the FEM would introduce several numerical problems within the optimization. The first is that the finite element mesh cannot capture the exact structural geometry, which will considerably lower the numerical precision [34, 35]. The second is the lower-order (C0 ) continuity between the neighboring finite elements, also existed in the higher-order elements. The final drawback is the low efficiency of the FEA in achieving a high quality of the finite element mesh. It can be seen that these above shortcomings significantly affect the numerical precision. The isogeometric analysis (IGA) proposed by Hughes and his coworkers [34, 35] might be viewed as a logical extension and generalization of the finite element method to solve the numerical analysis. The central idea is that basis functions employed in the construction of structural geometry model are simultaneously applied to develop the finite-dimensional solution space in numerical analysis, so that the geometrical model and numerical analysis model can be consistent. In IGA, several different B-splines techniques can be used, for example, the initial B-splines basis functions [34, 35]. Although the B-splines are flexible for modeling many geometries, some important shapes cannot be exactly presented, like the circles and ellipsoids. Hence, nonuniform rational B-splines (NURBS) working as a generalization of B-splines are introduced, where NURBS basis functions are formed by assigning a positive weight to each B-spline basis functions [36, 37]. Recently, other candidate computational geometry technologies have been applied into IGA, like the T-splines [38]. In terms of the B-splines, they possess several useful mathematical properties, such as the ability to be refined by knot insertion, the higher-order continuity for the geometry and analysis, the variation diminishing to remove the local oscillatory and the convex hull properties.

1.3 Isogeometric Topology Optimization (ITO) An earlier work developed an ITO design framework with considering trimmed spline surfaces and IGA [39, 40]. In previous works, the complex spline surfaces using the trimming information are applied into the trimmed surface analysis which is applied to calculate unknown structural responses and also the calculation of sensitivity analysis in topology optimization. The spline surface and the trimming curves are

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applied present structural boundaries in geometrical model. In the optimization, the physical coordinates of control points act as design variables. The proposed ITO framework can implement the inner front creation and also merging. However, if structural geometry is complex, the numbers of trimming curves are required to cause a prohibitive computational demand. Later, the main focuses of the development of the ITO design method are located at the classic Top-opt methods and the IGA, and the critical issue is how to consider the IGA in the Top-opt methods. Based on the discussions about classifying Top-opt methods into two branches, namely MDMs-based and BDMs-based, the descriptions about the developments of the ITO methods can be divided into two branches, also including the Material Description Model (MDM)-based and Boundary Description Model (BDM)-based. In the first branch of the MDMs-based ITO methods, the developments of the ITO methods are strongly dependent on the artificial density, also named by the density-based ITO design methods. As far as the second branch with the BDMs is considered, the ITO methods are developed on the boundary, especially for the LSM and MMC/V. Hence, the latter descriptions will be divided into subtypes, namely the density-based ITO and the boundary-based ITO. The readers who have the interest can refer to [41] with the detailed discussions.

1.3.1 Density-Based ITO Methods As already discussed in [41], the branch of the MDMs-based Top-opt methods mainly stems from the classic homogenization approach. Later, a powerful alternative that refers to solid isotropic material with penalization (SIMP) method is proposed with several improvements in its conceptual clarity and easy numerical implementation [24, 25]. Meanwhile, the ESO method with the density also has gained a wide of attentions [42]. In the density-based Top-opt methods, an identical feature that the intention of the Top-opt to find the continuous material distribution is effectively converted into searching for the spatial distribution of a family of finite elements by introducing the “artificial densities to denote the existence of materials”. In the density-based Top-opt methods, several numerical problems exist in the optimized topologies during the transformation, like poor checkerboards, the zig-zag or wavy structural boundary features and the mesh-dependency [43–45]. Several works [46– 48] point out the main reason that the optimization is strongly dependent on finite elements of densities-based methods. Later, several variations of the density-based methods are developed to remove the above numerical issues, such as the construction of the elementary nodes [49–51]. The detailed discussions about the SIMP method can refer to several reviews [3, 52]. The earlier works that develop the ITO method on the concept of the artificial density might go back to [53]. In the works, B-splines are innovatively applied to construct finite elements that are employed to present the density distribution and construct the field of structural displacement in the analysis. In the analysis of numerical examples, we can easily find that the previously mentioned numerical issues in

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density-based Top-opt methods can be successfully eliminated. The authors manifest that B-splines with the local support feature can remove the mesh dependency, and it has the similar feature of the density filtering scheme in the SIMP. Later, an ITO method was also developed in [54], in which the densities at control points are defined, also named by control densities, and NURBS basis functions combined with the defined control densities to develop a density response surface to present the layout of materials in the design domain. In this work, the provided numerical examples can also show the considerations of IGA into the conventional densitybased Top-opt methods can offer more benefits for removing several numerical artifacts in the optimization. However, several critical problems are also occurred in the development of the ITO method on solving the design problems, like the pointwise distribution materials to lead to the blur and wavy structural boundaries in the optimization. In the later, Qian [55] constructed a B-spline space for the Top-opt, where the tensor-product feature of the B-spline was applied to present arbitrarily shaped design domain by an embedding representation. An innovative concept of the B-spline field of the density was proposed for the representation of the structural topology in the optimization. Moreover, the work [55] revealed the inherent filter of the B-spline space for the density filter owing to the local compact of the B-spline, which is similar to the density filter to effectively remove numerical artifacts and also control minimal feature of structural length in the optimized topologies. Additionally, a critical characteristic that the B-spline space can decouple the density field to present structural topology and the latter finite element analysis, and the strong dependence of finite elements in previous density-based Top-opt methods can be extensively alleviated in the current work. In the optimized topologies, structural terrible features exist owing to a fact that the presentation of the structural topology is still based on the development of finite element densities. To sufficiently employ the positive features of the B-splines, the work [56] constructed the enhanced density response NURBS surface, also termed by the density distribution function (DDF), by considering two important factors: (1) the smoothness to ensure the form of structural boundaries in the optimized topologies; (2) the continuity stemming from the NURBS parameterization using the higher-order basis functions to ensure the distinct interfaces. Based on the given numerical examples, we can find the effectiveness of the DDF to present structural topology and facilitate the optimization. However, an essential issue is that structural boundaries heuristically refer to the isocontour (value is equal to 0.5) of the DDF. The authors point out the idea coming from the immersed representation model of the LSM but with the intrinsically different optimization mechanism. In the proposed ITO, the immersed representation of structural boundaries by the DDF is only a simple numerical scheme in the optimization, where the reasonability of the numerical scheme has been extensively discussed in the work [56]. In recent years, the development of the ITO methods on the basis of the density has been sufficiently discussed, and several optimization problems have been addressed using the developed ITO methods. Lieu and Jaehong [57] studied the multiresolution design problems using topology optimization by replacing the FEM using the IGA to develop the ITO design framework also on the density, and the multiresolution

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ITO method was applied to study the multi-material topology optimization problems [58], where the alternating active-phase algorithm [59] in the multi-material topology designs is used. In [60], Alireza and Suresh also discussed the multimaterial ITO designs and also realized the design of functionally graded materials. In this work, a multi-material interpolation scheme proposed in [61] is directly applied to present the distribution of discrete multiple materials. To effectively resolve the multi-material designs, a new multi-material ITO method [62] is developed, and an N-MMI (NURBS-based Multi-Material Interpolation) model is innovatively defined by the NURBS parameterization. After that, the ITO method proposed in [56] and the multi-material ITO method proposed in [62] are applied to discuss many design problems, like the design of micro-architected materials [63,64], the rational design of auxetic meta-materials [65] and the systematic design of auxetic meta-composites [66,67]. An educational MATLAB code for the implementation of the ITO method in [56] was also given in [68]. The ITO method on the density was also applied to optimize the spatially graded hierarchical structures [69]. Lie et al. [70] studied the stress-constrained topology optimization design problems in the plane stress and also discussed the optimization of bending of thin plates. Xie et al. [71] developed a new ITO method with the use of the truncated hierarchical B-spline and discussed the optimization problems of structural stiffness and compliant mechanism. Zhao et al. [72] also employed the T-spline to replace the known NURBS and developed an ITO method for the optimization of arbitrarily design domains with the superior effectiveness.

1.3.2 Boundary-Based ITO Methods The level-set method (LSM) is a well-known boundary-based topology optimization method, which is earlier applied to track the variations of the interface and shape in many engineering disciplines. In the LSM, the core is to define a level-set function with a higher-dimension to determine the existence of materials in the design domain, and the isocontour of the level-set function describes the structural boundaries in the optimization. Compared with the density-based methods, the LSM has many merits in the optimization: (1) a smooth and distinct boundary description for the optimized topology; (2) the shape fidelity and topology flexibility during the optimization; (3) the shape and topology optimization are performed; (4) a physical meaning solution of the H-J PDEs in the analysis. The earlier work that considered the IGA into the LSM to develop a level-setbased ITO method for the design of structures might go back to [73]. Later, wang et al. [74] proposed a level-set-based ITO method with a parameterization, where the same NURBS basis functions are applied to construct the level-set function and the solution space; and then the proposed parametric level-set-based ITO was applied for the design of geometrically constrained domains [75], and a point-inpolygon algorithm and also the trimmed elements were considered. The work [76] also studied the level-set-based ITO method, and then it was applied to discuss the

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optimization of flexoelectric materials to improve the energy conversion efficiency. The flexoelectric effect was firstly modeled using the NURBS-based IGA elements. Jahangiry et al. [77] also proposed a new level-set-based ITO design framework, and the control mesh was gradually evolved in the optimization, which has been employed to discuss the design of the concentrated heat flow and the uniformly distributed heat in [78]. In [79], the IGA into the shape optimization to realize the topological changes was also addressed with the use of level sets and the dual evolution by the boundary integral equation. The level-set-based ITO method proposed in [74] was also applied to discuss the design problems of vibrating structures, and it aims to enhance the fundamental eigenfrequency and also avoid the resonance [80]. The multiscale design for cellular structures was also implemented by the level-setbased ITO method [81], where the offsetting scheme is applied to obtain a series of prototype microstructures. A level-set-based method was proposed for the control of the high-frequency electromagnetic wave propagation in [82]. Gao et al. [83] developed a parametric level-set-based isogeometric topology and shape optimization for composite structures with the multiple materials in the design. The MMC/V method can be also regarded as a kind of the boundary-based Topopt method with the promising capabilities [32], which has accepted more and more interests in recent years. Later, several powerful alternatives are proposed for the MMC/V Top-opt method to improve capabilities of several design problems, like the work [84] with the ersatz material model, the works [85,86] for developing the framework of MMVs. The earlier work that studied the MMC-based ITO design method can go back to Hou et al. [87] in 2017. In this work, NURBS basis functions to construct a NURBS patch are applied to present the structural geometries of components with some design parameters, which are then applied in the latter analysis. In the latter discussions of [87], several numerical examples can manifest that the proposed MMC-based ITO method cannot only inherit the advantages of the MMC method, but also embrace the positive features of the IGA. Moreover, the direct connection of the MMC with structural geometries can provide benefits for the IGA into MMC. Xie et al. [88] developed a MMC-based ITO design method, where R-functions are applied to develop the functions for the description of structural topology and overcome the lower-order continuity issue in several overlapping regions of structural components. In the discussions of numerical examples in [88], the proposed MMCbased ITO method can effectively improve the numerical efficiency with a range of 17–60% for several design benchmarks, which was employed to design symmetric structures by using energy penalization method [89]. Later, the hierarchical B-spline was applied to develop the MMC-based ITO method in [90], which can implement the adaptive IGA to improve numerical accuracy of the optimization. As far as the MMV Top-opt method, Zhang et al. [91] developed the MMV-based ITO framework that can seamlessly integrate IGA using trimming surface analysis technique, which can flexibly control the complex structural geometries. Moreover, the critical issue of the self-intersection and the jagged boundaries can be removed during the optimization. Later, Gai et al. [92] developed a MMC-based ITO design method but with the closed B-spline boundary curves to model the MMVs that are applied to

1.4 Applications of Topology Optimization

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present structural topology. In [93], the MMC-based ITO design framework for the multiresolution design problems was also discussed.

1.4 Applications of Topology Optimization 1.4.1 Multi-material Structures The key intention of the multi-material topology optimization is to find the optimized distribution of multiple materials in the design domain that can have the expected structural performance. Recently, the multi-material topology optimization has gained a wide of discussions, and the first work might go back to [94], in which the homogenization method for the design of multiple materials was studied. Later, the SIMP method for the distribution of multiple materials was also discussed for the problems of material microstructures with extreme thermal expansion [95], micro unit cells with extreme bulk modulus [96] and also the design of multiphysics actuators [97], in which a known mixture scheme to evaluate three-phase material properties was proposed. The discrete material optimization (DMO) on the basis of the classic SIMP method was developed in [61] to realize the multi-material topology optimization of the composite laminate shell structures. After that, the mixture rule and also the DMO were sufficiently discussed for the design problem with the total mass constraint in [98], where two effective alternatives are developed for the multi-material interpolation models, including the UMMI (uniform multiphase materials interpolation) model and the RMMI (recursive multiphase materials interpolation) model. In [99], the peak function with the intention of decreasing design variables was proposed for the multi-material topology optimization designs. An alternating active-phase algorithm was proposed in [100], where the initial multimaterial topology optimization has been split into an array of the binary designs. An ordered SIMP interpolation was also provided for the optimization of multiple materials in the design domain to present the distribution of multiple materials by a kind of design variables. In the framework of the LSM, several multi-material LSMs were performed, such as the color level-set method [101] and the MM-LS model [102] for multiple materials. A continuum topology optimization framework for a multimaterial design with the considerations of the arbitrary volume and mass constraints was developed in [103].

1.4.2 Stress-Related Problems In the engineering applications, the extreme requirement of structural strength to avoid the failure is a critical problem in the design of structures, where the stress should be considered in the development of topology optimization models. The

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seminar work [104] has studied the topology optimization with several local stress constraints. Currently, the stress-related topology optimization can be divided into two main branches, namely the stress-constrained designs [105–109] and also the stress-minimization [110–112]. In the stress-related topology optimization, three main obstacles are involved, consisting of the singular topologies, the local nature and the nonlinear stress behavior. In the case of the first obstacle for singular topologies, the earlier works were discussed for the design of the truss [113, 114]. It is mainly because that the related degenerate subspaces cannot be identified by the mathematical programming algorithms. In the classic density-based Top-opt methods, it has a similar feature that finite elements with the low artificial densities but having the high stress, so that the optimization algorithms cannot remove these elements during the optimization. Until now, several remedy algorithms have been developed, and a critical feature that element artificial densities and the corresponding stress can decrease simultaneously exists to ensure the degenerate regions that can be eliminated during the optimization, such as the qp relaxation [107], the epsilon-relaxation [104, 115] and the stress penalization [109, 116]. It is noticed that the critical problem of the singular topologies can only occur in the density-based works with the intermediate densities, and other boundary-based Top-opt methods or the discrete methods can naturally remove this problem, such as the LSM-based works [10, 117–120] and MMC/V-based works [121]. As far as the second issue of the local nature feature in stress is concerned, the stress-related designs aim to satisfy the requirement of strength in each designable point, and it means that the related stress constraint should be imposed at each point, which will lead to a large number of local constraints in the optimization and a prohibitive computational cost. Currently, an effective scheme that aggregates all local stress constraints by a or some combined stress constraints has gained a wide of attentions, such as the active-set strategy [104, 122], a global aggregation [106–111] and the block aggregation [109, 123]. The classic p-norm function and Kreisselmeier–Steinhauser (KS) function has been broadly adopted. However, the high nonlinearity and poor approximation of these two functions are always occurred in the optimization, which extensively damage the effectiveness of the stress evaluation. Hence, some variations of the global aggregation functions have been developed and employed in the stress-related topology optimization works, like a multi-p-norm formulation [124], p-norm correction [125] and the integration of relaxation and aggregation [126]. Moreover, a critical problem of the aggregation is the mesh dependency in stress evaluation [127], and the critical influence on topology optimization is not addressed until now. In the case of the critical problem of the stress nonlinear behavior, the main cause is that the stress is computed by the displacement gradient. However, in the FEM, the computation of the displacement gradient has the lower numerical precision, particularly for the critical areas. Hence, the accuracy of stress evaluation has a significant effect on the optimization, like the iterative instabilities. In practical, the problem of numerical instabilities in the stress-related designs comes from many factors,

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like the stress nonlinear behavior [109, 110], the nonlinear of aggregation functions [106–111, 122] and the lower stress computation precision [128, 129]. Several numerical schemes are proposed to improve maintain the optimization stability, such as the consistent density filtering [109]. Other finite element methods are applied to compute the stress and improve numerical accuracy, such as the adaptive finite element mesh [119], the extended FEM [128], and finite cell method [129].

1.4.3 Piezoelectric Structures Piezoelectric materials that have the ability of generating an electric charge in response to mechanical deformation have been broadly applied to design the actuators, energy harvesters and sensors [130]. The piezoelectric actuators with the large output force, the high displacement resolution and dynamic response can perfectly serve for the nanopositioning and micromanipulation instruments [131, 132]. In previous works for the design of piezoelectric structures, analytical and numerical methods were discussed with considering the change of structural sizes, geometries and layer numbers [133–135]. The special feature of topology optimization can offer more freedoms for the design of piezoelectric actuators. In the earlier work, the optimization of piezocomposite microstructures using the Top-opt and the homogenization was performed [136], which has been applied to the design of flexotensional piezoelectric actuators and also piezoelectric transducers [137, 138]. Based on the classic SIMP, the piezoelectric material with penalization (PEMAP) model was proposed to evaluate the corresponding piezoelectric and dielectric properties with respect to the artificial densities in the optimization [139]. The OC method and sequential linear programming combined into topology optimization can serve for the design of compliant mechanical amplifiers using piezoelectric materials [140]. In [141], the Continuous Approximation of Material Distribution (CAMD) was used to optimize microtools with multiple piezoceramics. Later, the PEMAP-P model with the considerations of the polarization was developed for optimizing the piezoelectric shell and plate actuators [142], in which a kind of design variables should be introduced for the piezo-actuator designs [143–145]. Until now, the idea of Top-opt for the design of piezoelectric structures has attracted many researchers, and several types of the piezoelectric actuators, such as functionally graded piezoelectric actuators [146], multilayers bending piezoelectric actuators [147], bi-material actuators [148], piezoelectric laminated actuators [149, 150], and also considering the layout of actuation voltage [151, 152] are presented. In previous works, a critical problem is the existence of the localized hinges with one-node connection in the optimization of piezoelectric actuators [153], which can lead to stress concentrations when the structural deformations are large. Moreover, it is difficult to meet the manufacturing-tolerant. As we know, the piezoelectric actuators are mostly used to process several complex and subtle operations, like the micromanipulation and nanopositioning. The related structural deformations of

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piezoelectric actuators should result from the flexible structural members, rather than the localized pins or joints. Moreover, several uncertainties always exist in the manufacturing of piezoelectric actuators when the localized hinges in the optimized topologies [52, 154–157]. The manufacturing tolerance can lead to the degradation or loss of structural functionalities.

1.4.4 Architected Materials Architected materials that can be viewed as a kind of human-designed artificial materials have several desirable properties, like the stiffness and strength, by rationally their micro unit cells with the specific topologies [158]. In recent years, architected materials have gained a wide of applications, like energy absorbers [159], body implants [160] and sandwich panels [161], and a comprehensive review about their state of the art can refer to [162–164]. As we know, a critical feature of architected materials is that the effectively macroscopic properties of architected materials are mostly dependent on the micro topologies of the periodically unit cells (PUCs), also named by microstructures, rather than material constituent properties [158, 162, 164]. Until now, several previous works are mostly located at how to modify geometrical shapes or sizes of PUCs to enhance their properties, which are strongly dependent on the experiments or human insights. As we know, the homogenization theory can be applied to evaluate effective macroscopic properties of architected materials based on the micro information from unit cells [165]. An inverse design idea that is developed by topology optimization and the homogenization, and the topology optimization is applied seek for the reasonable topology of micro unit cells until the expected properties can be found. Later, a family of novel architected materials with innovative properties are presented [166], like the extreme elastic properties of bulk and shear moduli [12, 13, 96, 167, 168]. In [169, 170], maximizing the structural stiffness and the fluid permeability in the optimization was discussed. Hollow structures [171, 172] and functional graded materials [173–175] were also respectively studied. However, an essential problem that the ultra-lightweight feature is not considered in previous works for the design of architected materials exists.

1.4.5 Auxetic Meta-Materials/composites Auxetic meta-materials are a family of architected materials with the negative Poisson’s ratios (NPRs), which are characterized the counterintuitive dilatational behavior, expanding laterally when stretched and then contracting laterally when compressed. This special property has a wide of potential merits in engineering applications, like indentation resistance, shear resistance and fracture toughness [176]. In the earlier works, several different structures with the NPRs are reported, such as

1.5 Implementations of Topology Optimization

13

the earliest foam [177], chiral-type auxetics [178, 179], re-entrant structures [180, 181] and also rotating-type structures [182]. A detailed description about auxetics can refer to [183, 184]. In recent years, topology optimization has gradually applied to discuss the optimization of architected materials with the NPR feature [185–192]. In the earlier works [187, 190, 193], architected materials with the nonlinear feature are also discussed to ensure the generation of the auxetic behavior, and the shape optimization was employed to tailor the NPR values in a preoptimized topology of micro unit cells [190]. A two-step design flowchart that was developed by the SIMP and the LSM methods was applied for the design of auxetic microstructures with the NPR feature [191]. In [186], the parametric LSM was also applied to optimize 2D and 3D auxetic microstructures. Although the optimization of 3D auxetic microstructures has been studied in many works, several unresolved challenges still exist in the design. For example, the denser finite element mesh (1003 ) is required for the optimization of micro unit cells to ensure the generation of the NPR feature in the 3D micro-topology, which cause a large number of iterative steps (overall 3000). This critical problem strongly limits the effectiveness of the topology optimization for the design of micro unit cells to seek for the novel 3D topologies. On the other side, the other critical problem of auxetic meta-materials is the soft feature, which leads to the design structure without the enough stiffness in engineering applications. Recently, composite materials with preferably physical or mechanical properties [194] have been widely employed in the engineering. Hence, a novel concept that design the composite with the NPRs has emerged in [195], which has been gradually attracted many researchers in several disciplines [196]. The design of the auxetic meta-composites using the topology optimization has been limitedly discussed in recent years. In [197], a series of 2D auxetic micro unit cells with multiple materials and only a 3D micro unit cell are presented in the optimization. The work [198] also studied the design of chiral auxetic composites with two materials.

1.5 Implementations of Topology Optimization The implementations of topology optimization in different platforms with the related languages have been presented, and a comprehensive review can refer to [199], which can provide an entry point for many newcomers that can be quickly familiar with the field of the Top-opt. The seminar work on the implementation of topology optimization can go back to [200], in which a 99-line MATLAB code of the SIMP method was presented. Recently, a powerful alterative of the 99-line code for the SIMP method was given in [201, 202] with the 88-line code and the new generation of the 99line code, respectively. Up to now, several educational papers with MATLAB codes are presented, which considerably promote the developments and applications of topology optimization, mainly including the LSM with MATLAB codes [203–206], the pareto-optimal tracing codes in topology optimization [207], ESO/BESO codes [208, 209], unstructured polygonal elements in topology optimization [210, 211], 3D

14

1 Introduction

SIMP with a 169-line MATLAB code [212], the design of architected materials [192], concurrent topology optimization of porous structures [7], the discrete topology optimization with binary structures [213], a 101-line MATLAB code using Canonical relaxation algorithm [214], the MMC MATLAB code in [84], the MATLAB code for geometrically nonlinear topology optimization [215]. Moreover, the implementations of the IGA were also presented, like “GeoPDEs” [216, 217], several in-house codes in [218] and a detailed comparison between the IGA and FEM in [219]. An IgaTop implementation framework for the ITO method proposed in [56] was also given in [68].

1.6 Main Focus of the Current Monograph In the current monograph, the main focus is to systematically overview of the method and applications of the isogeometric topology optimization (ITO). In the Chap. 2, the key purpose is to present the development of the ITO method with the concept of the density. Compared to previous density-based topology optimization methods, the main contribution is to develop a Density Distribution Function (DDF) with the higher-order continuity and smoothness to present structural topology, rather than the distribution of a family of discrete elementary/nodal densities in structural design domain. Moreover, the same nonuniform rational B-splines (NURBS) basis functions are employed to model structural geometry and construct the space of structural responses in analysis, which can effectively guarantee numerical precision and ensure the numerical stability during the optimization. In Chap. 3, the main purpose is to present the development of the multi-material isogeometric topology optimization (M-ITO) method, which can effectively determine the optimized layouts of multiple materials in structural design domain. In the M-ITO method, a Multi-Material Interpolation model using NURBS (N-MMI) with powerful effectiveness is developed, which has a formulation in a manner of the decoupled expression and serial evolving for topology variables and design variables. This positive feature can extensively improve the effectiveness of the M-ITO method on solving the problems of the multi-material topology optimization, particularly for the problems with a total mass constraint. In Chap. 4, the main intention is to present the effectiveness of the ITO method on solving the stress-minimization design problems, which can search for the better designs to remove the critical issue of stress concentration. Moreover, to remove the influence of mesh dependency in conventional aggregation functions on the optimization, induced aggregation functions are applied to measure the global stress and improve the approximation accuracy of the maximum von Mises stress, which offer more benefits for enhancing iterative stability of stress-related topology optimization and improving the credibility of designs. In Chap. 5, the key intention is to develop a robust ITO (RITO) method for the optimization of piezoelectric actuators, which can effectively avoid the appearance of one-node connected pins and joints and maintain the flexible deformations to

1.6 Main Focus of the Current Monograph

15

transmit the micrometer or nanometer displacements subject to an applied electrical load. In the RITO method, an erode-dilate operator is applied into the development of topology representation model with the construction of the eroded, intermediate and dilated DDFs. The ITO and RITO formulations are both developed for the systematic optimizations of piezoelectric actuators, which can present their difference and validate the indispensability of the RITO to ensure the uniform manufacturability by removing the localized hinges. In Chap. 6, the main focus is located at optimizing the ultra-lightweight architected materials with the higher stiffness-mass ratio by the ITO method with the homogenization, which can present a rational design of architected materials. The numerical homogenization is implemented by IGA to evaluate the effective macroscopic property, with the imposing of the periodic boundary formulation on the micro-architecture. Moreover, the integration of the CAD model and CAE model for the micro-architecture can be beneficial to improve numerical precision and enhance the effectiveness of the topology optimization for the ultra-lightweight architected materials. In Chaps. 7 and 8, the main purpose is to discuss the design of auxetic metamaterials and auxetic meta-composites using the ITO method and M-ITO method, respectively. In the last chapter, the key intention is to present the detailed descriptions about the in-house MATALB codes for the ITO, also named by the “IgaTop” framework. Besides its educational purposes for newcomers, it can also serve as an entry-level tutoring to lower the barrier for researchers who have an interest in familiarization of the ITO and applications to other problems. The basic framework of the current monograph is also shown in Fig. 1.3.

Fig. 1.3 Main framework of the current monograph

Chapter 2

Density-Based ITO Method

The key purpose of the current chapter is to develop a new framework of topology optimization using isogeometric analysis (IGA) to replace the classic finite element method (FEM) that is employed to solve unknown responses, termed by the isogoemetric topology optimization (ITO) method. In the development of the ITO method, one critical issue should be resolved: In previous FEM-based Top-opt methods using the density, a series of discrete elements are applied not only in the FEM solve the unknown responses, but also present structural topology by defining artificial densities to determine the existence of materials, and zero denotes the voids and one indicates the solids in the design domain. When applying the IGA into Top-opt, the construction of geometrical model has a strong connection with the latter analysis, and a unified mathematical model of basis functions is used. How to construct material description model to present structural topology and then adopts the direct connection within the latter analysis also the optimization when considering the IGA into Top-opt. In actual, it is not a simple replacement of the FEM, the special features of the IGA, such as the NURBS, the higher-order and continuity response can offer more benefits for the whole process of the Top-opt. Hence, in the current chapter, a detailed description on how to develop the ITO method with the density is given, and several numerical examples will be presented to show its effectiveness and efficiency. Moreover, the comparisons between the ITO method and the classic SIMP method are clearly presented to show the unique advantages for the optimization.

2.1 NURBS-Based IGA In IGA, a critical feature that the same basis functions are employed in the constructions of the models of CAD and CAE to ensure their integration exists. As we know, the non-uniform rational B-splines (NURBS) [36] are mostly employed in IGA, which is also considered here.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gao et al., Isogeometric Topology Optimization, Engineering Applications of Computational Methods 7, https://doi.org/10.1007/978-981-19-1770-7_2

17

18

2 Density-Based ITO Method

2.1.1 NURBS Basis Functions NURBS is a logical extension of the B-spline. In the definition of NURBS, a knot vector in an ordered set of values should be given, expressed as  =  ξ1 , ξ2 , · · · , ξn+ p+1 and ξi ∈ R (ξi ≤ ξi+1 ) is the ith knot. n is the total number of basis functions, and p is the polynomial order of basis functions in the first parametric direction. The knot vector can partition the parametric space into a series of subintervals, generally named by knot spans, and ξ1 , ξn+ p+1 refers to the interval, also termed by a patch of IGA. Knot spans correspond to the finite elements in IGA, namely IGA elements. If knot spans are equally spaced in the definition, the knot vector is the uniform and otherwise non-uniform. When the first and last knot appear p + 1 times, it is open. The Cox-deBoor formula [36] with easy implementations is broadly adopted to define B-spline basis functions, starting with piecewise constants (p = 0):  Ni,0 (ξ ) =

1 if ξi ≤ ξi+1 0 0 otherwise.

(2.1)

when p ≥ 1, the B-spline basis functions can be defined by: Ni, p (ξ ) =

ξi+ p+1 − ξ ξ − ξi Ni, p−1 (ξ ) + Ni+1, p−1 (ξ ) ξi+ p − ξi ξi+ p+1 − ξi+1

(2.2)

In Eq. (2.2), the fractions with a form of 0/0 are equal to 0. We can easily find that the B-spline basis functions with p = 0, 1 are same as the piecewise constant and the linear functions, respectively. If the degree p ≥ 2, the B-splines basis functions are different from basis functions employed in the FEM. As shown in Fig. 2.1, a simple illustration for the quadrature B-spline basis function is given, and the important features can be given as follows:

Fig. 2.1 Quadratic B-spline basis functions for the knot vector {0, 0, 0, 0.2, 0.4, 0.6, 0.8, 1, 1, 1}

2.1 NURBS-Based IGA

19

• Nonnegativity: Ni, p (ξ ) ≥ 0 • Local support:  The support of each basis function Ni, p is contained in the interval ξi , ξi+ p+1 . Also, at most p +  1 number of B-spline basis functions are nonzero in a given knot span ξi , ξi+1 , namely the Ni− p, p , · · · , Ni, p.  • Partition of unity: For an arbitrary knot span ξi , ξi+1 , ∀ξ ∈  i  ξi , ξi+1 , j=i− p N j, p (ξ ) = 1. • Continuity: The continuity within knot spans is C p−k where k is the multiplicity of the knots. Based on the definition of B-spline basis functions, the corresponding NURBS basis functions can be defined as: ⎧ ⎨ Ri, p (ξ ) = n Ni, p (ξ )ωi in 2D j=1 N j, p (ξ )ω j (2.3) Ni, p (ξ )M j,q (η)ωi j p,q ⎩ i, j (ξ, η) = n m N (ξ )M (η)ω in 3D ˆ ˆ ˆˆ ˆ i=1

ˆ j=1

i, p

j,q

ij

where M j,q is the definedB-spline basis function in the other parametric direction  with a knot vector H = η1 , η2 , · · · ηm+q+1 . q is the polynomial order, m is the number of basis functions, and ωi j is the related weight value assigned to the tensor product Ni, p (ξ )M j,q (η). NURBS basis functions can inherit all favorable properties of B-spline functions, which can also be effective in constructing circles and ellipsoids. The related details can refer to [34].

2.1.2 Galerkin’s Formulation for Elastostatics The linearly elastic problems are studied here for the sake of numerical simplicity but without losing the generality for the presentation of the IGA. A strong form of the boundary value problem for design domain  bounded by the boundary  is formally stated. Given f i :  → R, gi :  Di → R, and h i :  Ni → R, find u i :  → R such that ⎧ ⎨ σi j, j + f i = 0 in  u = gi on  Di ⎩ i σi j n j = h i on  Ni

(2.4)

where σ is stress tensor, f i is the body force, and gi is the prescribed boundary displacement on the Dirichlet boundary  D . h i denotes the boundary traction on the Neumann boundary  N , and n j is the component of a unit normal vector to . Defining a trial solution space S = {u|ui ∈ Si } and a weight space by V = {w|wi ∈ Vi }, where each trial solution u i satisfies the Dirichlet condition u i = gi on  Di and each weight function wi should be zero on the Dirichlet boundary  Di . A variational form can be obtained: Given f = { f i }, g = {gi } and h = {h i }, find u ∈ S

20

2 Density-Based ITO Method

such that for all w ∈ V    T ε(w) Dε(u)d = fwd + hwd N 



(2.5)

N

where ε corresponds to the strain matrix, D is the elastic tensor matrix. In Galerkin’s method, the finite-dimensional approximants of the spaces S and V are constructed by NURBS basis functions, denoted by S h and V h , which consist of all linear combinations of the corresponding NURBS basis functions. The finite-dimensional nature of the function converts the weak form of the problem into a system of linear algebraic equations. Galerkin’s approximation can be expressed by: finding uh = vh +gh ∈ S h such that for all wh ∈ V h    T ε wh Dε vh d = wh fd + wh hd N 



N



T ε wh Dε gh d

−

(2.6)



Introducing a set containing all NURBS basis functions, denoted by A, and a subset containing the basis functions that are equal to 0 on the Dirichlet boundary  D , symbolled by B (B ⊂ A). The trial solution uh ∈ S h and weight wh ∈ V h can be stated as a function of basis functions:    uh = j∈A\B R j g j + i∈B Ri di  (2.7) wh = j∈A\B R j c j where c is the arbitrary to hold for all wh ∈ V h . wh Substituting Eq. (2.7) into Eq. (2.6), Galerkin’s form is transformed into the next equation, given as: ⎛ ⎞   

⎝ ε(Ri )T Dε R j d⎠di = R j fd + R j hd N 





−

T

N

ε R j Dε g h d

(2.8)



Proceeding to define ⎧  T ⎪ ⎨ K i j = ε(Ri ) Dε R j d    T ⎪ F = R j fd + R j hd N − ε R j Dε g h d j ⎩ 

N



(2.9)

2.2 Density Distribution Function (DDF) for Material …

21

Hence, Eq. (2.9) can be expanded a more compact matrix form, as: Kd = F

(2.10)

    K = K i j ; d = {di }; F = F j

(2.11)

where

2.2 Density Distribution Function (DDF) for Material Description Model 2.2.1 NURBS for Structural Geometry In the construction of structural geometry using NURBS, the NURBS basis functions combined with a series of control points are applied to develop the corresponding model of the NURBS. The related mathematical formula of the NURBS surface for a 2D geometry can be expressed as the following, where a control net of points Pi, j ∈ R2 is given. S(ξ, η) =

n  m 

p,q

Ri, j (ξ, η)Pi, j

(2.12)

i=1 j=1

p,q

where Ri, j (ξ, η) are bivariate NURBS basis functions defined in Eq. (2.3). As shown in Fig. 2.2, an example of the NURBS surface for a quarter annulus is given, where control points are plotted in red dots. A critical feature that control points are not necessary located at design domain is the uniqueness of the NURBS, namely the non-interpolation property. Several important properties of NURBS are given as follows: (1) Strong convex hull property; (2) The differentiability: The NURBS surface is p − k times differentiable with respect to the ξ parametric direction and the q − k times differentiable with respect to the η direction at the current knot with a multiplicity k; (3) Local modification: the support domain of each  control point   Pi, j in the NURBS surface corresponds to the interval ξi , ξi+ p+1 × η j , η j+q+1 . It is important to note that the above important properties are important for constructing the density distribution function (DDF) to describe structural topology in material description model. As shown in Fig. 2.3, the corresponding IGA mesh of the quarter annulus is provided, and the discretized quarter annulus with IGA elements and control points is shown in Fig. 2.3 (b). As we can easily see, the IGA mesh can be kept consistent with the initial structural geometry of quarter annulus, and the elementary nodes in IGA do not coincide with control points in NURBS.

22

2 Density-Based ITO Method

Fig. 2.2 NURBS surface for a quarter annulus with knot {0, 0, 0, 0.15, 0.3, · · · , 0.9, 1, 1, 1} and H = {0, 0, 0.25, 0.5, 0.75, 1, 1}

vectors



=

Fig. 2.3 IGA Mesh for the quarter annulus

2.2.2 Density Distribution Function (DDF) It is known that one essential problem of topology optimization is how to determine the existence of materials in structural design domain. In the classic SIMP method, the problem has been resolved by seeking a reasonable distribution of finite elements with artificial densities to determine the existence of materials. In the above manner to show structural topology using the artificial densities, two basic conditions that ensure the validity of the density should be satisfied [49–51]: (1) the nonnegativity; (2) the strict bounds from 0 to 1. If the development of material description model uses the NURBS, the artificial densities should be defined at control points, rather than the element densities. After introducing the densities at control points (control

2.2 Density Distribution Function (DDF) for Material …

23

densities), the construction mechanism of NURBS can be used to construct a response surface for the density in the design domain. The density response surface can be applied to determine the existence of materials in the design domain, rather than the initial discrete element densities to determine the existence of elements. The combined layout of element densities is applied to represent structural topology. The above density response surface using NURBS also can be named by the DDF, and the whole construction of the DDF mainly contains the following steps: Step 1: Define a series of discrete densities at control points, namely control densities φi, j ∈ [0, 1]

(2.13)

Step 2: Develop a smoothing mechanism to improve the overall smoothness of control densities ⎧ ⎫ n  n  n˜  m m ⎨ m˜  ⎬    ψ φi, j φi, j = ϕ φi,ˆ jˆ φi, j (2.14) ϕ φi, j / φ˜ i, j = ⎩ ⎭ i=1 j=1

i=1 j=1

ˆ ˆ i=1 j=1

where φ˜ is the smoothed control design variable, and ψ is the Shepard function which is defined by C4 compactly supported radial basis functions ϕ. Step 3: Use NURBS to construct the DDF for the whole design domain

(ξ, η) =

n  m 

p,q Ri, j (ξ, η)φ˜ i, j

(2.15)

i=1 j=1

where is the DDF, also a NURBS density response surface that has a higher dimension in design domain. As we can see, the mathematical formula of the DDF is same as the NURBS in Eq. (2.12). A critical difference is located at the corresponding physical meanings of coefficients at control points (control coefficients). In the NURBS, control coefficients denote the physical coordinates, whereas the control coefficients in the DDF correspond to the control densities. The density can be regarded as a property of control points, which will be gradually evolved. Meanwhile, the properties 1–3 in NURBS basis functions can effectively ensure the basic two conditions of artificial densities (strict bounds and the nonnegativity). Hence, the DDF that can have the justified physical meanings can be reasonably applied to determine the existence of materials in each designable point of design domain. It should be noted that the DDF that represents the material distribution in the design domain, but it has no capability of determining the existence or not of control points. The superior characteristic of the variation diminishing property of NURBS can eliminate the oscillatory of Lagrange polynomials [34, 35], which can be beneficial to improve the smoothness and continuity of the DDF. Additionally, it should be

24

2 Density-Based ITO Method

Fig. 2.4 Construction of the DDF

noted that the current work inherits the idea of the previous work [54], but extensively improves it considering the smoothness and continuity of the DDF and also extensively discusses the physical meanings. A flowchart of the construction of the DDF is shown in Fig. 2.4. Step 4: The DDF can determine the existence of materials in design domain. However, the values of the DDF vary from 0 to 1, and the intermediate densities exist in the DDF. It is assumed that structural boundaries of the topology are described by the isocontour (2D)/surface (3D) of the DDF. The solids are expressed by the DDF with the higher values, and the lower values of the DDF present the voids. The corresponding formula can be expressed as: ⎧ ⎨ (ξ, η) > iso solids

(ξ, η) = iso boundaries ⎩

(ξ, η) < iso voids

(2.16)

where iso is the value of the isocontour/surface of the DDF, and the value is equal to 0.5. As shown in Fig. 2.5, the immersed description of structural boundaries of the topology by the DDF is provided. As we can easily see, it is similar to the level-set implicitly description model, but the evolving of the DDF is intrinsically different from the classic LSM. In the LSM, the isocontour/surface of the level-set function is gradually evolved by the H-J PDEs and then structural topology can be changed with several numerical schemes. In the current ITO method, the evolvement of the DDF is evolved by the change of control densities, and the isocontour/surface of the DDF to

Fig. 2.5 Immersed representation of structural boundaries using the DDF

2.2 Density Distribution Function (DDF) for Material …

25

present structural boundaries is a simple heuristic scheme to achieve the optimized topology from the DDF.

2.2.3 Material Interpolation Model As shown in Fig. 2.6, the quarter annulus is discretized by a series of IGA elements. In physical space, the IGA element stiffness matrix is calculated by:  Ke =

BT DBde

(2.17)

e

where e is the physical domain of the IGA element. B is the strain–displacement matrix calculated by the derivatives of NURBS basis functions with respect to parametric coordinates. In the IGA, the isoparametric formulation is adopted to compute ˆ e → e maps the paraelement stiffness matrices. As displayed in Fig. 2.6, X :  ˆ e from the bi-unit ˜ metric space to physical space and an affine mapping Y : e →  parent element to the parametric element are defined. The numerical integral for the IGA element stiffness matrices is pulled back first onto the parametric element and then onto bi-unit parent element. The detailed form is given by:

Fig. 2.6 Numerical integration in IGA and FEM

26

2 Density-Based ITO Method

 Ke =

˜e BT DB|J1 ||J2 |d 

(2.18)

˜e 

where J1 and J2 are Jacobi matrices of two mappings, respectively. The numerical computation of the element stiffness matrix is given by:



3  3  3  T  B ξi , η j , ζk DB ξi , η j , ζk  Ke =



  J1 ξi , η j , ζk J2 ξi , η j , ζk wi w j wk

(2.19)

i=1 j=1 k=1

where wi , w j and wk are the corresponding quadrature weights. In material description model, the elasticity property of isotropic materials is a power function of the density with a penalty parameter. The IGA element stiffness matrix corresponds to a function of material densities at Gauss quadrature points, rather than the constant element density, as follows:



γ 3  3  3  T  B ξi , η j , ζk ξi , η j , ζk Ke =

 D0 B ξi , η j , ζk |J1 ||J2 |wi w j wk i=1 j=1 k=1

(2.20)



where ξi , η j , ζk is the density of the ξi , η j , ζk Gauss quadrature point, and γ corresponds to the penalization parameter. D0 is the elastic tensor matrix for the solid density.

2.3 ITO Formulation for Stiffness-Maximization Here, the classic compliance-minimization design problems are considered to act as the benchmarks that are applied to present the effectiveness and efficiency of the proposed ITO methods. In the ITO formulation for the stiffness, the structural compliance is computed and acts as the optimized objective function, and material volume fraction is the constraint. The detailed formula is stated as: Find : φi, j (i = 1, 2, . . . , n; j = 1, 2, . . . , m)  1 Min : J (u, ) = ε(u)T D( )ε(u)d 2  ⎧ 1 ⎨ a(u, δu) =  l(δu), u| D = g, ∀δu ∈ H () S.t. : G( ) =  (ξ, η)d − V0 ≤ 0 ⎩ 0 < φmin ≤ φi, j ≤ 1

(2.21)

2.4 Numerical Implementations

27

where φi, j is the control density to act as design variable. i, j indicate the indices for two parametric directions, respectively. n, m are the numbers of control design variables in two parametric directions ξ, η, respectively. J is the objective function defined by the structural mean compliance. G is the volume constraint, where V0 denotes the maximum material volume fraction. u is the displacement field in structural design domain , and g is the prescribed displacement vector on the Dirichlet boundary  D . δu is the virtual displacement field belonging to the Sobolev space H 1 (). a is the bilinear energy function, and l is the linear load function. The equilibrium state equation established by the principle of virtual work can be defined as: ⎧  T ⎪ ⎨ a(u, δu) = ε(u) D( )ε(δu)d    (2.22) ⎪ ⎩ l(δu) = f δud + hδud N 

N

where f is the body force and h is the boundary traction on the Neumann boundary N . The detailed derivations about the sensitivity analysis of the objective and constraint functions with respect to control design variables are given in Appendix, and the final first-order derivatives of the objective and constraint functions with respect to control design variables are expressed as: ⎞ ⎛ 

∂J 1 p,q = −⎝ ε(u)T γ γ −1 D0 ε(u)d⎠ Ri, j (ξ, η)ψ φi, j ∂φi, j 2

(2.23)



Similarly, the derivative of volume constraint with respect to control design variables is given by: ⎛ ⎞  ∂G p,q = ⎝ 1d⎠ Ri, j (ξ, η)ψ ρi, j ∂φi, j

(2.24)



2.4 Numerical Implementations The OC is adopted here to solve the formulation due to its superior characteristic for the optimization problems with a large number of design variables but only a few constraints, defined as:

28

2 Density-Based ITO Method         ⎫ ⎧ (κ) (κ) ϑ (κ) (κ) ⎪ ⎪ φi, j − m , φmin ⎪ ⎪ ⎪ ⎪ max φi, j − m , φmin if ⎧i, j φi, j ≤ max ⎪ ⎫ ⎪     ⎪ ⎪ ⎪ (κ) ϑ (κ) ⎬ ⎪ ⎬ ⎨ max φ (κ) − m , φ ⎨ ϑ φi, j min < i, j (κ+1) (κ) (κ) i, j    i, j φi, j , φi, j = if (κ) ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ < min φi, j + m , φmax ⎪ ⎪ ⎪ ⎪ ⎪ ⎪         ⎪ ⎪ ⎭ ⎩ (κ) (κ) (κ) ϑ (κ) min φi, j + m , φmax , if min φi, j + m , φmax ≤ i, j φi, j

(2.25)

where m, ϑ are the move limit and the damping factor, respectively. i,(κ)j is the updating factor for the (i, j)th design variable at the κth iteration step. i,(κ)j

  ∂J (κ) ∂G =− /max μ,  ∂φi, j ∂φi, j

(2.26)

where μ is a small positive constant to avoid the zero term in the denominator. (κ) is the Lagrange multiplier at the κth step, which is updated by a bisectioning algorithm. In Fig. 2.7, the flowchart of the ITO is given, and the main parts within red box in the flowchart are different from the traditional topology optimization methods.

Fig. 2.7 Basic flowchart of the ITO

2.5 Numerical Examples

29

2.5 Numerical Examples In the latter numerical examples, Young’s modulus for the solid materials are set to be 1, and Poisson’s ratio is equal to 0.3. The value of the imposed load is set as 1. In each IGA element, 3 × 3 (2D) or 3 × 3 × 3 (3D) Gauss quadrature points are chosen. In the initial designs, the values of control densities are to be 1, and the corresponding penalty parameter γ is set as 3. In the optimization, the terminal criterion is that the L ∞ norm of the difference of the nodal densities between two consecutive iterations is less than 1% or the maximum 150 iteration steps are reached.

2.5.1 Several Numerical Examples in 2D Several different design domains with the corresponding loads and boundary conditions are presented in Fig. 2.8, including the classic cantilever beam, MBB beam, Michell-type structure and L beam. A same feature is that these four design domains are the rectangle, and they can be modeled by a single patch in IGA. In these four design domains, structural scales are described by two indices L (length) and H (height). As shown in Fig. 2.9, a complex design domain with the curve geometry is presented. In the case of the current design domain, multiple patches in IGA should be required to exactly model structural design domain. The related structural scales are defined as: L = 9; H = 5; and 2r = 3.75. Figure 2.9c presents a simple IGA mesh with four patches shown in Fig. 2.9b. As we can easily see, the IGA mesh is same as the initial structural design domain.

Fig. 2.8 Design domains with the rectangle geometry

Fig. 2.9 A design domain with the complex geometry, reprinted from Ref. [56], copyright 2019, with permission from Wiley

30

2.5.1.1

2 Density-Based ITO Method

The Classic Cantilever Beam

In the current subsection, the classic cantilever beam with L = 10 and H = 5 is discussed to demonstrate the effectiveness and efficiency of the ITO method. The value of maximum material consumption is set to be 30%. Knot vectors  = {0, 0, 0, 0, 0.01, · · · , 0.99, 1, 1, 1, 1}, H = {0, 0, 0, 0, 0.02, · · · , 0.98, 1, 1, 1, 1} are considered in NURBS and the IGA analysis model with 5000 elements. As shown in Fig. 2.10, the initial definitions of control densities with equal to 1 and the DDF are both provided. The optimized solutions of control densities, the DDF and also the corresponding optimized topology for the design of cantilever beam are displayed in Fig. 2.11. It can be found that the optimized DDF is characterized with the sufficient smoothness and continuity, which can be beneficial to show the corresponding topology. As already discussed in the construction of the DDF, we can know that the key cause stems from the smoothing mechanism to improve the smoothness degree and the higher-order NURBS basis functions to control the continuity of the DDF.

Fig. 2.10 Initial definitions of control design variables and the DDF

Fig. 2.11 Optimized solutions of control design variables and the DDF

2.5 Numerical Examples

31

As presented in Fig. 2.12, several intermediate designs of the DDF are provided. It can be easily found that the advancement of the DDF is characterized with the superior stability during the optimization. Figure 2.13 also presents the convergent histories of the objective function and material volume fraction with several intermediate topologies, which can clearly demonstrate the high efficiency with only 85 steps. Based on some intermediate topologies and the final optimized topology shown in Fig. 2.11c, we can easily observe that several known numerical artifacts, like checkerboards, the zigzag or wavy structural boundaries, the islanding or layering structures, can be completely avoided in the optimized topology. Hence, the currently optimized topology with the related positive features of the cantilever beam and the quick convergence can clearly demonstrate the effectiveness and efficiency of the ITO method for the stiffness design, respectively.

Fig. 2.12 Convergent histories of the DDF

Fig. 2.13 Convergent histories of the objective and constraint functions, and also with several intermediate topologies

32

2 Density-Based ITO Method

2.5.1.2

The Simple Structures with the Single-Patch IGA

Other three design domains shown in Fig. 2.8, including the MBB beam, Michelltype structure and L beam, are discussed in the current subsection to further present the effectiveness and efficiency of the ITO method on the stiffness design by several benchmarks. The corresponding NURBS details in the models of these three design domains are listed in Table 2.1. In the case of the optimization of the MBB beam, the optimized designs are given in Table 2.2 and the convergent histories of the objective function and volume fraction are shown in Fig. 2.14a. Table 2.3 presents the final optimized results of the Michelltype structure, mainly including the DDF and the related topology. As presented in Fig. 2.14b, the corresponding iterations for structural stiffness and material volume fractions are given. The optimized DDF and the corresponding topology of L beam are listed in Table 2.4, and Fig. 2.14c shows the convergent curves for structural stiffness and material volume fractions in the optimization. As we can see, the optimized topologies for three benchmarks are all free of the above numerical artifacts, and the iteration is very stable and quickly arrives at the predefined convergent conditions. Hence, the ITO method can perfectly serve for structural topology optimization by successfully eliminating numerical issues. Table 2.1 Data of NURBS for the latter three design domains Knot vectors 1

L = 18, H = 3; order: 3; IGA mesh: 180 × 30; control points: 183 × 33; design variables: 6039  = {0, 0, 0, 0, 0.0056, · · · , 0.9944, 1, 1, 1, 1}, H = {0, 0, 0, 0, 0.0333, · · · , 0.9667, 1, 1, 1, 1}

2

L = 10, H = 4; order: 3; IGA mesh: 100 × 40; control points: 103 × 43; design variables: 4429  = {0, 0, 0, 0, 0.0100, · · · , 0.9900, 1, 1, 1, 1}, H = {0, 0, 0, 0, 0.0250, · · · , 0.9750, 1, 1, 1, 1}

3

L = 10, H = 5; order: 3; IGA mesh: 100 × 50; control points: 103 × 53; design variables: 5459  = {0, 0, 0, 0, 0.0100, · · · , 0.9900, 1, 1, 1, 1},H = {0, 0, 0, 0, 0.0200, · · · , 0.9800, 1, 1, 1, 1}

Table 2.2 Optimized results of the MBB beam DDF

Topology

Obj: 212.11

Num: 57

2.5 Numerical Examples

33

(a) MBB beam

(b) Michell-type structure

(c) L beam

Fig. 2.14 Convergent histories

Table 2.3 Optimized results of the Michell-type structure DDF

Obj: 38.52

2.5.1.3

Topology

Num: 36

The Complex Structures with the Multi-patches IGA

In the current subsection, the proposed ITO method is applied to optimize a complex structural design domain shown in Fig. 2.9 to demonstrate its effectiveness and efficiency, where the value of material volume fraction in this example is defined as

34

2 Density-Based ITO Method

Table 2.4 Optimized results of the L beam DDF

Topology

Obj

Num

181.51

44

30%. As shown in Table 2.5, the optimized solutions that include the DDF, the corresponding topology, the objective function and the total iterations are listed. Meanwhile, Fig. 2.15 also presents the corresponding convergent histories of structural stiffness and material volume fraction within the optimization with only 75 steps. Table 2.5 Optimized results DDF

Obj: 244.05

Fig. 2.15 Convergent histories

Topology

Num: 75

2.5 Numerical Examples

35

Based on the above results, we can easily obtain that the current ITO method has the powerful capabilities of optimizing the complex structural design domain even with the curve geometry.

2.5.2 Several Numerical Examples in 3D In the current subsection, the optimization of several 3D numerical benchmarks is sufficiently studied by the ITO method to present its powerful effectiveness and efficiency. As shown in Fig. 2.16, a 3D Michell structural with the related loads and boundary conditions is provided, and the corresponding NURBS details for the 3D design domain are given in Fig. 2.16. Figure 2.17 displays a 3D Bridge-type structure with the curved geometry, where the related loads and boundary conditions are already defined. In the optimization of 3D Michell structure, material consumption is defined as 15%, and it is equal to 20% in the optimization of 3D Bridge-type

Fig. 2.16 3D Michell structure: IGA elements 30 × 30 × 20;  = {0, 0, 0, 0.0333, · · · , 0.9667, 1, 1, 1}, H = {0, 0, 0, 0.0333, · · · , 0.9667, 1, 1, 1}, Z = {0, 0, 0, 0.05, · · · , 0.95, 1, 1, 1}; p = q = r = 2, reprinted from Ref. [56], copyright 2019, with permission from Wiley

Fig. 2.17 3D Bridge-type structure: IGA elements 64 × 14 × 14; p = 3, q = r = 2;  = {0, 0, 0, 0, 0.0156, · · · , 0.4844, 0.5, 0.5, 0.5, 0.5156, · · · , 0.9844, 1, 1, 1, 1}, H = {0, 0, 0, 0.0714, · · · , 0.9286, 1, 1, 1}, Z = {0, 0, 0, 0.0714, · · · , 0.9286, 1, 1, 1}, reprinted from Ref. [56], copyright 2019, with permission from Wiley

36

2 Density-Based ITO Method

Table 2.6 Optimized results of 3D Michell-type structure Topology

Cross-sectional view

Table 2.7 Optimized results of the 3D Bridge-type structure Topology

Cross-sectional view

structure. In initial designs, the values of control densities in two cases are all equal to 1. In Table 2.6, the optimized results in the design of 3D Michell structure are provided, mainly including the optimized topology and the corresponding crosssectional view to present the interior geometry. Table 2.7 presents numerical solutions that also include the optimized topology and the related cross-sectional view for the design of 3D Bridge-type structure. It can be easily seen that the optimized 3D topologies in two cases have smooth structural boundaries and unique interfaces within the solids and voids, which can effectively present the merits of the construction of the DDF. As shown in Figs. 2.18 and 2.19, the convergent histories of structural stiffness, material volume fraction and the change of control densities are all presented for the optimization of 3D Michell structure and the 3D Bridge-type structure, respectively. As we can easily see, the total number of the first design is only equal to 34 and the evolving of the changing of control densities between two consecutive iterations is smooth, which can demonstrate the superior ability of the ITO for the design of 3D Michell structure.

2.6 Discussions on the Indispensability of ITO

37

Fig. 2.18 Convergent histories for 3D Michell-type structure

Fig. 2.19 Convergent histories for 3D Bridge-type structure

2.6 Discussions on the Indispensability of ITO 2.6.1 Extension of the DDF As already presented in the above sections, the DDF to represent the structural topology should maintain three requirements: nonnegativity, strict bounds and the nonnegativity of its derivatives. Moreover, the smoothness and continuity of the DDF have a significant effect on the optimized solutions. That is, the development of smoothness mechanism by Shepard function and maintaining the higher-order continuity using NURBS basis functions are both imperative in the optimization.

38

2 Density-Based ITO Method

Fig. 2.20 Construction of an extended DDF

However, although the optimized designs can have smooth structural boundaries and distinct material interfaces, and the transition area from 0 to 1 in the optimized designs has a number of intermediate densities [41, 56, 62], which can add numerical errors in the evaluation of material elastic properties. Hence, an extension of DDF, namely the version 2 of the DDF, should be developed by introducing threshold projection, which can reduce the number of intermediate densities in the optimization. The construction of the extended DDF involves three components: (1) smoothness; (2) continuity; (3) binary property, and the detailed construction of the DDF contains five steps, and only a difference is located at the introducing of threshold projection after the step 2, namely φˆ i, j =

   tanh(βη) + tanh β φ˜ i, j − η tanh(βη) + tanh(β(1 − η))

(2.27)

where β and η are the related parameters in threshold projection. The DDF will be constructed by the NURBS basis functions linearly combined with projected control design variables in Eq. (2.27). A detailed process with the comparison is shown in Fig. 2.20. As we can easily see, the extended DDF can extensively reduce the number of intermediate densities in the final layout.

2.6.2 Comparisons Between ITO and FEM-Based Three-Field SIMP As shown in Fig. 2.21, two flowcharts of the ITO method and the FEM-based threefield SIMP are provided. The distinct differences mainly contain the following points:

2.6 Discussions on the Indispensability of ITO

39

Fig. 2.21 Flowcharts: a the ITO; b the FEM-based three-field SIMP

• The extended DDF in the ITO method offers a promising manner for topology representation using a smooth and continuous function, rather than a series of discrete element densities in three-field SIMP method; • The same NURBS basis functions adopted in the extended DDF and numerical analysis can effectively ensure their integration in topology optimization. However, the linear function in the construction of three-field element densities and the shape function in numerical analysis by FEM are completely different, which can result in numerical deviations; • In numerical analysis, geometry information of design domain and the topology information of the DDF can be exactly considered in IGA, whereas only the topology information stemming from discrete element densities is utilized in the FEM. The difference is discussed in the last paragraph of Sect. 2.2.3, shown in Fig. 2.6. In actual, the development of the ITO method is based on the FEM-based threefield SIMP method. For example, the filtering scheme and threshold projection in the

40

2 Density-Based ITO Method

Table 2.8 Increasing criterion of parameter beta

FEM-based three-field SIMP are inherited in the construction of the DDF, where only the difference is NURBS parameterization to ensure the features of the continuity. Overall speaking, the considerations of IGA in the optimization can improve numerical precision. However, the effectiveness of the higher numerical precision under the same finite elements in IGA and the indispensability of IGA are still unavailable in topology optimization.

2.6.3 Numerical Examples In the threshold projection, the parameter η is set to be 0.5, and β increases within the optimization, and the increasing criterion is listed in Table 2.8. The convergent condition is that the L ∞ norm of the difference of control design variables between two consecutive iterations is less than 1% within the maximum 400 steps.

2.6.3.1

The Effectiveness of the Extended DDF

In this subsection, the key intention is to study the effectiveness of the extended DDF and shows the necessity of the initial DDF. Material penalization model is used in this subsection, and the penalty parameter is set to be 3. The detailed design parameters of three benchmarks are defined in Table 2.9, including structural scales, the degrees of NURBS basis functions in two parametric directions, the total numbers of control points (also equal to the total number of control densities), the total numbers of IGA elements and the maximum of material consumption. Table 2.9 Related parameters of three benchmarks L

H

p

q

Control points

IGA elements

V max

(a)

10

5

1

1

161 × 81

160 × 80

0.2

(b)

10

4

1

1

151 × 61

150 × 60

0.2

(c)

12

4

1

1

181 × 61

180 × 60

0.2

2.6 Discussions on the Indispensability of ITO

41

As shown in Fig. 2.22, the optimized solutions of cantilever beam by the ITO method with the initial DDF and the extended DDF are provided, including the optimized distributions of control densities, the DDF and the corresponding topologies. Figures 2.23 and 2.24 display the optimized results of Michell structure and MBB beam, respectively. As we can easily find, the optimized values of control densities in Fig. 2.22a1, Fig. 2.23a1 and Fig. 2.24a1 are mostly equal to 0 or 1, namely the binary distribution of control densities. However, a few numbers of intermediate values of control densities exist in the optimized distributions shown in Fig. 2.22a2, Fig. 2.23a2 and Fig. 2.24a2. As shown in Fig. 2.25, the data layouts of control densities in Fig. 2.22a1, a2, Fig. 2.23a1, a2 and Fig. 2.24 (a1, a2) are all provided. As we can easily see, considering the threshold projection in the construction of the

Fig. 2.22 Optimized designs of cantilever beam: a1–c1 the ITO method using the extended DDF; a2–c2 the ITO method using the initial DDF

Fig. 2.23 Optimized designs of Michell structure: a1–c1 the ITO method using the extended DDF; a2–c2 the ITO method using the initial DDF

42

2 Density-Based ITO Method

Fig. 2.24 Optimized designs of MBB beam: a1–c1 the ITO method using the extended DDF; a2–c2 the ITO method using the initial DDF

Fig. 2.25 Data layout of density values of control design variables in the optimized designs presented in Fig. 2.22a1, a2; Fig. 2.23a1, a2 and Fig. 2.24a1, a2

extended DDF can be beneficial to reducing the number of intermediate densities and lowering the grayscale in the optimized DDF. On the other side, if the optimized layouts of control densities become a binary manner, the optimized topologies have wavy structural boundaries, as shown in Fig. 2.22c1, Fig. 2.23c1 and Fig. 2.24c1. The wavy feature of structural boundaries can result in difficulties in the latter manufacturing, and it is similar to zigzag boundaries in the densities-based method. Hence, the post-processing scheme to smooth structural boundaries is required. In actual, if the number of intermediate densities of control densities can be reasonably adjusted and then reduced to an extent. Moreover, only a limited number of intermediate values of control densities are still retained in the extended DDF, which can maintain the generation of a narrow transition area

2.6 Discussions on the Indispensability of ITO

43

for smooth boundaries in the optimized topologies. Hence, a reasonable criterion should be defined in the ITO method, such that the smooth or binary designs can be achieved based on the engineering requirements.

2.6.3.2

The Indispensability of the IGA to Replace FEM in Topology Optimization

In this subsection, the key intention is to discuss the indispensability of the IGA to replace FEM in topology optimization. Firstly, the FEM-based three-field SIMP method is applied to optimize above three examples with same design parameters, including the numbers of finite elements, the maximum of volume fraction and a same increasing criterion of parameters in threshold projection. As displayed in Fig. 2.26, the optimized topologies of three examples are provided. The topologies in the first row are optimized by the FEM-based three-field SIMP method with material penalization model, and the second row consists of the designs optimized by the FEM-based three-field SIMP method with ersatz material model. As we can easily see, the FEM-based three-field SIMP method with ersatz material model cannot achieve a reasonable topology with a full loading-transmission path. Secondly, the proposed ITO method is employed to optimize three benchmark examples. A critical difference is ersatz material model which is used in the current subsection. As already discussed in Sect. 2.3, there is no penalization mechanism in ersatz material model. As shown in Fig. 2.27, three topologies optimized by the

Fig. 2.26 Numerical results of three benchmark examples

Fig. 2.27 Optimized designs of three benchmark examples using the ITO method with ersatz material

44

2 Density-Based ITO Method

Fig. 2.28 Convergent histories of structural compliance and parameter Beta of Fig. 2.27a–c

ITO method with ersatz material model are given, which are similar to the optimized designs shown in Figs. 2.22, 2.23 and 2.24. Meanwhile, the convergent histories of structural compliance of three benchmark examples shown in Fig. 2.28, which are plotted by the red color. It can be easily seen that iterative curves of three benchmarks are all characterized with a clear and smooth variation in the optimization. Additionally, some slight fluctuations emerge in the convergent process, such as the iteration 29, 70 in the iterative curve. It results from the increasing of parameter Beat in threshold projection during the optimization, which intends to push control design variables toward 0 or 1. Hence, as we can conclude, the extended DDF with threshold projection in the current ITO method can gradually tend to approach the binary layout with a reasonable topology when the parameter Beta of threshold projection increases during the optimization. Additionally, several intermediate topologies of cantilever beam are presented in Fig. 2.29. We can easily see that the earlier iterations with the increasing of parameter Beta facilitate the generation of the key structural members in design domain, until a full loading-transmission path is generated in the final topology. In the latter iterations, the increasing of the parameter Beta aims to push all control design variables toward 0 or 1 and form a nearly pure 0–1 distribution of the extended DDF. However, the FEM-based three-filed SIMP method with ersatz material model has no capability of seeking for a reasonable topology in a stable process, presented in Fig. 2.26a2–c2. With the increasing of finite elements in FEM to improve numerical precision, it can enhance the stability of the optimization to find several terrible designs. Hence, it can reveal that the higher numerical precision of IGA to replace the FEM can maintain the effectiveness of the ITO method with ersatz material model to find an appropriate design. Hence, it is indispensable to develop the framework of the ITO to seek for the optimal material distribution in design domain.

2.6 Discussions on the Indispensability of ITO

45

Fig. 2.29 Several intermediate topologies of the cantilever beam using the ITO method with ersatz material model

2.6.3.3

Stiffness-Maximization Designs in 3D

In this subsection, the key focus is to study the utility of the proposed ITO method with the extended DDF and ersatz material model in the optimization of 3D structures. As displayed in Fig. 2.30, three different 3D structures with loads and boundary conditions are defined, including Michell structure, L-type structure and Bridge-type structure. The related optimization parameters of three benchmarks are provided in Table 2.10.

Fig. 2.30 Three structures with boundary and load conditions

Table 2.10 Related parameters of three benchmarks p

q

r

Control points

IGA elements

V max

(a)

1

1

1

31 × 31 × 21

30 × 30 × 20

0.2

(b)

2

1

1

76 × 19 × 19

72 × 18 × 18

0.2

(c)

2

1

1

73 × 19 × 19

70 × 18 × 18

0.2

46

2 Density-Based ITO Method

As shown in Fig. 2.31, the optimized designs of Michell structure, L-type structure and Bridge-type structure are provided, including the smooth structural topologies that can be manufactured and the nearly pure 0–1 designs, which can clearly present the effectiveness of the current ITO method with the extended DDF. Convergent histories of structural compliance, parameter Beat, the degree of grayscale and numerical error of Michell structure are provided in Fig. 2.32. As we can see, iterative curves have smooth variations and stable convergence. Several intermediate topologies of Michell structure are shown in Fig. 2.33. It can be easily seen that the role of threshold projection in the earlier iterations focuses on generating a complete loading-transmission path. In the latter iterations, the increasing of parameter Beta

Fig. 2.31 Optimized designs of three structures using the ITO method

Fig. 2.32 Convergent histories of numerical results in the optimization of 3D Michell structure

2.7 Appendix for Sensitivity Analysis

47

Fig. 2.33 Intermediate designs of 3D Michell structure

in threshold projection can transform the smooth design into a wavy topology with a pure 0–1 density layout.

2.7 Appendix for Sensitivity Analysis In the ITO formulation, the design variables are control densities, and the first-order derivative of the objective function with respect to the DDF can be derived as: ∂J = ∂

 ˙ T D( )ε(u)d + ε(u) 

1 2

 ε(u)T 

∂D( ) ε(u)d ∂

(2.28)

where u˙ is the first-order of the displacement field with respect to the DDF. Performing the first-order derivative on both sides of the equilibrium state equation, given by: ⎧   ∂a ⎪ T ⎪ = ε(P u) D( )ε(δu)d + ε(u)T D( )ε(δP u)d ⎪ ⎪ ⎨ ∂    ∂D( ) ε(δu)d + ε(u)T ⎪ ⎪ ∂  ⎪  ⎪ ⎩ ∂l =  fδP ud + hδP ud 

∂

N

(2.29)

N

where δ u˙ is the derivative of the virtual displacement field with respect to the DDF. Considering that δ u˙ ∈ H 1 (), the corresponding equilibrium state equation is given as:    ˙ ˙ ˙ N ε(u)T D( )ε(δ u)d = fδ ud + hδ ud (2.30) 



N

48

2 Density-Based ITO Method

Substituting Eq. (2.30) into Eq. (2.29) and eliminating all the terms containing ˙ and a much more compact form of Eq. (2.30) can be obtained, as: δ u, 

 ˙ T D( )ε(δu)d = − ε(u) 

ε(u)T 

∂D( ) ε(δu)d ∂

(2.31)

It is known that compliance optimization design problems are self-adjointed. Hence, Eq. (2.31) can be expanded as a form by: 

 ˙ D( )ε(u)d = − ε(u)

ε(u)T

T





∂D( ) ε(u)d ∂

(2.32)

Substituting Eq. (2.32) into Eq. (2.28). The first-order sensitivity of the mean compliance with respect to the DDF is gained, explicitly as: ∂J 1 =− ∂ 2

 ε(u)T 

∂D( ) ε(u)d ∂

(2.33)

As given in Sect. 2.2.3, the elastic tensor of the isotropic material is assumed to be an exponential function of the DDF. Equation (2.33) can be expanded as a new form, by: 1 ∂J =− ∂ 2





ε(u)T γ γ −1 D0 ε(u)d

(2.34)



Similarly, the first-order derivative of the volume constraint is given as: 1 ∂G = || ∂

 υ0 d

(2.35)



Firstly, the first-order sensitivity analysis of the DDF is derived by: ∂ (ξ, η) p,q = Ri, j (ξ, η) ∂ φ˜ φi, j

(2.36)

p,q

where Ri, j (ξ, η) denotes the NURBS basis function at the computational point (ξ, η). Based on the Shepard function, the first-order derivatives of the smoothed control design variables with respect to control design variables are described as: ∂ φ˜ φi, j = ψ φi, j ∂φi, j

(2.37)

2.8 Summary

49

where ψ φi, j is the value of the Shepard function at the current control point (i, j). It is noted that the computational point (ξ, η) is different from control point (i, j). In the current formulation, the computational points are the Gauss quadrature points. The first-order derivatives of the objective, and constraint functions with respect to the nodal densities are derived based on the chain rule, stated by:   ⎧  γ −1 p,q ⎪ T ∂J 1 ⎪ ⎨ ∂φi, j = − 2 ε(u) γ D0 ε(u)d Ri, j (ξ, η)ψ φi, j     ⎪ p,q ∂G ⎪ 1d Ri, j (ξ, η)ψ ρi, j ⎩ ∂φi, j =

(2.38)



It can be seen that the derivatives of the objective function are negative owing to the nonnegativity of the NURBS basis functions and the Shepard function.

2.8 Summary In this chapter, an isogeometric topology optimization (ITO) method with powerful capabilities is proposed, where the same NURBS basis functions are simultaneously applied to construct structural geometry, the space of numerical analysis and topology description model. The NURBS-based IGA is considered here to replace the FEM and solve the unknown structural responses. In the construction of material description model using the DDF, two different versions are proposed. The first version intends to construct a higher-order smooth and continuous DDF to represent structural topology, and the second version further considers the binary characteristic in the density distribution to remove a number of intermediate densities. Several numerical examples are, respectively, studied to present the applicability, the high optimization efficiency and the robustness of the current ITO method. It can be found that the new method is effective in removing numerical artifacts, including checkerboards, the zigzag boundaries, the mesh-dependency and the islanding or layering structures. The optimized topologies have smooth boundaries and the distinct interfaces between the solids and voids. We can conclude that the NURBS-based IGA can perfectly serve for the optimization of structures. Moreover, three critical problems in previous works of SIMP-based topology optimization are also extensively reconsidered and then discuss how to remove them using the current ITO method with the DDF of version 2, and several important remarks can be achieved: (1) the necessity of isogeometric analysis to replace FEM in the optimization; (2) the overestimation problem of structural stiffness in density-based methods using material penalization model; (3) the segmented representations of the smooth and binary topologies; (4) the ITO method with ersatz material model, the threshold projection in the extended DDF can effectively push the generation of an appropriate topology.

Chapter 3

Density-Based Multi-material ITO (M-ITO) Method

In Chap. 2, the main focus is to develop a promising ITO method to seek for the optimal layout of single material in a given design domain. However, in the practice use, the considerations of many types of materials generally emerge in the engineering applications. Hence, how to distribute multiple kinds of materials in a given design domain to achieve the expected structural performance still poses more challenges than the single-material topology optimization. In multi-material designs, a critical issue is how to develop a rational description model to present the distribution of multiple materials in design domain.

3.1 NURBS-Based Multi-material Interpolation (N-MMI) As an example, a design domain should be rationally distributed by  distinct materials that can have an expected structural performance. Overall speaking, a multimaterial description model is required to represent the layout of multiple kinds of materials, which is mainly involved into two files, namely topology variables and design variables. In a multi-material description model,  fields of topology variables (TVFs) φ ϑ (ϑ = 1, 2, . . . , ) are required to define the layout of  distinct materials, and  fields of design variables (DVFs) ϕ ϑ (ϑ = 1, 2, . . . , ) are also needed to define all TVFs, and each of TVFs is defined by a combination of all DVFs.

3.1.1 Field of Design Variables (DVF) In the construction of the DVF, the basic flowchart is consistent with the development of the DVF in Chap. 2, and mainly containing three steps: (1) the definition of control design variable at control points ρi, j (i = 1, 2, . . . , n; j = 1, 2, . . . , m), where n © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gao et al., Isogeometric Topology Optimization, Engineering Applications of Computational Methods 7, https://doi.org/10.1007/978-981-19-1770-7_3

51

52

3 Density-Based Multi-material ITO (M-ITO) Method

and m are total numbers of control design variables in two parametric directions, also equal to the numbers of control points; (2) the smoothing mechanism is applied to improve the smoothness of control design variables; (3) the NURBS parameterization is applied to construct the DVF. In the current multi-material description model,  different DVFs will be defined for constructing the TVFs that represent the overall layout of multiple kinds of materials in the design domain, and the mathematical formula of the DVF can be written as: ⎞ ⎛ m M n  N      p,q Ri, j (ξ, η)⎝ ψ ρi, j ρi, j ⎠ (3.1) ϕ(ξ, η) = i=1 j=1

i=1 j=1

On the basis of the above steps,  kinds of DVFs can be defined respectively, ϕ ϑ (ϑ = 1, 2, . . . , ).

3.1.2 Field of Topology Variables (TVF) Each of TVF in the multi-material description model is applied to determine the overall layout of the distinct material in the design domain, and ϑ which

(φ (ϑ = 1, 2, . . . , )) is defined by a summation of all DVFs (ϕ = ϑ ϕ , (ϑ = 1, 2, . . . , )), stated by: φϑ =

ϑ  λ=1

ϕλ

  

 1 − ϕλ ,

(ϑ = 1, 2, . . . , )

(3.2)

λ=ϑ+1

The detailed forms in three cases with  = 1, 2, 3 are clearly defined as: ⎧ ⎨  = 1 : φ1 = ϕ1   = 2 : φ 1 = ϕ 1 1 − ϕ 2 ; φ 2 = ϕ1 ϕ 2   ⎩  = 3 : φ1 = ϕ1 1 − ϕ2 1 − ϕ3 ; φ2 = ϕ1ϕ2 1 − ϕ3 ; φ3 = ϕ1ϕ2ϕ3

(3.3)

As shown in Fig. 3.1, a clear description for three cases is given. We can easily see that  DVFs are combined in the description for each distinct material , and a unique material is plotted by a distinct color, namely material 1 plotted with the black, material 2 filled with the red and material 3 with the green. Hence, a critical feature is that each TVF is related to all DVFs, and only change a DVF in the optimization can effectively affect the variation of the corresponding material.

3.1 NURBS-Based Multi-material Interpolation (N-MMI)

53

Fig. 3.1 Multi-material topology description in the N-MMI model, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

3.1.3 Multi-material Interpolation Model In the single-material optimization, the discrete densities of the DDF at Gauss quadrature points are penalized with the corresponding parameter in a power function, and then which are interpolated with the constitutive elastic property of materials in the computation of all IGA element stiffness matrices. In the current multi-material description model, the corresponding discrete densities of TVFs at Gauss quadrature points should be considered in the latter computation, mainly because they determine the distribution of materials, rather than the DVFs. Hence, the corresponding multimaterial interpolation model can be computed by a summation of all TVFs interpolated with the corresponding constitutive elastic tensors of materials, expressed as:  ϑ        γ   ϑ γ ϑ    λ λ γ D= φ ϕ 1−ϕ D0 = Dϑ0 (3.4)    ϑ=1 ϑ=1 λ=1 λ=ϑ+1 wϑ    wϑ

where Dϑ0 denotes the constitutive elastic tensor matrix of the ϑth distinct material, and γ indicates the penalty parameter. γ refers to the penalty parameter. Recently, several multi-material description models are developed, mainly including the mixture rule [95,96], the DMO scheme [61], the extensions RMMI and UMMI (UMMI-1 and UMMI-2) models of the DMO scheme [98]. The detailed

54

3 Density-Based Multi-material ITO (M-ITO) Method

Fig. 3.2 Expression and evolving mechanisms of design variables and topology variables, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

comparisons about the DMO, RMMI, UMMI and the current N-MMI models can refer to [62]. A distinct feature of many previous models (DMO scheme, mixture rule, RMMI and UMMI models) is that the design variables and topology variables are integrated by a mathematical language, so that a parallel optimization mechanism exists in the evolvement of design variables and topology variables, clearly shown in Fig. 3.2a. However, several numerical troubles occur in the optimized topologies, such as the unrealistic designs with the mixed materials, the local solution, etc., and the detailed discussions are given in [60, 61, 98]. In the case of the developed N-MMI model, it can be still regarded as a branch of the DMO scheme, but several improvements are considered, mainly including: (1) the decouple description manner is used in the construction of design variables and topology variables; (2) design variables and topology variables are advanced in a serial optimization mechanism, shown in Fig. 3.2b. These above two advantages can be beneficial to omit these previously mentioned numerical troubles extensively. In the viewpoint of the TVFs, the related mathematical equation of the N-MMI model is similar to the UMMI-1 model, and it is also analogous to the UMMI-2 model in the viewpoint of the DVFs.

3.2 Multi-material ITO (M-ITO) In the current work, the classic compliance-minimization design problems are still considered here to present the effectiveness and efficiency of the M-ITO method and also demonstrate the unique merits of the N-MMI model in the optimization. In the multi-material description model, structural stiffness is the concerned performance to act as the objective function in the optimization, multiple volume constraints are considered in the formulation, and the detailed formula can be given as: Find : ρi,ϑ j (ϑ = 1, 2, . . . , ; i = 1, 2, . . . , m; j = 1, 2, . . . , n)

3.2 Multi-material ITO (M-ITO)

55

    1 Min : J (u, φ) = D φ ρi,ϑ j ε(u)ε(u)d 2 ⎧ a(u, δu) = l(δu), u| D = g, ∀δu ∈ H 1 ( ) ⎪ ⎪ ⎪  ⎨    1 ˆ ˆ ϑˆ φ ϑ ρi,ϑ j υ0 d − V0ϑ ≤ 0, ϑˆ = 1, 2, . . . ,  S.t. : G v = ⎪ | | ⎪ ⎪ ⎩ 0 ≤ ρi,ϑ j ≤ 1

(3.5)

where ρi,ϑ j is the (i, j)th control design variable for the ϑth DVF. J indicates the objec  ˆ tive function defined by structural compliance. G ϑv ϑˆ = 1, 2, . . . ,  is the volume ˆ constraint for the ϑˆ th unique material. V0ϑ is the corresponding maximum consumpˆ tion and υ0 is the volume fraction of the solid. φ ϑ is the TVF to present the layout of the ϑˆ th material. u is the displacement field in the domain . g is the prescribed displacement vector on  D , and δu is the virtual displacement field belonging to the space H 1 ( ). a and l are the bilinear energy and linear load functions, given as:



    a(u, δu) = D φ ρi,ϑ j ε(u)ε(δu)d   l(δu) = fδud +  N hδud N

(3.6)

where f is the body force and h is the boundary traction on  N . The powerful utility and capability of the M-ITO method are also discussed, and a practical problem to seek for the optimized topologies with the total mass constraint is also considered, and the alternative formula is stated as: Find : ρi,ϑ j (ϑ = 1, 2, . . . , ; i = 1, 2, . . . , m; j = 1, 2, . . . , n)     1 D φ ρi,ϑ j ε(u)ε(u)d Min : J (u, φ) = 2 ⎧ a(u, δu) = l(δu), u| D = g, ∀δu ∈ H 1 ( ) ⎪ ⎪ ⎪ ⎪ ⎪    ⎨  ˆ ˆ G ϑv ϑ0 − M0 ≤ 0, ϑˆ = 1, 2, . . . ,  S.t. : G m = ⎪ ⎪ ˆ ⎪ ϑ=1 ⎪ ⎪ ⎩ 0 ≤ ρi,ϑ j ≤ 1 

(3.7)

where 0ϑ is the mass density of the ϑ th unique material, and M0 is the maximal value of structural mass in the constraint G m . Compared with the M-ITO formulation with multiple volume constraints, the current M-ITO formulation in the above equation can effectively study the influence of structural mass and material volume fractions on the distribution of different materials during the optimization. 

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3 Density-Based Multi-material ITO (M-ITO) Method

3.3 Design Sensitivity Analysis The sensitivity of structural compliance, volume constraints and a mass constraint are required in the latter algorithms. The derivatives of structural compliance with ˆ respect to the TVS φ ϑ are derived as: ∂J ∂φ ϑˆ

 =

   ∂D φ ϑˆ     1 ˆ ˆ = 1, 2, . . . ,  ˙ + D φ ϑ ε(u)ε(u)d ε(u)ε(u)d , ϑ 2 ∂φ ϑˆ

(3.8) where u˙ is the first-order derivative of the displacement field with respect to the TVS. Performing the derivatives on both sides of the equilibrium state equation, by:   ⎧ ∂a ⎪ ⎪ = Dε(P u)ε(δu)d + Dε(u)ε(δP u)d ⎪ ⎪ ϑˆ ⎪ ⎪   ⎨ ∂φ  ∂D φ ϑˆ ⎪ ε(u)ε(δu)d + ⎪ ⎪ ⎪ ∂φ ϑˆ  ⎪  ⎪ ⎩ ∂l ˆ = fδP ud +  N hδP ud N ∂φ ϑ

(3.9)

where δ u˙ is the derivative of the virtual displacement field with respect to the TVS, Considering that δ u˙ ∈ H 1 ( ), the equilibrium state equation is given as: 

    ϑˆ ˙ ˙ ˙ N = fδ ud D φ ε(u)ε(δ u)d + hδ ud





(3.10)

N

˙ Substituting Eq. (3.10) into Eq. (3.9) and eliminating all the terms containing δ u, and a much more compact form of Eq. (3.9) can be obtained with considering the self-adjoint, stated as: 

   ∂D φ ϑˆ   ˆ ˙ D φ ϑ ε(u)ε(u)d ε(u)ε(u)d =− ∂φ ϑˆ

(3.11)



Hence, the derivatives of the compliance with respect to the TVS are obtained, explicitly expressed as: ∂J

1 =− ˆ ϑ 2 ∂φ

   ∂D φ ϑˆ

∂φ ϑˆ

ε(u)ε(u)d

(3.12)

3.3 Design Sensitivity Analysis

57

The sensitivity of structural compliance can be solved by calculating the derivative of material elastic tensor with respect to the VTS. Based on the N-MMI model, sensitivity of the multi-material elastic tensor can be computed, stated as: ∂J ∂φ ϑˆ

=−

   ∂D φ ϑˆ 1 2



∂φ ϑˆ

ε(u)ε(u)d

(3.13)

ˆ

As discussed in the development of the N-MMI model, the VTS φ ϑ to display the layout of the ϑˆ unique material is expressed as a combination of all DVFs, and each DVF is developed by NURBS basis functions with the nodal design variables ˆ ρ ϑ . We can firstly derive the derivative of the VTS φ ϑ with respect to the DVF ϕ ϑ , given as: ˆ

∂φ ϑ = ∂ϕ ϑ

 ϑˆ

   λ 1 − ϕλ ˆ λ=1,λ=ϑ ϕ λ=ϑ+1    ˆ 1 − ϕλ − ϑλ=1 ϕ λ  ˆ λ=ϑ+1,λ =ϑ

if ϑ ≤ ϑˆ , (ϑ = 1, 2, . . . , ) if ϑ > ϑˆ (3.14)

Then, the derivative of the DVF with respect to control design variables can be expressed as:   ∂ϕ ϑ p,q = Ri, j (ξ, η)ψ ρi,ϑ j ϑ ∂ρi, j

(3.15)

p,q

where Ri, j (ξ, η) is the NURBS basis function at the computational point (ξ, η).   ψ ρi,ϑ j is the Shepard function at the current control point (i, j). The derivative of the structural compliance with respect to control design variables can be derived, and the final form is explicitly expressed as:  ˆ  ∂J ∂ J ∂φ ∂ϕ ϑ ∂ J ∂φ ϑ ∂ϕ ϑ = = = ··· ∂φ ∂ϕ ϑ ∂ρi,ϑ j ∂ρi,ϑ j ∂φ ϑˆ ∂ϕ ϑ ∂ρi,ϑ j ˆ ϑ=1 ⎧ ⎧    ϑ γ −1 ϑˆ  λ γ   γ  ⎫ ⎨−1 ⎬ ⎪ γ ϕ ϕ 1 − ϕλ ˆ ⎪ λ=1,λ = ϑ 2 λ= ϑ+1 ⎪   ⎪ l if ϑ ≤ ϑˆ ⎪ ϑ ⎭  ⎪ ⎨ ⎩ Dϑ0ˆ · · · Ri,p,q η)ψ ρ ε(u)ε(u)d (ξ,  i, j j ⎧       ⎫ ϑˆ  λ γ  γ −1 ⎪ 1 ϑ λ γ ⎬ ⎨− ⎪ ϕ 1 − ϕ −γ 1 − ϕ ⎪ ˆ ˆ λ=1 ϑ=1 2 λ= ϑ+1,λ = ϑ ⎪   ⎪ if ϑ > ϑˆ ⎪ ⎩ ⎩ Dϑˆ · · · R p,q (ξ, η)ψ ρ ϑ ε(u)ε(u)d ⎭ 0

i, j

i, j

(3.16) In the case of multiple volume constraints, the first-order derivatives of all volume constraints with respect to control design variables can be derived as:

58

3 Density-Based Multi-material ITO (M-ITO) Method ˆ

ˆ

ˆ

∂G ϑv ∂G ϑv ∂φ ϑ ∂ϕ ϑ = = ··· ∂ρi,ϑ j ∂φ ϑˆ ∂ϕ ϑ ∂ρi,ϑ j ⎧  ϑˆ   p,q  ϑ   λ λ ⎨ 1 1 − ϕ Ri, j ψ ρi j υ0 d if ϑ ≤ ϑˆ ϕ ˆ λ=1,λ = ϑ | | λ=ϑ+1  ˆ    p,q    J  λ ⎩− 1 1 − ϕ λ Ri, j ψ ρiJj υ0 d if ϑ > ϑˆ λ=1 ϕ | | λ= Jˆ+1,λ1J

(3.17)

Based on the development of the total structural mass, the derivative of the total mass constraint with respect to control design variables can be expressed by:  ˆ  ∂G ϑv ϑˆ ∂G m =  = ··· ∂ρi,ϑ j ∂ρi,ϑ j 0 ˆ ϑ=1 ⎧ ⎫ ⎞ ⎛  ϑˆ  ⎪ ⎪   ⎪ ⎪     1 ⎪ ⎪ p,q λ λ ϑ ⎠ ϑˆ ⎪ ⎪ ⎝ ˆ 1 − ϕ R  ϕ ψ ρ υ d if ϑ ≤ ϑ ⎪ ⎪ 0 0 i j i, j ⎪ ⎪ ⎪ ⎪ | |  ⎨ ⎬  λ=1,λ=ϑ ˆ λ=ϑ+1 ⎞ ⎛  ⎪ ⎪ ϑˆ  ⎪ ⎪     p,q  ϑ  ⎪ ⎪ ˆ 1 ϑ=1 ϑˆ λ λ ⎪ ⎪ ⎠ ⎝ ˆ ⎪ ⎪ 1 − ϕ R  − ϕ ψ ρ υ d if ϑ > ϑ 0 ⎪ ⎪ 0 ij i, j ⎪ ⎪ ⎩ | | ⎭ λ=1

ˆ λ=ϑ+1,λ =ϑ

(3.18) Hereto, the sensitivity analysis for the above two M-ITO formulations is derived in detail. In the M-ITO method, the method of moving asymptotes (MMA) [220] is applied to solve the multi-material topology optimization design problems with different constraints. In the latter subsection, numerical examples in 2D and 3D are all considered by the M-ITO method to demonstrate the effectiveness and efficiency. In all numerical examples, the linearly elastic materials are only considered. In examples of 2D cases, structural design domains will be discretized by several IGA element with only one unit edge thickness. The imposed loads with the value equal to 1 are considered in all numerical examples, and 3 × 3 (2D) or 3 × 3 × 3 (3D) number of Gauss quadrature points are considered in the computation of all IGA element stiffness matrices. In the initial designs, the values of all control design variables are equal to 0.5, and the penalty parameter is equal to 3 which is same as the singlematerial topology optimization. The iteration will be terminated if the maximum change of the DVFs is lower than 1% within 300 iterations. Four virtual isotropic solid materials will be considered in next examples, and the corresponding details are listed in Table 3.1.

3.4 Numerical Examples in 2D In the current section, the M-ITO formulation 1 with multiple kinds of material volume constraints is firstly discussed, and the MBB beam displayed in Fig. 3.3

3.4 Numerical Examples in 2D

59

Table 3.1 Four “virtual” isotropic solid materials, reprinted from Ref. [62], copyright 2020, with permission from Elsevier i

Materials

Young’s modulus:E 0i

Poisson’s ratio υ

Mass density:i0

Stiffness-to-mass ratio:Rio

1

M1

10

0.3

2

5

2

M2

10

0.3

5

2

3

M3

5

0.3

2

2.5

4

M4

3

0.3

2

1.5

Fig. 3.3 MBB beam: IGA elements 180 × 30;  = {0, 0, 0, 0.00565, · · · , 0.99444, 1, 1, 1}, H = {0, 0, 0, 0.0333, · · · , 0.9667, 1, 1, 1}; n = 182, m = 32; p = q = 2, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

with the corresponding loads and boundary conditions is considered here. The corresponding structural scales of L and H are set to be 18 and 3, respectively. The classic cantilever beam given in Fig. 3.4 with the corresponding loads and boundary conditions is also discussed but by the developed M-ITO formulation 2 to study the positive features of the N-MMI model on the design problem with the total mass constraint. The related structural sizes of the cantilever beam are equal to 10 (L) and 5 (H), respectively. The related NURBS details of these two examples are expressed in Fig. 3.4.

Fig. 3.4 Cantilever beam: IGA elements 100 × 50;  = {0, 0, 0, 0.01, · · · , 0.99, 1, 1, 1}, H = {0, 0, 0, 0.02, · · · , 0.98, 1, 1, 1}; n = 102, m = 52; p = q = 2, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

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3 Density-Based Multi-material ITO (M-ITO) Method

3.4.1 Two-Material Design 3.4.1.1

Messerschmitt-Bolkow-Blohm (MBB) Beam

In this subsection, the multi-material optimization of MBB beam considers two unique materials (M2 and M3), which are available in the latter topology optimization. In order to present the distribution of two materials, two TVFs should be required in the N-MMI model, namely φ 1 andφ 2 . In the M-ITO formulation, the maximum material consumptions for M2 and M3 materials are defined as 20% and 8%, respectively. In initial definitions, the values of control design variables are initially set to be 0.5, and the corresponding initial distributions of two TVFs can be directly computed. As shown in Fig. 3.5, the initial TVFs for M2 and M3 materials are given. Figure 3.6 presents the optimized designs of the two-material topology in the MBB beam. It can be easily seen that the values of the optimized TVFs are mostly distributed in the lower and upper bounds. On the basis of the optimized TVFs, the topology of each distinct material is defined by the isocontour of the corresponding TVF, and the value of the isocontour is equal to 0.5. As already discussed in Chap. 2 or Refs [56, 65] for the single-material optimization, 0.5 is an extremely suitable value in the representation of the topology using the isocontour of the density response surface using the NURBS parameterization. In the multi-material topology optimization, the value equal to 0.5 is only a reasonable definition in the multi-material representation. The main reason is that no overlaps and no redundant phases should be effectively avoided in the N-MMI model. If the value of the iso in the TVFs is equal to be

Fig. 3.5 Initial designs of two TVFs, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

Fig. 3.6 Optimized design of the MBB beam, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

3.4 Numerical Examples in 2D

61

Table 3.2 Optimized results of the MBB beam with two materials, reprinted from Ref. [62], copyright 2020, with permission from Elsevier M2 material

The topology The TVFs

1

and

2

in the design domain

M3 material

The final topology

0.3, and the corresponding values of the TVFs at structural boundaries of M2 and M3 materials are equal to 0.6, and their summation is 0.6. Hence, the requirement that the summation of the values of all TVFs in each point should be equal to 1 cannot be effectively maintained. We can also conclude that 0.5 is the only value to effectively guarantee the physical meanings of the N-MMI model in the multimaterial representation. It cannot be randomly defined. As listed in Table 3.2, the optimized designs of the MBB beam are provided, including the optimized topology of M2 material, the optimized topology of M3 material, the combined distributions of two TVFs φ 1 andφ 2 , and the final topologies of MBB beam with two materials. The corresponding volume fractions of M2 and M3 materials in the optimized topologies are equal to 19.8% (nearly 20%) and 7.9% (nearly 8%). As we can easily see, the final distributions of two materials in the MBB beam are suitable for the loading the force to offer the higher stiffness, and each unique material can be formed into several independent structural members owing to the comparable of the elastic tensors in two materials. Hence, it can be easily concluded that the M-ITO formulation 1 is effective in the current design problems to seek for the optimal distribution of two materials, and a pure distribution that each point only has one phase can clearly demonstrate the effectiveness of the N-MMI model.

3.4.1.2

Cantilever Beam

In the current work, M2 and M3 materials are still available in the M-ITO formulation 2 for the multi-material design of the classic cantilever beam. The corresponding value of the total structural mass is defined as 30. Other design parameters keep consistent with the above settings. As clearly shown in Table 3.3, the optimized distributions of two TVFs and their corresponding topologies are all provided. We can easily find that the optimized topology of the cantilever beam with M2 and M3 materials has distinct interfaces and smooth boundaries between the solids and voids, which can successfully show the effectiveness of the proposed N-MMI model for the latter multi-material topology optimization. Moreover, similar to the above example, several independent structural members can be also formed by M2 and M3 materials

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3 Density-Based Multi-material ITO (M-ITO) Method

Table 3.3 Optimized results of cantilever beam with two materials, reprinted from Ref. [62], copyright 2020, with permission from Elsevier 1

in the design domain

The topology of M2 material

2

in the design domain

The topology of M3 material

1

and

2

in the design domain

The topology of cantilever beam

in the final topology, where the black areas contain the stronger M2 material and the red regions include the weaker M3 material. According to the overall distribution in the final topology, the critical regions with the high possibility of the occurrence of stress concentrations are mainly filled by M2 material to afford the higher structural stiffness for the cantilever beam, and the weaker M3 material is mainly dominant in the topology to afford the load. The main reason is that a larger stiffness-to-mass ratio of M2 and M3 materials exists in the optimization. The above essential feature can clearly present the M-ITO formulation 2 to improves structural stiffness considering from two factors, namely the stiffness and mass. As shown in Fig. 3.7, the convergent histories of structural stiffness are also presented, which show the superior stability of the M-ITO formulation 2. Moreover, the structural mass of the distinct material is gradually evolved to improve structural stiffness as much as possible, rather than a monotonous manner in the single-material topology optimization.

Fig. 3.7 Convergent histories, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

3.4 Numerical Examples in 2D

63

3.4.2 Three-Material Design 3.4.2.1

Messerschmitt-Bolkow-Blohm (MBB) Beam

In the current subsection, the M-ITO formulation 1 is applied to discuss the optimization of the MBB beam with three materials, and M2, M3 and M4 materials are available during the optimization. In the M-ITO formulation 1 for three-material design, the corresponding volume fractions for M2, M3 and M4 materials are equal to be 20%, 12% and 3%, respectively. In the initial designs, the values of control design variables in the DVFs keep consistent with the above sections, and the corresponding TVFs for M2, M3 and M4 materials are shown in Fig. 3.8. Figure 3.9 presents the optimized of three TVFs for M2, M3 and M4 materials. As we can observe, the optimized TVFs are all characterized with the sufficient smoothness and continuity, which can show the positive features of the N-MMI model. As clearly listed in Table 3.4, the corresponding numerical results of M2, M3 and M4 materials in the MBB beam are provided, and also including the optimized topologies of three materials, the overall distributions of three TVFs and the final topology of three-material design of the MBB beam. The essential feature that the three-material topology of the MBB beam has the unique structural interfaces within multiple materials and the voids can clearly present the superior merits of the proposed N-MMI model, and several basic requirements that can effectively ensure the physical meanings of the N-MMI in the representation of multiple materials are perfectly maintained in the M-ITO formulation 2, namely no overlaps in multiple materials and no redundant phases. Hence, we can conclude that the proposed NMMI can have the powerful capabilities in the multi-material topology optimization by the currently proposed M-ITO method.

Fig. 3.8 Initial designs of three TVFs, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

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3 Density-Based Multi-material ITO (M-ITO) Method

Fig. 3.9 Optimized design of the MBB beam, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

Table 3.4 Optimized results of the MBB beam with three materials M2 material The topology

M4 material

The TVFs

3.4.2.2

M3 material

1

,

2

and

3

in the design domain

The optimized topology

Cantilever Beam

The main focus of the current subsection is to discuss the effectiveness and efficiency of the proposed M-ITO formulation 2 for the three-material optimization of the classic cantilever beam, and also M2, M3 and M4 materials are considered in the current case. The representation of three materials is still consistent with Sect. 3.4.2.1. The corresponding maximum structural mass is set to be 35, and other design parameters are same as the above examples. As listed in Table 3.5, the optimized results of the cantilever beam with three materials are presented, including the distributions of three TVFs for M2, M3 and M4 materials, the corresponding topologies of M2, M3 and M4 materials in the design domain, the combined distribution of three TVFs for M2, M3 and M4 materials and the final topology of the cantilever beam with M2, M3 and M4 materials. As we can easily find, the optimized three-material topology of the cantilever beam

3.4 Numerical Examples in 2D

65

Table 3.5 Optimized results of cantilever beam with three materials, reprinted from Ref. [62], copyright 2020, with permission from Elsevier 1

in the design domain

The topology of M2

1

,

2

and

3

2

in the design domain

The topology of M3

in the design domain

3

in the design domain

The topology of M4

The topology of cantilever beam

is characterized with the smooth boundaries and unique interfaces between the solids and voids. Meanwhile, each of material (M2, M3 and M4 materials) can be reasonably generated into several independent structural members in the final topology to afford the corresponding boundary conditions as much as possible, which can effectively demonstrate the current M-ITO formulation 2 for the optimization of the three-material design problems with the total mass constraint, and the corresponding iterative histories can refer to Fig. 3.10.

Fig. 3.10 Convergent histories, reprinted from ref. [62], copyright 2020, with permission from Elsevier

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3 Density-Based Multi-material ITO (M-ITO) Method

3.4.3 Discussions on the Stiffness-To-Mass Ratio In the current example, the main intention is to address the influence of the stiffnessto-mass ratio on the multi-material topology optimization. A distinct different point is that M1 and M3 materials will be available in the current example compared with previous numerical cases, where M1 material have the larger values of the Young’s modulus and stiffness-to-mass ratio over than M3 material. The mass constraint is set to be 30, which is consistent with Sect. 3.4.1.2. Other design parameters are same as the above examples, mainly including the initial designs of control design variables, the TVFs of two materials, the representation of two materials, etc. Table 3.6 provided the optimized results of the cantilever beam with two materials (M1 and M3), also including the distributions of two TVFs for M1 and M3 materials, a combined distribution of M1 and M3 materials, the optimized topologies of M1 and M3 materials and the optimized topology of the cantilever beam with M1 and M3 materials. As we can easily see, the currently optimized topology of the cantilever beam with M1 and M3 materials is completely different the final topology presented in Sect. 3.4.1.2. In the current example, the final optimized topology of the cantilever beam only has the M1 material, even if M3 material is available in the initial setting of the optimization. Figure 3.11 presents the convergent curves of the iterations for the structural mass of M1 and M3 materials in the optimization, which also shows the structural mass of M3 material equal to zero in the final iteration. Hence, we can conclude that the material with the larger Young’s modulus and stiffness-to-mass ratio is mostly employed in the optimization to afford structural stiffness as much as possible. Moreover, the optimized design also reveals that the currently proposed M-ITO method can effectively maintain physical meanings of the N-MMI model, namely no overlaps and no redundant phases in the multi-phase representation, and there are no mixture materials (unrealistic designs) in the optimization. It can manifest the effectiveness of the proposed M-ITO method and also the positive features of the N-MMI model for multi-material topology optimization. Table 3.6 Optimized results of cantilever beam with two materials, reprinted from Ref. [62], copyright 2020, with permission from Elsevier 1

in the design domain

The topology of M1 material

2

in the design domain

The topology of M3 material

1

and

2

in the design domain

The topology of cantilever beam

3.4 Numerical Examples in 2D

67

Fig. 3.11 Convergent histories, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

3.4.4 Quarter Annulus The main purpose of the current subsection is to address the effectiveness of the M-ITO method on the design of the curved structures. Figure 3.12 presents the quarter annulus with the corresponding loads and boundary conditions, in which structural scales with two indices r and R are defined as 5 and 10, respectively. Two cases with the two-material and three-material designs are both discussed, and the corresponding volume fractions of M2 and M3 materials are set as 25% and 12%, respectively, and the related maximum material consumptions of M2, M3 and M4 materials are respectively set to be 25%, 10% and 5%. Other design parameters keep consistent with the above examples, like the initial designs of control design variables, the TVF, etc. As provided in Table 3.7, the optimized results of the quarter annulus with two materials in case 1 (M2 and M3 materials are available) and three materials in case 2 (M2, M3 and M4 materials) are clearly shown, namely including the optimized Fig. 3.12 Quarter annulus: IGA elements 100 × 50;  = {0, 0, 0, 0, 0.01, · · · , 0.99, 1, 1, 1, 1}, H= {0, 0, 0, 0, 0.02, · · · , 0.98, 1, 1, 1, 1}; n = 103, m = 53; p = q = 3, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

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3 Density-Based Multi-material ITO (M-ITO) Method

Table 3.7 Optimized results of quarter annulus in cases 1 and 2, reprinted from Ref. [62], copyright 2020, with permission from Elsevier 1

in the design domain

The topology of M2

2

1

in the design domain

The topology of M3

1

in the design domain

The topology of M2

3

in the design domain

The topology of M4

2

and

2

The topology of M2 and M3

in the design domain

The topology of M3

and

The topology

1

,

2

3

distributions of all TVFs for multiple materials, the combined distribution of all TVFs, the corresponding topologies of multiple materials and the final topology of the quarter annulus with multiple materials. As we can see, an essential feature of smooth boundaries and distinct interfaces exist in the optimized designs, which can clearly show the effectiveness of the proposed M-ITO method. Meanwhile, no overlaps between multiple materials and no redundant of multiple phases are occurred in the final topologies, which can clearly demonstrate the effectiveness of the NMMI model. Moreover, each of material can be successfully generated into several structural members that can afford the loads as much as possible. Hence, we can conclude that the proposed M-ITO method also has the powerful capabilities for the curved structures with multiple materials.

3.5 Numerical Examples in 3D The current subsection focuses on studying the effectiveness of the M-ITO method on the design of 3D structures with multiple materials. Figure 3.13 shows an example of 3D Michell-type structure with the corresponding loads and boundary conditions. The corresponding NURBS details for 3D Michell-type are given in Fig. 3.13. M2

3.5 Numerical Examples in 3D

69

Fig. 3.13 3D Michell structure: IGA elements 30 × 30 × 18;  = H = {0, 0, 0, 0.0333, · · · , 0.9667, 1, 1, 1}, Z= {0, 0, 0, 0.0556, · · · , 0.9444, 1, 1, 1};n = m = 32, l = 20; p = q = r = 2, reprinted from Ref. [62], copyright 2020, with permission from Elsevier

and M3 materials will be available in the optimization, and the related material volume fractions are set as 14% and 6%, respectively. M2 and M3 materials are shown in the black, red colors, respectively. Other design parameters keep consistent with above examples. As shown in Table 3.8, the optimized results of 3D Michell structure with two materials (M2 and M3) are provided. As already discussed in Chap. 2, the density response surfaces are 4D functions, and it is hard to present 4D function in the Table 3.8 Optimized results of 3D Michell structure, reprinted from Ref. [62], copyright 2020, with permission from Elsevier M2 material

M3 material

The topology

View 1 of the topology

View 2 of the topology

View 3 of the topology

70

3 Density-Based Multi-material ITO (M-ITO) Method

platform of MATLAB. Hence, the isocontours of the TVFs with the higher values are directly presented. Similar to the above examples, the optimized topologies of M2 and M3 materials have the smooth structural boundaries and unique interfaces between multiple materials and the voids. The suitable distributions of M2 and M3 materials in the final topology can provide the higher capability for affording the imposed loads, which also constitute several structural members in the design domain. Hence, the effectiveness of the current M-ITO method on the design of 3D structures can be explicitly presented.

3.6 Summary In the current chapter, the main purpose is to develop a M-ITO method on the multimaterial designs of 2D and 3D structures, where two different topology optimization formulations are developed and the corresponding multiple material volume constraints and the total structural mass constraint are all discussed. In the developed M-ITO method, an N-MMI model is constructed for the representation of multimaterial distributions and the evaluation of the corresponding material properties. An essential feature of the decoupled expression and serial evolving mechanism exist in the N-MMI model for the definition of the DVFs and TVFs. In the latter numerical examples, 2D and 3D structures are both discussed to present the effectiveness of the proposed M-ITO method and also the N-MMI model, and we can gain the following remarks: (1) The N-MMI model has the superior advantages for the description for multi-material layout, namely no overlaps and no redundant phases; (2) the optimized topologies have smooth structural boundaries and distinct interfaces between the solids of multiple materials and voids; (3) the material properties, including the Young’s modulus and the stiffness-to-mass ratio, have the critical influence on the latter optimization; (4) the design problems with different constraints, namely multiple volume constraints and the total structural mass, have different optimized topologies; (5) the effectiveness and efficiency of the M-ITO method with the N-MMI model are sufficiently addressed.

Chapter 4

ITO for Structures with Stress-Minimization

In Chaps. 2 and 3, the classic stiffness-maximization design problems are concerned in the discussions of the effectiveness and efficiency of the proposed ITO and M-ITO methods, due to the corresponding positive features, like the stable convergence and quickly arrive at the expected values. In practical use, the stress is a more critical factor in the design of engineering structures, where stress concentrations are mostly occurred in the practical use, rather than the lower stiffness. However, the considerations of the stress in the topology optimization will introduce several numerical difficulties in the optimization due to the special features of the stress, namely the highly nonlinear behavior, the local naturality and the singular topologies during the optimization. Hence, the stress-related topology optimization poses more challenge than the known compliance-minimization topology optimization designs. In the current chapter, the main intention is to develop a stress-related ITO formulation for the minimization of the global stress to eliminate the occurrence of stress concentrations in structures, which is applied to show the effectiveness and efficiency of the proposed ITO method on the design of stress-related optimization problems. In the ITO method for the stress-related designs, the critical aspects should be considered: (1) how to develop an IGA model to replace the traditional finite element method for stress computation to improve numerical precision; (2) how to develop a stress evaluation model to ensure the computational accuracy of the global stress by aggregating all local stress in the design domain; (3) how to develop the ITO formulation for the stress-minimization designs. The ITO formulation for the stress-minimization also contains some similar aspects: (1) the material description model using the DDF to represent structural topology; (2) the NURBS-based IGA. Meanwhile, in the current chapter, the main intention is to show the effectiveness and efficiency of the ITO method on the stress-minimization design problems, and the related positive features or merits of the ITO on the stress-related design problems have been already submitted to the journal.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gao et al., Isogeometric Topology Optimization, Engineering Applications of Computational Methods 7, https://doi.org/10.1007/978-981-19-1770-7_4

71

72

4 ITO for Structures with Stress-Minimization

4.1 Topology Description Model In Chaps. 2 and 3, the core of material description model is to develop a DDF to represent structural topology in the optimization. Then, the same NURBS basis functions are applied to develop the solution space in analysis. However, a critical limitation will occur in the stress-related topology optimization design problems, owing to a feature that the constructed DDF still has a certain number of intermediate densities resulting from a smoothness mechanism. A peculiarity that the penalization mechanism cannot push the intermediate densities towards the lower or upper bounds exists in stress-related topology optimization. The existence of intermediate densities also offers the high possibility for the generation of degenerate space in the optimization, so that a reasonable distribution with the structural members cannot be found. Additionally, the intermediate densities can also result in a larger error in numerical measure of von Mises stress, which extensively hurt the numerical accuracy of the maximum von Mises by the conventional aggregation formulations.

Fig. 4.1 a1 DDF in 2D view; b1 the 2D view of stress distribution of a1; c1 the optimized topology; d1 the 2D view of stress distribution of c1; e1 numerical deviations of stress measure between b1 and d1; f1 data analysis of e1; a2 The DDF in 2D view; b2 the 2D view of stress distribution of a2; c2 the optimized topology; d2 the 2D view of stress distribution of c2; e2 numerical deviations of stress measure between b2 and d2; f2 data analysis of e2

4.1 Topology Description Model

73

As an example, Fig. 4.1 presents two optimized topologies of the classic L beam by the ITO method in Chap. 2. It can be easily seen that a certain number of intermediate densities of the DDF exist in the final optimized topologies, clearly displayed in Fig. 4.1a1, a2. As already discussed in the above paragraph, the existence of intermediate densities in the optimization will extensively damage the effectiveness of the stress-related ITO designs, so that a reasonable topology without a complete loading-transmission path is generated, and an example is shown in Fig. 4.1c2. As shown in Fig. 4.1e1, the corresponding stress evaluation errors in the optimized DDF (Fig. 4.1b1) and the optimized topology (Fig. 4.1d1) are provided. The relative stress evaluation errors for example 2 are also displayed in Fig. 4.1e2. The data analysis in a histogram for the stress evaluation errors is shown in Fig. 4.1f1, f2. As we can easily see, the larger numerical deviations of stress measurement are occurred in the ITO method for the stress-minimization designs. The worst case is that a stress-related design with an uncomplete topology. Hence, a critical feature of the current material description model should carefully control the grayscale and reasonably adjust the number of intermediate densities in the optimized DDF. Hence, the version 2 of the DDF in Chap. 1 is considered here for the stressminimization topology optimization. A clear illustration for the construction of the version 2 of the DDF is shown in Fig. 4.2, where a detailed flowchart about the construction of the DDF is given. Based on the construction of NURBS, 4D homogeneous coordinates that include 3D Cartesian physical coordinates and weights in NURBS basis functions are located in control points. In the construction of the DDF using NURBS parameterization, the core is to introduce the fifth homogeneous coordinate in control points, namely the density. In Chaps. 2 and 3, the densities at control points, also named by control densities, acting as design variables are gradually evolved, and other four homogeneous coordinates at control points keep unchanged in the optimization. The main intention of the current construction focuses on producing a nearly binary distribution but still having an extremely narrow transition area.

Fig. 4.2 Topology description model using the DDF

74

4 ITO for Structures with Stress-Minimization

4.2 NURBS-Based IGA for Stress Computation The linear elasticity is considered here: a design domain  with boundary , if given f i :  → R, gi :  Di → R, and h i :  Ni → R, find u i :  → R, and a Galerkin form can be stated by: ⎛ ⎝



⎞    ε(Ri )T Dε R j d⎠u i = R j fd + R j hd N 







−

N

 T   ε R j Dε g h d

(4.1)



The detailed derivations of the Galerkin form can refer to [34], and proceeding to define: ⎧

  T ⎪ ⎨ K i j = ε(Ri ) Dε R j d



 T  h  (4.2) ⎪ ⎩ F j = R j fd + R j hd N − ε R j Dε g d 

N



A more compact form of the above equation can be stated by: Ku = F

(4.3)

    K = K i j and F = F j . K denotes the global stiffness matrix, given as: K=

Nel 

Ke

(4.4)

e=1

where e is the number of IGA element, and Nel denotes the total number of IGA elements. Ke is the stiffness matrix of eth IGA element, which can be calculated by Gauss quadrature method with two inverse mappings from the bi-unit parent space to physical space, expressed by:  Ke =

˜e BTe DBe |J1 ||J2 |d

(4.5)

˜e 

˜ e denotes the bi-unit parent space. Be is the strain–displacement matrix. where  J1 and J2 are Jacobi matrices of two inverse mappings, respectively. D denotes material elastic tensor matrix, which is computed by a power-law relationship, and a detailed form is given by:

4.3 Induced Aggregation Formulations of p-Norm and KS

D =  pd D0

75

(4.6)

where D0 is the constitutive matrix of material elastic tensor in the plane stress. pd denotes the penalty parameter of material elastic tensor, which is generally set to be 3. u = {u i } denotes the displacement field in design domain, which is defined by a linear combination of all NURBS basis functions with displacements at control points, and a detailed form is given by: u=

n  m 

p,q

Ri, j (ξ, η)ui, j

(4.7)

i=1 j=1

As far as the plane stress problem is concerned, the stress vector at GQPs can be expressed by: σel.k = epl.kσ D0 Bel.k uel.k

(4.8)

where σel.k denotes the stress vector at the (l.k)th GQP of the eth IGA element. el.k is the value of the DDF at the corresponding GQP. pσ is the penalty parameter of stress [107–109], and a general value is equal to 0.5 to remove singular topology in the optimization. The von Mises stress in the corresponding GQP can be evaluated by: σevm = l.k

 1/2 T σel.k Vσ el.k

(4.9)

where V is the constant matrix for a plane stress state, and a detailed form is given by: ⎡

⎤ 1 −0.5 0 V = ⎣ −0.5 1 0 ⎦ 0 0 3

(4.10)

4.3 Induced Aggregation Formulations of p-Norm and KS In previously mentioned works, the classic p-norm and Kreisselmeier–Steinhauser (KS) functions are mostly applied to aggregate all local stress, and infinitedimensional constraints are converted into a global constraint. The general forms of both the p-norm aggregation function, and the corresponding discrete form can be given as:

76

4 ITO for Structures with Stress-Minimization

⎛ σpn = ⎝



⎞1 ppn (σ vm ) ppn d⎠

1 ppn  Nel 3 3   ppn σevm = l.k

(4.11)

e=1 l=1 k=1



where ppn is the stress norm parameter. It is known that the p-norm aggregation function gives the average stress if ppn = 1, and it gradually approaches the maximum vm with the increasing of the stress norm parameter ppn → ∞. The above stress σmax vm . The KS aggregation function and its discrete form can equation has the σpn ≥ σmax be expressed by: ⎛ ⎞   Nel 3 3     p σ vm 1 ⎝ 1 vm pks σ ks el.k ⎠ σks = ln e d = ln e pks pks e=1 l=1 k=1

(4.12)



where pks is the stress norm parameter of KS function. The above equation has vm vm , and σks will gradually approach the maximum stress σmax when the σks ≥ σmax pks → ∞. According to the definition of the p-norm and KS functions, we can see if the finite element mesh becomes more and more coarser, aggregation functions diverge. The mesh dependency will determine the approximation accuracy of aggregation functions for the maximum von Mises stress, which affects the effectiveness of topology optimization. Hence, the induced aggregation functions for the p-norm and KS are developed to remove mesh dependency [127]. The induced p-norm aggregation function can be given as:

σ Ipn =

 Nel 3 3  Nel 3 3      (σ vm ) p I pn +1 d vm p Ipn +1 vm p Ipn = σel.k σel.k vm ) p Ipn d  (σ e=1 l=1 k=1 e=1 l=1 k=1



(4.13) where p Ipn denotes the stress norm parameter of the induced p-norm aggregation function. As far as the induced aggregation for the KS function is concerned, the corresponding equation is given by:  Nel 3 3  Nel 3 3  vm p Iks σ vm     σ e d vm vm = =  p I σ vm σevm e p Iks σel.k e p Iks σel.k l.k ks d e e=1 l=1 k=1 e=1 l=1 k=1

σ Iks

(4.14) where p Iks denotes the stress norm parameter of the induced KS aggregation function. In order to demonstrate the positive features of the induced aggregation functions over conventional aggregation functions, a benchmark of L beam shown in Fig. 4.3 is used. Sixteen cases with different IGA meshes are performed, and the numbers of IGA elements in sixteen cases are listed in Table 4.1.

4.3 Induced Aggregation Formulations of p-Norm and KS

77

Fig. 4.3 Stress analysis of a L beam with 200 × 100 IGA elements: a the loads and boundary conditions; b Stress distribution in 2D view; c Stress distribution in 3D view

Table 4.1 Numbers of IGA elements in sixteen cases Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

Case 7

Case 8

50 × 25

60 × 30

70 × 35

80 × 40

90 × 45

100 × 50

110 × 55

120 × 60

Case 9

Case 10

Case 11

Case 12

Case 13

Case 14

Case 15

Case 16

‘130 × 65

140 × 70

150 × 75

160 × 80

170 × 85

180 × 90

190 × 95

200 × 100

The basic boundary and loads conditions of L beam are presented in Fig. 4.3. As already discussed in previous works [106–111], the stress norm parameter has a significant effect on the optimization. If the value is smaller, the approximation accuracy of aggregation functions to the maximum stress is worse. However, when the value of the stress norm parameter increases, the high nonlinearity of aggregation functions leads to numerical instabilities, such that the optimizer cannot find a better design. Hence, a good choice for norm parameter of stress should offer the adequate smoothness for the optimizer that can perform well and should have the adequate approximation of maximum stress to remove the violation of local constraints. Currently, in order to show the specific features of different aggregation functions, the corresponding stress norm parameters are: ppn , p Ipn = [6, 7, 8] and pks , p Iks = [4, 5, 6]. Numerical results of the related comparison between the p-norm and the induced p-norm aggregation functions are given, and the detailed discussions about their own characteristics can refer to [127]. Figure 4.4a shows the detailed stress measure values by aggregation functions, and Fig. 4.4b displays measure errors of p-norm and induced p-norm aggregation functions over maximum stress values in different IGA meshes. Moreover, stress distributions in 2D and 3D of a L beam with 200 × 100 IGA elements and p Ipn = 6 are displayed in Fig. 4.3b, c, where a stress concentration exists at the reentrant corner of L beam. The detailed comparisons between the induced aggregation functions and conventional aggregation functions are performed in [127], which can present the positive features of induced p-norm and KS aggregation functions to improve the approximation accuracy of maximum von Mises stress and eliminate the problem of mesh dependency. Several important remarks which have significant effect on the latter optimization are provided as follows: (1) Induced aggregation functions of the p-norm and KS have the higher accuracy of the approximation over conventional p-norm and

78

4 ITO for Structures with Stress-Minimization

Fig. 4.4 Results of stress measure and numerical errors of P-norm and induced P-norm aggregations in sixteen cases: a Stress measure; b Measure errors

KS aggregation functions; (2) Measure errors of p-norm and KS aggregation functions become smaller and smaller as the number of IGA elements increases, which present a mesh dependency; (3) Measure errors of induced aggregation functions keep unchanged and nearly approach 0, which can present the superior ability of removing mesh dependency in conventional p-norm and KS aggregation functions; (4) the KS aggregation function has a higher numerical precision over the p-norm aggregation function, and the same conclusion also exists in induced aggregation functions.

4.4 ITO for Stress-Minimization Designs 4.4.1 Stress-Minimization Design Formulation Currently, the stress-minimization design problem is considered here to discuss the effectiveness and efficiency of the proposed ITO method with the induced aggregation formulations. In the formulation using the ITO method, the optimizer focuses on the optimization of the DDF by gradually evolving control design variables, until structural performance reaches the expected value. In the stress-related ITO design, the aggregated value of the global stress works as the objective function, and the optimizer aims to find the minimal value. Material consumption acts as the volume constraint. Hence, the ITO for the stress-minimization problem with a material usage constraint can be formulated as: Find : φi, j (i = 1, 2, . . . , n; j = 1, 2, . . . , m) Min : J (u, ) = f (σ vm )

4.4 ITO for Stress-Minimization Designs

79

⎧ 1 ⎨ a(u, δu) =

l(δu), u| D = g, ∀δu ∈ H () S.t. : G() =  (ξ, η)d − Vmax ≤ 0 ⎩ 0 < φmin ≤ φi, j ≤ 1

(4.15)

where J denotes the objective function, defined by stress aggregation functions, namely σpn , σks , σ and σ Iks . φi, j denotes the discrete density at the (i, j)th control Ipn point. G is material usage constraint, in which Vmax is the maximum value of material consumption and  denotes the DDF. The linearly elastic equilibrium equation is developed by a = l, where a is the bilinear energy function and l indicates the linear load function. u denotes numerical response, namely the global displacement field, in design domain, and δu indicates the virtual displacement field, belonging to the Sobolev space H 1 (). g denotes the prescribed displacement vector at the Dirichlet boundary  D . A variational weak form of the linear elastic equilibrium equation can be stated, where f is the body force and h is the boundary traction on the Neumann boundary  N . ⎧

T ⎪ ⎨ a(u, δu) = ε(u) D()ε(δu)d

 ⎪ ⎩ l(δu) = fδud + hδud N 

(4.16)

N

4.4.2 Design Sensitivity Analysis In the stress-related ITO design problems, the gradient-based mathematical programming algorithm MMA (the method of moving asymptotes) [220] is employed here, where sensitivity analysis of the objective function and constraint function with respect to control design variables are required. The first-order derivatives of the global stress with respect to control design variables can be achieved by the chain rule, expressed as: ∂ f (σ vm ) ∂σevm ∂J ∂σel.k ∂el.k l.k = ∂φi, j ∂σevm ∂σ ∂ e el.k ∂φi, j l.k l.k

(4.17)

In Sect. 4.3, four different aggregation functions, including p-norm, KS, induced p-norm and induced KS are presented. Therefore, there are four different objective and σ Iks . In the case of the p-norm aggregation functions, namely σpn , σks , σ Ipn function σpn , the first part of the above equation can be given by:

80

4 ITO for Structures with Stress-Minimization

⎛ ⎞ 1−p ppn pn   vm  ppn −1  1− ppn  vm  ppn −1 ∂σpn vm ppn ⎝ ⎠ σel.k σel.k = = σpn (σ ) d vm ∂σel.k

(4.18)



Based on the von Mises stress, the derivative of the von Mises stress can be stated as: ∂σevm l.k ∂σel.k

=

− 1 −1 T  1 T σel.k Vσ el.k 2 2σTel.k V = σevm σel.k V l.k 2

(4.19)

The third part of Eq. (4.17) can be derived according to the definition of stress vector, given as: ∂u ∂σel.k = pσ epl.kσ −1 D0 Bel.k uel.k + epl.kσ D0 Bel.k Lel.k ∂el.k ∂el.k

(4.20)

where the matrix Lel.k collects the displacements of the (el.k )th control design variable in the global displacement field u, and it satisfies uel.k = Lel.k u. It is noted that the predefined global force vector is independent with the structural topology. Meanwhile, differentiating both sides of the equilibrium equation, a detailed form of the above equation can be given by: ∂σel.k ∂K = pσ epl.kσ −1 D0 Bel.k uel.k − epl.kσ D0 Bel.k Lel.k K−1 u ∂el.k ∂el.k

(4.21)

According to the definition of the DDF, the first-order derivatives of the DDF with respect to control design variables can be expressed by: ∂ φˆ i, j   ∂el.k p,q = Ri, j (ξ, η) ψ φi, j ∂φi, j ∂ φ˜ i, j

(4.22)

Based on the above derivations, a general form of the first-order derivative of the p-norm aggregation function with respect to control design variables can be stated as:     ∂σ pn ∂σ pn ∂σ pn = + (4.23) ∂φi, j ∂φi, j 1 ∂φi, j 2 where

4.4 ITO for Stress-Minimization Designs

81

 ⎧  1− ppn  vm  ppn −2 ∂σpn ⎪ ⎪ σel.k = σpn pσ epl.kσ −1 σTel.k ⎪ ⎪ ∂φ ⎪ i, j 1 ⎪ ⎪ ⎪ ∂ φˆ i, j   ⎪ p,q ⎪ ⎪ VD0 Bel.k uel.k Ri, j (ξ, η) ψ φi, j ⎨ ∂ φ˜ i, j    1− ppn  vm  ppn −2 pσ T ∂σpn ⎪ ⎪ σel.k = − σpn el.k σel.k V ⎪ ⎪ ⎪ ∂φi, j 2 ⎪ ⎪ ⎪ ⎪ ∂ φˆ i, j   ∂K ⎪ p,q ⎪ uRi, j (ξ, η) ψ φi, j D0 Bel.k Lel.k K−1 ⎩ ∂el.k ∂ φ˜ i, j

(4.24)

with the solution λ1 of the following adjoint problem, T  1− ppn  vm  ppn −2 pσ  σel.k el.k D0 Bel.k Lel.k Vσ el.k Kλ1 = − σpn

(4.25)

A detailed form of Eq. (4.24) is expressed as: ⎧   1− ppn  vm  ppn −2 ∂σpn ⎪ ⎪ σel.k = σpn pσ epl.kσ −1 σTel.k ⎪ ⎪ ⎪ ∂φi, j 1 ⎪ ⎨ ∂ φˆ i, j   p,q VD0 Bel.k uel.k Ri, j (ξ, η) ψ φi, j ⎪ ⎪ ∂ φ˜ i, j ⎪  ⎪   ⎪ ∂σpn ∂ φˆ p,q ⎪ ⎩ = λT1 ∂∂Ke uRi, j (ξ, η) ∂ φ˜i, j ψ φi, j ∂φi, j 2

(4.26)

i, j

l.k

In the case of KS aggregation function, the sensitivity of stress aggregation function with respect to control design variables can be expressed as: ∂σks = ∂φi, j



∂σks ∂φi, j



 + 1

∂σks ∂φi, j

 (4.27) 2

where ⎧  −1  −1 vm  ∂σks ⎪ vm p σ ⎪ ks ⎪ = e d e pks σel.k σevm pσ epl.kσ −1 σTel.k ⎪ l.k ⎪ ∂φ ⎪ i, j  1 ⎨ ∂ φˆ i, j   p,q VD0 Bel.k uel.k Ri, j (ξ, η) ψ φi, j ⎪ ⎪ ⎪ ∂ φ˜ i, j ⎪  ⎪   ⎪ ⎩ ∂σks = λT ∂K uR p,q (ξ, η) ∂ φˆi, j ψ φi, j ∂φi, j

2

2 ∂el.k

i, j

(4.28)

∂ φ˜ i, j

with the solution λ2 of the following adjoint problem, ⎛ Kλ2 = −⎝





⎞−1

−1 pσ  T vm  e pks σ d⎠ e pks σel.k σevm el.k D0 Bel.k L el.k V σel.k l.k vm

(4.29)

82

4 ITO for Structures with Stress-Minimization

As far as the induced p-norm function is concerned, the related first-order derivative with respect to control design variables can be given by: ∂σ Ipn = ∂φi, j



∂σ Ipn ∂φi, j



 +

1

∂σ Ipn ∂φi, j



 +

2

∂σ Ipn ∂φi, j



 +

3

∂σ Ipn ∂φi, j

 (4.30) 4

where ⎧  ∂σ    p Ipn −1  Ipn ⎪ ⎪ = (A2 )−1 p Ipn + 1 σevm pσ epl.kσ −1 ⎪ l.k ⎪ ∂φ i, j 1 ⎪ ⎪ ⎪ ⎪ ∂ φˆ i, j   ⎪ p,q ⎪ ⎪ σTel.k VD0 Bel.k uel.k Ri, j (ξ, η) ψ φi, j ⎪ ⎪ ∂ φ˜ i, j ⎪ ⎪  ⎪   ⎪ ⎨ ∂σ Ipn = λT3 K−1 ∂K uR p,q (ξ, η) ∂ φˆi, j ψ φi, j i, j ˜ i, j ∂φi, j 2 ∂el.k ∂ φ     p Ipn −2 ∂σ Ipn ⎪ ⎪ = −A1 (A2 )−2 p Ipn σevm pσ epl.kσ −1 ⎪ ⎪ l.k ⎪ ∂φ i, j ⎪ 3 ⎪ ⎪ ⎪ ∂ φˆ i, j   ⎪ p,q T ⎪ σel.k VD0 Bel.k uel.k Ri, j (ξ, η) ψ φi, j ⎪ ⎪ ⎪ ∂ φ˜ i, j ⎪  ⎪   ⎪ ∂σ Ipn ∂ φˆ p,q ⎩ = λT4 K−1 ∂∂Ke uRi, j (ξ, η) ∂ φ˜i, j ψ φi, j ∂φi, j 4

(4.31)

i, j

l.k

with two solutions λ3 and λ4 of the following adjoint problems, !

  p Ipn −1 pσ  T  el.k D0 Bel.k Lel.k Vσ el.k Kλ3 = −(A2 )−1 p Ipn + 1 σevm l.k   p Ipn −2 pσ  T Kλ4 = A1 (A2 )−2 p Ipn σevm el.k D0 Bel.k Lel.k Vσ el.k l.k

(4.32)

where  A1 = 

(σ vm ) p Ipn +1 d; A2 =

 (σ vm ) p Ipn d

(4.33)



In the case of the induced KS aggregation function, the corresponding first-order derivative with respect to control design variables can be stated as:        ∂σ Iks ∂σ Iks ∂σ Iks ∂σ Iks + + + ∂φi, j 1 ∂φi, j 2 ∂φi, j 3 ∂φi, j 4     ∂σ Iks ∂σ Iks + + ∂φi, j 5 ∂φi, j 6

∂σ Iks = ∂φi, j



(4.34)

where three solutions λ5 , λ6 and λ7 of the following adjoint problems should be solved, given by:

4.4 ITO for Stress-Minimization Designs

83

⎧  vm −1 pσ  T −1 p I σevm ⎪ el.k D0 Bel.k Lel.k Vσ el.k ⎨ Kλ5 = −B2 e ks l.k σvmel.k T p I σe p  Kλ6 = −B2−1 p Iks eσ ks l.k el.kσ D0 Bel.k Lel.k Vσ el.k ⎪ −1 pσ  T vm  ⎩ Kλ7 = −B1 (B2 )−2 p Iks e p Iks σel.k σevm el.k D0 Bel.k Lel.k Vσ el.k l.k

(4.35)

where  B1 =

σ

vm p Iks σ vm

e

 d; B2 =



e p Iks σ d vm

(4.36)



and the details of the corresponding six parts of Eq. (4.34) can refer to:  ⎧ −1 vm  ∂σ Iks ⎪ ⎪ = B2−1 e p Iks σel.k σevm pσ epl.kσ −1 σeTl.k ⎪ l.k ⎪ ∂φ ⎪ i, j 1 ⎪ ⎪ ⎪ ∂ φˆ i, j   ⎪ p,q ⎪ ⎪ VD0 Bel.k uel.k Ri, j (ξ, η) ψ φi, j ⎪ ⎪ ∂ φ˜ i, j ⎪ ⎪  ⎪   ⎪ ∂σ Iks ∂ φˆ p,q ⎪ = λT5 ∂∂Ke uRi, j (ξ, η) ∂ φ˜i, j ψ φi, j ⎪ ⎪ ∂φ i, j i, j l.k ⎪ 2 ⎪ ⎪ vm ∂σ Iks ⎪ ⎪ = B2−1 p Iks e p Iks σel.k pσ epl.kσ −1 σTel.k ⎪ ⎪ ⎪ ∂φi, j 3 ⎪ ⎨ ∂ φˆ i, j   p,q VD0 Bel.k uel.k Ri, j (ξ, η) ψ φi, j ⎪ ⎪ ∂ φ˜ i, j ⎪  ⎪  ⎪ ∂σ Iks ∂ φˆ i, j  p,q ⎪ T ∂K ⎪ ⎪ ⎪ ∂φi, j 4= λ6 ∂el.k uRi, j (ξ, η) ∂ φ˜i, j ψ φi, j ⎪ ⎪ ⎪ −1 vm  ⎪ ∂σ Iks ⎪ = B1 (B2 )−2 p Iks e p Iks σel.k σevm ⎪ l.k ⎪ ⎪ ∂φi, j 5 ⎪ ⎪ ⎪ ⎪ ∂ φˆ i, j   p,q ⎪ pσ −1 T ⎪ η) p  σ VD B u R ψ φi, j (ξ, ⎪ σ 0 e e l.k l.k e e i, j l.k l.k ⎪ ⎪ ∂ φ˜ i, j ⎪  ⎪   ⎪ ⎩ ∂σ Iks = λT ∂K uR p,q (ξ, η) ∂ φˆi, j ψ φ ∂φi, j

6

7 ∂el.k

i, j

∂ φ˜ i, j

(4.37)

i, j

The derivatives of global stiffness matrix with respect to control design variables can be given by:   p −1 ∂K ∂Ke = = BeT (ξl , ηk ) pd el.k d ∂el.k ∂el.k D(ξl , ηk )Be (ξl , ηk )|J1 (ξl , ηk )||J2 (ξl , ηk )|wl wk

(4.38)

Similarly, the derivative of volume constraint with respect to control design variables is given by: ⎛ ⎞  ∂ φˆ i, j   ∂G p,q = ⎝ 1d⎠ Ri, j (ξ, η) ψ ρi, j ∂φi, j ∂ φ˜ i, j 

(4.39)

84

4 ITO for Structures with Stress-Minimization

4.4.3 Numerical Implementations The third challenge of the highly nonlinear stress behavior of stress-related design problems can result in numerical instabilities in the optimization [106–111], so that the optimizer cannot seek the preferable designs with the expected performance. In practice, the critical problem of unstable iterations stems from many aspects, like the high nonlinearity of stress aggregation functions and the lower accuracy in stress measurement using FEM. Hence, the stability transformation method (STM) [70, 221] from the nonlinear dynamic using a reasonable linear transformation [222] is applied to keep the numerical stability of the optimization, and a general form can be given by: φϑ+1 = φϑ + q[g(φϑ ) − φϑ ]

(4.40)

where φϑ is a vector including all control design variables at the ϑth iteration, and q is a parameter in [0, 1]. If it is equal to 1, the dynamical system is not controlled. g(·) is an implicit iterative function.

4.5 Numerical Examples In the current section, several numerical examples are performed to demonstrate the effectiveness of the developed ITO method for stress-related design problems, and also to present the superior features of the induced p-norm and KS aggregation functions relative to previous p-norm and KS aggregation functions. It is noted that all 2D numerical examples are considered in the state of plane stress. In the first example, a benchmark of a L beam is considered, where the loads and boundary conditions are shown in Fig. 4.5. In order to present the advantages of IGA for curve structures, an inverse L-type structure having a curve design domain shown in Fig. 4.6 with the loads and boundary conditions will be discussed. Numerical example 3 focuses on

Fig. 4.5 A L beam: a loads and boundary conditions; b 2D stress distribution; c 3D stress distribution

4.5 Numerical Examples

85

Fig. 4.6 An inverse L-type structure with curved design domain: a loads and boundary conditions; b 2D stress distribution; c 3D stress distribution

realizing the optimization of a MBB beam with one preexisting crack notch, and its loads and boundary conditions are presented in Fig. 4.7. A half annulus with a square hole, displayed in Fig. 4.8, is further studied to present the effectiveness of the ITO for stress-minimization designs. The initial values of control design variables are all set to be 0.6 in four numerical examples, and the corresponding stress distributions

Fig. 4.7 A MBB beam with one preexisting crack notch: a loads and boundary conditions; b 2D stress distribution; c 3D stress distribution

Fig. 4.8 A half annulus with a square hole: a loads and boundary conditions; b 2D stress distribution; c 3D stress distribution

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4 ITO for Structures with Stress-Minimization

Table 4.2 Design parameters in four numerical examples Example

NURBS orders

Control points

IGA elements

q

V max

pd

pσ

1

p = 3, q = 3

n = 165, m = 83

160 × 80

0.3

0.30

3

0.5

2

p = 4, q = 3

n = 167, m = 83

160 × 80

0.3

0.28

3

0.5

3

p = 3, q = 3

n = 209, m = 63

200 × 80

0.3

0.20

3

0.5

4

p = 4, q = 3

n = 167, m = 83

160 × 80

0.3

0.30

3

0.5

in both 2D and 3D are presented in Figs. 4.5, 4.6, 4.7 and 4.8. As we can see, stress concentrations are mostly distributed at critical regions, such as the sharps and reentrant corners in the design domain. In numerical examples, Young’s moduli of the solid and void materials are set to be 1 and 1e − 9, respectively, to avoid numerical singularity. Poisson’s ratio is equal to 0.3. If the imposed regions of Dirichlet and Neumann boundary conditions generate stress concentrations, numerical instabilities will emerge in iterations to force the optimizer cannot find a good solution, mainly originating from a fact that the optimizer cannot eliminate stress concentrations by removing materials at the regions of Dirichlet or Neumann boundaries. Hence, the distributed force with the total value equal to 1 versus the concentrated force is imposed at the boundaries, and Dirichlet boundary conditions should be also forced at regions, rather than the single point. The regions for the imposing of the distributed force generally contain three IGA elements. Finally, design parameters of four numerical examples are provided in Table 4.2. In each IGA element, Gauss quadrature points (GQPs) with a number 3 × 3 are chosen for the numerical integration. The convergent condition is that the L ∞ norm of the relative difference of the objective function in two consecutive iterations is less than 1% and the constraint should be satisfied within the maximum 300 steps.

4.5.1 Discussions on Aggregation Formulations of p-Norm and the Induced p-Norm 4.5.1.1

Stress-Minimization Design

The setting of optimization problem and design parameters is shown in Fig. 4.5 and the second row of Table 4.2, respectively. As discussed in Sect. 4.3, a good choice of stress norm parameter not only offer the adequate approximation of maximum von Mises stress, but also provide the sufficient smoothness for the optimizer to avoid the numerical instabilities resulting from the higher nonlinear behavior. In the current example, the values of stress norm parameters in the p-norm and the induced p-norm aggregation formulations are both set as 6. Meanwhile, the classic compliance-minimization design of the L beam by the ITO method [56] is presented in Fig. 4.9, including the structural topology, stress distributions in 2D and 3D. As we can see, a clear stress concentration with the maximum von Mises stress equals

4.5 Numerical Examples

87

Fig. 4.9 Compliance-minimization design of L beam: a structural topology; b stress distribution in 2D; c stress distribution in 3D

Fig. 4.10 Stress-minimization designs of L beam: a1–c1 p-norm; a2–c2 induced p-norm

to 7.376 emerges in the reentrant corner of the topology in L beam, which will lead to the safety problem of structure failure in the engineering. In Fig. 4.10a1–c1, the optimized designs, including the structural topology, stress distributions in 2D and 3D, of L beam by the ITO formulation with the p-norm aggregation formulation are provided. Based on 3D stress distribution, the high stress concentration in the reentrant corner of the stiffness-maximization design is strongly alleviated. The topology in Fig. 4.10a1 of the stress-minimization design takes small circle boundaries to replace the reentrant corner, where the value of maximum von Mises stress is reduced to be 2.634. Figure 4.10a2–c2 displays the optimized structural topology and the corresponding stress distributions in 2D and 3D. It can be easily seen that the optimized topology in Fig. 4.10a2 also has the smooth circle boundaries at the region of the reentrant corner. Moreover, the smooth feature in the current topology has a higher radius over the topology presented in Fig. 4.10a1, and the corresponding distributions of stress shown in Fig. 4.10b2, c2 are more uniform compared to the results displayed in Fig. 4.10 b1, c1. The maximum value of von Mises stress in the optimized topology in Fig. 4.10a2 is equal to 1.798, which is smaller than the maximum von Mises stress value 2.634 in the optimized topology

88

4 ITO for Structures with Stress-Minimization

displayed in Fig. 4.10a1. In actual, the 3D stress distribution in Fig. 4.10c1 optimized by the ITO with the p-norm aggregation formulation still has a slight stress concentration at the reentrant corner. Figure 4.11 provides convergent histories of the objective function, von Mises stress and volume fraction for the topology in Fig. 4.10a1, and the iterative histories of the objective function, von Mises stress and volume fraction for the topology in Fig. 4.10a2 are shown in Fig. 4.12. Firstly, the optimization designs displayed in Fig. 4.10a1, a2 come from a same initial design with a full distribution of solid

Fig. 4.11 Convergent histories of stress-minimization design using the p-norm aggregation formulation

Fig. 4.12 Convergent histories of stress-minimization design using the induced p-norm aggregation formulation

4.5 Numerical Examples

89

materials, in which a stress concentration with a high von Mises stress value exists. Secondly, the induced p-norm aggregation can take a lower value of von Mises stress over the conventional aggregation, namely 1.798 < 2.634. Thirdly, " " p-norm vm " the measure error  Ipn = "σ Ipn − σmax = 0.743 in the" induced p-norm aggregation " vm " = 3.19, which presents the formulation is much smaller than pn = "σpn − σmax higher numerical accuracy of the approximation using the induced p-norm aggregation formulation. Finally, the convergent curve of the objective function presented in Fig. 4.11 has a slight fluctuation in the latter iterations from 200th to the last, whereas the trajectory of the objective function shown in Fig. 4.12 has a stable and smooth convergence to find a good topology with even distribution of stress. Hence, the numerical accuracy of the approximation using aggregation formulations has a significant impact on the optimization. A conclusion that the induced p-norm aggregation formulation has the superior ability to remove stress concentrations in the optimization relative to conventional p-norm aggregation function can be obtained.

4.5.1.2

The Effect of Mesh Dependency on the Optimization

In this subsection, the main focus is to discuss the mesh dependency of p-norm aggregation functions and its influence on the latter optimization. Twelve cases are performed with different IGA meshes by the ITO method with the p-norm and induced p-norm aggregation formulations, respectively. The setting of twelve cases with the corresponding IGA elements is provided in Table 4.3. Another design parameters keep consistent with the definition in the second row of Table 4.2. Stress norm parameters σpn and σ Ipn are still set to be 6. The optimized results including the structural topology and 2D stress distribution using the ITO method with the p-norm aggregation function are shown in Fig. 4.13, and Fig.4.14 presents the optimized topology and 2D stress distribution using the ITO method with the induced p-norm aggregation formulation. We can easily observe that all optimized topologies using the p-norm and induced p-norm aggregation formulations can effectively lower the maximum value of von Mises stress 7.376 in the compliance-minimization design in Fig. 4.9. Firstly, the induced p-norm aggregation formulation can provide the more even distribution of the stress in the optimized topology, and the boundaries in the optimized topology have the circle with a larger radius at the reentrant corner which can effectively eliminate the problem of stress concentrations. Secondly, as the IGA mesh is denser, the measure error becomes lower and lower, and a clear mesh dependency arises Table 4.3 Numbers of IGA elements in twelve cases p-norm Induced p-norm

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

100 × 50

120 × 60

140 × 70

160 × 80

180 × 90

200 × 100

Case 7

Case 8

Case 9

Case 10

Case 11

Case 12

100 × 50

120 × 60

140 × 70

160 × 80

180 × 90

200 × 100

90

4 ITO for Structures with Stress-Minimization

Fig. 4.13 Optimized stress-minimization designs using the p-norm aggregation formulation in six cases with different IGA meshes

Fig. 4.14 Optimized stress-minimization designs using the induced p-norm aggregation formulation in six cases with different IGA meshes

in the optimization with the p-norm aggregation formulation. In the case of the induced p-norm aggregation formulation, the value of measure error almost has no fluctuation, nearly 0.742, if the number of IGA elements is larger than 160 × 80. Hence, the induced p-norm aggregation function can remove the problem of mesh dependency. As presented in Fig. 4.13, all the optimized topologies have the slight stress concentration, even if the number of elements in IGA mesh increases to enhance numerical precision of the optimization. Moreover, the mesh dependency significantly influences numerical accuracy of the approximation to the maximum value of von Mises stress in the p-norm aggregation function, which can damage the capability of eliminating the problem of stress concentration. In the case of the optimization by the ITO with the induced aggregation formulation, mesh dependency is removed to improve approximation accuracy of the maximum value of von Mises stress, which can sufficiently maintain numerical stabilities and better convergence to find a good solution without stress concentration. As presented in Fig. 4.15, a numerical case of L beam is optimized by the stressbased ITO method using the p-norm induced aggregation formulation but with a norm

4.5 Numerical Examples

91

Fig. 4.15 Optimized stress-minimization designs using the induced p-norm aggregation formulation in six cases with different IGA meshes

stress parameter equal to 5. As we can see, the optimized topology has the worse stress distribution with a slight stress concentration at the position of the reentrant corner relative to the design displayed in Fig. 4.10c2. However, the maximum value of von Mises stress is equal to 2.139, which is still smaller than the value in Fig. 4.10c1. The induced p-norm aggregation function has the superior capability to remove the problem of stress concentration in the stress-based ITO topological design to ensure the engineering safety and avoid structural failure.

4.5.2 Discussions on Aggregation Formulations of KS and the Induced KS 4.5.2.1

Stress-Minimization Design

In this subsection, the intention is to discuss the effectiveness of the KS and induced KS aggregation functions. According to numerical results, the approximation accuracy of the KS aggregation function is higher than the p-norm aggregation function. The same conclusion also exists in the comparison between the induced KS and the induced p-norm aggregations. Hence, stress norm parameters are set to be 5, which is smaller than the above example. Another design parameters keep the same in the second row of Table 4.2. As shown in Fig. 4.16a1–c1, the stress-minimization of the L beam is optimized by the ITO method with the KS aggregation formulation, and the numerical results with the optimized topology, stress distributions in 2D and 3D are all provided. The optimized results of the L beam using the ITO method with the induced KS aggregation formulation to minimize stress are displayed in Fig. 4.16a2–c2. According to the optimized results, some important conclusions can be achieved: (1) the induced KS aggregation formulation can give a lower value of maximum von Mises stress (equal to 1.889) over the KS aggregation formulation (equal to 1.967); (2) The boundaries of the optimized topology shown in Fig. 4.16b2 have the circle with a higher radius compared to the topology in Fig. 4.16b1, which can effectively avoid the stress concentration in the corner of L beam; (3) the optimized topology in Fig. 4.16a2 has an evener uniform stress distribution relative to the optimized topology displayed

92

4 ITO for Structures with Stress-Minimization

Fig. 4.16 Stress-minimization designs of L beam: a1–c1 KS; a2–c2 induced KS

in Fig. 4.16a1 which has a slight stress concentration at the position of the reentrant corner in L beam. We can achieve a conclusion that the induced KS aggregation formulation has the superior capability of removing stress concentration relative to the conventional KS aggregation formulation in the stress-based topology optimization. Convergent histories of the objective function, the maximum von Mises stress and volume fraction of the topology shown in Fig. 4.16a1 are presented in Fig. 4.17, and Fig. 4.18 displays the iterative histories of the objective function, the maximum value of von Mises stress and volume fraction of the topology as shown in Fig. 4.16a2. The iterative curves in Figs. 4.17 and 4.18 are characterized with the stable and smooth convergence to find the good solutions in the optimization.

Fig. 4.17 Convergent histories of stress-minimization design using the KS aggregation formulation

4.5 Numerical Examples

93

Fig. 4.18 Convergent histories of stress-minimization design using the induced KS aggregation formulation

4.5.2.2

The Effect of Mesh Dependency on the Optimization

In the subsection, we intend to study mesh dependency of the KS and induced KS aggregations. The design parameters keep consistent with the above numerical case, expect for the IGA mesh. Twelve cases with different IGA meshes for the KS and induced KS aggregation formulations are discussed, where the corresponding numbers of IGA elements are listed in Table 4.3. The stress norm parameters are still set to be 5. As presented in Fig. 4.19, the optimized results, namely the structural topology and stress distribution in 2D, using the ITO method with the KS aggregation function are provided, and Fig. 4.20 presents the optimized results optimized by the ITO with the induced KS aggregation formulation, including structural topology and 2D stress distribution.

Fig. 4.19 Optimized stress-minimization designs using the KS aggregation formulation in six cases with different IGA mesh

94

4 ITO for Structures with Stress-Minimization

Fig. 4.20 Optimized stress-minimization designs using the induced KS aggregation formulation in six cases with different IGA meshes

Firstly, all the optimized topologies shown in Figs. 4.19 and 4.20 using the ITO method with the KS and the induced KS aggregation formulations can effectively reduce the maximum value of von Mises stress in the stiffness-maximization design shown in Fig. 4.9 (equal to 7.376). Secondly, the boundaries of the optimized topologies shown in Fig. 4.20 have the circle with a large radius over the optimized topologies presented in Fig. 4.19, which can successfully avoid the generation of stress concentration at the locations of the reentrant corner. Moreover, the stress distributions displayed in Fig. 4.20 are evenly uniform in the L beam compared with stress distributions presented in Fig. 4.19. Thirdly, as the number of IGA elements increases, measure error of stress in Fig. 4.19 becomes lower and lower, which shows an explicit feature of mesh dependency in the formulation of the conventional KS aggregation. In the case of the induced KS aggregation, the value of measurement error gradually approaches 0.84 with the increasing of the number of IGA elements. The elimination of the problem of mesh dependency in the induced KS aggregation can effectively improve approximation accuracy of the maximum value of von Mises stress, which can enhance numerical stabilities and the smooth convergence during the optimization to find the better designs.

4.5.3 An Inverse L-Type Structure with the Curved Design Domain The main intention of this subsection is to discuss the effectiveness and utility of the proposed ITO method with aggregation formulations to minimize the stress of an inverse L-type structure with the curved design domain. The basic setting of structural domain and the design parameters is presented in Fig. 4.6 and the third row of Table 4.2. The stress norm parameters of the aggregation functions are defined as: σpn , σ Ipn = 6 and σks , σ Iks = 5, respectively, to maintain the adequate smoothness for the optimization to remove numerical instabilities and

4.5 Numerical Examples

95

keep the sufficient approximation of maximum value of von Mises stress to avoid the violence of all local stress constraints. The stiffness-maximization design of the inverse L-type structure is performed by the standard ITO method [56], where an explicit stress concentration occurs in the reentrant corner of design domain and the maximum value of von Mises stress is equal to 10.7615. In Fig. 4.22a1–c1, the optimized results by the ITO with the p-norm aggregation formulation are provided, including the structural topology, 2D stress distribution and 3D stress distribution. Figure 4.22a2–c2 displays the structural topology, 2D and 3D stress distributions of the optimized design by the ITO with the induced p-norm aggregation formulation. In Fig. 4.23a1–c1, the optimized results of the inverse L-type structure using the ITO method with KS aggregation formulation are provided. Figure 4.23a2–c2 shows the optimized design of an inverse L-type structure using the ITO with the induced KS aggregation formulation, including the structural topology and stress distributions in 2D and 3D. Firstly, the maximum values of von Mises stress in four optimized topologies in Figs. 4.22 and 4.23 are equal to 4.016, 3.361, 2.862 and 2.786, respectively. Compared with the maximum value 10.7615 in Fig. 4.21, the ITO method with four aggregation formulations can effectively reduce the maximum value of von Mises stress, which shows the effectiveness of the ITO method with a single global stress constraint by aggregating all stresses. Meanwhile, the formulations of the induced KS and induced p-norm aggregations can achieve the lower values of maximum von Mises stress in the optimized topology compared to the KS and p-norm aggregation formulations, which can show the effectiveness of the induced aggregation formulations in the proposed ITO method. Based on the optimized topologies shown in Figs. 4.22a1–a2 and 4.23a1–a2, we can easily observe that the circle boundaries of the optimized topologies can be generated in the reentrant corner to eliminate the occurrence of stress concentration. In addition, stress distributions of the optimized topologies also gradually become even uniform with the decreasing of the maximum von Mises stress, except for the design in Fig. 4.22a1. It mainly results from a fact that the ITO with the p-norm aggregation function has the worse capability of removing stress concentration, the slight stress concentration still exists in the reentrant corner and regions that Dirichlet boundary conditions are imposed. Hence, it can manifest a fact that the induced aggregation formulations are crucial in the development of the ITO method for the stress-minimization design.

Fig. 4.21 Compliance-minimization design of an inverse L-type structure: a structural topology; b stress distribution in 2D; c stress distribution in 3D

96

4 ITO for Structures with Stress-Minimization

Fig. 4.22 Stress-minimization designs of an inverse L-type structure: a1–c1 p-norm; a2–c2 induced p-norm

Fig. 4.23 Stress-minimization designs of an inverse L-type structure: a1–c1 KS; a2–c2 induced KS

4.5.4 A MBB Beam with One Preexisting Crack Notch In this section, the classic MBB beam but with one preexisting crack notch is optimized by the ITO method with the induced p-norm aggregation formulation to study its effectiveness and utility. The basic setting of the optimization problem with the structural design domain is shown in Fig. 4.7, where the structural sizes of the preexisting crack notch are equal to 3 (Length) and 1 (Height), respectively. The design parameters for the optimization are defined in the fourth row of Table 4.2, and the value of stress norm parameter in the induced p-norm aggregation formulation is set as 6. The corresponding compliance-minimization design of the MBB beam with one

4.5 Numerical Examples

97

Fig. 4.24 Compliance-minimization design of a MBB beam with one preexisting crack notch: a structural topology; b stress distribution in 2D; c stress distribution in 3D

Fig. 4.25 Stress-minimization designs of a MBB beam with one preexisting crack notch using the induced p-norm aggregation formulation: a structural topology; b stress distribution in 2D; c stress distribution in 3D

preexisting crack notch is displayed in Fig. 4.24, including the structural topology, stress distributions in both 2D and 3D. In Fig. 4.24, we can easily find that the optimized design with the maximized stiffness has an explicit feature of stress concentration in the preexisting crack corner, and the value of maximum von Mises stress is equal to 24.66. Figure 4.25 shows the optimized design of the MBB with one preexisting crack notch using the ITO method with the induced p-norm aggregation formulation. As we can easily see, the stress-minimization design using the induced p-norm aggregation can extensively lower the value of maximum von Mises stress, equal to 4.521. The corresponding optimized topology can remove the problem of stress concentration by adding materials to generate a structural member over the corner of preexisting crack notch. The stress distributions in both 2D and 3D are shown in Fig. 4.25a2, a3, which are even uniform in design domain compared with the stress distribution in stiffnessmaximization design. The iterative curves of the objective function, the maximum von Mises stress and volume fraction are shown in Fig. 4.26, which have the smooth and stable convergence in the optimization process. Moreover, several intermediate distributions of stress are also attached in Fig. 4.26. It can be easily found that the initial stress concentration in the corner of the crack notch is gradually removed with the increasing of materials over the crack notch, until a complete structural member

98

4 ITO for Structures with Stress-Minimization

Fig. 4.26 Convergent histories of stress-minimization design using the induced p-norm aggregation formulation

is formed to afford the transmission of the loading. The corresponding stress distribution arrives at the best uniform state that has the superior capability of avoiding material failure in the application of engineering.

4.5.5 A Half Annulus with a Square Hole In this section, a more complex design domain, namely a half annulus with a square hole, is optimized by the ITO method with the induced KS aggregation formulation to study the effectiveness and utility. The basic setting of structural design domain is shown in Fig. 4.8, where structural dimensions of the square hole are equal to 3 (Length) and 2 (Height), respectively. The design parameters for the stress-based ITO method to minimize stress are defined in the last row of Table 4.2, and the value of stress norm parameter is defined in the induced KS aggregation formulation. The corresponding stiffness-maximization design of a half annulus with a square hole is shown in Fig. 4.27, namely the optimized topology, 2D stress distribution and 3D

Fig. 4.27 Compliance-minimization design of a half annulus with a square hole: a structural topology; b stress distribution in 2D; c stress distribution in 3D

4.5 Numerical Examples

99

Fig. 4.28 Stress-minimization designs of a half annulus with a square hole using the induced KS aggregation formulation: a structural topology; b stress distribution in 2D; c stress distribution in 3D

stress distribution. As we can easily observe, the optimized design with complianceminimization has an explicit feature of stress concentration in the first corner of the square hole, and the maximum value of von Mises stress is equal to 13.76. Figure 4.28 gives the optimized design of a half annulus with a square hole with the stress-minimization by the proposed ITO method with the induced KS aggregation formulation, including the structural topology, 2D and 3D stress distributions. Firstly, materials in the first corner with stress concentration in the square hole are reconfigured, where an arch-type member is formed within the optimization to avoid the generation of stress concentration, and the maximum value of von Mises stress is reduced to be 2.99. The corresponding stress distributions in 2D and 3D are both presented in Fig. 4.28a2, a3. In comparison with stress distributions shown in Fig. 4.27a2, a3, the distribution of stress of the optimized topology shown in Fig. 4.28a1 is evener homogeneous, which can have a higher ability to ensure the nonoccurrence of stress concentration. Additionally, several intermediate designs of the structural topology are displayed in Fig. 4.29. As we can see, the regions with the high possibility to have stress concentration gradually form a completely different distribution of materials relative to the stiffness design, which intends to generate the feature with circle boundaries that have the uniform stress distribution. Finally, the

Fig. 4.29 Iterative histories of structural topology for the design of a half annulus with a square hole using the induced KS aggregation formulation

100

4 ITO for Structures with Stress-Minimization

intermediate designs of the structural topology can also present the perfect feature of numerical stabilities and smooth convergence in the ITO.

4.6 Summary In the current chapter, the main intention is to develop a stress-related ITO formulation that has the promising effectiveness and efficiency, where two induced aggregation formulations are innovatively constructed for the global stress measure. In the ITO method, the DDF is parametrized by the NURBS, where threshold projection is adopted to control the grayscale of intermediate densities to avoid the generation of singular topologies. The DDF with superior capability can maintain smooth boundaries to eliminate terrible features that have the high possibility of the occurrence of stress concentrations. The IGA offers more benefits for improving numerical precision in stress measurement. The effective STM is applied to stable optimization process, so that the optimizer can search for the better solutions. Several numerical examples are studied to show the effectiveness and efficiency of the stress-related ITO formulation with the induced aggregation formulations for stress-minimization designs. Based on the optimized results, we can achieve several important conclusions: (1) the induced aggregation formulations can effectively improve the approximation accuracy of the maximum von Mises and remove mesh dependency of conventional aggregations; (2) the proposed ITO method can effectively find the better solutions with the stress-minimization, and the optimized topologies have the evener uniform stress distributions relative to previous designs; (3) in the case of stress concentrations, the optimizer tends to form the boundaries with circle shapes to remove the feature of the reentrant corner, so that stress concentrations can be successfully eliminated; (4) the well-known challenges in stress-related designs are well-solved in the ITO method with the induced aggregation formulations; (5) the optimized topologies in the current work have the smaller value of maximum von Mises stress and the evener uniform stress distributions in design domain.

Chapter 5

ITO for Piezoelectric Structures with Manufacturability

In previous chapters, a similar feature is that the linearly elastic problems are considered in the related works. In the current chapter, the piezoelectric materials are considered, and the main intention is to discuss the effectiveness and efficiency of the ITO method on the piezoelectric materials, and it also focuses on seeking for the expected designs of piezoelectric actuators. Moreover, the critical problem of the pins and joints in previous designs is extensively discussed, and a robust ITO (RITO) method is applied to study the optimization of piezoelectric structures but with considering manufacturability. In the RITO method, an erode–dilate operator is introduced.

5.1 NURBS-Based IGA for Piezoelectric Materials 5.1.1 Piezoelectric Constitutive Relations The linearly constitutive relations of piezoelectric materials can be expressed as [223], 

T = c E S − eT E D = eS + ε S E

(5.1)

where T and S denote mechanical stress and strain, respectively, and D and E indicate electrical displacement and electrical field, respectively. c E is the stiffness tensor in constant electrical field. e is the piezoelectric coefficient matrix. ε S is the permittivity coefficient matrix subject to a constant mechanical strain. T is the matrix transpose. In the current work, lead zirconate titanate (PZT) is the basic material considered in the latter numerical examples, and the related material properties of PZT are listed in [144], including mechanical stiffness tensor, piezoelectric property © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gao et al., Isogeometric Topology Optimization, Engineering Applications of Computational Methods 7, https://doi.org/10.1007/978-981-19-1770-7_5

101

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5 ITO for Piezoelectric Structures with Manufacturability

and the permittivity matrices. In [144], several basic assumptions are defined in the 2D case of piezoelectric plate, and it also corresponds to the classic structure of the piezoelectric actuator. In the 2D piezoelectric plate, it is poled along the corresponding thickness direction. The whole piezoelectric plate can be viewed in the state of the plane stress due to the smaller thickness of the plate than the in-plane scales. Moreover, the electric field only exists in the thickness direction, and a matrix form of Eq. (5.1) can be stated as ⎡

⎤ ⎡ ∗E T1 c11 ⎢ T2 ⎥ ⎢ c∗E ⎢ ⎥ = ⎢ 12 ⎣ T3 ⎦ ⎣ 0 ∗ D3 e31

∗E c12 ∗E c11 0 ∗ e31

⎤ ∗ ⎤⎡ 0 −e31 S1 ∗ ⎥⎢ ⎥ 0 −e31 ⎥⎢ S2 ⎥ ∗E ⎣ ⎦ c33 0 S3 ⎦ ∗ 0 ε33 E3

(5.2)

where ⎧ 2 2 ⎨ c∗E = c E − (c13E ) ; c∗E = c E − (c13E ) ; c∗E = c E E E 11 11 12 12 33 66 c33 c33 ⎩ e∗ = e31 − e33 c13EE ; ε∗ = ε S + (e33E )2 31

c33

33

33

(5.3)

c33

are the corresponding transformed material constants for the plane stress design problems.

5.1.2 IGA Formulation for Piezoelectric Materials The weak form of the electromechanical coupled equilibrium equations can be written as

E   T T T T T c δu f d + δu f d = δ S T d = δ S S − e E d v s    

 (5.4) T T S  δφqd =  δ E Dd =  δ E eS + ε E d where f v and f s denote the imposed body force and surface traction, respectively, and u denotes the displacement field. q is the surface charge density on the electrodes, and φ is the electric potential. The mechanical strain and electrical field of each element can be expressed by a linear interpolation of mechanical and electrical degrees of freedom, given as 

S = Bu u E = Bφ φ

(5.5)

5.2 DDF with the Erode–Dilate Operator

103

where B u is the strain–displacement matrix and B φ =1/ h. h indicates the thickness of the piezoelectric plate. In IGA, the displacement field is represented by a linear combination of NURBS basis functions and mechanical degrees of the freedom at control points. The field of electric potential has the same principle, and the detailed mathematical form is given as u=

n  m 

p.q

Ri, j (ξ, η)ui, j

(5.6)

i=1 j=1

where n and m denote the total numbers of control points at two parametric directions ξ and η, respectively. ui, j is the displacement at the (i, j)th control point. p.q Ri, j (ξ, η) are 2D NURBS basis functions. Based on Hamilton’s variational principle without considering the damping effect, the coupled equation governing the single piezoelectric element can be expressed as 

kuu kuφ T kuφ −kφφ



ue φe



 =

fe qe

 (5.7)

where kuu , kuφ , kφφ are mechanical stiffness matrix, piezoelectric coupling matrix, and dielectric stiffness matrix, respectively, and the corresponding formulas can be stated as: ⎧ T E T E ⎪  B u c B u | J 1 || J 2 |dξ dη ⎨ kuu = h e B u c B u d = h  T T kuφ = h e B u e B φ d = h  B T eT B | J || J |dξ dη (5.8)  uT S φ 1 2 ⎪ ⎩ T S | || |dξ kφφ = h e B φ ε B φ d = h  J J dη B ε B φ 1 2  φ  denotes the bi-unit parent element where e denotes the element domain, and  space in IGA. J 1 and J 2 are Jacobi matrices of mappings from the parametric space to the physical space and from the bi-unit parent element to parametric element in IGA [35]. The current work considers piezoelectric material, after assembling elemental matrices, the global equation for the actuation can be given as K uu u + K uφ φ = f

(5.9)

5.2 DDF with the Erode–Dilate Operator In the current work, the main intention is to show the effectiveness and efficiency of the ITO for the design of piezoelectric actuators, and then discuss how to eliminate the critical issue of the joints and pins in previous piezoelectric actuators by the ITO

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5 ITO for Piezoelectric Structures with Manufacturability

considering the uniform manufacturability. In the ITO design for piezoelectric actuators, the DDF is still needed to present the distribution of materials in the design domain, and in order to remove the number of intermediate densities, the second version of the DDF is considered here. Moreover, to consider the uniform manufacturability in the design of piezoelectric actuators, a known erode–dilate operation is adopted here, which is originally applied to realize automatic inspection in the morphology field [52, 155]. In the later robust ITO formulation for the design of piezoelectric actuators, the corresponding eroded, intermediate and dilated designs will be simultaneously considered during each iteration. The robust ITO formulation that can be effective in the elimination of the one-node connected pins or joints is developed. In the erode–dilate operator, three values η (ηeo , ηid , ηdo ) in the threshold projection should be defined to generate three topologies, namely the eroded, intermediate and dilated topologies. As shown in Fig. 5.1, a detailed illustration for the erode–dilate operation for the DDF to achieve the eroded, intermediate and dilated topologies is provided, and the corresponding ηeo , ηid and ηdo are set to be 0.7, 0.5 and 0.3, respectively. The details for the control design variables, the DDF and the topologies are all shown in Fig. 5.1. As we can see, the erode–dilate operator in actual is employed to process the control design variables which are already smoothed by the corresponding mechanism. After this operation, the related eroded, intermediate and dilated versions of control design variables can be obtained, which will be applied to construct the corresponding eroded, intermediate and dilated DDFs, termed by eo , id and do . Meanwhile, the corresponding eroded, intermediate and dilated topologies can be id do also defined by the corresponding isocontours, named by eo top , top and top . In the viewpoint of the authors, the details of the erode–dilate operator have the similar principle of the offset operation in the level-set function to define a family of micro unit cells with the similar topologies [224] or develop the coating or shell features in structures [225, 226].

Fig. 5.1 Erode–dilate operator for the DDF

5.3 Piezoelectric Materials Interpolation Schemes

105

5.3 Piezoelectric Materials Interpolation Schemes In the previous works for the design of piezoelectric actuators using the classic densities-based works, a power-law function should be defined for the relationship within piezoelectric material properties and the densities. A known PEMAP model for piezoelectric materials is developed in [139], in which the elastic, piezoelectric and dielectric material properties can be stated by the corresponding power functions with the densities. After that, the electrical polarization direction (positive or negative) of the piezoelectric material is also considered in the optimization, and the extension of PEMAP model is developed, named by the PEMAP-P model [142]. In PEMAP-P model, a kind of design variables i, j ∈ [0, 1] are required to the representation of the polarization profile in the optimization, and the details of the PEMAP-P model can be expressed by: 

⎧ p ⎨ kuu = E min + ( (ξ, η)) uu (E 0 − E min) kuu k = e + ( (ξ, η)) puφ (e0 − emin ) (2 (ξ, η) − 1) ppo kuφ ⎩ uφ min kφφ = εmin + ( (ξ, η)) pφφ (ε0 − εmin ) kφφ

(5.10)

where E min , emin and εmin denote the minimum values of stiffness, electromechanical coupling and dielectric matrices of the voids, respectively, and they are taken to be 1e-9 to avoid the occurrence of numerical singularity during the optimization. E 0 , e0 and ε0 are maximum values of the stiffness, electromechanical coupling and dielectric matrices of the piezoelectric solids. is the continuous function for the second type of design variables ϕi, j , which is defined by a same manner including five steps shown in Sect. 3.1 to construct the DDF. puu , puφ , pφφ and p po are the corresponding penalization parameters for the stiffness, piezoelectricity, dielectric and polarization, respectively. Hence, the topology optimization of piezoelectric actuators will contain four penalization parameters, which have the significant effect on the latter optimization, and the suitable definitions of them can be beneficial to effectively improve the numerical stability [141, 142, 146], so that parameters cannot be defined arbitrarily or simply set to be the value in linearly elastic design problems. As also already studied in [145], the extensive discussions for the definition of four penalization parameters are given, and the corresponding necessary definition conditions are offered for many different design problems to ensure the stable convergence that can be beneficial to find the suitable designs. As already given in the above discussions, the main focus of the current work is to develop the RITO formulation for the design of piezoelectric actuators without the critical issue of the one-node connected hinges, and there is no need to further study the influence of four penalization parameters. Hence, in the current work, the values of the above three penalty parameters puu , puφ and p po are set as 3, 4 and 1 in all numerical examples, respectively, which keep consistent with the previous work [142]. Meanwhile, it is noticed that the penalty parameter pφφ is not involved in the design of piezoelectric actuators to maximize the output displacement.

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5 ITO for Piezoelectric Structures with Manufacturability

5.4 ITO and RITO for Piezoelectric Actuators In the current work, the ITO formulation for the design of piezoelectric actuators is firstly developed to present the effectiveness and efficiency of the ITO method, and then the RITO formulation for the design of piezoelectric actuators is developed to eliminate the critical problem of the pins and joints occurred in the design of piezoelectric actuators. In the RITO formulation for piezoelectric actuators, the uniform manufacturability is considered by the considerations of the erode–dilate operation.

5.4.1 ITO Formulation Without Uniform Manufacturability In previous works for the design of piezoelectric actuators, several different objective functions have been already defined, mainly including maximizing the mean transduction of piezoelectric actuators [142], the maximization of the output displacement [144, 145]. As we know, the formulations for the design of piezoelectric actuators can be developed by considering structural performance. The design problems of piezoelectric materials have a similar feature of the optimization of compliant mechanism, and the main intention is to seek for a suitable distribution of materials to achieve the maximal output displacement in the imposed locations or direction of the design domain, which can be reasonably in response to the imposed excitation of an electrical load. The structural mean transduction working as the objective function is to have a better electromechanical conversion. Compared to the maximizing the output displacement, it might be difficult and introduces several numerical difficulties in the latter optimization. Hence, in the current work, the maximization of the output displacement at the specific locations is chosen as the optimized objective function in the ITO formulation, and the corresponding mathematical formula can be stated as: Find : ψ i, j , ϕi, j (i = 1, 2, . . . , n; j = 1, 2, . . . , m) Min : J id , id = −uout = − f dt u ⎧ ⎨ Kuu id u + Kuφ id , id φ = f S.t. : G( ) =  id (ξ, η)d − Vmax ≤ 0 ⎩ 0 < ψmin ≤ ψi, j ≤ 1; 0 ≤ ϕi, j ≤ 1

(5.11)

where J is the objective function, and it is defined by the output displacement with a negative sign at the specific location of the design domain. u out is the output displacement at the specified locations of the design domain. f td is a dummy load, and it is equal to 1 at the related output displacement node and 0 otherwise. G( ) is the volume constraint, where Vmax is the maximum material consumption. ψmin is a positive integer to avoid the occurrence of numerical singularity, generally equal to 1e-9. As we can easily see, only the intermediate design presented by the DDF id and id is considered in the optimization formulation, and the eroded and dilated designs in

5.4 ITO and RITO for Piezoelectric Actuators

107

the erode–dilate operation are not considered in the ITO formulation for the current design.

5.4.2 RITO Formulation with Uniform Manufacturability Based on previous designs [141, 142] for piezoelectric actuators, although several innovative designs can be effectively created using topology optimization, the essential issue of the one-node connected pins and joints still exist. In practical use, this critical problem of the existence of the hinges can have a negative effect on the latter manufacturing and also the practical applications. In the earlier works, minimizing structural stiffness is also considered as a subcomponent in the definition of the objective function, whereas it can only add a number of materials at the hinges, and the critical problem cannot be removed [141, 142]. Hence, the RITO formulation with the erode–dilate operator is developed for the design of piezoelectric actuators, which involves the optimization of the eroded, intermediate and dilated designs with control design variables, the DDF and also the topologies within the optimization, and the detailed formula can be expressed as: Find : ψi, j , ϕi, j (i = 1, 2,

. . . , n; j = 1,

2, . . . , m)  Min : J ( eo , eo ) + J id , id + J do , do ⎧ eo eo eo Kuu ( ⎪

id )u + Kuφ ( id , id )φ = f ⎪ ⎪ ⎪ Kuu u + Kuφ , φ ⎪ ⎪  = f ⎪ do do do ⎪ u + K φ= f ,

K ⎪ uu uφ ⎪  ⎪ ⎪

eo id do  ⎨ G1 , , = eo d S.t. :     ⎪ ⎪ ⎪ id do ⎪ ⎪ d /3 − Vmax ≤ 0 + d + ⎪ ⎪  ⎪

id   id ⎪ ⎪ ⎪ G =  d − Vmax ≤ 0 ⎪ ⎩ 2 0 < ψmin ≤ ψi, j ≤ 1; 0 ≤ ϕi, j ≤ 1

(5.12)

where G 1 denotes the volume constraint for the eroded, intermediate and dilated topologies. G 2 is the volume constraint of the intermediate design, which corresponds to the blueprint design. The robust topology optimization for compliant mechanism has been studied in previous researches to consider the manufacturing-tolerant [155], where many different objective functions are developed, including the sum of three objective functions of the eroded, intermediate and dilated topologies, the min–max formulation and also the sum of two objective functions chosen from three designs randomly [52, 227].

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5 ITO for Piezoelectric Structures with Manufacturability

5.5 Design Sensitivity Analysis In the current work, the OC method is employed in the ITO formulation and the MMA is applied for the computation of the RITO formulation owing to their special features in the related formulations. The sensitivity analysis of the objective function with respect to control design variables is given as: 

∂J ∂ψi j ∂J ∂ϕi, j

K uu = λT ∂∂ψ u + λT ij

= λT

∂ K uφ φ ∂ϕi, j

∂ K uφ φ ∂ψi j

(5.13)

where λ is the adjoint vector of a dummy load f d , and it can be solved by f d = λT K uu . According to Eq. (5.8), the derivatives of mechanical stiffness matrix and the piezoelectric coupling matrix with respect to two kinds of design variables can be stated as: ⎧ ∂ kuu p,q ⎪ ⎪ = puu ( (ξ, η))( puu −1) (E 0 − E min )Ri, j (ξ, η) ⎪ ⎪ ∂ψ ⎪ i j ⎪  ⎪ ⎪ ∂ ψˆ i j ∂ ψ˜ i j ⎪ ⎪ ⎪ B tu c E B u | J 1 || J 2 |dξ dη ⎪ ⎪ ˜ ∂ψ  ∂ ψ ⎪ ij  ij ⎪   ⎪ ⎪ ∂ kuφ p,q ⎪ ⎪ = puφ ( (ξ, η))( puφ −1) (e0 − emin ) (2 (ξ, η) − 1) ppo Ri, j (ξ, η) ⎨ ∂ψi j (5.14)  ⎪ ∂ ψˆ i j ∂ ψ˜ i j ⎪ t t ⎪ | || |dξ J J dη B e B φ 1 2 ⎪ u ⎪  ⎪ ∂ ψ˜ ∂ψi j  ⎪ ⎪ ∂k ij ⎪ ⎪ uφ p,q ⎪ = 2 ppo (e0 − emin )(2 (ξ, η) − 1)( ppo −1) Ri, j (ξ, η) ⎪ ⎪ ⎪ ∂ϕ i, j ⎪  ⎪ ⎪ ∂ ϕˆi, j ∂ ϕ˜i, j ⎪ ⎪ ⎩ B tu et B φ | J 1 || J 2 |dξ dη ∂ ϕ˜i, j ∂ϕi, j   The derivatives of volume constraints with respect to two types of design variables can be given by: ⎧ ⎪ ⎪ ⎨

∂G 1 ∂ψi j

=



 p,q  1d Ri, j (ξ, η)

 p,q ⎪ ∂G 2 ⎪ = 1d Ri, j (ξ, η) ⎩ ∂ψ  ij

 

∂ ψˆ ieoj ∂ ψ˜ i j ∂ ψˆ iidj ∂ ψ˜ i j

+ 

∂ ψˆ iidj ∂ ψ˜ i j

∂ ψ˜ i j ∂ψi j

+

∂ ψˆ idoj ∂ ψ˜ i j



2 ; ∂G =0 ∂ϕi j

∂ ψ˜ i j ∂ψi j

;

∂G 1 ∂ϕi j

=0 (5.15)

Based on the above general forms of sensitivity analysis, the first-order derivatives of the objective and constraint functions with respect to two types of design variables for the eroded, intermediate and dilated topologies can be solved. Two volume constraint functions are developed in the RITO method for the optimization of piezoelectric actuators, where the MMA method [220] is required to evolve design variables and solve the problem. The OC method is enough to solve the ITO for the design of piezoelectric actuators with only a constraint [2].

5.6 Numerical Examples

109

5.6 Numerical Examples In the current section, the main intention is to address the effectiveness and efficiency of the ITO and RITO formulations for the design of piezoelectric actuators by some numerical benchmarks and also present the effectiveness in the elimination of one-node connected features in the optimized designs of piezoelectric actuators, which can clearly display the extreme requirement of the considerations of the manufacturing-tolerant in the optimization of piezoelectric actuators. As shown in Fig. 5.2, three different structural design domains with the relative loads and boundary conditions are provided. In all numerical examples, PZT-4 is chosen in the latter optimization, and the related material properties can refer to [144]. The thickness of the piezoelectric plate is equal to 1e − 5 in all numerical examples, and there are no basis materials only with the piezoelectric material. Each IGA element with a number 3 × 3 of Gauss quadrature points is considered in the computation id of all element stiffness matrices. In the threshold projection, the values ( do top , top eo and top ) for the definition of the eroded, intermediate and dilated topologies are defined as 0.3, 0.5 and 0.7, respectively, for all numerical examples, except for the specific description. The applied voltage is set to be 1 in all numerical examples. In the initial designs, the values of all control design variables ψi, j are set to be 1, and the values of the other kind of control design variables ϕi, j are set as 0.1. The corresponding NURBS details for three structural design domains are listed in Table 5.1. The convergent condition is: the L ∞ norm of the difference of the objective function in two consecutive iterations should be less than 1% with the maximum step being 400 steps.

Fig. 5.2 Boundary and load conditions of a structural design domain 1; b structural design domain 2; c structural design domain 3

Table 5.1 Related parameters in three structural design domains Domain

L

H

NURBS orders

Control points

IGA elements

V max

1

1e − 2

1e − 2

[3, 3]

102 × 102

100 × 100

0.26

2

1e − 2

1e − 2

[3, 3]

102 × 102

100 × 100

0.20

3

1e − 2

1e − 2

[3, 3]

102 × 102

100 × 100

0.30

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5 ITO for Piezoelectric Structures with Manufacturability

5.6.1 Design of Piezoelectric Actuators Using ITO The main focus of the current subsection is to discuss the effectiveness of the ITO formulation on the design of piezoelectric actuators shown in Fig. 5.2. In the optimization of each design domain, three numerical cases with different settings of the spring stiffness at the corresponding output regions are discussed. As shown in Fig. 5.3, the corresponding representations for the DDF, the solids and voids, the polarity profile are provided. In next examples, the corresponding representations for them keep unchanged expect for the specific definition. In the case of the first structural design domain, the corresponding settings of spring stiffness are 0.01, 0.005 and 0.0015 in three numerical cases, respectively. As clearly shown in Fig. 5.4, the optimized results of the first design domain are all provided, including the optimized DDFs id , the optimized topologies id top and id the optimized polarization layouts of id and

in the optimized topologies. The top top id id polarization profile is defined as: (1) the positive polarity is defined by top · top = 1,   id and (2) the id · 1 −

top top = 1 corresponds to the negative polarity in the design domain, and (3) id top = 0 defines the null polarity. According to the above designs in three cases, we can easily find that the objective function of the output displacement s defined at the increases with the decreasing of the value of the spring stiffness kuu corresponding location. Several lumped structural members are formed in the final topology when the value of the spring stiffness is equal to 0.01. Moreover, most of PZT-4 materials have the negative polarity. However, more PZT-4 materials have the positive polarization profile if the value of the spring stiffness is equal to 0.0015. Additionally, it can be easily observed that there are two one-node connected pins or joints in the optimized topology shown in Fig. 5.4b3, which are clearly displayed by two arrows with the red color. The output displacement at the relative position is equal to 3.51e − 6, which is much larger than the first case with 1.13e − 6. The main cause is that the existence of the localized hinges with one-node connection can extensively magnify the displacement of the piezoelectric actuator by the extreme deformations in these critical areas. However, the extreme deformations in these critical regions, like hinges, can induce the critical problem of the occurrence of stress concentrations. Moreover, the existence of the localized hinges in the piezoelectric actuators also introduces several difficulties in the latter manufacturing. Hence, we

Fig. 5.3 Color setting in the DDF, the topology and the polarization

5.6 Numerical Examples

111

Fig. 5.4 Optimized designs of the first domain for three different cases: a1 the optimized DDF of structural design domain 1 with stiffness equal to 0.01; b1 the optimized topology of structural design domain 1 with stiffness equal to 0.01; c1 the layout of the polarization in the optimized topology in b1; a2 the optimized DDF of structural design domain 1 with stiffness equal to 0.005; b2 the optimized topology of structural design domain 1 with stiffness equal to 0.005; c2 the layout of the polarization in the optimized topology in b2; a3 the optimized DDF of structural design domain 1 with stiffness equal to 0.0015; b3 the optimized topology of structural design domain 1 with stiffness equal to 0.0015; c3 the layout of the polarization in the optimized topology in in b3

can conclude that eliminating the issue of the localized hinges in the design of piezoelectric actuators is extremely required to ensure the manufacturing robustness. As shown in Fig. 5.5, the iterative curves of the objective function of the output displacement and the volume fraction of PZT-4 materials are given, where some intermediate topologies of piezoelectric actuators are also provided. It can be easily seen that the iterative histories are very stable and quickly arrive at the expected value with satisfaction of the convergent condition, which can clearly show the effectiveness of the ITO method on the design of piezoelectric actuators. The ITO designs for structural design domains presented in Fig. 5.2b, c are also extensively discussed, in which three cases with different settings of the spring stiffness are also defined for each design domain. Table 5.2 lists the corresponding values for setting the spring stiffness in the design of the first structural domain, and the corresponding settings for the second structural design domain are also listed in Table 5.3. In Tables 5.2 and 5.3, the corresponding output displacements in three cases for each structural domain are both given. As we can observe, a similar feature that the case with the lowest value of the spring stiffness can achieve the maximum output displacement as expected.

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5 ITO for Piezoelectric Structures with Manufacturability

Fig. 5.5 Iterative histories of the objective function and volume fraction in the optimization of Fig. 5.4b3

Table 5.2 Spring stiffness and optimized displacement of piezoelectric actuator in Fig. 5.2 (b)

Table 5.3 Spring stiffness and optimized displacement of piezoelectric actuator in Fig. 5.2 (c)

Case 1

Case 2

Case 3

s1 , k s2 , k s3 , k s4 kuu uu uu uu

0.01

0.005

0.001

u out

4.54e − 7

5.57e − 7

2.30e − 6

Case 1

Case 2

Case 3

s kuu

0.01

0.005

0.002

u out

1.48e − 6

1.96e − 6

3.14e − 6

Figure 5.6 presents the final optimized results for the second design domain, and the optimized results for the third design domain are shown in Fig. 5.7. In the representation of the final results, the related optimized topologies and the polarization profile in the corresponding optimized topologies are given. In the case of the results shown in Fig. 5.6a1, b1 for the second domain, we can easily observe that a uniform polarization distribution exists in the optimized topology that has no PZT-4 materials with the positive polarity when the values of spring stiffness are set to be 0.01 and 0.005, and the relative distributions are shown in Fig. 5.6a2, b2. Meanwhile, several structural lumps are generated in final topologies displayed in Fig. 5.6a1, b1. However, the third design of the piezoelectric actuator has a dramatic variation in the final topology, where the spring stiffness is the smallest. In the optimized topology, eight one-node connected hinges are generated. The corresponding output displacement is equal to 2.30e − 6, and it is much larger than the values in the former two cases (4.54e − 7 and 5.57e − 7). As far as the third design domain is concerned, we can see

5.6 Numerical Examples

113

Fig. 5.6 Optimized designs of the second design domain for three different cases: a1 the optimized topology of structural design domain 2 with stiffness equal to 0.01; a2 the layout of the polarization in the optimized topology in a1; b1 the optimized DDF of structural design domain 2 with stiffness equal to 0.005; b2 the layout of the polarization in the optimized topology in b1; c1 the optimized DDF of structural design domain 2 with stiffness equal to 0.001; c2 the layout of the polarization in the optimized topology in c1

Fig. 5.7 Optimized designs of the third design domain for three different cases: a1 the optimized topology of structural design domain 3 with stiffness equal to 0.01; a2 the layout of the polarization in the optimized topology in a1; b1 the optimized DDF of structural design domain 3 with stiffness equal to 0.005; b2 the layout of the polarization in the optimized topology in b1; c1 the optimized DDF of structural design domain 3 with stiffness equal to 0.002; c2 the layout of the polarization in the optimized topology in c1

that the optimized topologies in three cases suffer from the critical issue of the onenode connected pins or joints, as seen in Fig. 5.7. The values of the output displacement are increased when the values of spring stiffness are decreased, as shown in Table 5.3. Finally, it should be noted that currently optimized designs are similar to the previous designs of the optimized topologies in previous works [142, 144] with

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5 ITO for Piezoelectric Structures with Manufacturability

the same spring stiffness being adopted, which can clearly show the effectiveness of the ITO formulation for the optimization of piezoelectric actuators. Similar to the result shown in Fig. 5.4, the existence of the localized hinges in the optimized topologies can also lead to the degradation of loss of the corresponding structural performance, which can present the extreme requirement for the development of the RITO formulation with considering the high manufacturing-tolerant.

5.6.2 Design of Piezoelectric Actuators Using RITO In the above example, the ITO formulation for the design of piezoelectric actuator only considers the intermediate design including the DDF and the topology. In the current subsection, the RITO method with considering the eroded, intermediate and dilated designs is applied to develop the optimization formulation for the piezoelectric actuators with the manufacturing-tolerant. Three design domains in Fig. 5.2a–c are, respectively, discussed, and the corresponding values of the spring stiffness for three design domains are set to be 0.0015, 0.001 and 0.002, respectively. It can be seen that these settings of the spring stiffness in the ITO designs of three structural domain can lead to the occurrence of critical issue of the localized hinges in the optimization. As shown in Fig. 5.8, the optimized results of the first structural domain are provided, including the eroded, intermediate and dilated designs of three optimized id do DDFs ( eo , id and do ), three optimized topologies ( eo top , top and top ) and also the optimized polarization distributions in the corresponding topologies. Firstly, as we can observe, the above three optimized topologies displayed in Fig. 5.8b1, b2,and b3 are completely free of the occurrence of the one-node connected pins or joints. It can also manifest that the uniform and flexible structural deformations can be occurred in the currently design of the optimized topology, rather than the localized hinges to trigger the extreme deformations in the design. The critical issue of stress concentrations originated from localized hinges can be effectively removed. Secondly, the currently optimized topologies in three designs with the eroded, intermediate and dilated are all robust to manufacturing uncertainties, which can effectively avoid the degradation or loss of structural performance stemming from the wear of some machining instruments and under- or over-etching devices. According to the final output displacement listed in Table 5.4, we can achieve that the current RITO formulation aims to optimize structural performance of both the eroded and the intermediate designs but sacrifice the performance of the dilated design, namely the values of output displacement at the former two cases larger than that of the final case for the dilated design. As shown in Fig. 5.9, the convergent histories of the robust designs are also provided, which can clearly present the superior stability in the optimization and also quick convergence. Hence, the above example can successfully present the effectiveness and efficiency of the RITO formulation for the optimization of the piezoelectric actuator with the manufacturing-tolerant. The RITO formulations for the optimization of other two structural domains are also performed, and the corresponding results are also given. As far as the

5.6 Numerical Examples

115

Fig. 5.8 Robust designs of the first domain using the RITO formulation: a1 the optimized DDF of the eroded design; b1 the optimized topology of the eroded design; c1 the corresponding layout of the polarization in the optimized topology in b1; a2 the optimized DDF of the intermediate design; b2 the optimized topology of the intermediate design; c2 the corresponding layout of the polarization in the optimized topology in b2; a3 the optimized DDF of the dilated design; b3 the optimized topology of the dilated design; c3 the corresponding layout of the polarization in the optimized topology in b3

Table 5.4 Output displacement and volume fraction of three designs

Eroded design

Intermediate design

Dilated design

Output displacement

2.38e − 6

3.23e − 6

1.68e − 6

The volume fraction

0.22

0.26

0.30

robust designs for the second structural domain are concerned, Fig. 5.10 presents the optimized robust design, including the eroded, intermediate and dilated designs id do of three optimized topologies ( eo top , top and top ) and also the optimized polarization distributions in the corresponding topologies. In the case of the third structural domain by the RITO, the corresponding results for the robust design are shown in Fig. 5.11, where the eroded, intermediate and dilated designs of the optimized id do topologies ( eo top , top and top ) and the optimized polarization distributions in the corresponding topologies are included. As we can find, the optimized topologies presented in Figs. 5.10 and 5.11 are all free of the localized hinges with only onenode connection. Compared to the previous designs given in Fig. 5.6c1 and Fig. 5.7c1,

116

5 ITO for Piezoelectric Structures with Manufacturability

Fig. 5.9 Convergent histories of the objective function and volume fraction in the RITO optimization for the first domain

Fig. 5.10 Robust designs of the second domain using the RITO formulation: a1 the optimized topology of the eroded design; a2 the corresponding layout of the polarization in the optimized topology in a1; b1 the optimized topology of the intermediate design; b2 the corresponding layout of the polarization in the optimized topology in b1; c1 the optimized topology of the dilated design; c2 the corresponding layout of the polarization in the optimized topology in c1

we can easily find that the initially localized pins or joints in the deterministic optimization are removed and replaced by some flexible structural bars in the currently robust designs shown in Figs. 5.10 and 5.11. In the robust designs, structural deformations will occur in the corresponding areas by varying flexible members, rather than the extreme deformations of the localized hinges. Hence, by removing one-node connected pins or joints, the robustly eroded, intermediate and dilated topologies are effective in removing the manufacturing errors that have the high possibility of

5.6 Numerical Examples

117

Fig. 5.11 Robust designs of the third domain using the RITO formulation: a1 the optimized topology of the eroded design; a2 the corresponding layout of the polarization in the optimized topology in a1; b1 the optimized topology of the intermediate design; b2 the corresponding layout of the polarization in the optimized topology in b1; c1 the optimized topology of the dilated design; c2 the corresponding layout of the polarization in the optimized topology in c1

degrading or damaging structural functionalities stemming from subtle structural members in final optimized topologies. The numerical results of the robust designs in Figs. 5.10 and 5.11 are listed in Tables 5.5 and 5.6, respectively, including the final output displacements and the volume fractions of PZT-4 materials. As we can easily observe, it is similar to the first case for the robust design of the first structural domain, where the intermediate topology has the largest value of the output displacement and the value of the output displacement is the smallest in the dilated design. Hence, a critical feature is that the RITO formulation optimizes the structural performance of Table 5.5 Numerical results of Fig. 5.10

Table 5.6 Numerical results of Fig. 5.11

Optimized designs

Eroded design

Intermediate design

Dilated design

Output displacement

1.72e − 6

1.99e − 6

1.35e − 6

The volume fraction

0.15

0.20

0.25

Optimized designs

Eroded design

Intermediate design

Dilated design

Output displacement

3.3e − 6

3.74e − 6

2.75e − 6

The volume fraction

0.25

0.30

0.35

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5 ITO for Piezoelectric Structures with Manufacturability

both the eroded topology and the intermediate topology, but sacrifices the structural performance of the dilated topology. As already shown in the above examples, we can easily find that the robust design for the piezoelectric actuator can achieve the eroded topology, the intermediate topology and also the dilated topology for each structural domain. In the development of the RITO formulation, it is noted that volume fraction of the intermediate topology is same as the deterministic design for the piezoelectric actuator by the ITO method. Hence, the final intermediate design can refer to the final blueprint design that can be applied in the manufacturing for the latter applications. Figure 5.12 presents the final robustly blueprint designs for three structural domains and also the corresponding deterministic designs by the ITO for the corresponding design domains. As we can find, the robustly blueprint designs in three domains have no extensive difference compared to the corresponding deterministic topologies. Only a critical difference is located at the localized hinges in the deterministic topologies, which are replaced by the flexible structural members in the robustly blueprint designs. Hence, the robustly blueprint designs of the optimized topologies can be effective in avoiding structural failure when the manufacturing or practical use is considered. Moreover, the corresponding values of the final output displacement for the deterministic designs and the robustly blueprint designs are listed in Table 5.7. We can achieve a conclusion that the RITO formulation can maintain the manufacturing-tolerant in the optimization of the piezoelectric actuator, and it not sacrifice structural performance of the deterministic designs. In Table 5.7, we can find that the RITO formulation can gain a better design of the third structural domain with the larger value of the output displacement.

Fig. 5.12 Comparison between the deterministic design by the ITO and the robustly blueprint design by the RITO for three domains: a1 the deterministic design of the first design domain; a2 the blueprint design of the first design domain; b1 the deterministic design of the second design domain; b2 the blueprint design of the second design domain; c1 the deterministic design of the second design domain; c2 the blueprint design of the second design domain

5.7 Summary

119

Table 5.7 Comparison of the output displacements between deterministic designs and robustly blueprint designs for three domains Domain

First domain

Second domain

Third domain

Deterministic design

3.51e − 6

2.30e − 6

3.14e − 6

Blueprint design

3.23e − 6

1.99e − 6

3.74e − 6

5.7 Summary In the current chapter, the key focus is to apply the ITO method developed in Chap. 2 for the design of piezoelectric actuators, and then the RITO formulation is developed for the design of piezoelectric actuators to eliminate the critical problem of the one-node connected hinges and effectively guarantee the uniform manufacturingtolerant. In the RITO formulation, the erode–dilate operation is introduced in the construction of material description model, in which the eroded design, the intermediate design and also the dilated design are all considered in the optimization of the piezoelectric actuators. Three structural domains are optimized by the ITO formulation and also the RITO formulation. According to the presented results, we can gain the following remarks: (1) IGA can be beneficial to optimize the piezoelectric actuators by enhancing numerical accuracy of analysis; (2) the ITO method is effective in the deterministic design of the piezoelectric actuators, and the convergent curves are superior stable and can quickly arrive at the expected values; (3) the RITO formulation is effective in eliminating the critical issue of the localized hinges and the high possibility of the occurrence of stress concentrations by reasonably distribute PZT-4 materials to form flexible structural members.

Chapter 6

ITO for Architected Materials

The main intention of topology optimization for architected materials is to find the expected material microstructural topologies that can have the predefined/specific material properties, like the extreme bulk/shear modulus. The customized design of architected materials using topology optimization has gradually attracted a wide of attentions in many researchers. In topology optimization for architected materials, it mainly contains the following components: (1) The homogenization is applied to evaluate macroscopic effective properties of architected materials considering the micro information including the sizes, shapes and topologies; (2) the topology optimization formulation for the customized design of architected materials is developed, and the special objective functions containing the maximization of the bulk/shear modulus should be constructed based on the homogenization evaluation.

6.1 Numerical Implementations of the Homogenization Using IGA As we know, the classic homogenization theory is proposed for predicting the macroscopic effective properties of architected materials using their micro unit microstructures [228, 229]. In the theory of the homogenization, two critical assumptions should be ensured: (1) The micro unit cell with specific topology should be periodically distributed in the bulk area of architected material; (2) the micro scales of unit cell should be much smaller than that of the bulk scale in architected materials [165, 228–230]. As shown in Fig. 6.1, an example of architected material with a specific micro unit cell is provided, and the microstructure is defined in Y = [0, y1 ] × [0, y2 ] (2D) or Y = [0, y1 ] × [0, y2 ] × [0, y3 ] (3D). In the case of linearly elastic materials, a Y-periodic function is applied to define the elastic property function E, and the detailed formula can be expressed as: E (x) = E(x, y) = E(x, y + Y ) © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gao et al., Isogeometric Topology Optimization, Engineering Applications of Computational Methods 7, https://doi.org/10.1007/978-981-19-1770-7_6

(6.1) 121

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6 ITO for Architected Materials

Fig. 6.1 Architected material composed of a kind of micro-architecture with a large number, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

where  indicate the aspect ratio of structural scales between the micro unit cell and the bulk material, namely y = x/. Hence, it can be found that material elastic properties in the bulk can be periodically changed with respect to the micro unit cell. The global displacement field u (x) in the bulk material can be calculated according to the asymptotic expansion theory [165, 228–230], stated as: u (x) = u0 (x, y) + u1 (x, y) +  2 u2 (x, y) + · · · , y = x/ 

(6.2)

According to Eq. (6.2), it can be also seen that the global displacement field is periodically dependent on the micro scale y at the macroposition. As far as the linearly elastic materials are considered, the dispersive behavior in Eq. (6.2) is not necessary to consider and only the first-order term of variation u1 (x, y) should be taken into account in the latter computation. If the limit  → 0, we can have that:  f (x, y)d =

lim

→0



1 |Y|

  f (x, y)dYd, y = x/

(6.3)

 Y

where f is the function of interest, and it is a function of architected materials with the periodicity in the micro scale y. Another new form of the above equation for the definition of the macroscopic effective properties E iHjkl of architected material can be stated as:   j)  i j  0(kl)   1 εr s − εr s u kl dY (6.4) E pqr s ε0(i E iHjkl = pq − ε pq u |Y | Y

6.1 Numerical Implementations of the Homogenization Using IGA

123

where Y denote the domain of microstructural unit cell. |Y | indicates the corresponding area (2D) or volume (3D). E pqr s denotes the locally elastic material prop0(i j) erty in the  i jbulk  material. ε pq denotes the unit test strain with linear independency, and ε pq u indicates the unknown strain field within the micro unit cell. The corresponding linearly elastic equilibrium equation in the bulk material with Y-periodic boundary conditions can be applied to solve the unknown response in above equation, and the detailed equation can be expressed by:  Y

    E pqr s ε pq u i j εr s δu i j dY 

=

 ij   j) dY, ∀δu ∈ H 1 Y, Rd E pqr s ε0(i pq εr s δu

(6.5)

Y

where δu is the virtual displacement belonging to the admissible displacement space Hper . In several pervious works [165, 228–230] for the design of architected materials, the classic FEM is applied to numerically implement the homogenization. In the current work, the proposed ITO method is considered here, and the IGA should be applied to the latter analysis of the homogenization. In the micro unit cell, the displacement field should be constructed by a linear combination of the NURBS basis functions with the corresponding control responses, namely the displacements at control points. The mathematical formulation is similar to the construction of NURBS, but the physical meanings of control coefficients are different. In the homogenization, a critical problem is located at how to impose the periodic boundary conditions at the micro unit cell. In the FEM, the periodic boundary conditions can be directly imposed at the corresponding elementary nodes. The elementary nodes are located at the boundaries of micro unit cells due to the interpolation in the FEM. In IGA, it is difficult to impose the periodic boundary conditions at the elementary nodes. However, an alternative scheme is that the periodic boundary conditions can be defined at the control points, due to a fact that control points are the basis of the construction of NURBS and also the later response space in analysis. The development of periodic boundary condition can be defined by the displacement relationship among control pints at structural boundaries of micro unit cells, and the corresponding mathematical formulation can refer to [192] for 2D and [7, 13] for 3D, and a general mathematical formula of 2D and 3D periodic boundary formulations can be expressed by: − u+ k − uk = ε(u0 )k

(6.6)

where k denotes the normal direction of boundaries in micro unit cell. u+ k is the displacement field of the boundaries with the normal direction k, where the normal direction corresponds to the positive direction of the physical coordinate axis. u− k are

124

6 ITO for Architected Materials

Fig. 6.2 IGA mesh for unit cell with Gauss quadrature points, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

the displacements of points at the opposite structural boundaries. k indicates the micro scale of material unit cell along the normal direction k. As shown in Fig. 6.2, a detailed illustration about the IGA mesh for micro unit cell with a series of IGA elements having the corresponding Gauss quadrature points is provided.

6.2 ITO for Micro-Architected Materials The basic flowchart of the ITO method for the design of architected materials is shown in Fig. 6.3. In the whole process, two main components are involved, namely (1) the density distribution function is constructed for the representation of the topology of micro unit cell; (2) the IGA is applied to develop the numerical implementation of energy-based homogenization method. These two components can be viewed as material description model and numerical analysis model for unit cells, respectively. Based on above derivations, we can easily obtain that macroscopic effective properties of architected materials are mainly dependent on the homogenized elastic tensor which is computed by the energy-based homogenization method using the micro information. The corresponding mathematical formula of macroscopic effective properties can be expressed by the function of subterms in the homogenized elastic tensor. As we know, the main intention of the ITO formulation to design architected materials is to achieve the ultra-lightweight micro unit cells. Hence, the prescribed material volume fraction in unit cell should be much smaller than previous works [13, 167, 168]. In the ITO formulation, the DDF is applied to present the topology of micro unit cell during the optimization, and the IGA-based energy-based homogenization method is employed to evaluate the homogenized elastic tensor of micro unit cell. Hence, the corresponding mathematical formula for the design of ultra-lightweight architected materials can be defined as:

6.2 ITO for Micro-Architected Materials

125

Fig. 6.3 Basic idea of the ITO for architected materials, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

  Find: φ φi, j , φi, j,k   Min: J (u, ) = f E iHjkl ⎧ ⎪ a(u, δu, ) = l(δu, ), ∀δu ∈ H 1 () ⎪ ⎪ ⎪ G() = 1  (ξ, η, ζ )υ dY − V ≤ 0 ⎨ 0 0 |Y | Y S.t. ⎪ ⎪ 0 < φmin ≤ φ ≤ 1 ⎪ ⎪ ⎩ i = 1, 2, . . . , n; j = 1, 2, . . . , m; k = 1, 2, . . . , l

(6.7)

where φ indicates the vector that includes a family of control design variables. J corresponds to the objective function which should be defined by the homogenized elastic tensor E iHjkl . G is material volume constraint for the design of micro unit cells in architected materials, which should be small to guarantee the generation of the truss-like bars in micro unit cells. υ0 indicates the volume fraction of solid materials, equal to 1.  corresponds to the DDF, and V0 is allowable material consumption. φmin denotes the minimal value of control densities to avoid numerical singularity. u denotes the field of structural displacement response in micro unit cells, which should maintain the periodic boundary formulation. δu is the virtual displacement field, which belongs to the admissible displacement space H 1 () with the Y-periodicity. The linearly elastic equilibrium equation is applied to solve unknown structural responses. a and l are the bilinear energy function and linear load function, respectively. The detailed mathematical formula for them is explicitly expressed by: ⎧   ij  ij ⎪ ⎨ a(u, δu, ) = E pqr s ()ε pq u εr s δu dY  Y  ij 0(i j) ⎪ ⎩ l(δu, ) = E pqr s ()ε pq εr s δu dY Y

(6.8)

126

6 ITO for Architected Materials

6.3 Design Sensitivity Analysis In the above development of the ITO formulation for the design of architected materials, the DDF is applied to present structural topology and control densities act as design variables in the optimization. The information of sensitivity analysis of objective function and constraint function should be given to derive the evolvement of structural topologies using the optimization algorithm. In order to achieve the derivations of the objective function with respect to control densities, the derivatives of objective function with respect to the DDF should be firstly computed, expressed by: ∂J = ∂

∂ f E iHjkl ∂ E H

i jkl

∂ E iHjkl

∂

(6.9)

In the above equation, we can easily find that the derivatives of the homogenized elastic tensor with respect to the DDF should be computed, and the corresponding derivatives can be calculated based on its definition and expressed by: ∂ E iHjkl ∂

=

1 |Y |



 j)  i j  0(kl)   εr s − εr s u kl dY γ γ −1 E 0pqr s ε0(i pq − ε pq u

(6.10)

Y

The details of the derivations for the above computation can be referred to [13]. E 0pqr s indicates material constituent elastic tensor. Based on the construction of the DDF, the first-order derivative of the DDF with control densities can be computed and stated as:   ∂(ξ, η) p,q = Ri, j (ξ, η)ψ φi, j ∂φi, j

(6.11)

p,q

where Ri, j (ξ, η) is the NURBS basis function at the (ξ, η) computational point, and   ψ φi, j is the Shepard function at control point (i, j). Based on the above derivations, the first-order derivatives of the homogenized elastic tensor with respect to control densities can be expressed as: ∂ E iHjkl ∂φi, j

1 = |Y |     j)  i j  0(kl)   p,q εr s − εr s u kl Ri, j (ξ, η)ψ ρi, j dY γ γ −1 E 0pqr s ε0(i pq − ε pq u Y

(6.12)

6.5 Numerical Examples

127

Similarly, the first-order derivatives of material volume constraint can be given by: ∂G 1 = |Y | ∂φi, j



  p,q Ri, j (ξ, η)ψ φi, j υ0 dY

(6.13)

Y

6.4 Optimality Criteria (OC) As we know, the OC method has powerful capabilities of solving the topology optimization problems with only one constraint but having a large number of design variables. Hence, in the current work, the OC method is employed, and the final updating scheme is given as:

φi,(κ+1) j,k

⎧



 

ζ ⎫ (κ) (κ) (κ) (κ) ⎪ ⎪ ⎪ − m , φ φ ≤ max φ − m , φ max φ , if  ⎪ ⎪ min min ⎪ i, j i, j i, j i, j ⎪ ⎪ ⎪ ⎪





⎪ ⎪ ζ ζ ⎪ (κ) (κ) (κ) (κ) (κ) ⎪ ⎨ if max φi, j − m , φmin < i, j φi, j ⎬ i, j φi, j , 

 = ⎪ ⎪ ⎪ ⎪ < min φi,(κ)j + m , 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪

 

  ζ ⎪ ⎪ ⎪ ⎪ (κ) (κ) (κ) (κ) ⎭ ⎩ min φ + m , 1 , if min φi, j + m , 1 ≤ i, j φi, j i, j (6.14)

where m, ζ denote the move limit and the damping factor, respectively. i,(κ)j is the updating factor, which is defined by the derivatives of the objective function and the constraint function, given by: i,(κ)j

   ∂J 1 ∂G = (κ) − (κ) / max , (κ)  ∂φi, j ∂φi, j

(6.15)

where  is a small positive constant to avoid the fraction with a form of 0/0.

6.5 Numerical Examples In the current section, several numerical examples in 2D and 3D are discussed to sufficiently present the effectiveness and efficiency of the ITO method for the rational design of architected materials but with the ultra-lightweight micro unit cells. In the latter, the rational design of architected materials in 2D is mainly applied to present the superior characteristics of the proposed ITO formulation, and then the

128

6 ITO for Architected Materials

Fig. 6.4 A cubic:  = H = Z = {0, 0, 0, 0.1111, . . . , 0.8889, 1, 1, 1}, n = 11, m = 11, l = 11; p = 2, q = 2, r = 2, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

ITO formulation is then employed to effectively seek for a series of new 3D ultralightweight unit cells. Moreover, the comparisons between the currently optimized 3D micro unit cells and the classic TPMS are presented. In all numerical examples, the Young’s moduli and Poisson’s ratio for basis materials are defined to be 1 and 0.3, respectively. In the IGA for the homogenization, 3 × 3 and 3 × 3 × 3 Gauss quadrature points are defined in each IGA element in 2D and 3D cases, respectively. For numerical simplicity, the structural normal scales of micro unit cells are all defined as 1. The penalization parameter γ in material penalization model is set as 3. The convergent criterion is that the L ∞ norm of the difference of control densities in two consecutive iterations is less than 1% within the maximum 100 iteration steps. As displayed in Fig. 6.4, the design domain of 3D micro unit cells is provided, and the structural geometry is shown in Fig. 6.4a, and the corresponding NURBS solid is shown in Fig. 6.4b, and the related IGA mesh is displayed Fig. 6.4c.

6.5.1 2D Micro-Architected Materials In the current subsection, the main intention is to discuss the effectiveness and efficiency of the ITO formulation for the design of 2D micro unit cells in architected materials. The corresponding design domain is as simple square with sizes of 1 × 1. In the construction of the NURBS model for a square, the quadrature NURBS

6.5 Numerical Examples

129

Fig. 6.5 Initial design of the micro-architecture, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

basis functions are adopted, and  = H = {0, 0, 0, 0.01, · · · , 0.99, 1, 1, 1} of knot vectors are defined in the construction of NURBS model. In the IGA, the corresponding mesh of the square has 10,000 finite elements, and control points are 101 × 101 in control net. Moreover, the ITO formulation focus on achieving the unltralightweight micro unit cells, the maximal material volume fraction is much smaller than previous works [12, 168, 192], namely 10% in the current work. In the latter optimization, the corresponding initial design of micro unit cells is shown in Fig. 6.5, including control densities, discrete densities at Gauss quadrature points and the DDF in design domain. In the illustration of Fig. 6.5, the height direction corresponds to the value of densities. As we can easily see, the initial values of control densities are equal to 0 or 1, respectively, and an inhomogeneous layout of densities can avoid the uniform distribution of the sensitivity in micro unit cells.

6.5.1.1

The Maximal Shear Modulus

The main focus of the current subsection is to design micro unit cells and ensure architected materials can have the maximum shear modulus, and the corresponding objective function can be defined as:   H J = f E iHjkl = −E 1212

(6.16)

As displayed in Fig. 6.6, the optimized designs of micro unit cell in architected materials are provided, including the optimized distribution of control densities presented in Fig. 6.6a, the optimized layout of discrete densities at Gauss quadrature points shown in Fig. 6.6b and the optimized DDF displayed in Fig. 6.6c. Firstly, it can be easily seen that the optimized DDF that can be viewed as a response surface

130

6 ITO for Architected Materials

Fig. 6.6 Optimized design of the micro-architecture, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

of the density in design domain has many superior features, namely the sufficient smoothness and higher-order continuity, although a clear characteristic of the break from the lower bound to the upper bound exist the initial design of control densities. As already discussed in Chap. 2, the main reason is that the smoothing mechanism and NURBS parameterization can effectively make sure the above superior features in the DDF during the optimization. Meanwhile, in order to present the details of the variation of the DDF in the optimization, several intermediate solutions of the DDF are shown in Fig. 6.7. As we can see, the advancement of the DDF has the features of the superior stability and the convergence can quickly arrive at the predefined terminal condition, only 25 iterations. Secondly, the final optimized results of the 2D micro unit cell in architected materials with maximum shear modulus are listed in Table 6.1, including the 2D view of discrete densities with the higher values (0.5) at Gauss quadrature points, the isocontour (0.5) of the DDF that corresponds to the optimized topology, the homogenized elastic tensor E H of the optimized micro unit cell and material volume fraction of the final topology. Thirdly, we can easily find that material consumption of the optimized topology is nearly equal to 10% (the predefined volume fraction), which can effectively demonstrate the rationally of the construction of the DDF in material description model to determine the existence of materials and present the topology. According to the optimized topology of the micro unit cell, an example of the corresponding architected material with 10 × 5 periodically distributed micro unit cells is provided in the last row of Table 6.1. As we can clearly observe, the final architected material with the optimized micro unit cells is similar to the design in previous works [12, 168, 192]. However, a critical feature of the current design is the extremely low relative density, which can basically satisfy the ultra-lightweight requirement in the design of micro unit cells. Additionally, the optimized topology of micro unit cell can be simply viewed as a combination of two-force structural

6.5 Numerical Examples

131

Fig. 6.7 Intermediate designs of the DDF for the micro-architecture, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

members having the inclination angle ± 45◦ , which can demonstrate the effectiveness of the current ITO formulation for the design of architected materials. It is known that the current kind of micro unit cells can offer the higher stiffness for the corresponding structure that can strongly afford the imposed loads but with the minimal material consumption. Hence, the effectiveness and efficiency of the proposed ITO method for the design of micro unit cells with ultra-lightweight in 2D architected material with the extreme shear modulus can be clearly demonstrated.

6.5.1.2

The Maximal Bulk Modulus

The key purpose of the current subsection is to address the rational design of micro unit cells with the ultra-lightweight in 2D architected materials with the maximal bulk modulus, and the corresponding objective function can be explicitly stated as: 2    E iiHj j J = f E iHjkl = −

(6.17)

i, j=1

In this example, the initial definitions of control densities, discrete densities at Gauss quadrature points and the DDF are same as the above example, also shown in Fig. 6.4, and the corresponding maximum material volume fraction of micro unit cells is still defined to be 10%. As clearly displayed in Fig. 6.8, the optimized

Architected material

2D view of densities

Topology

EH ⎤ ⎡ 0.0274 0.0272 0 ⎥ ⎢ ⎢ 0.0272 0.0274 0 ⎥ ⎦ ⎣ 0 0 0.0258 0.098

V0

Table 6.1 Optimized results of the 2D micro-architecture with the maximum shear modulus, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

132 6 ITO for Architected Materials

6.5 Numerical Examples

133

Fig. 6.8 Optimized designs of the micro-architecture, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

results of the micro unit cell are provided, namely the optimized distribution of control densities displayed in Fig. 6.8a, the optimized layout of discrete densities at Gauss quadrature points indicated in Fig. 6.8b and the optimized design of the DDF shown in Fig. 6.8c. As we can easily see, the currently optimized distributions of control densities, discrete densities at Gauss quadrature points and the DDF have the similar superior merits in the final representation of structural topology, namely the expected smoothness and continuity. Additionally, the final values of the DDF are mostly located in the upper and lower bounds (0 and 1), and a narrow transition region emerges in the DDF for the existence of intermediate densities that can form smooth boundaries. As clearly provided in Table 6.2, the related numerical results of the current example are provided, also including the 2D view of discrete densities at Gauss quadrature points with values higher than 0.5, the isocontour of the DDF with the isovalue equal to 0.5 that corresponds to the optimized topology, the corresponding homogenized elastic tensor of micro unit cell and the final volume fraction of the optimized topology. It can be easily found that the value of material volume fraction is mostly equal to 0.099 (the prescribed material consumption), which can clearly demonstrate the reasonability of the current definition scheme and also the isovalue of the DDF. Meanwhile, the optimized topology of the micro unit cell have smooth boundaries and also distinct interfaces between solids and voids. These superior features of the optimized topology can offer more benefits for the manufacturing. In the last row of Table 6.2, an architected material with 10 × 5 periodically arranged micro unit cells is provided. It can be easily seen that the currently optimized micro unit cell has a different topology compared to previous works [12, 167, 168, 192]. The main cause is that the bulk modulus is defined by four terms in the homogenized elastic tensor, and the optimization of four terms does not have the unique solution. Moreover, even if the micro unit cells have same value of the homogenized elastic tensor, and the corresponding topologies might be different. In the above example,

Architected material

2D view of densities

Topology

EH ⎡ ⎤ 0.0293 0.0282 0 ⎢ ⎥ ⎢ 0.0282 0.0293 0 ⎥ ⎣ ⎦ 0 0 0.0002 0.099

V0

Table 6.2 Optimized results of the 2D micro-architecture with the maximal bulk modulus, reprinted from ref. [64], copyright 2020, with permission from Elsevier

134 6 ITO for Architected Materials

6.5 Numerical Examples

135

H the problem for the design of extreme shear modulus is only related to one term E 1212 of the homogenized elastic tensor in the bulk material, and the similar micro unit cell can be easily found using topology optimization. However, the design of micro unit cells with the extreme bulk modulus is involved four terms of the homogenized elastic tensor in 2D, which have a significant effect on the latter designs that can have many different topologies but with the same value of the bulk modulus. Finally, convergent curves are shown in Fig. 6.9, and it can show that the optimization is stable. In the current example, the aim is to study the influence of the initial design on the latter optimization, and a different initial design of micro unit cell is displayed in Fig. 6.10, also including control densities at control points, discrete densities at Gauss quadrature points, the DDF in structural domain and also the initial topology of the micro-architecture. As we can easily see, the wavy structural boundaries exist in the initial micro-topology of unit cell, because the initial DDF is not characterized with the sufficient smoothness and continuity. Compared with the first design shown by Fig. 6.4, the current design of micro unit cell has a critical feature of the anisotropic. In next case, the optimization of micro unit cell in 2D architected material with the maximal bulk modulus will be addressed using the initial design shown in Fig. 6.10, and other design parameters keep unchanged. As listed in Table 6.3, the final optimized results of the micro unit cell in architected material with the maximal bulk modulus are provided, including the 2D view of discrete densities at Gauss quadrature points with values higher than 0.5, the

Fig. 6.9 Iterative curves of 2D architected material with the extreme bulk modulus, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

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6 ITO for Architected Materials

Fig. 6.10 Initial design 2 of the micro-architecture, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

isocontour equal to be 0.5 of the DDF that refers to optimized topology, the corresponding homogenized elastic tensor and material volume of the final topology. It can be easily observed that the optimized micro-topology of the unit cell is characterized with smooth boundaries and distinct interfaces between the solids and voids, similar to the above two examples. A different feature is that the current topology of the micro unit cell is the anisotropic based on the layout of materials in structural domain, and the corresponding homogenized elastic tensor can also show this characteristic. Hence, we can confirm that the initial definition of control densities can have a significant influence on the final micro-topology. However, it should be noted that the optimized objective functions in different cases are mostly identical, even if the topologies are different.

6.5.2 3D Micro-Architected Materials In the current section, the main purpose is to seek for a series of new 3D ultra-lightweight micro unit cells in architected materials with extreme bulk/shear modulus. In the case of 3D example, the design domain of a cube is defined and discretized by a number of 24 × 24 × 24 IGA elements, and the related NURBS model for the cube contains 26 × 26 × 26 control points. Hence, the ITO formulation for the design of 3D architected materials has 26 × 26 × 26 design variables, namely control densities. Three different initial designs of 3D micro unit cell in architected materials are defined to search for many kinds of 3D novel ultra-lightweight micro unit cells with the extreme bulk/shear modulus, shown in Fig. 6.11. Meanwhile, it is noticed that the DDF, namely a NURBS response surface of the density, corresponds to a 4D (four-dimensional) function, and it is hard to clearly display a 4D density

Architected material

2D view of densities

Topology

EH ⎤ ⎡ 0.0292 0.0284 0.0066 ⎥ ⎢ ⎢ 0.0284 0.0292 0.0066 ⎥ ⎦ ⎣ 0.0066 0.0066 0.0021 0.099

V0

Table 6.3 Optimized results of the 2D micro-architecture with the maximal bulk modulus, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

6.5 Numerical Examples 137

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6 ITO for Architected Materials

Fig. 6.11 Three initial designs for the 3D micro-architectures, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

function in a 3D figure by the platform of MATLAB. Hence, the corresponding topology of the 3D micro unit cell is directly presented in the final representation of the optimized results. Moreover, in order to show the details of the topologies, the cross-sectional views of the optimized topologies are also provided to clearly present the interior information of structural geometry. As shown in Fig. 6.11, three different initial topologies with their cross-sectional views for micro unit cells are given. Similar to the above example in 2D, several holes should be distributed in the initial designs of the 3D micro unit cells to avoid the uniform distribution of sensitivity analysis in the whole design domain. In the next designs, the corresponding volume fraction V0 is set to be 5% within the optimization, which is much smaller than several previous works [12, 168] to guarantee the generation of lattice micro unit cells.

6.5.2.1

The Maximal Shear Modulus

The current example intends to achieve the novel 3D ultra-lightweight micro unit cell of architected materials with the maximal shear modulus, and the corresponding objective function is given as: 3    E iHjkl (i = j & J = f E iHjkl = − i, j,k,l=1

k = l & i j = kl)

(6.18)

6.5 Numerical Examples

139

Fig. 6.12 Topologically optimized micro-architectures in three cases, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

Figure 6.12 shows the optimized topologies of 3D micro unit cells in architected materials with extreme shear modulus are provided, and the corresponding crosssectional views of the optimized topologies are also presented to show their interior information. Firstly, a similar feature is that these optimized topologies of 3D micro unit cells are characterized with smooth boundaries and unique interfaces in the solids and voids also exist in the current example. Secondly, we can directly view that the related 2D views of the 3D micro unit cells No. 1 and No. 2 in three normal physical directions are same as the optimized design of the 2D micro unit cell listed in Table 6.1. It manifests that the currently optimized 3D micro unit cell No. 1 and No. 2 have a critical feature of the maximal shear modulus in three norm physical directions, which can clearly demonstrate the effectiveness of the current ITO formulation for the design of 3D micro unit cells. Meanwhile, these two optimized topologies can be also viewed as the combination of several two-force structural members to provide the maximal stiffness for the imposed load as much as possible. The optimized topology of 3D micro unit cell No. 3 can be regarded as a novel finding of architected materials with maximum shear modulus, where some twoforce line bars can be reasonably arranged in structural design domain, which is similar to the f.c.c. structure in [162]. Moreover, the 3D micro unit cell No. 2 is extremely analogous to the IWP ligament-based TPMS in [231], and this type of lattice structure with merits has been broadly applied into many engineering fields, such as the stiffest sandwich structure [162]. Overall speaking, the current design framework is in an effective and also scientific manner, which can sufficiently seek for a large number of findings of 3D micro unit cells, rather than strongly relying on the insight and intuitions of human. Additionally, the current designs of 3D micro unit cells are all ultra-lightweight, which can effectively guarantee the requirements in the practical use. As shown in Fig. 6.13, some intermediate topologies in the iteration of the 3D micro unit cells are also presented. According to the variations

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6 ITO for Architected Materials

Fig. 6.13 Intermediate topologies of the micro-architecture no. 2, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

of the topologies, we can easily see that the ITO formulation for 3D micro unit cells in architected materials with extreme shear modulus is very stable and can quickly arrive at the prescribed convergent condition in a few numbers of iterations, which can successfully present the effectiveness of the current ITO method on the design of 3D micro unit cells in architected materials with extreme shear modulus. Based on the presentation of the optimized 3D micro unit cells shown in Fig. 6.12, the corresponding architected materials can be formed by periodically distributed the micro unit cells along the normal physical directions. Figures 6.14, 6.15 and 6.16 display the optimized architected materials with the extreme shear modulus, where a number of 6 × 3 × 3 3D micro unit cells are included in the bulk area. As we can easily see, the optimized 3D architected materials can be regarded as the combination of a large number of lattice members with a two-force feature, which can not only offer the strong stiffness but also have the ultra-lightweight (the relative density is nearly 0.05). Hence, it might be concluded that the developed ITO formulation has the promising effectiveness and efficiency for the design of micro unit cells of 2D and 3D architected materials with extreme shear modulus.

6.5.2.2

The Maximal Bulk Modulus

In the current subsection, the main focus is to achieve the 3D micro unit cells of architected materials with extreme bulk modulus, and the corresponding objective function can be defined as:

6.5 Numerical Examples

141

Fig. 6.14 3D architected material no. 1: The relative density is 0.0498, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

Fig. 6.15 3D architected material no. 2: The relative density is 0.0501, reprinted from Ref. [64], copyright 2020, with permission from Elsevier 3    J = f E iHjkl = − E iHjkl (i = j i, j,k,l=1

&

k = l)

(6.19)

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6 ITO for Architected Materials

Fig. 6.16 3D architected material no. 3: The relative density is 0.0499, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

Figure 6.17 shows the final optimized 3D micro unit cells of architected materials and the relative cross-sectional views, namely the 3D micro unit cells No. 4 to No. 6. It can be easily seen that a feature of the smooth boundaries and distinct interfaces between the solids and voids exists in the current designs, similar to all previous examples. As shown in Fig. 6.18, the 3D architected material no. 1 is also given, and a number of 6 × 3 × 3 3D micro unit cells are periodically distributed in the

Fig. 6.17 Topologically optimized micro-architectures in three cases, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

6.6 Summary

143

Fig. 6.18 3D architected material no. 4: the relative density is 0.0503, reprinted from Ref. [64], copyright 2020, with permission from Elsevier

bulk area. It can be seen that the 2D views of the 3D micro unit cells No. 5 and No. 6 are all similar to the corresponding 2D designs in [167, 192] along three normal physical directions. It can manifest that the 3D micro unit cells No. 5 and No. 6 have the maximal bulk modulus in three normal physical directions to successfully present the effectiveness of the current ITO formulation for 3D architected materials with the extreme bulk modulus. Additionally, the 3D micro unit cell No. 4 is also a novel finding of architected material with the extreme bulk modulus, also displayed in Fig. 6.18. Meanwhile, the currently optimized 3D micro unit cells No. 4 to No. 6 have similar structural members compared with the b.c.c. structure listed in [162]. Hence, we can conclude that the current ITO formulation can have the powerful capabilities of finding the novel architected materials in 3D with the extreme bulk modulus.

6.6 Summary In the current chapter, we intend to develop the design formulation for architected materials with the extreme elastic moduli using the proposed ITO method in the Chap. 2. In the development of the ITO formulation for architected materials with the extreme elastic moduli, material description model using the DDF to present the structural topology of micro unit cells and the IGA-based energy-based homogenization method to effectively predict macroscopic effective properties are both developed. In later examples, the corresponding 2D and 3D micro unit cells are both

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6 ITO for Architected Materials

discussed. According to the presented numerical examples in 2D and 3D, we can conclude that: (1) a series of new and interesting 2D and 3D micro unit cells of architected materials with the extreme bulk modulus can be achieved; (2) a series of 2D and 3D micro unit cells of architected materials with the extreme shear modulus can be also found; (3) the optimized topologies of 2D and 3D micro unit cells are all characterized with a critical feature of smooth boundaries and critical interfaces in the solids and voids due to the merits of the DDF; (4) the promising and powerful capabilities of finding the novel architected materials with extreme elastic moduli can be clearly demonstrated in the 2D and 3D ITO formulations based on the above discussions in numerical examples.

Chapter 7

ITO for Auxetic Metamaterials

In the above chapter, the ITO method is applied to address the design of micro unit cells in architected materials and intends to find a series of novel and interesting microstructures. In the current chapter, the ITO method is applied to study the design of auxetic metamaterials, and the main intention is to seek for a family of new and interesting micro unit cells but with the auxetic behavior. Comparatively speaking, the design of auxetic metamaterials and the design of architected materials can be putted in a same category of topology optimization problem. The main cause is that auxetic metamaterials can be also viewed as a subtype of architected materials but with a special property, namely the negative Poisson ratio (NPR). In the design of auxetic metamaterials using topology optimization, the whole process is similar to the design of architected materials, and a critical difference lies in the objective function. In Chap. 6, the objective function in the design of architected materials refers to extreme elastic moduli. In the current work, the corresponding objective function is also defined by the homogenized elastic tensor, but it can effectively push the optimizer can find micro unit cells with the NPR, namely the auxetic behavior. Hence, in the design of auxetic metamaterials using the ITO method, material description model to present structural topology using the DDF and the IGA-based energy-based homogenization method should be required in the development of the ITO formulation for the design of auxetic metamaterials, and the details for the development of these two components can be referred to Chaps. 2 and 6, respectively.

7.1 ITO Formulation for Auxetic Metamaterials It is known that the definition of the Poisson’s ratio of materials results from the aspect ratio between the transverse contraction strain and the longitudinal extension strain when the force is stretched along one normal direction. Considering from the homogenized elastic tensor, the values of Poisson’s ratios of 2D materials can be set to be υ12 = D1122 /D1111 (one direction) and υ21 = D1122 /D2222 (the other direction). © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gao et al., Isogeometric Topology Optimization, Engineering Applications of Computational Methods 7, https://doi.org/10.1007/978-981-19-1770-7_7

145

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7 ITO for Auxetic Metamaterials

In the intention of generating the NPRs feature in materials, several pervious works with many different objective functions have been discussed [185–187, 191], like minimizing the weighted square difference between the evaluated elastic tensor and the expected elastic tensor, minimizing the difference between the predicted NPR and its target [189], the minimization of the combination of the elastic tensor [192, 232] and so on [190]. As already discussed, the NPR behavior of materials is highly dependent on the rotating effect of mechanisms in micro unit cells [166, 232]. Here, the current formulation considers the combination of the homogenized  elastic tensor DH to achieve the optimization of auxetic metamaterials. The terms d i , j =1,i = j i i j j  and d DH are simultaneously considered in the corresponding formui , j =1,i = j i i j j lation, and the optimizer will have the capability of finding the micro unit cells with the NPRs feature, and the detailed form is given as: 





Find: φ















     φi, j 2D φi, j,k 3D

Min: J (u, ) =

d 

DiˆHiˆ jˆ jˆ (u, ) − β

ˆ j=1, ˆ i, iˆ = jˆ

d 

DiˆHiˆ jˆ jˆ (u, )

ˆ j=1, ˆ ˆ jˆ i, i=

 ⎧ a(u, δu) = l(δu), ∀δu ∈ Hper , Rd ⎪ ⎪ ⎪  ⎪ ⎨ 1 S.t. : G() = || (φ)υ0 d − V0 ≤ 0 ⎪ ⎪  ⎪ ⎪ ⎩ 0 < ρmin ≤ ρ ≤ 1, (i = 1, 2, . . . , n; j = 1, 2, . . . , m; k = 1, 2, . . . , l) (7.1) where φ denotes control densities assigned to control points, working as the design variables. J is the objective function. β is a weighting parameter to denote the importance of the corresponding terms. d is the spatial dimension of materials. G is the volume constraint, in which V0 is the maximum value and v0 is the volume fraction of the solid.  is the DDF in Eq. 2.15. u is the unknown displacement field in material microstructure, which have to satisfy the PBCs given in Eq. 6.6. δu is the virtual displacement field belonging to the admissible displacement space Hper with y-periodicity, which is calculated by the linearly elastic equilibrium equation. a and l are the bilinear energy and linear load functions, stated as:  ⎧ γ ⎨ a(u, δu) = ((φ)) D0 ε(u)ε(δu)d   ⎩ l(δu) = ((φ))γ D0 ε0 ε(δu)d

(7.2)



Material elastic tensor is assumed to be an exponential function with respect to the DDF, and γ is the penalization parameter. D0 is the constitutive elastic tensor of the basic material.

7.2 Design Sensitivity Analysis

147

7.2 Design Sensitivity Analysis The first-order derivatives of the objective function with respect to the DDF are firstly derived before obtaining the sensitivity analysis with respect to design variables, as: ∂J = ∂

∂ DiˆHiˆ jˆ jˆ (u, )

d  ˆ j=1, ˆ i, iˆ = jˆ

∂

−β

d 

∂ DiˆHiˆ jˆ jˆ (u, )

ˆ j=1, ˆ ˆ jˆ i, i=

∂

(7.3)

In the above equation, the derivatives of the homogenized elastic tensor with respect to the DDF are required, and the details for derivations of the homogenized stiffness tensor with respect to the DDF can be referred to [166, 186, 232], and the final form is given by: ∂ DiˆHiˆ jˆ jˆ ∂

1 = ||

       ˆ ˆ     0 ii 0 jˆ jˆ iˆiˆ jˆ jˆ γ −1 0 − εr s u ε pq − ε pq u γ () D pqr s εr s d 

(7.4) The first-order derivatives of the DDF with respect to control densities can be derived by:

 ∂(ξ, η) p,q = Ri, j (ξ, η)ψ φi, j ∂φi, j

(7.5)

p,q

where Ri, j (ξ, η) is the NURBS basis function at the computational point (ξ, η).

 ψ ρi, j is the value of the Shepard function at the control point (i, j). Based on the chain rule, the form of the derivatives of the homogenized elastic tensor with respect to control densities can be computed by: ∂ DiˆHiˆ jˆ jˆ ∂

1 = ||

       ˆ ˆ     0 ii 0 jˆ jˆ γ −1 0 iˆiˆ jˆ jˆ ε pq − ε pq u γ () D pqr s εr s − εr s u



 p,q Ri, j (ξ, η)ψ φi, j d

(7.6)

Hence, the sensitivity analysis of the objective function J with respect to control design variables can be given. Similarly, the first-order derivatives of the volume constraint can be expressed by: 1 ∂G = || ∂φi, j

 

 p,q Ri, j (ξ, η)ψ φi, j υ0 d

(7.7)

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7 ITO for Auxetic Metamaterials

Meanwhile, it should be noted that the above derivations are developed for 2D materials, which can be directly extended to 3D scenario.

7.3 A Relaxed OC Method In previous works, the classic OC method [233] has the superior capabilities of the optimization for several design problems that all have a single constraint but with a large number of design variables [2]. A critical feature of the monotonicity property should be maintained in the corresponding design problems, where the derivatives of the objective function and constraint function with respect to the design variables have the same sign. However, the optimization of auxetic metamaterials has no this feature of the monotonicity property, and the corresponding signs of the derivatives in the objective and constraint functions are different. Currently, a relaxed OC method [234] is adopted here to update design variables, where a Lagrange function is constructed, stated as: L(u, ) = J (u, ) − μG() + ( + μ)G()

(7.8)

A new form the optimality criteria can be expressed by: ∂ J (u, ) ∂G() ∂ L(u, ) = −μ ∂φi, j ∂φi, j ∂φi, j ∂G() + ( + μ) , φmin ≤ φi, j ≤ 1 ∂φi, j

(7.9)

A new form of the above equation can be gained, and the corresponding relax form is given: i, j

  1 ∂G ∂J = / μ− +μ ∂φi, j ∂φi, j

(7.10)

Hence, the updating factor for design variables at each iteration can be given as: i,(κ)j

   1 ∂J ∂G (κ) = (κ) / max , (κ) μ −  + μ(κ) ∂φi,(κ)j ∂φi, j

(7.11)

where  is a small positive constant to avoid the fraction with a form of 0/0. As we can easily find, the updating factor i,(κ)j can be positive during the optimization process, by choosing an appropriate value of the shift parameter μ(κ) , namely:

7.4 2D Auxetic Metamaterials

 (κ)

μ

≥ max

∂J ∂φi,(κ)j

149

 / max ,

∂G



∂φi,(κ)j

(i = 1, 2, . . . , n; j = 1, 2, . . . , m) (7.12)

Hence, the final updating scheme is given as:     ⎫     ⎧ (κ) (κ) ζ (κ) (κ) ⎪ max φi, j − m , φmin , if i, j φi, j ≤ max φi, j − m , φmin ⎪ ⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪      ⎪ ⎪ ⎪ (κ) ζ ⎬ ⎪ ⎨ max φ (κ) − m , φ ⎨ ⎬ ζ <  min (κ+1) (κ) (κ) i, j i, j    i, j φi, j , φi, j = if (κ) (κ) ⎩ ⎪ ⎭⎪ ⎪ ⎪ φ < min φ + m , 1 ⎪ ⎪ i, j i, j ⎪ ⎪ ⎪ ⎪         ⎪ ⎪ ⎩ ⎭ (κ) (κ) (κ) ζ (κ) if min φi, j + m , 1 ≤ i, j φi, j min φi, j + m , 1 ,

(7.13)

where m, ζ are the move limit and the damping factor, respectively. The detailed derivations of the OC algorithm can be referred to [24]. The Lagrange multiplier at the κth iteration step (κ) is updated by a bi-sectioning algorithm [2]. In the latter, several numerical examples in 2D and 3D are discussed to show the effectiveness and efficiency of the ITO method on the design of auxetic metamaterials, and the critical purpose is to find a series of new and interesting micro unit cells with the NPRs features. Firstly, the examples for the design of 2D auxetic metamaterials are studied to demonstrate the merits of the developed ITO formulation for the NPRs designs. Secondly, the 3D examples are innovatively discussed to find a series of novel 3D micro unit cells with the NPR behavior, which can clearly show the superior effectiveness of the proposed ITO method. In all numerical examples, the linear elasticity is considered in the latter optimization, and 2D micro unit cells are discretized by a series of 2D IGA elements with a unit thickness in the state of the plane stress. The material properties of the Young’s moduli E 0 and the Poisson’s ratio υ0 are set to be 1 and 0.3, respectively. In the computation of IGA elementary matrices, Gauss quadrature points with a number 3 × 3 (2D) or 3 × 3 × 3 (3D) are chosen. The corresponding penalty parameter is defined to be 3, and the constant parameter β in all numerical examples for the optimization is set to be 0.03, expect for a specific illustration. The terminal criterion is that the L ∞ norm of the difference of control densities between two consecutive iterations is less than 1% or the maximum iterative 100 steps are reached.

7.4 2D Auxetic Metamaterials In the case of 2D micro unit cells, the corresponding design domain is a basic square with the scales of 1 × 1. The NURBS is applied to model structural design domain, where the knot vectors are defined as  = H = {0, 0, 0, 0.01, . . . , 0.99, 1, 1, 1}, the total number of all IGA elements is 100 × 100, and 101 × 101 (10,202) control points are contained in the control net, also equal to the total number of design variables in the latter optimization. In the formulation for the design, the maximum material

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7 ITO for Auxetic Metamaterials

Fig. 7.1 Initial design of material microstructure, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

volume fraction V0 is set to be 30%, and the developed ITO formulation with the intention of optimizing the micro unit cells gradually changes the micro-structural topology, until the NPR feature can be gained. As shown in Fig. 7.1, the initial designs of 2D micro unit cells are provided, including the originally distribution of control design variables, the overall distribution of the initial discrete densities at Gauss quadrature points and the corresponding DDF. As we can easily see, the inhomogeneous distribution of the values of control design variables can ensure the initial topology with several holes to eliminate the uniform sensitivity field, owing to the periodic boundary conditions imposed in the energy-based homogenization of micro unit cells. As shown in Fig. 7.2, the optimized results of 2D micro unit cells are provided, mainly including the optimized distribution of discrete densities at Gauss quadrature points in Fig. 7.2a and the optimized DDF in Fig. 7.2b. As we can easily see, a similar feature of the sufficient smoothness and continuity exist in the optimized distributions of discrete densities at Gauss quadrature points and also the final DDF. Meanwhile,

Fig. 7.2 Optimized designs of material microstructure, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

7.4 2D Auxetic Metamaterials

151

Fig. 7.3 also presents several intermediate designs of the DDF in the optimization. As we can easily observe, the initial DDF has several breaks from the lower bound to the upper bound, shown in Figs. 7.1c and 7.3a. In the latter optimization, the smoothness is gradually improved in the DDF. Moreover, the values of the optimized density distribution function (DDF) are mostly equal to 0 or 1, and only a limited number of intermediate densities in a narrow transition area to construct smooth structural boundaries. As clearly shown in Table 7.1, the corresponding numerical results of the 2D micro unit cell are given, mainly including the 2D-view of discrete densities at Gauss

Fig. 7.3 Intermediate density response surfaces of the DDF, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

Table 7.1 Optimized 2D auxetic metamaterial, reprinted from Ref. [65], copyright 2019, with permission from Elsevier 2D view of densities

Topology

DH υ Vf ⎤ ⎡ 0.088 −0.054 0 ⎥ ⎢ ⎥ − 0.61 29.88% ⎢ −0.054 0.088 0 ⎦ ⎣ 0 0 0.0027

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7 ITO for Auxetic Metamaterials

Fig. 7.4 Rotating mechanisms in the optimized 2D auxetic metamaterial, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

quadrature points with the values higher than 0.5, the isocontour (value equal to 0.5) of the optimized DDF, namely the optimized topology, the related homogenized elastic tensor D H of the optimized topology, the Poisson’s ratio υ = −0.61 and the final volume fraction of the optimized topology Vf = 29.88%. It can be easily achieved that the optimized micro unit cell has the feature of the NPR, and the auxetic behavior exists in the related design of 2D micro unit cell, which can show the effectiveness of the current ITO formulation for the design of auxetic metamaterials. Moreover, the final volume fraction is mostly equal to the predefined value of material consumption 30%, which can present the ITO optimization can meet the convergent condition. Additionally, according to the optimized topology, we can easily observe that two rotating mechanisms (shown in Fig. 7.4) that are beneficial to form the auxetic feature in micro unit cells can be generated, which can show the reasonability of the objective function. According to the representation of the optimized topology in Table 7.1, we can easily see that smooth boundaries and clear interfaces between the solids and voids are generated in the final topology, and this feature can lower several difficulties in the latter manufacturing. Although the ITO formulation for the design of auxetic metamaterials is constructed using the density, and the core is to evolve the continuous and smooth function and then it applied to present structural topology. Finally, as shown in Fig. 7.5, the convergent curves of the objective and constraint functions are given, in which several intermediate topologies of the 2D auxetic micro unit cell are also provided. We can easily see that the iterative trajectories are very smooth, and the optimization can quickly satisfy the iterative condition within only 38 steps, which can show the perfect effectiveness of the developed ITO formulation for the design of 2D auxetic metamaterials with micro unit cells.

7.5 Discussions of the Weight Parameter

153

Fig. 7.5 Iterative curves of 2D auxetic metamaterial, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

7.5 Discussions of the Weight Parameter The current section aims to discuss the effect of the constant parameter β in the objective function on the latter design of auxetic micro unit cells. There are 15 numerical cases that will be performed, and the corresponding values are 0.03 (the above example), 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10, 0.15, 0.20, 0.25, 0.30, 0.0001, 0.0005, 0.02. Other design parameters keep consistent with the above design example, such as the details of NURBS parametrization, the maximum material volume fraction, the initial designs of control design variables, etc. The corresponding numerical solutions in the former 12 cases with the parameter equal to 0.03 to 0.3 are presented in Fig. 7.6. As we can easily observe, the values of the Poisson’s ratio increase in these 12 cases with the increasing of the constant parameter. As shown in Fig. 7.7, the corresponding micro unit cells are provided. Based on the given NPR values and the auxetic mechanism in the optimized topologies, we can intuitively and numerically obtain that the NPR behavior becomes the smaller and smaller when the weight parameter increases. In the last design, the weight parameter with the value 0.3 cannot push the optimizer to find a micro unit cell with the NPR feature. The first case with the value equal to 0.03 of the constant parameter can find the auxetic micro unit cell with the minimum value of the NPR − 0.614. The numerical results of the later three cases with the values of the constant parameter equal to 0.02, 0.0005 and 0.0001, respectively are all provided in Table 7.2. As we can easily see, the ITO formulation intends to minimize the NPR value in one direction if the constant parameter decrease to a great extent with the value equal to 0.0005. As clearly shown in the third row of Table 7.2, the relative results are

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7 ITO for Auxetic Metamaterials

Fig. 7.6 Numerical results of the former twelve cases, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

Fig. 7.7 Auxetic microstructures in twelve cases, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

given, where the optimized auxetic micro unit cell has the feature of the orthotropic and the value of υ21 is smaller than υ12 . In the last numerical case, the optimized design of micro unit cell with auxetic behavior is the anisotropic, and the chiral deformation mechanism is generated in the final topology of the auxetic micro unit cell. The main cause is that the constant parameter controls the weight (effect degree) d  of the corresponding term DiˆHiˆ jˆ jˆ in the objective function. Additionally, we ˆ j=1, ˆ ˆ jˆ i, i=

can find that the total iteration increases with the decrease of the constant parameter. Hence, in the case of the design of auxetic micro unit cells with the equal NPR values

7.6 3D Auxetic Metamaterials

155

Table 7.2 Numerical results of three cases, reprinted from ref. [65], copyright 2019, with permission from Elsevier β

0.02

Topology

Homogenized elastic tensor D H ⎤ ⎡ 0.0762 −0.038 0 ⎥ ⎢ ⎥ ⎢ −0.038 0.0702 0 ⎦ ⎣ 0 0 0.0008

⎡ 0.0005

0.1204 −0.053

⎢ ⎥ ⎢ −0.053 0.0392 0 ⎥ ⎣ ⎦ 0 0 0.0011

⎤ 0.084 −0.057 0.013 ⎥ ⎢ ⎢ −0.057 0.085 −0.013 ⎥ ⎦ ⎣ 0.013 −0.013 0.0028 ⎡

0.0001

⎤ 0

υ 

Iterations υ12 = −0.498 υ21 = −0.541



υ12 = −0.442 υ21 = −1.352



υ12 = −0.678 υ21 = −0.671

117

101

157

in two directions, the weight parameter 0.3 is a suitable value for the ITO formulation for the design of auxetic micro unit cells.

7.6 3D Auxetic Metamaterials In the current subsection, the ITO formulation for the design of 3D auxetic micro unit cells is studied in several numerical cases to find new and interesting microstructures with the auxetic feature. In the design of 3D auxetic microstructure, the corresponding design domain is a basic cube with the scales of 1 × 1 × 1, as shown in Fig. 7.8a. In the NURBS parameterization for the solid cube, the quadratic NURBS basis functions are adopted and knot vectors  = H = Z = {0, 0, 0, 0.417, . . . , 0.9583, 1, 1, 1} are adopted in three parametric directions. The corresponding NURBS solid and the related IGA mesh for the cube are shown in Fig. 7.8b, c, respectively. In the IGA mesh, 24 × 24 × 24 IGA elements and 26 × 26 × 26 control points in the control net (also equal to the number of control design variables) are included for the latter optimization.

156

7 ITO for Auxetic Metamaterials

Fig. 7.8 3D material microstructure, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

Fig. 7.9 Four initial designs for 3D material microstructure, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

In order to seek for novel microstructures of 3D auxetic micro unit cells, four different initial designs are defined and four numerical cases are discussed in the latter. Similar to previous chapters for the design of 3D structures, it is difficult to present the corresponding 4D DDF, and the relative 3D isocontours to present micro-structural topologies are given, and the initial designs are shown in Fig. 7.9. As far as the case 1 is concerned, the corresponding initial design is presented in Fig. 7.9a, in which the maximum value of material consumption is defined as 30%. As shown in Fig. 7.10a, the final optimized topology of the 3D micro unit cell with the auxetic behavior is given. The corresponding cross-sectional view of the optimized 3D micro unit cell is also presented in Fig. 7.10b to display the interior information of the optimized microstructure. Figure 7.10c displays a auxetic metamaterial having 3 × 3 × 3 repetitive micro unit cells. Similar to the design of 2D micro unit cells, the optimized micro unit cell also has smooth boundaries and distinct interfaces within the solids and voids. As we can easily see, the counterintuitive dilatational behavior can be clearly exhibited in the optimized 3D auxetic micro unit cell, if a force is imposed at one normal direction of the current microstructure. As clearly provided in Table 7.3, the corresponding homogenized elastic tensor of the 3D micro unit cell is given, and the Poisson’s ratio is equal to − 0.047. Hence, we can obtain that the optimized 3D micro unit cell shown in Fig. 7.10a has the auxetic feature that can be confirmed by the qualitative analysis and also the quantitative calculation.

7.6 3D Auxetic Metamaterials

157

Fig. 7.10 3D auxetic microstructure no. 1, reprinted from ref. [65], copyright 2019, with permission from Elsevier

As far as case 2 is concerned, the maximum value of material volume fraction is defined as 30%, and the ITO formulation optimizes the current case by starting from the initial design 2 shown in Fig. 7.9b. The initial design 3 presented in Fig. 7.9c is available in numerical case 3 with the requirement of material volume fraction equal to 30%, and numerical case 4 with the maximum material volume fraction equal to 24% optimize the 3D micro unit cell by starting from the initial design 3 presented in Fig. 7.9d. The corresponding optimized results in numerical cases 2, 3 and 4 are clearly presented in Figs. 7.11, 7.12 and 7.13, respectively. In the final representation, the optimized topologies, the corresponding cross-sectional views of the optimized topologies to present the interior geometry in detail and the related auxetic metamaterial with 3 × 3 × 3 periodically arranged 3D micro unit cells are all included. The final homogenized elastic tensors in three numerical cases and the corresponding values of the NPRs are listed in Table 7.3, namely − 0.082, − 0.12, − 0.11. Thereby, the above results can successfully show the effectiveness of the ITO formulation for the design of 3D micro unit cells with the auxetic behavior. As shown in Fig. 7.14, the corresponding 2D views of the topologically optimized 3D auxetic micro unit cells are presented. As we can easily observe, these 2D views have the similar structural features compared to the previously reported 2D auxetic micro unit cells in previous works [186, 192]. In the current work, the corresponding ITO formulation for the design of 3D auxetic micro unit cell that can rationally seek for the corresponding micro-topologies with the auxetic behavior, rather than directly extend 2D case to 3D micro unit cells. As shown in Fig. 7.15. The corresponding convergent curves of the objective and constraint functions for the DDF and the topological variations in two adjacent steps of numerical cases 1 and 2 are provided. A similar feature of the smooth and quick convergence emerges in current two cases, namely 34 iterations in numerical case 1 and 51 steps in numerical case 2. As presented in Figs. 7.16 and 7.17, several intermediate topologies of the 3D micro unit cells in numerical cases 1 and 2 are also provided. Hence, the above results can show the effectiveness and efficiency of the ITO formulation for the design of 3D micro unit cells with the auxetic behavior. As shown in Table 7.3, the final optimized

0.045

0

⎤

0

υ = −0.082

0

0

υ = −0.12

0

⎢ ⎢ −0.0094 ⎢ ⎢ −0.0094 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎣0

0.0789

0

0

0

0

0

0

0

0

0

0.006

0.006 0

0.006 0

0

0

0

0

−0.0094 0.0789

0

0

−0.0094 0

0.0789

0

−0.0094 −0.0094 0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

υ = −0.11

0

⎢ ⎢ −0.0038 ⎢ ⎢ −0.0038 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎣0

0.0331

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

0.0052

0.0024

0.0024 0 0

0

0

0

0

0.0052 0

0.0052 0

0.0024 0

0

−0.0038 0

−0.0038 0.0331

0.0331

−0.0038 −0.0038 0

⎡

0

0.0031

0

0

⎡

0

0

0

0

−0.0065 0

−0.0065 0.0788

0.0788

−0.0065 −0.0065 0

3D auxetic microstructure 4

0

0

⎢ ⎢ −0.0065 ⎢ ⎢ −0.0065 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎣0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0.0788

⎡

⎤

3D auxetic microstructure 2

3D auxetic microstructure 3

0

0

0

0

0

0

0.0031 0

0.0031 0

0

0

0

0

−0.0021 0.045

0

0

−0.0122 0

0.045

0

−0.0021 −0.0021 0

υ = −0.047

0

⎢ ⎢ −0.0021 ⎢ ⎢ −0.0021 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎣0

⎡

3D auxetic microstructure 1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

Table 7.3 Homogenized elastic tensors of four 3D auxetic microstructures, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

158 7 ITO for Auxetic Metamaterials

7.6 3D Auxetic Metamaterials

159

Fig. 7.11 3D auxetic microstructure no. 2, reprinted from ref. [65], copyright 2019, with permission from Elsevier

Fig. 7.12 3D auxetic microstructure no. 3, reprinted from ref. [65], copyright 2019, with permission from Elsevier

Fig. 7.13 3D auxetic microstructure no. 4, reprinted from ref. [65], copyright 2019, with permission from Elsevier

160

7 ITO for Auxetic Metamaterials

Fig. 7.14 2D views for four auxetic microstructures, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

Fig. 7.15 Convergent histories of cases 1 and 2, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

Fig. 7.16 Intermediate results of case 1, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

7.6 3D Auxetic Metamaterials

161

Fig. 7.17 Intermediate results of case 2, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

values of the NPR in 3D micro unit cells are much smaller than previous works for 3D auxetic micro unit cells [185, 190, 191]. The main cause is that the optimization of auxetic micro unit cells is extensively dependent on the definition of the objective function. In the current formulation, the combination of the homogenized elastic tensor is the objective function, and it can only offer a suitable direction for the ITO formulation to find the micro unit cells with the NPR feature, and it is impossible to gain the expected NPR feature with a larger value. In practice, it is not a critical problem in the design of auxetic metamaterials, particularly for the 3D case. The main cause is that the critical difficult in the design of 3D micro unit cells is located at how to achieve the micro unit cells with the NPR, rather than how to achieve a higher NPR value in micro unit cells. The ITO formulation can find several novel micro unit cells with the NPR feature, and then shape optimization can be applied to design the topologically optimized micro unit cells to achieve the any given values of the NPR [190]. Hence, the currently proposed ITO formulation has the powerful capability for the design of the auxetic micro unit cells with the NPR feature. According to the above discussion about the effect of the weight parameter, other two numerical cases with the values 0.02 and 0.0001 are also performed for the design of 3D micro unit cells. As shown in Fig. 7.18, the optimized topologies of 3D micro unit cells are provided and also the corresponding cross-sectional views to clearly present the interior geometrical information. We can easily find that the optimized 3D micro unit cell given in Fig. 7.18a has a similar topology compared to

162

7 ITO for Auxetic Metamaterials

Fig. 7.18 3D auxetic microstructures no. 5 and no. 6, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

the reported microstructure in [191]. The topology of the optimized 3D micro unit cell No. 6 is the anisotropic and has the chiral deformation mechanism to generate the NPR feature. The corresponding values of the NPR in two 3D micro unit cells are in Table 7.4, and the minimum values of the NPR in two cases are equal to − 0.257 and − 0.188, respectively.

7.7 Summary In the current chapter, the main intention is to apply for the ITO method to develop the formulation for the design of 2D and 3D micro unit cells but with the auxetic behavior. In the ITO formulation, a rational definition of the objective function is constructed by considering the combination of several sub terms in the homogenized elastic tensor, which can effectively control the generation of the NPR feature during the optimization. Several numerical examples in 2D and 3D are performed, and several important remarks can be gained: (1) The effectiveness and efficiency of the ITO formulation for the design of 2D and 3D auxetic micro unit cells can be clearly demonstrated; (2) the weight parameter in the objective function has an extensive influence on the design of auxetic micro unit cell with different deformation mechanism; (3) the current ITO method can only seek for a series of novel and interesting novel micro unit cells in 2D and 3D, whereas the higher values of the NPR cannot be determined in the optimization due to the choice of the objective function; (4) a series of novel and interesting micro topologies of the auxetic unit cell to generate auxetic metamaterials can be achieved.

0.0483

0

0

0

0

0

0

0

0

0.0047

0.0048 0

0.0047 0

0

0

0

0

0

0

−0.0122 0.0505

0

0

−0.0122 0

0.0633

0

−0.0124 −0.0049 0

υmin = −0.257

0

⎢ ⎢ −0.0124 ⎢ ⎢ −0.0049 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎣0

⎡

3D auxetic microstructure 5

0.0067

⎤

υmin = −0.188

0.0038

⎥ 0.0426 −0.0062 −0.0032 −0.0062 −0.0004 ⎥ ⎥ −0.0062 0.053 −0.0003 0.0045 −0.0053 ⎥ ⎥ ⎥ −0.0032 −0.0003 0.004 −0.0002 −0.0002 ⎥ ⎥ ⎥ −0.0062 0.0045 −0.0002 0.0038 0.0004 ⎦

0.0009

0.0067 −0.0004 −0.0053 −0.0002 0.0004

⎢ ⎢ −0.0028 ⎢ ⎢ −0.008 ⎢ ⎢ ⎢ 0.0031 ⎢ ⎢ ⎣ 0.0009

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

0.0457 −0.0028 −0.008 0.0031

⎡

⎤

3D auxetic microstructure 6

Table 7.4 Homogenized elastic tensors of 3D auxetic microstructures no. 5 and 6, reprinted from Ref. [65], copyright 2019, with permission from Elsevier

7.7 Summary 163

Chapter 8

M-ITO for Auxetic Meta-Composites

The design of 2D and 3D auxetic micro unit cells has been addressed in the above chapter, and many novel and interesting microstructures with the NPR feature can be found. Although the NPR behavior has several potential advantages in the engineering applications, a critical issue of the auxetic micro unit cells is the weaker with the enough stiffness in affording the load in practical use. Hence, a basic requirement of the composite can provide more benefits for the engineering applications, and a novel concept of meta-composites with the auxetic behavior in the composite has gained a wide of attentions among many researchers. In the current work, the main intention is to discuss the systematical design of auxetic meta-composites by the developed M-ITO method with the energy-based homogenization to construct the corresponding M-ITO formulation. In the M-ITO formulation for the optimization of the auxetic meta-composites, a computational design framework is developed.

8.1 Computational Design Framework The current chapter mainly focuses on how to rationally distribution multiple materials in the micro unit cell that can have the auxetic behavior. As shown in Fig. 8.1, a computational design framework is innovatively developed, and it is mainly involved into five components, including the initialization and preparation, the multimaterial description model, the M-ITO formulation for the optimization of the auxetic meta-composites, solving and output the final designs, and the details can refer to [83].

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gao et al., Isogeometric Topology Optimization, Engineering Applications of Computational Methods 7, https://doi.org/10.1007/978-981-19-1770-7_8

165

166

8 M-ITO for Auxetic Meta-Composites

Fig. 8.1 Flowchart of the computational design framework for auxetic composites, reprinted from Ref. [67], copyright 2021, with permission from Elsevier

8.2 M-ITO Formulation for Auxetic Meta-Composites Based on the discussions on the definition of the objective function in the ITO formulation, the related influence mechanism of the weight parameter in the objective function has been extensively studied in the about chapter for the design of 2D and 3D auxetic micro unit cells. In the current formulation, the main focus is to design 2D and 3D auxetic micro unit cells but with multiple materials (two, three or more). Hence, the predefined objective function in the above chapter is still considered here in the development of the M-ITO formulation for the design of the auxetic metacomposite micro unit cells, and it corresponds to the combination of subterms in the corresponding homogenized elastic tensor of the meta-composite unit cell, and the details of the M-ITO formulation can be stated by:   Find: ρ ϑ ρi,ϑ j , ρi,ϑ j,k    ϑ  d   ϑ  d H H + Min: J (u, φ) = −β ˆ j=1, ˆ ˆ jˆ Diˆiˆ jˆ jˆ u, φ ρ ˆ j=1, ˆ ˆ = jˆ Diˆiˆ jˆ jˆ u, φ ρ i, i= i, i ⎧   a(u, δu) = l(δu), ∀δu ∈ Hper , Rd ⎪ ⎪

⎪  ⎪ ⎨ G ϑˆ = 1 φ ϑˆ υ0 d − V ϑˆ ≤ 0, ϑˆ = 1, 2, . . . , v 0 || S.t.  ⎪ ⎪ 0 < ρmin ≤ ρ ϑ ≤ 1 ⎪ ⎪ ⎩ ϑ = 1, 2, . . . , ; i = 1, 2, . . . , m; j = 1, 2, . . . , n; k = 1, 2, . . . , l (8.1) where ρ ϑ denotes control design variables of the ϑth set, including ρi,ϑ j in 2D and ρi,ϑ j,k in 3D. J is the objective function, in which β is a weight parameter. It can be seen

8.3 Design Sensitivity Analysis

167

that the optimizer intends to increase the first term and decrease the second term in the optimization, so that the objective function can arrive at the minimal. As already discussed in [65, 166, 192], the decrease of second term can push its value smaller than 0 in the optimization, namely Dii j j < 0 (i = j), which facilitates the generation of the mechanism-type features in composite microstructures. The mechanism-type members can be beneficial to the generation of the auxetic behavior in the micro unit cell. Meanwhile, the increase of the first term can push its value larger than 0 in the optimization, namely Dii j j > 0 (i= j). Hence, the objective function can effectively ensure the diagonal terms and off-diagonal components with the opposite sign, so that the auxetic composite micro unit cells can be found using the M-ITO method. d is ˆ the spatial dimension of composites. G ϑv is the volume constraint for the ϑˆ th distinct ˆ material, where V0ϑ is the maximum material consumption and υ0 is the volume fraction of solids. u denotes the displacement field in the microstructure, which should meet the periodic boundary conditions. δu denotes the virtual displacement field in the admissible space Hper with y-periodicity, which is calculated by the linearly elastic equilibrium equation. a and l are the bilinear energy function and the linear load function, respectively, given as:    ϑ  ⎧ ⎨ a(u, δu) = D φ ρ ε(u)ε(δu)d     ⎩ l(δu) = D φ ρ ϑ ε0 ε(δu)d

(8.2)



where D is the elastic tensor of composites defined by the N-MMI model.

8.3 Design Sensitivity Analysis ˆ

The derivatives of the objective function with respect to the TVFs φ ϑ can be stated by: ∂J ∂φ ϑˆ

= −β

⎧ d ⎨ 

   ⎫ ∂ DiˆHiˆ jˆ jˆ u, φ ρ ϑ ⎬

⎩ ⎭ ∂φ ϑˆ ˆ j=1, ˆ ˆ jˆ i, i= ⎧ ⎫    d ⎨ 

∂ DiˆHiˆ jˆ jˆ u, φ ρ ϑ ⎬ ˆ = 1, 2, . . . , + , ϑ ⎩ ⎭ ∂φ ϑˆ ˆ ˆ ˆ ˆ

(8.3)

i, j=1,i= j

In the above equation, we can see that the derivatives of the objective function require the derivatives of the homogenized elastic tensor DiˆHiˆ jˆ jˆ with respect to the ˆ

TVFs φ ϑ . The derivations in detail of them can refer to [186,198], and the final formula can be stated as:

168

8 M-ITO for Auxetic Meta-Composites

∂ DiˆHiˆ jˆ jˆ ∂φ ϑˆ

1 = ||



   γ −1 0 iˆiˆ 0 jˆ jˆ ϑˆ ϑˆ iˆiˆ jˆ jˆ εr s d γ φ D0, pqr s ε pq − ε pq u − εr s u



(8.4) ˆ

According to the above derivations, the first-order derivatives of the TVF φ ϑ with respect to the DVF ϕ ϑ can be written as: ˆ

∂φ ϑ = ∂ϕ ϑ

 ϑˆ

   λ 1 − ϕλ ˆ λ=1,λ=ϑ ϕ λ=ϑ+1    ˆ 1 − ϕλ − ϑλ=1 ϕ λ ˆ λ=ϑ+1,λ =ϑ

if ϑ ≤ ϑˆ , (ϑ = 1, 2, . . . , ) (8.5) if ϑ > ϑˆ

Then, the derivatives of the DVF with respect to control design variables ρ ϑ can be gained, and the detailed form with control design variables ρi,ϑ j,k is given by:   ∂ϕ ϑ p,q,r = Ri, j,k (ξ, η, ζ )ψ ρi,ϑ j,k ϑ ∂ρi, j,k

(8.6)

p,q,r

where Ri, j,k (ξ, η, ζ ) is the NURBS basis function at the computational point

(ξ, η, ζ ). ψ ρi,ϑ j,k is the Shepard function at the control point (i, j, k). Finally, the derivatives of the homogenized elastic tensor with respect to control design variables can be derived, and the final equation can be explicitly expressed as: ∂ DH ∂ DH ∂ DH ˆ  iˆiˆ jˆ jˆ ∂φ ϑ ∂ϕ ϑ iˆiˆ jˆ jˆ iˆiˆ jˆ jˆ ∂φ ∂ϕ ϑ = ∂φ = = ··· ϑ ϑ ϑ ˆ ϑ=1 ∂φ ϑˆ ∂ϕ ϑ ∂ρ ϑ ∂ϕ ∂ρ ∂ρi, j,k i, j,k  i, j,k ⎫ ⎧⎧ ⎫ γ 

γ  γ −1   ˆ ⎪ ϑ 1 ϑˆ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ϕλ 1 − ϕλ ··· ⎪ γ ϕϑ D0, ⎪ ⎪ ⎪ ⎪ λ=1,λ = ϑ pqr s || ˆ ⎪ ⎪ ⎪ ⎪ λ= ϑ+1 ⎨ ⎬ ⎪ ⎪  ⎪ ⎪ ⎛ ⎞⎛ ⎞ ⎪ ˆ ⎪ ⎪ if ϑ ≤ ϑ ⎪ ⎪ ⎪ ˆiˆ ˆ jˆ





0 i 0 j ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ˆiˆ ⎠⎝ ˆ jˆ ⎠ p,q,r ϑ ⎪ ⎪ i j ⎪ ⎪ ⎝ ⎪ − εr s u d ⎪ Ri, j,k (ξ, η, ζ )ψ ρi, j,k ε pq − ε pq u εr s ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎨ ⎬    ⎧ ⎫





γ −1 γ γ  ˆ ϑˆ  ϑ=1 ˆ ⎪ ⎪ ⎪ ⎪ 1 ϑ ϑ λ λ ⎪ ⎪ ⎪ −γ 1 − ϕ D0, pqr s · · · ⎪ 1−ϕ ⎪ ⎪ ⎪ ⎪ ⎪ λ=1 ϕ ˆ ⎪ ⎪ λ=ϑ+1,λ =ϑ ⎪⎪ ⎪ ⎨ ||  ⎬ ⎪ ⎪ ⎛ ⎞⎛ ⎞ ⎪ ⎪ ⎪ ⎪ ˆ ⎪ ⎪ if ϑ > ϑ ˆ ˆ ˆ ˆ





⎪ ⎪ 0 ii 0 jj ⎪ ⎪ ⎪ ⎪ ˆ ˆ ˆ ˆ p,q,r ⎪ ⎪ ϑ i i ⎠⎝ε j j ⎠d ⎪ ⎪ ⎝ ⎪ ⎪ ⎪ ⎪ η, ζ ρ ε u u − ε − ε R (ξ, )ψ ⎪ ⎪ pq rs pq rs ⎪ i, j,k ⎩⎪ ⎭ ⎩ i, j,k ⎭

(8.7)

Similarly, the derivatives of volume constraint with respect to design variables can be expressed by: ˆ

∂G ϑv ∂ρi,ϑ j,k

⎧ 1 ⎪ ⎨ ||

ˆ

ϑˆ

∂G ϑ

= ∂φ ϑvˆ ∂φ ∂ϕ ϑ  ϑˆ 

1 ⎪ ⎩ − ||

∂ϕ ϑ ∂ρi,ϑ j,k

λ=1,λ=ϑ

 ϑˆ 

λ=1

ϕ

= ··· 

λ



  p,q,r 1 − ϕ λ Ri, j,k (ξ, η, ζ )ψ ρi,ϑ j,k υ0 d if ϑ ≤ ϑˆ

  p,q,r ϑ λ υ0 d if ϑ > ϑˆ 1 − ϕ R η, ζ ρ (ξ, )ψ ˆ i, j,k i, j,k λ=ϑ+1,λ=ϑ

ϕ  λ

ˆ λ=ϑ+1

(8.8)

8.4 Numerical Implementations

169

Hereto, the derivatives of the objective function and constraint function with respect to control design variables can be achieved. In the latter numerical optimization, the mathematical optimization method of the moving asymptotes (MMA) [220] is used to evolve design variables.

8.4 Numerical Implementations A flowchart for the M-ITO formulation for the rational design of 2D and 3D auxetic meta-composite micro unit cells is shown in Fig. 8.2, and it mainly involves the following steps: the parameterization of structural geometry using NURBS and then discretization to achieve the IGA mesh for the analysis, the construction of both the DVFs and TVFs for the representation of multiple materials, the energy-based homogenization to evaluate the macroscopic effective properties, the computation of the related objective function and material volume constraint function, the calculation of their sensitivity analysis and the final evolvement of the DVFs and TVFs, and outputting the optimized designs of the auxetic meta-composite micro unit cells with the NPR feature.

Fig. 8.2 Flowchart of the ITO formulation for auxetic composites, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

170

8 M-ITO for Auxetic Meta-Composites

8.5 Numerical Examples The current section focuses on discussing the effectiveness and efficiency of the M-ITO formulation for the design of 2D and 3D auxetic meta-composite micro unit cells with the novel micro-topologies. Firstly, the optimization of 2D auxetic meta-composite micro unit cells is performed in several cases with two and three materials, which intends to demonstrate the effectiveness and efficiency of the M-ITO formulation for auxetic meta-composite. Secondly, the M-ITO formulation for the design of 3D auxetic meta-composite micro unit cells is extensively discussed to seek for a series of novel findings. In all numerical examples, the corresponding structural design domain is a square in 2D and a cube in 3D, and the corresponding structural sizes are equal to be 1 in all normal directions. Three virtual isotropic materials will be available in next examples, and the related details are given in Table 8.1.

8.5.1 2D Auxetic Meta-Composites with Two Materials In the current subsection, the key intention is to study the optimization of 2D auxetic meta-composites with two materials, in which the re-entrant and chiral deformation mechanisms are addressed. In the case of the 2D design domain, the corresponding NURBS parameterization is employed and the knot vectors are  = H = {0, 0, 0, 0.01, . . . , 0.99, 1, 1, 1}, and the total number of IGA elements is equal to 100 × 100, and the control net has a number 102 × 102 of control points, also equal to the number of control design variables. In the design of auxetic meta-composites with two materials, M1 and M2 materials defined in Table 8.1 will be available in the optimization, the corresponding volume fractions for M1 and

ϑˆ ˆ M2 materials V0 ϑ = 1, 2 are set to be 20% and 20%, respectively. As clearly shown in Fig. 8.3, the original designs of control design variables ρ ϑ (ϑ = 1, 2), two DVFs χ ϑ (ϑ = 1, 2) and two TVFs φ ϑ (ϑ = 1, 2) for the distributions of M1 and M2 materials are provided. Table 8.1 Three “virtual” isotropic solid materials, reprinted from Ref. [67], copyright 2021, with permission from Elsevier i

Materials

Young’s modulus:E 0i

Poisson’s ratio v

1

M1

10

0.3

2

M2

5

0.3

3

M3

3

0.3

8.5 Numerical Examples

171

Fig. 8.3 Initial designs of control design variables, DVFs and TVFs for two materials, respectively, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

8.5.1.1

2D Auxetic Meta-Composite with the Re-Entrant Deformation

In the above chapter, the effect of the weight parameter β in the definition of the objective function on the optimization of auxetic micro unit cells has been studied. Currently, the value of the constant parameter is set to be 0.03 for the latter optimization that can achieve auxetic meta-composites with the re-entrant deformation mechanism. As shown in Fig. 8.4, the optimized TVFs to determine the distributions of M1 and M2 materials for the auxetic meta-composite micro unit cell are provided. A similar feature of the extreme smoothness and continuity exist in the optimized TVFs, and the final values of the TVFs are mostly equal to zero or one where only a limited region emerges for the transition from zero to one for the construction of the smooth boundaries. The corresponding numerical results for the optimized design given in Fig. 8.4 are provided, mainly including the optimized topologies of M1 and M2 materials (the isocontour of the TVFs), the combined topology for the overall distribution of M1 and M2 materials, the evaluated elastic tensor by the homogenization and the corresponding NPR value. According to the optimized topology, we can see that several structural members with the re-entrant deformation mechanism are formed in the final topology. Additionally, Fig. 8.5 also presents the

Fig. 8.4 Optimized TVFs, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

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8 M-ITO for Auxetic Meta-Composites

Fig. 8.5 Optimized re-entrant auxetic composite with two materials, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

corresponding re-entrant auxetic meta-composite with a periodical distribution of 5 × 5 repetitive auxetic micro unit cells (Table 8.2). As displayed in Fig. 8.6, the corresponding convergent histories of the objective function and volume fractions of M1 and M2 materials are provided. It can be easily seen that the iterative curves have the dramatic variations, and the reason is that volume fractions of M1 and M2 materials are not equal to the prescribed values. After they reach the prescribed values, the iteration of auxetic meta-composite is in a superior stable process and the optimization can quickly arrive at the expected condition of the convergence. The objective function decreases and is convergent during the whole optimization, and the efficiency and effectiveness of the M-ITO formulation for auxetic meta-composite can be shown. Table 8.2 Optimized 2D re-entrant auxetic composite microstructure, reprinted from Ref. [66], copyright 2021, with permission from Elsevier M1 material

M2 material

The topology

DH ⎤ ⎡ 0.66 −0.40 0 ⎥ ⎢ ⎥ ⎢ −0.40 0.66 0 ⎦ ⎣ 0 0 0.0202 υ − 0.606

8.5 Numerical Examples

173

Fig. 8.6 Convergent histories of the objective function and volume fractions, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

8.5.1.2

2D Auxetic Meta-Composite with the Chiral Deformation

In the current subsection, the main focus is to achieve the auxetic meta-composite micro unit cell with the chiral deformation mechanism, and the corresponding value of the weight parameter is defined as 0.0001. Other design parameters keep consistent with the above subsection. As listed in Table 8.3, the optimized results are provided, including the optimized topologies of M1 and M2 materials (the isocontour of two TVFs), the optimized topology of the combined distribution of M1 and M2 materials, the homogenized elastic tensor and the predicted NPR values in two directions. Based on the results of the topology and their values of the NPR, we can gain that the chiral deformation mechanism can be generated in the optimized topology by rationally distribute M1 and M2 materials. Moreover, an explicit illustration of the auxetic meta-composite having 5 × 5 auxetic micro unit cells is displayed in Fig. 8.7, and it also manifests that the chiral behavior stems from the combination of M1 and M2 materials in four corners of the auxetic micro unit cell. Hence, an important remark that the developed M-ITO formulation with powerful capabilities for the optimization of auxetic meta-composites with two materials can be gained.

8.5.2 2D Auxetic Meta-Composites with Three Materials In the current subsection, the main focus is to address the optimization of 2D auxetic meta-composites with three materials, where the re-entrant and chiral deformation mechanisms are both studied. In the design of three-material micro unit cell, the

M1 material

M2 material

The topology

υ12 − 0.662

υ11 − 0.689

⎤ 0.514 −0.354 0.0122 ⎢ ⎥ ⎢ −0.354 0.535 −0.0114 ⎥ ⎣ ⎦ 0.0122 −0.0114 0.0128

DH ⎡

Table 8.3 Optimized 2D chiral auxetic composite microstructure, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

174 8 M-ITO for Auxetic Meta-Composites

8.5 Numerical Examples

175

Fig. 8.7 Optimized chiral auxetic composite with two materials, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

Fig. 8.8 Initial DVFs 1, 2 and 3, reprinted from Ref. [67], copyright 2021, with permission from Elsevier

corresponding initial designs of three DVFs are shown in Fig. 8.8 with the values equal to 0.5 or 0.

8.5.2.1

2D Auxetic Meta-Composite with the Chiral Deformation

In the current subsection, the key purpose is to achieve the chiral-type deformation mechanism in the design of 2D auxetic meta-composite micro unit cell. The related volume fractions of three materials (V01 of M1, V02 of M2 and V03 of M3) are defined as 15%, 15% and 10%, respectively. The details of the NURBS parametrization keep consistent with the above subsection.

176

8 M-ITO for Auxetic Meta-Composites

Fig. 8.9 Iterative curves of objective and constraint functions, reprinted from Ref. [67], copyright 2021, with permission from Elsevier

As displayed in Fig. 8.9, convergent histories of the objective function and volume fractions of three materials are provided. We can easily find that the optimization of three-material auxetic micro unit cells is more difficult than the design of twomaterial designs, due to the three-material optimization has dramatic variations in earlier steps to seek for the tri-material topology with the NPR feature. If the NPR feature emerges in the topology, the latter iterations become very stable and quickly arrive at the predefined condition with a highly numerical efficiency. As presented in Fig. 8.10, the optimized TVFs 1, 2, 3 and the combined TVF are provided. The related topologies that are defined by the isocontours of the optimized TVFs are shown in Fig. 8.11. As we can easily see, the optimized topologies of three materials are characterized with smooth boundaries and distinct interfaces between multiple materials and voids. Moreover, the corresponding evaluated elastic tensor of the optimized tri-material topology is also given in Fig. 8.11 also with the values of the NPR in two directions (equal to − 0.738 and − 0.732). Based on the optimized tri-material topology and the evaluated NPR values in two directions, we can easily obtain that the chiral-type deformation mechanisms can be generated in the final optimized tri-material topology, which can clearly present the effectiveness and efficiency of the developed M-ITO formulation for the design of auxetic meta-composites with three materials that having the chiral deformation feature.

8.5 Numerical Examples

177

Fig. 8.10 Optimized TVFs for three materials of auxetic micro-architecture 1 in a design domain, reprinted from Ref. [67], copyright 2021, with permission from Elsevier

Fig. 8.11 Auxetic composite with the tri-material, reprinted from Ref. [67], copyright 2021, with permission from Elsevier

8.5.2.2

2D Auxetic Meta-Composite with the Re-Entrant Deformation

In the current subsection, the key intention is to achieve the re-entrant deformation mechanism in the optimization of 2D auxetic meta-composite with three materials. The corresponding volume fractions of three materials are set to be: V01 = 18% of M1, V02 = 14% of M2 and V03 = 8% of M3 materials. Other design parameters keep consistent with the settings in the above example. As shown in Fig. 8.12, the optimized TVFs for M1, M2 and M3 materials and the combined TVFs for M1, M2 and M3 materials are all presented. Meanwhile, the corresponding numerical solutions are listed in Table 8.4, including the optimized topologies for M1, M2 and M3 materials, the topology for the whole tri-material distributions, the related volume fractions of the optimized topologies, the homogenized elastic tensor, the values of the NPRs in two directions and the

178

8 M-ITO for Auxetic Meta-Composites

Fig. 8.12 Optimized TVFs for three materials of auxetic micro-architecture 2 in a design domain, reprinted from Ref. [67], copyright 2021, with permission from Elsevier

auxetic meta-composite. Based on the optimized topology and the evaluated NPRs in two directions, we can easily obtain that the optimized topology of the tri-material auxetic meta-composite micro unit cell has the re-entrant deformation mechanism, similar to previous works [13, 186, 193, 198, 235, 236]. The discussions can clearly demonstrate the effectiveness of the M-ITO formulation for the optimization of the tri-material auxetic meta-composite micro unit cells with different deformation mechanisms.

8.5.3 3D Auxetic Meta-Composites with Two Materials In the current subsection, the key focus is to discuss the design of 3D auxetic metacomposite micro unit cells with different deformation mechanism. Here, only M1 and M2 materials will be considered in the latter optimization, and the corresponding values of materials consumptions are 25% and 20%, respectively. In the representation of 3D topology, red and green colors are applied to describe M1 and M2 materials, respectively. In the NURBS parametrization, the corresponding knot vectors are defined as:  = H = Z = {0, 0, 0, 0.0417, . . . , 0.9583, 1, 1, 1}, and 24 × 24 × 24 IGA elements and 26 × 26 × 26 control design variables are contained in the NURBS. Figure 8.13 presents three different initial designs of control design variables ρ ϑ (ϑ = 1, 2), the related two DVFs and also two TVFs, where the values of control design variables are set to be 0.5 or 1. In the representation, the isosurface (value equal to 0.25) of the TVFs are provided, and the latter isovalues of the TVFs are still equal to 0.5 to define structural topology.

υ11 = υ12 = −0.608

V0 = 40% ⎡ ⎤ 0.622 −0.378 0 ⎢ ⎥ ⎥ DH = ⎢ ⎣ −0.378 0.622 0 ⎦ 0 0 0.0144

V03 = 8%

V02 = 14%

V01 = 18%

M1 material

M2 material

M3 material

Topology

Table 8.4 Numerical results of auxetic micro-architecture 2, reprinted from Ref. [67], copyright 2021, with permission from Elsevier

8.5 Numerical Examples 179

180

8 M-ITO for Auxetic Meta-Composites

Fig. 8.13 Initial designs of control design variables, DVFs and TVFs for two materials, respectively, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

8.5.3.1

3D Auxetic Meta-Composite with the Re-Entrant Deformation

In the current subsection, the key intention is to achieve the re-entrant deformation mechanism in the optimized auxetic meta-composite micro unit cell, where the value of the weight parameter is defined to be 0.03. The first initial design presented in Fig. 8.13a is considered in case 1, and the optimized 3D auxetic meta-composite micro unit cell with M1 and M2 materials is shown in Fig. 8.14, including the optimized topologies of M1 and M2 materials, their cross-sectional views and the final topology of the 3D auxetic micro unit cell and the corresponding cross-sectional view. Based on the optimized topology, it can be easily seen that the re-entrant deformation mechanism exists in the current design, it mainly results from the structural deformation of M1 material plotted with red color, and the corresponding homogenized elastic tensor of the current microstructure is given in Table 8.5, and the same conclusion can be obtained from the values of the NPRs in three directions. As shown in Fig. 8.15, the iterative curves of the objective function and volume constraint functions of M1 and M2 materials are provided. As we can easily see, a similar feature that the former iterations (nearly 40 steps) in the optimization are mainly applied to seek for the auxetic feature exists, so that the topology of micro unit cell has undergone dramatically change. After the auxetic feature is formed in the micro topology, the latter iterations are very stable and quickly arrive at the expected value of the convergent condition. Hence, the convergent state can clearly present the efficiency of the M-ITO formulation and the optimized re-entrant micro unit cell can demonstrate the effectiveness of the M-ITO formulation for the design of 3D auxetic meta-composites.

8.5 Numerical Examples

181

Fig. 8.14 Optimized topology of 3D auxetic composite microstructure no. 1 with two materials, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

As we know, the optimization of micro unit cells is strongly dependent on the initial designs owing to the nonuniqueness of the micro unit cell [13, 166, 192]. In the current work, in order to find other novel 3D auxetic meta-composite micro unit cells with re-entrant deformation mechanism, other two cases with the considerations of the second and third initial designs are performed, and the optimized results in these two numerical cases are presented in Figs. 8.16 and 8.17, respectively, including the optimized topologies of M1 and M2 materials, their cross-sectional views and the final topology of the 3D auxetic micro unit cell and the corresponding crosssectional view. A similar feature that smooth boundaries and distinct interfaces exist in the optimized topologies. As listed in Table 8.5, the homogenized elastic tensors of the optimized topologies are provided and the corresponding values of the NPRs are also given. As we can see, the re-entrant deformation mechanism can be formed by the corresponding structural members filled by M1 materials.

8.5.3.2

3D Auxetic Meta-Composite with the Chiral Deformation

In the current subsection, the key intention is to obtain the chiral deformation mechanism in the design of 3D auxetic meta-composite micro unit cell with M1 and M2 materials, where constant parameter is set to be 0.0001 in the current example. The

0

0

0.496

0

⎢ ⎢ −0.038 ⎢ ⎢ −0.038 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎣0

⎡

0

0

0

0

0

0

0

0

0

0

0

0.024

0.0318

0.0318 0

0.0318 0

0

0

0

0

−0.038 0.496

0

0

−0.038 0

0.496

0

−0.038 −0.038 0

0

0

0

0

0.024 0

0.024 0

0

0

0

0

0

0

−0.0272 0.3192

0

0

−0.0272 0

0.3192

3D auxetic composite no. 3

0

⎢ ⎢ −0.0272 ⎢ ⎢ −0.0272 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎣0

0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

0

⎢ ⎢ −0.033 ⎢ ⎢ −0.033 ⎢ ⎢ ⎢0 ⎢ ⎢ ⎣0

0.422

0

0

0 0

0

0

0

0

0

0

0

0

0

0

0

0

0.034

0.034 0

0.034 0

0

−0.033 0

−0.033 0.422

0.422

−0.033 −0.033 0

⎡

0

−0.0272 −0.0272 0

⎡

0.3192

3D auxetic composite no. 2

3D auxetic composite no. 1

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

⎤

Table 8.5 Homogenized elastic tensors of auxetic composites no. 1–no. 3, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

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8.5 Numerical Examples

183

Fig. 8.15 Iterative curves of the objective function and volume fractions of two materials, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

Fig. 8.16 Optimized topology of 3D auxetic composite microstructure no. 2 with two materials, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

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8 M-ITO for Auxetic Meta-Composites

Fig. 8.17 Optimized topology of 3D auxetic composite microstructure no. 3 with two materials, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

initial design 1 will be considered in the current design. Other design parameters keep consistent with above examples. As shown in Fig. 8.18, the optimized results of the current case are provided, including the optimized topologies of M1 and M2 materials, their cross-sectional views and the final topology of 3D auxetic micro unit cell and the corresponding cross-sectional view. The corresponding homogenized elastic tensor is given in Table 8.6. As we can easily see, the chiral deformation mechanism exists in the current micro unit cell, which can be from the deformation of structural members filled with M2 materials. Moreover, the current 3D micro unit cell is anisotropic, and the values of the NPRs in three directions are not identical. Finally, the 2D cross-sectional views of the above optimized auxetic metacomposites micro unit cells are also shown in Fig. 8.19. As we can easily see, the optimized micro-topologies in 3D auxetic meta-composites No. 1–No. 3 have the similar feature that the re-entrant deformation mechanism exists in the final designs stemming from M1 material. In the case of the 3D auxetic meta-composite micro unit cell No. 4, structural members filled with M1 material have the re-entrant deformation mechanism and the chiral-type deformation mechanism stems from M2 material. Hence, the auxetic behavior of the auxetic meta-composite micro unit cell No. 4 originates from a combination of the re-entrant and chiral deformation mechanisms.

8.6 Summary

185

Fig. 8.18 Optimized topology of 3D auxetic composite microstructure no. 4 with two materials, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

Table 8.6 Homogenized elastic tensors of auxetic composite no. 4, reprinted from Ref. [66], copyright 2021, with permission from Elsevier 3D auxetic composite no. 4 ⎡

0.3953

⎢ ⎢ −0.1112 ⎢ ⎢ −0.0575 ⎢ ⎢ ⎢ −0.0138 ⎢ ⎢ ⎣ −0.0007

−0.112 −0.575 −0.0138 −0.0007 −0.0028 0.382

−0.0481 0.0215

−0.048 0.4360

0.0019

−0.005 0.003

0.0215 −0.005 0.0752

0.0018

0.002

0.0728

0.0003

−0.0028 −0.003 0.0124

0.0018

−0.0026 0.001

⎤

⎥ −0.0031 ⎥ ⎥ 0.0124 ⎥ ⎥ ⎥ −0.0026 ⎥ ⎥ ⎥ 0.001 ⎦

0.0757

8.6 Summary The main focus of the current chapter is to address the systematic design of auxetic meta-composites micro unit cells with the re-entrant or chiral deformation mechanisms by the M-ITO method proposed in Chap. 3, the M-ITO formulation for the design of auxetic meta-composites is developed, and it is mainly involved into the

186

8 M-ITO for Auxetic Meta-Composites

Fig. 8.19 Deformation mechanisms of four auxetic composites, reprinted from Ref. [66], copyright 2021, with permission from Elsevier

following components: (1) the multi-material description model to show the microtopology with multiple materials; (2) the energy-based homogenization method to evaluate the effective properties of the composite. Several numerical examples are performed in the latter, and we can obtain several important remarks: (1) A series of novel 2D auxetic meta-composites micro unit cells with the re-entrant or chiral deformation mechanism can be found; (2) a series of novel 3D auxetic meta-composites micro unit cells with the re-entrant or chiral deformation mechanism can be found; (3) structural deformation mechanism in the optimization of micro unit cells can be rationally adjusted by the definition of the objective function; (4) the currently developed M-ITO formulation has superior capabilities of optimizing auxetic meta-composites micro unit cells.

Chapter 9

An In-House MATLAB Code of “IgaTop” for ITO

In the implementation of the ITO in MATLAB, a critical aspect is how to consider the IGA into topology optimization. Before programming the ITO using MATLAB, the related descriptions about three spaces in IGA are presented in the following, namely the parametric space, physical space and bi-unit parent space. In the authors’ viewpoint, the physical space corresponds to the representation of the structural geometry using NURBS, and the parametric space offers a systematic mathematical language for the development of NURBS as far as structural geometries in the physical space are concerned. The bi-unit space is applied to compute all IGA element stiffness matrices. The detailed descriptions about these three spaces for NURBS and IGA can refer to [35]. Parametric space: Knot vectors with an ordered set of increasing manner should be originally defined for the NURBS. The parametric space corresponds to these nonzero intervals of knot vectors, which can be also viewed as a pre-image of structural geometries by a NURBS mapping. In the practice use, knot vectors are normalized, and the parametric space can be transformed into a unit interval in 1D, square in 2D or cube in 3D. The corresponding mathematical language is parametric coordigiven as follows:  ∈ Rd denotes    the parametricspace, three nates (, H, Z)3D where  = ξ1 , ξ2 , . . . , ξn+ p+1 , H = η1 , η2 , . . . ηm+q+1 and Z = {ζ1 , ζ2 , . . . , ζl+r +1 },the corresponding   mathematical   formula for the parametric space is  = ξ1 , ξn+ p+1 ⊗ η1 , ηl+r +1 ⊗ ζ1 , ζm+q+1 . Physical space: The physical space provides a visualized look of structural geometries defined by the NURBS mappings, where a series of control points should be appropriately chosen with the previous definition of knot vectors. Hence, a control mesh is naturally formed, which can be viewed as a local area containing structural geometries. In the mathematical language,  denotes the physical space, physical coordinates (x, y, z). Noting that a critical feature is the noninterpolation of control points at structural geometries, which is completely different from conventional Lagrangian mesh in FEM. 



© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022 J. Gao et al., Isogeometric Topology Optimization, Engineering Applications of Computational Methods 7, https://doi.org/10.1007/978-981-19-1770-7_9

187

188

9 An In-House MATLAB Code of “IgaTop” for ITO

Fig. 9.1 A MATLAB implementation framework of ITO: IgaTop2D, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

Biparent space: The biparent space is an additional space, which is mainly applied into numerical integration in the IGA for the computation of all IGA element stiffness  = [−1, 1]Rd denotes the biparent space matrices. In the mathematical language,   and ξ˜ , η, ˜ ζ˜ are the corresponding coordinates. Mappings: In the IGA, NURBS basis functions that act as shape functions to approximate structural geometries are also applied to construct the space of unknown structural responses, and three spaces are, respectively, defined. Compared to the FEM, two mappings are required in the latter computation of IGA, namely a mapping X:  →  from the parametric space to the physical space, and an affine mapping  →  from the parent space to the parametric space. Y:  As displayed in Fig. 9.1, the systematic flowchart for the MATLAB implementation of the ITO method is provided, and the corresponding main function IgaTop2D having a 56-line MATLAB code is applied to implement the ITO method for 2D structures, and nine components are involved into the related implementation: (1) a subfunction Geom_Mod with a 27-line MATLAB code called in line 5 is defined for the construction of geometrical model of structures by NURBS; (2) a subfunction Pre_IGA with a 39-line MATLAB code called in line 7 is developed for preparing IGA; (3) a subfunction Boun_Cond with a 38-line MATLAB code implemented in line 9 intends to impose Dirichlet and Neumann boundary conditions; (4) lines 11–20 of the main function IgaTop2D are applied to initialize design variables 



9 An In-House MATLAB Code of “IgaTop” for ITO

189

and the DDF at Gauss quadrature points; (5) the subfunction Shep_Fun with a 22-line MATLAB code called in line 22 is developed for the implementation of the smoothing mechanism; (6) a subfunction Stiff_Ele2D with a 33-line MATLAB code called in line 28, a subfunction Stiff_Ass2D with a 18-line MATLAB code called in line 29 and a subfunction Solving with a 14-line MATLAB code called in line 30 are applied to realize the NURBS-based IGA to solve the unknown structural responses; (7) lines 32–46 of the main function IgaTop2D are adopted to compute the objective function and also sensitivity analysis; (8) a subfunction Plot_Data with a 16-line MATLAB code called in line 25 and a subfunction Plot_Topy with a 20-line MATLAB code is implemented in line 47 are developed for the latter representation of numerical solutions, including the optimized DDF, topology and convergent curves of objective function and volume fractions; (9) a subfunction OC with a 14-line MATLAB code implemented in line 52 is applied to update design variables and solve design problems. The detailed MATLAB codes of the main function IgaTop2D are attached in the following: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

function IgaTop2D(L, W, Order, Num, BoundCon, Vmax, penal, rmin) %% Material properties path = genpath(pwd); addpath(path); E0 = 1; Emin = 1e-9; nu = 0.3; DH=E0/(1-nu^2)*[1 nu 0; nu 1 0; 0 0 (1nu)/2]; NURBS = Geom_Mod(L, W, Order, Num, BoundCon); close all %% Preparation for IGA [CtrPts, Ele, GauPts] = Pre_IGA(NURBS); Dim = numel(NURBS.order); Dofs.Num = Dim*CtrPts.Num; [DBoudary, F] = Boun_Cond(CtrPts, BoundCon, NURBS, Dofs.Num); %% Initialization of control design variables X.CtrPts = ones(CtrPts.Num,1); GauPts.Cor = [reshape(GauPts.CorU',1,GauPts.Num); reshape(GauPts.CorV',1,GauPts.Num)]; [GauPts.PCor,GauPts.Pw] = nrbeval(NURBS, GauPts.Cor); GauPts.PCor = GauPts.PCor./GauPts.Pw; [N, id] = nrbbasisfun(GauPts.Cor, NURBS); R = zeros(GauPts.Num, CtrPts.Num); for i = 1:GauPts.Num, R(i,id(i,:)) = N(i,:); end R = sparse(R); [dRu, dRv] = nrbbasisfunder(GauPts.Cor, NURBS); X.GauPts = R*X.CtrPts; %% Smoothing mechanism [Sh, Hs] = Shep_Fun(CtrPts, rmin); %% Start optimization in a loop change = 1; nloop = 150; Data = zeros(nloop,2); Iter_Ch = zeros(nloop,1); [DenFied, Pos] = Plot_Data(Num, NURBS); for loop = 1:nloop %% IGA to evaluate the displacement responses

190 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

9 An In-House MATLAB Code of “IgaTop” for ITO [KE, dKE, dv_dg] = Stiff_Ele2D(X, penal, Emin, DH, CtrPts, Ele, GauPts, dRu, dRv); [K] = Stiff_Ass2D(KE, CtrPts, Ele, Dim, Dofs.Num); U = Solving(CtrPts, DBoudary, Dofs, K, F, BoundCon); %% Objective function and sensitivity analysis J = 0; dJ_dg = zeros(GauPts.Num,1); for ide = 1:Ele.Num Ele_NoCtPt = Ele.CtrPtsCon(ide,:); edof = [Ele_NoCtPt,Ele_NoCtPt+CtrPts.Num]; Ue = U(edof,1); J = J + Ue'*KE{ide}*Ue; for i = 1:Ele.GauPtsNum GptOrder = GauPts.Seque(ide, i); dJ_dg(GptOrder) = -Ue'*dKE{ide}{i}*Ue; end end Data(loop,1) = J; Data(loop,2) = mean(X.GauPts(:)); dJ_dp = R’*dJ_dg; dJ_dp = Sh*(dJ_dp./Hs); dv_dp = R’*dv_dg; dv_dp = Sh*(dv_dp./Hs); %% Print and plot results fprintf(' It.:%5i Obj.:%11.4f Vol.:%7.3fch.:%7.3f\n',loop,J,mean(X.GauPts(:)),change); [X] = Plot_Topy(X, GauPts, CtrPts, DenFied, Pos); if change < 0.01, break; end %% Optimality criteria to update design variables X = OC(X, R, Vmax, Sh, Hs, dJ_dp, dv_dp); change = max(abs(X.CtrPts_new(:)-X.CtrPts(:))); Iter_Ch(loop) = change; X.CtrPts = X.CtrPts_new; end end

9.1 Geom_Mod: Construct Geometrical Model Using NURBS As already pointed out, the subfunction Geom_Mod is applied to model structural geometry using the NURBS. As shown in Fig. 9.2, an example of a quarter annulus is provided, where structural geometry is shown in Fig. 9.2a with structural sizes, the corresponding NURBS surface in the physical space with a control net is presented in Fig. 9.2b–d shows the corresponding NURBS basis functions in two parametric directions, respectively. In the implementation of the subfunction Geom_Mod, five input parameters L, W, Order, Num and BoundCon are contained, which will be called from the command in line 5 of the main function IgaTop2D to prompt the implementation of this subfunction, given as: NURBS = Geom_Mod(L, W, Order, Num, BoundCon)

9.1 Geom_Mod: Construct Geometrical Model Using NURBS

191

Fig. 9.2 A NURBS surface for a quarter annulus, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

The corresponding out parameter in this subfunction is NURBS, In MATLAB, it corresponds to a simple structure array, which consists of six fields related to the detailed information in the construction of the NURBS, namely form, dim, number, coefs, knots and order. In the case of the structure shown in Fig. 9.2, the corresponding input parameters are given as: L = 10, W = 10, Order = [0 1], Num = [11 5] and BoundCon = 5, and the output parameter NURBS is stated as: NURBS

form: ‘B-NURBS’ dim: 4 number: [12 6] coefs: [4 × 12 × 6 double] knots: {[0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 1] [0 0 0 0.25 0.50 0.75 1 1 1]} order: [3 3]

In the subfunction Geom_Mod, the initial knot vectors in two parametric directions are firstly defined and the related control points with the homogeneous coordinates (ωx, ωy, ωz, ω) are also provided. The initial definition is applied to construct structural geometry, which only contains a few numbers of IGA elements. Based on the initial definition, the p/h refinement with an order elevation or a knot insertion is applied to gain an analysis model with a suitable number of IGA elements.

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Fig. 9.3 Three different NURBS surfaces for the quarter annulus, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

In the process, a NURBS toolbox developed by Spink [237] is employed, and the related numerical algorithms for the implementations of the NURBS in detail can refer to [238]. In NURBS toolbox programmed by MATLAB, the function nrbmak is firstly employed to construct the NURBS surface 1 presented in Fig. 9.3b, and then the order of NURBS basis functions is elevated by the function nrbdegelev, and the corresponding NURBS surface 2 is shown in Fig. 9.3c. Later, a series of new knots with a uniform span are inserted in the interval, which is implemented by the function nrbkntins. After the insertion process, the corresponding NURBS surface 3 is displayed in Fig. 9.3d, which will be used for the latter analysis in IGA. The related MATLAB calls in the subfunction Geom_Mod are given as follows: NURBS = nrbmak(coefs, knots); NURBS = nrbdegelev(NURBS,Order); NURBS = nrbkntins(NURBS,{setdiff(iknot_u,NURBS.knots {1}),…setdiff(iknot_v,NURBS.knots{2})}); It should be noted that the above process of the knot insertion and order elevation corresponds to the k-refinement, namely elevating the order firstly and then inserting knots. Compared to previous the p-refinement, the generated NURBS surface can have a few numbers of NURBS basis functions and also control points, which can reduce the computation time in the latter analysis but still with similar numerical accuracy. The details about the k-refinement in IGA can refer to [34]. The detailed MATLAB codes of the Geom_Mod subfunction are given by: 1

function NURBS = Geom_Mod(L, W, Order, Num, BoundCon)

2 switch BoundCon 3 case {1, 2, 3} 4 knots{1} = [0 0 1 1]; knots{2} = [0 0 1 1]; 5 ControlPts(:,:,1) = [0 L; 0 0; 0 0; 1 1]; 6 ControlPts(:,:,2) = [0 L; W W; 0 0; 1 1]; 7 case 4 8 knots{1} = [0 0 0.5 1 1]; knots{2} = [0 0 1 1]; 9 ControlPts(:,:,1) = [0 0 L; L 0 0; 0 0 0; 1 1 1]; 10 ControlPts(:,:,2) = [W W L; L W W; 0 0 0; 1 1 1];

9.2 Pre_IGA: Preparation for IGA 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

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case 5 W = W/2; knots{1} = [0 0 0 1 1 1]; knots{2} = [0 0 1 1]; ControlPts(:,:,1) = [0 W W; W W 0; 0 0 0; 1 sqrt(2)/2 1]; ControlPts(:,:,2) = [0 L L; L L 0; 0 0 0; 1 sqrt(2)/2 1];

end coefs = zeros(size(ControlPts)); coefs(1,:,:) = ControlPts(1,:,:).*ControlPts(4,:,:); coefs(2,:,:) = ControlPts(2,:,:).*ControlPts(4,:,:); coefs(3,:,:) = ControlPts(3,:,:).*ControlPts(4,:,:); coefs(4,:,:) = ControlPts(4,:,:); NURBS = nrbmak(coefs, knots); NURBS = nrbdegelev(NURBS,Order); nrbplot(NURBS,[100 100],'light','on') iknot_u = linspace(0,1,Num(1)); iknot_v = linspace(0,1,Num(2)); NURBS = nrbkntins(NURBS,{setdiff(iknot_u,NURBS.knots{1}),setdiff(iknot_v,NURBS.kn ots{2})}); 27 end

9.2 Pre_IGA: Preparation for IGA In the Top-opt with the classic FEM, the preparation for the FEM is a critical component in the latter optimization, which can effectively use positive features of matrix operation in MATLAB to reduce the computational cost. Hence, in the implementation, the preparation for the IGA is also developed, and it mainly contains the computation of numbers of control points, IGA elements and Gauss quadrature points. The MATLAB code for the preparation of IGA is called to prompt the following line: [CtrPts, Ele, GauPts] = Pre_IGA(NURBS) In the above calling, only one input parameter (NURBS) with the sufficient information of structural geometry is required, and the output parameters are given as follows: CtrPts, Ele, and GauPts. CtrPts corresponds to a structural array, and it contains five fields, given as: CtrPts.Cordis

The Cartesian coordinates of control points in the physical space, namely (x, y, z, ω)

CtrPts.Num

The total number of control points

CtrPts.NumU

The total number of control points in the first parametric direction

CtrPts.NumV

The total number of control points in the second parametric direction

CtrPts.Seque

The numbers of all control points

In the numbering control points, the numbering manner of control points with parametric directions is presented in Fig. 9.4, and each control point is coined by the corresponding number, and it is ordered by the arc (the first parametric direction)-wise

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Fig. 9.4 Numbers of all control points, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

left-to-right and then bottom (the second parametric direction)-to-up, an example of the details for the numbers of control points are shown in Fig. 9.7a. In the case of the structure array Ele, it mainly includes eleven fields, namely (1) NumU denotes the number of all IGA elements in the first parametric direction of the parametric space; (2) NumV is the total number of all IGA elements in the second parametric direction of the parametric space; (3) Num denotes the total number of all IGA elements in the parametric space; (4) Seque contains the ordering numbers of all IGA elements in the physical space, and the numbering manner of IGA elements keeps consistent with control points, and it is the arc-wise left-to-right and bottomto-up, shown in Fig. 9.7b. (5) KnotsU is the knot span which corresponds to IGA element in the first parametric direction; (6) KnotsV indicates the knot span related to the IGA element in the second parametric direction of the parametric space; (7) CtrPtsNum is the total number of all control points that can have the effect on each IGA element in the parametric space, and it is equal to the total number of nonzero NURBS basis functions in each IGA element; (8) CtrPtsNumU is the total number of control points that can affect each IGA element in the first parametric direction; (9) CtrPtsNumV denotes the total number of control points that can have the effect on each IGA element in the second parametric direction; (10) CtrPtsCon includes the numbers of control points that have the influence on IGA elements, and an example is also displayed in Fig. 9.5, it refers to a fact that the i-th row of this matrix includes the six (also equal to CtrPtsNum) indices of control points that can have the effect on the i-th IGA element. It is similar to the matrix edofMat developed in 88-line

Fig. 9.5 Matrix of Ele.CtrPtsCon, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

9.2 Pre_IGA: Preparation for IGA

195

MATLAB code of the classic SIMP method [201]; (11) GauPtsNum is the total number of Gauss quadrature points for numerical integration in each IGA element. The structure array GauPts includes six fields, namely (1) QuaPts contains the coordinates of Gauss quadrature points in the bi-unit parent space; (2) Weigh refers to the corresponding weight parameters at each Gauss quadrature point in IGA elements; (3) Num denotes the total number of Gauss quadrature points in each IGA element; (4) CorU includes the corresponding knots of Gauss quadrature points in the first parametric direction if mapping Gauss quadrature points from the parent space to parametric space; (5) CorV indicates the corresponding knots of Gauss quadrature points in the second parametric direction when mapping Gauss quadrature points from parent space to parametric space; (6) Seque: the numbers of Gauss quadrature points, and the i-th row of this matrix has nine indices of all Gauss quadrature points in the i-th IGA element. As shown in Fig. 9.6, the details of matrix GauPts.Seque are provided, and the numbering mechanism of all Gauss quadrature points in IGA elements is also same as the numbering way of control points in structural geometry, shown in Fig. 9.7c. The subfunction Pre_IGA mainly focuses on computing the related data for numerical analysis. The lines 3–13 aim to compute structural array CtrPts that have five fields (Cordis, Num, NumU, NumV and Seque). Line 15–26 of MATLAB code is applied to compute structural array Ele that has eleven fields in the subfunction. In numerical integration, a subfunction Guadrature is defined and called to calculate all Gauss quadrature points and with the corresponding weights. The corresponding

Fig. 9.6 Matrix of GauPts.Seque, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

Fig. 9.7 Numbers of control points, IGA elements and all Gauss quadrature points, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

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MATLAB implementation of the GauPts array with six fields is implemented in lines 26–37. The details for the MATLAB code of the subfunction Pre_IGA are given as: 1 2 3 4 5 6 7 8 9 10 11

12

13 14

function [CtrPts, Ele, GauPts] = Pre_IGA(NURBS) %% the unique knots in two parametric directions Knots.U = unique(NURBS.knots{1})'; Knots.V = unique(NURBS.knots{2})'; %% the information of control points including the physical coordinates, numbers, sequence CtrPts.Cordis = NURBS.coefs(:,:); CtrPts.Cordis(1,:) = CtrPts.Cordis(1,:)./CtrPts.Cordis(4,:); % the X Cartesian coordinates of control points; CtrPts.Cordis(2,:) = CtrPts.Cordis(2,:)./CtrPts.Cordis(4,:); % the Y Cartesian coordinates of control points; CtrPts.Cordis(3,:) = CtrPts.Cordis(3,:)./CtrPts.Cordis(4,:); % the Z Cartesian coordinates of control points; CtrPts.Num = prod(NURBS.number); % the total number of control points or basis functions; CtrPts.NumU = NURBS.number(1); % the total number of control points or basis functions in the first parametric; CtrPts.NumV = NURBS.number(2); % the total number of control points or basis functions in the second parametric; CtrPts.Seque = reshape(1:CtrPts.Num,CtrPts.NumU,CtrPts.NumV)'; %% the information of the elements (knot spans) in the parametric space,

including the numbers, sequence 15 Ele.NumU = numel(unique(NURBS.knots{1}))-1; % the number of elements in the first parametric direction 16 Ele.NumV = numel(unique(NURBS.knots{2}))-1; % the number of elements in the second parametric direction 17 Ele.Num = Ele.NumU*Ele.NumV; % the number of all elements in the structure 18 Ele.Seque = reshape(1:Ele.Num, Ele.NumU, Ele.NumV)'; 19 Ele.KnotsU = [Knots.U(1:end-1) Knots.U(2:end)]; % the unique knots of the elements in the first parametric direction 20 Ele.KnotsV = [Knots.V(1:end-1) Knots.V(2:end)]; % the unique knots of the elements in the second parametric direction 21 Ele.CtrPtsNum = prod(NURBS.order); 22 Ele.CtrPtsNumU = NURBS.order(1); Ele.CtrPtsNumV = NURBS.order(2); 23 [~, Ele.CtrPtsCon] = nrbbasisfun({(sum(Ele.KnotsU,2)./2)', (sum(Ele.KnotsV,2)./2)'}, NURBS); 24 %% the information of the Gauss quadrature points in the parent space 25 [GauPts.Weigh, GauPts.QuaPts] = Guadrature(3, numel(NURBS.order)); 26 Ele.GauPtsNum = numel(GauPts.Weigh);

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27 GauPts.Num = Ele.Num*Ele.GauPtsNum; 28 GauPts.Seque = reshape(1:GauPts.Num,Ele.GauPtsNum,Ele.Num)'; 29 GauPts.CorU = zeros(Ele.Num,Ele.GauPtsNum); GauPts.CorV = zeros(Ele.Num,Ele.GauPtsNum); 30 for ide = 1:Ele.Num 31 [idv, idu] = find(Ele.Seque == ide); % The two idices in two parametric directions for an element 32 Ele_Knot_U = Ele.KnotsU(idu,:); % The knot span in the first parametric direction for an element 33 Ele_Knot_V = Ele.KnotsV(idv,:); % The knot span in the second parametric direction for an element 34 for i = 1:Ele.GauPtsNum 35 GauPts.CorU(ide,i) = ((Ele_Knot_U(2)Ele_Knot_U(1)).*GauPts.QuaPts(i,1) + (Ele_Knot_U(2)+Ele_Knot_U(1)))/2; 36 GauPts.CorV(ide,i) = ((Ele_Knot_V(2)Ele_Knot_V(1)).*GauPts.QuaPts(i,2) + (Ele_Knot_V(2)+Ele_Knot_V(1)))/2; 37 end 38 end 39 end

The 20-line MATLAB code of the subfunction Guadrature is given as: 1 2 3 4 5 6

function [quadweight,quadpoint] = Guadrature(quadorder, dim) quadpoint = zeros(quadorder^dim ,dim); quadweight = zeros(quadorder^dim,1); r1pt=zeros(quadorder,1); r1wt=zeros(quadorder,1); r1pt(1) = 0.774596669241483;

7 8 9 10 11 12 13

r1pt(2) r1pt(3) r1wt(1) r1wt(2) r1wt(3) n=1; for i =

14 15 16 17 18 19 end 20 end

=-0.774596669241483; = 0.000000000000000; = 0.555555555555556; = 0.555555555555556; = 0.888888888888889; 1:quadorder for j = 1:quadorder quadpoint(n,:) = [ r1pt(i), r1pt(j)]; quadweight(n) = r1wt(i)*r1wt(j); n = n+1; end

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9.3 Boun_Cond: Define Dirichlet and Neumann Boundary Conditions The subfunction Boun_Cond that has four input parameters (CtrPts, BoundCon, NURBS and Dofs.Num) is applied to define Dirichlet and Neumann boundary conditions, and the corresponding MATLAB calling line in the main function is given as: [DBoudary, F] = Boun_Cond(CtrPts, BoundCon, NURBS, Dofs.Num) Two parameters will be outputted in the calling, including DBoudary and F. The former parameter DBoudary is a structural array of MATLAB, which contains one field CtrPtsOrd. It mainly includes the locations of the force that is imposed at structural geometries in the physical space. In the case of the imposed areas of the force, if the total number of control points at the corresponding parametric direction is an even value, it means that no control point is located at the imposed locations of the force in structural geometries. Hence, it is impossible to impose the force at the exact control point. In [74], a simple method is defined, and the detailed implementations are given as follows: (1) compute parametric coordinates RS of the locations of the imposed force using NURBS toolbox; (2) evaluate the values of all nonzero NURBS basis functions at parametric coordinates RS ; (3) uniformly impose the force RS F at all corresponding control points. Line 3–29 implements Dirichlet and Neumann boundary conditions in five numerical cases, where the cantilever beam is defined in the lines 5–8 of MATLAB code, the lines 10–13 are called for the case of MBB beam, and Michell-type structure is called in the lines 15–18, and the MATLAB code for L beam is programmed in lines 20–23, and lines 25–28 are defined for quarter annulus with the curved design domain. The detailed MATLAB code of Boun_Cond is given as: 1 function [DBoudary, F] = Boun_Cond(CtrPts, BoundCon, NURBS, Dofs_Num) 2 %% boundary conditions 3 switch BoundCon 4 case 1 % Cantilever beam 5 DBoudary.CtrPtsOrd = CtrPts.Seque(:,1); 6 load.u = 1; load.v = 0.5; 7 [N, id] = nrbbasisfun([load.u; load.v], NURBS); 8 NBoudary.CtrPtsOrd = id'; NBoudary.N = N; 9 case 2 % MBB beam 10 DBoudary.CtrPtsOrd1 = CtrPts.Seque(1,1); DBoudary.CtrPtsOrd2 = CtrPts.Seque(1,end); 11 load.u = 0.5; load.v = 1; 12 [N, id] = nrbbasisfun([load.u; load.v], NURBS); 13 NBoudary.CtrPtsOrd = id'; NBoudary.N = N;

9.4 Initializing Control Densities and DDF at Gauss Quadrature Points 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

199

case 3 % Michell-type structure DBoudary.CtrPtsOrd1 = CtrPts.Seque(1,1); DBoudary.CtrPtsOrd2 = CtrPts.Seque(1,end); load.u = 0.5; load.v = 0; [N, id] = nrbbasisfun([load.u; load.v], NURBS); NBoudary.CtrPtsOrd = id'; NBoudary.N = N; case 4 % L beam DBoudary.CtrPtsOrd = CtrPts.Seque(:,1); load.u = 1; load.v = 1; [N, id] = nrbbasisfun([load.u; load.v], NURBS); NBoudary.CtrPtsOrd = id'; NBoudary.N = N; case 5 % A quarter annulus DBoudary.CtrPtsOrd = CtrPts.Seque(:,end); load.u = 0; load.v = 1; [N, id] = nrbbasisfun([load.u; load.v], NURBS); NBoudary.CtrPtsOrd = id'; NBoudary.N = N; end %% the force imposed on the structure F = zeros(Dofs_Num,1); switch BoundCon case {1,2,3,4} F(NBoudary.CtrPtsOrd+CtrPts.Num) = -1*NBoudary.N; case 5 F(NBoudary.CtrPtsOrd) = -1*NBoudary.N; end end

9.4 Initializing Control Densities and DDF at Gauss Quadrature Points In Chap. 1, the definitions, discussions and effectiveness of the DDF are already addressed, in which the smoothness and continuity are essential factors in the construction of the DDF. In the optimization, the DDF is applied to determine the existence or not of materials in design domain, and the isovalue of the DDF is adopted to present structural boundaries of the optimized topologies, and an immersed boundary representation model is developed. The detailed process of the construction of the DDF is involved into four steps, it can refer to Sect. 2.2 in Chap. 2. In the ITO optimization for problems, control densities are design variables, and which are gradually evolved to advance the DDF, until an expected material distribution with the optimized structural performance can be found. In the whole process of the optimization, the representation of the optimized designs needs the detailed information of control densities, the DDF at Gauss quadrature points. Hence, as far as initial definitions of control densities are concerned, the implementation is defined in line 11 of the main program IgaTop2D. an output parameter GauPts.Cor is computed in line 12, which includes the parametric coordinates of Gauss quadrature points in the parametric space, and then the MATLAB line nrbeval(NURBS, GauPts.Cor)

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is applied to compute physical coordinates of Gauss quadrature points, including the outputs GauPts.PCor and GauPts.Pw. The subfunction nrbbasisfun in NURBS toolbox is adopted to compute the detailed values of NURBS basis functions at parametric coordinates of Gauss quadrature points, and the outputs contain N and id. In the matrix of N, the i-th row has six values of NURBS basis functions at the corresponding parametric coordinate of the i-th Gauss quadrature point. In matrix id, the i-th row has six numbers of nonzero NURBS basis functions at the corresponding parametric coordinate of the i-th Gauss quadrature point. The firstorder derivatives of NURBS basis functions with respect to parametric directions are solved at the corresponding parametric coordinate of the i-th Gauss quadrature point using nrbbasisfunder in NURBS toolbox, denoted by dRu and dRv. The DDF at Gauss quadrature points can be evaluated by calling the MATLAB code of the main function in line 20. The corresponding MATLAB code for the initialization of control densities and the DDF at Gauss quadrature points is given as: X.CtrPts = ones(CtrPts.Num,1); GauPts.Cor = [reshape(GauPts.CorU',1,GauPts.Num); reshape(GauPts.CorV',1,GauPts.Num)]; [GauPts.PCor,GauPts.Pw] = nrbeval(NURBS, GauPts.Cor); GauPts.PCor = GauPts.PCor./GauPts.Pw; [N, id] = nrbbasisfun(GauPts.Cor, NURBS); R = zeros(GauPts.Num, CtrPts.Num); for i = 1:GauPts.Num, R(i,id(i,:)) = N(i,:); end R = sparse(R); [dRu, dRv] = nrbbasisfunder(GauPts.Cor, NURBS); X.GauPts = R*X.CtrPts;

9.5 Shep_Fun: Define the Smoothing Mechanism In the main function, the smoothing mechanism defined by the Shepard function is implemented by the line 22 of the MATLAB code by inputting two parameters (CtrPts and rmin), and two parameters (Sh and Hs) will be outputted. In the smoothing mechanism, the parameter rmin control the degree of the smoothness, which denotes the radius length of the circle in the normal parametric directions, generally equal to 2 in practical use. According to the MATLAB code, the MATLAB implementation of the smoothing mechanism is similar to the density filter defined in the 88-line SIMP MATLAB code [201]. However, several intrinsic different aspects exist and the detailed descriptions can refer to [56], and the MATLAB code of the subfunction Shep_Fun can be given in the following:

9.6 IGA to Solve Structural Responses 1 2 3 4 5 6 7 8 9

11 12 13

function [Sh, Hs] = Shep_Fun(CtrPts, rmin) Ctr_NumU = CtrPts.NumU; Ctr_NumV = CtrPts.NumV; iH = ones(Ctr_NumU*Ctr_NumV*(2*(ceil(rmin)-1)+1)^2,1); jH = ones(size(iH)); sH = zeros(size(iH)); k = 0; for j1 = 1:Ctr_NumV for i1 = 1:Ctr_NumU e1 = (j1-1)*Ctr_NumU+i1; for j2 = max(j1-(ceil(rmin)-1),1):min(j1+(ceil(rmin)1),Ctr_NumV) for i2 = max(i1-(ceil(rmin)-1),1):min(i1+(ceil(rmin)1),Ctr_NumU) e2 = (j2-1)*Ctr_NumU+i2; k = k+1; iH(k) = e1;

14 15

jH(k) = e2; theta = sqrt((j1-j2)^2+(i1-i2)^2)./rmin/sqrt(2);

16

sH(k) = (max(0, (1-theta)).^6).*(35*theta.^2 +

10

201

18*theta + 3); 17 end 18 end 19 end 20 end 21 Sh = sparse(iH,jH,sH); Hs = sum(Sh,2); 22 end

9.6 IGA to Solve Structural Responses As we know, the same NURBS basis functions used in the construction of structural geometries and the DDF keep unchanged and then are used to construct the solution space for the unknown structural responses, such as the displacement and stress, in analysis. In the IGA, a series of coefficients will be introduced at control points to develop the corresponding analysis space. The MATLAB code of the IGA is mainly involved three main aspects: (1) Gauss quadrature technique is applied to compute stiffness matrices of all IGA elements, implemented by the subfunction Stiff_Ele2D in line 28 of the main function; (2) all IGA element stiffness matrices are assembled into an IGA global stiffness matrix, and it is realized by the subfunction Stiff_Ass2D in the line 29 MATLAB code of the main program; (3) the unknown structural responses are computed by a subfunction Solving implemented in line 30.

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Fig. 9.8 Numbers of all control points, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

9.6.1 Stiff_Ele2D The isoparametric formulation is employed in the calculation of element stiffness matrix, and the detailed derivations in MATLAB language can refer to [7]. In the evaluation of element stiffness matrix, the Gauss quadrature method is employed, and the corresponding integration is performed in a bi-unit area, namely the bi-unit parent element space. As already given, two mappings X from the parametric space to physical space and Y from the parent space to parametric space should be defined in IGA. An illustration in detail of these two mappings is provided in Fig. 9.8. The calling code in the main function to solve all IGA element stiffness matrices is in line 28, given as: [KE, dKE, dv_dg] = Stiff_Ele2D(X, penal, Emin, DH, CtrPts, Ele, GauPts, dRu, dRv); As we can see, twelve input parameters are involved in the subfunction Stiff_Ele2D, namely (1) X is a structural array containing three fields to show the distribution of densities, including CtrPts field for control densities, GauPts field for the densities at Gauss quadrature points and DDF field for the densities in design domain; (2) penal is the penalty parameter, equal to be 3; (3) Emin is Young’s modulus of void materials to avoid numerical singularity; (4) DH is material constitutive elastic tensor matrix; (5–7) three structure arrays CtrPts, Ele and GauPts; (8–9) dRu and dRv corresponds to the first-order derivatives of NURBS basis functions with respect to two parametric directions, respectively. Stiff_Ele2D outputs three parameters, and KE is a cell matrix containing all IGA element stiffness matrices, dKE is a cell matrix containing the first-order derivatives of all IGA element stiffness matrices with respect to the densities of the Gauss quadrature points. dv_dg includes the first-order derivatives of volume constraint.

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203

Stiff_Ele2D: the initial definitions of KE, dKE and dv_dg are implemented in lines 2–4. In lines 6–32, a MATLAB loop is implemented to solve all IGA element stiffness matrices. The knots, numbers of control points and their corresponding physical coordinates of control points in each IGA element are firstly called in lines 7–11. Then the computation of the current IGA element stiffness matrix is completed in a subloop from lines 14–29. In this subloop, Jacobi matrix J 1 which denotes the first-order derivatives of physical space with respect to parametric space is firstly calculated in lines 15–18, including dPhy_dPara and J1. The strain–displacement matrix in each IGA element is calculated from lines 19–21, namely Be. In lines 22–24, the second Jacobi matrix J 2 which denotes the first-order derivatives of the parametric space with respect to parent space is calculated, namely J2. Then, the current IGA element stiffness matrix Ke, its derivatives dKe with respect to the densities of Gauss quadrature points in the current IGA element and the first-order derivatives of volume constraint dv_dg in the current IGA element are calculated in lines 26–28, respectively. In the MATLAB code, the calculation of IGA element stiffness matrices needs the information containing physical coordinates of control points and the densities at Gauss quadrature points. The MATLAB code of the subfunction Stiff_Ele2D subfunction is given as follows: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

function [KE, dKE, dv_dg] = Stiff_Ele2D(X, penal, Emin, DH, CtrPts, Ele, GauPts, dRu, dRv) KE = cell(Ele.Num,1); dKE = cell(Ele.Num,1); dv_dg = zeros(GauPts.Num,1); Nen = Ele.CtrPtsNum; for ide = 1:Ele.Num [idv, idu] = find(Ele.Seque == ide); % The two idices in two parametric directions for an element Ele_Knot_U = Ele.KnotsU(idu,:); span in the first parametric direction for an element Ele_Knot_V = Ele.KnotsV(idv,:); span in the second parametric direction for an element Ele_NoCtPt = Ele.CtrPtsCon(ide,:); of control points in an element Ele_CoCtPt = CtrPts.Cordis(1:2,Ele_NoCtPt); coordinates of the control points in an element Ke = zeros(2*Nen,2*Nen); dKe = cell(Ele.GauPtsNum,1); for i = 1:Ele.GauPtsNum GptOrder = GauPts.Seque(ide, i);

% The knot % The knot % The number % The

204 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

9 An In-House MATLAB Code of “IgaTop” for ITO dR_dPara = [dRu(GptOrder,:); dRv(GptOrder,:)]; dPhy_dPara = dR_dPara*Ele_CoCtPt'; J1 = dPhy_dPara; dR_dPhy = inv(J1)*dR_dPara; Be(1,1:Nen) = dR_dPhy(1,:); Be(2,Nen+1:2*Nen) = dR_dPhy(2,:); Be(3,1:Nen) = dR_dPhy(2,:); Be(3,Nen+1:2*Nen) = dR_dPhy(1,:); dPara_dPare(1,1) = (Ele_Knot_U(2)-Ele_Knot_U(1))/2; % the mapping from the parametric space to the parent space dPara_dPare(2,2) = (Ele_Knot_V(2)-Ele_Knot_V(1))/2; J2 = dPara_dPare; J = J1*J2; % the mapping from the physical space to the parent space; weight = GauPts.Weigh(i)*det(J); % Weight factor at this point Ke = Ke + (Emin+X.GauPts(GptOrder,:).^penal*(1Emin))*weight*(Be'*DH*Be); dKe{i} = (penal*X.GauPts(GptOrder,:).^(penal-1)*(1Emin))*weight*(Be'*DH*Be); dv_dg(GptOrder) = weight; end KE{ide} = Ke; dKE{ide} = dKe;

32 end 33 end

9.6.2 Stiff_Ass2D All IGA element stiffness matrices are assembled into a global stiffness matrix K by implementing a subfunction stiff_Ass2D in line 29 of the main function with five input parameters, namely (1) the cell matrix KE contains all IGA element stiffness matrices; (2–3) two structural arrays including the information of control points and all IGA elements, namely CtrPts and Ele. (4) Dim denotes the structural dimension, and (5) Dofs.Num is the total number of Degree of Freedoms (DOFs). The output parameter K is the global stiffness matrix. The corresponding MATLAB command called in the main function can be given as: [K] = Stiff_Ass2D(KE, CtrPts, Ele, Dim, Dofs.Num); Stiff_Ass2D: the indices II and JJ for two directions of the matrix KX which is another form of global stiffness matrix and a count ntriplets for the loop program are defined from lines 2–4. The MATLAB code of the loop to realize

9.6 IGA to Solve Structural Responses

205

the assembly of all IGA element stiffness matrices into KX is called in lines 5–16. A final assembly to obtain the global stiffness matrix K with a sparse form is performed in line 17 of the subfunction. The MATLAB code of Stiff_Ass2D is listed as: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

function [K] = Stiff_Ass2D(KE, CtrPts, Ele, Dim, Dofs_Num) II = zeros(Ele.Num*Dim*Ele.CtrPtsNum*Dim*Ele.CtrPtsNum,1); JJ = II; KX = II; ntriplets = 0; for ide = 1:Ele.Num Ele_NoCtPt = Ele.CtrPtsCon(ide,:); edof = [Ele_NoCtPt,Ele_NoCtPt+CtrPts.Num]; for krow = 1:numel(edof) for kcol = 1:numel(edof) ntriplets = ntriplets+1; II(ntriplets) = edof(krow); JJ(ntriplets) = edof(kcol); KX(ntriplets) = KE{ide}(krow,kcol); End end end K = sparse(II,JJ,KX,Dofs_Num,Dofs_Num); K = (K+K')/2; end

9.6.3 Solving The computation of the global displacement field is implemented by the subfunction Solving with six inputs in line 30 of the main program. In six input parameters, the detailed information of control points CtrPts, boundary conditions DBoudary and BoundCon, Dofs, the global stiffness matrix K and the load force F are included. The output corresponds to the global displacement field U in the design domain. The implementation of the subfunction solving is called by a MATLAB line: U = Solving(CtrPts, DBoudary, Dofs, K, F, BoundCon); Solving: In lines 2–9, five different numerical cases are defined with the displacements equal to 0 at Dirichlet boundary conditions, namely U_fixeddofs and V_fixeddofs. Then, the matrix U is solved in lines 10–13. The MATLAB code of the subfunction Solving is given as:

206 1 2 3 4 5 6 7 8 9 10 11 12 13 14

9 An In-House MATLAB Code of “IgaTop” for ITO function U = Solving(CtrPts, DBoudary, Dofs, K, F, BoundCon) switch BoundCon case {1, 4, 5} U_fixeddofs = DBoudary.CtrPtsOrd; V_fixeddofs = DBoudary.CtrPtsOrd + CtrPts.Num; case {2,3} U_fixeddofs = DBoudary.CtrPtsOrd1; V_fixeddofs = [DBoudary.CtrPtsOrd1; DBoudary.CtrPtsOrd2] + CtrPts.Num; end Dofs.Ufixed = U_fixeddofs; Dofs.Vfixed = V_fixeddofs; Dofs.Free = setdiff(1:Dofs.Num,[Dofs.Ufixed; Dofs.Vfixed]); U = zeros(Dofs.Num,1); U(Dofs.Free) = K(Dofs.Free,Dofs.Free)\F(Dofs.Free); end

9.7 Objective Function and Sensitivity Analysis Sensitivity analysis implemented by the MATLAB code is provided in lines 32–46 of the main function, and two steps are mainly involved on the basis of the derivations of sensitivity analysis. In the first step, the sensitivity analysis of objective function and constraint function with respect to the DDF at Gauss quadrature points are defined, and the MATLAB code is called by the following command: J = 0; dJ_dg = zeros(GauPts.Num,1); for ide = 1:Ele.Num Ele_NoCtPt = Ele.CtrPtsCon(ide,:); edof = [Ele_NoCtPt,Ele_NoCtPt+CtrPts.Num]; Ue = U(edof,1); J = J + Ue'*KE{ide}*Ue; for i = 1:Ele.GauPtsNum GptOrder = GauPts.Seque(ide, i); dJ_dg(GptOrder) = -Ue'*dKE{ide}{i}*Ue; end end Data(loop,1) = J; Data(loop,2) = mean(X.GauPts(:));

9.8 OC: Update Design Variables and DDF

207

Secondly, the first-order derivatives of the DDF with respect to design variables (control densities) are calculated in the second step, and the chain rule of the derivatives is utilized. The related MATLAB code of the sensitivity analysis of objective function and constraint functions with respect to control design variables is implemented by calling the following command: dJ_dp = R’*dJ_dg; dJ_dp = Sh*(dJ_dp./Hs); dv_dp = R’*dv_dg; dv_dp = Sh*(dv_dp./Hs);

9.8 OC: Update Design Variables and DDF Based on the first-order derivatives of the objective function and constraint function with respect to control design variables, the OC method with inputting the derivatives is applied to solve numerical optimization problems, and the MATLAB code of the OC is called in line 52 of the main function, and a subfunction OC is programmed by inputting seven parameters, namely the DDF at control densities and also Gauss quadrature points attached in a structural array X, the Shepard function and NURBS basis functions developed in the matrices Sh, Hs and R, respectively, the sensitivity analysis of the objective function and constraint function with respect to design variables in the matrices dJ_dp and dv_dp, and Vmax is the maximal material consumption in the optimization. The output parameter only contains the updated DDF at control densities and Gauss quadrature points. The MATLAB code of the OC method is performed in the following command: X = OC(X, R, Vmax, Sh, Hs, dJ_dp, dv_dp). OC: In line 2, the updated parameters l1, l2 and move are defined. The evolving of the design variables is implemented in a while loop from lines 3–13, until the constraint for the maximal material consumption is satisfied. It is noticed that the smoothing mechanism for control densities is active in each step of the optimization, and the DDF at Gauss quadrature points (densities) is applied to approximately calculate volume fraction. The MATLAB code of the subfunction OC is given as: 1 2 3 4 5 6 7 8 9 10 11 12 13 14

function X = OC(X, R, Vmax, Sh, Hs, dJ_dp, dv_dp) l1 = 0; l2 = 1e9; move = 0.2; while (l2-l1)/(l1+l2) > 1e-3 lmid = 0.5*(l2+l1); X.CtrPts_new = max(0,max(X.CtrPts-move, min(1, min(X.CtrPts+move,X.CtrPts.*sqrt(-dJ_dp./dv_dp/lmid))))); X.CtrPts_new = (Sh*X.CtrPts_new)./Hs; X.GauPts = R*X.CtrPts_new; if mean(X.GauPts(:)) > Vmax l1 = lmid; else l2 = lmid; end end end

208

9 An In-House MATLAB Code of “IgaTop” for ITO

9.9 Plot_Data and Plot_Topy: Representation of Numerical Results In the representation of the optimized results during the optimization, five related distributions in the optimization will be presented: (1) the discrete distribution of control densities: the representation of control design variables which corresponds to the densities at control points, namely control densities, in the design domain, shown in Fig. 9.9a. It is noted that the vertical direction denotes the values of densities. (2) the discrete distribution of densities at Gauss quadrature points, shown in Fig. 9.9b; in the optimization, all IGA element stiffness matrices are computed by Gauss quadrature method with nine Gauss quadrature points. Moreover, there is no need to map control densities into center densities of all IGA elements, mainly because that the mapping will introduce a large number of intermediate densities. The densities at Gauss quadrature points are more suitable to represent material distribution. (3) the continuous distribution of the DDF to present material layout in the design domain is shown in Fig. 9.9c, and a function surf in MATLAB is adopted here to approximately present the DDF using a family of samples. (4) The 2D view of the densities at Gauss quadrature points with the values larger than 0.5 is given to approximately describe the topology, shown in Fig. 9.9d. We can easily observe that it can be viewed as a discrete distribution of a series of point densities in design domain, similar to the layout of element densities. (5) It is assumed that the isocontour (the value 0.5) of the DDF represents structural boundaries, and the discussions about the rationality of the value equal to 0.5 can refer to [56, 62]. A function contourf in MATLAB is used here to approximately plot the structural topology, and the isocontour of the DDF is shown in Fig. 9.9e. The representation of numerical results involves two subfunctions Plot_Data and Plot_Topy. Firstly,

Fig. 9.9 Representation of the optimized designs: a the densities at control points, namely control densities; b the DDF at Gauss quadrature points; c the DDF; d the 2D view for the densities at Gauss quadrature points with the values higher than 0.5; e the isocontour of the DDF, namely the topology, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

9.9 Plot_Data and Plot_Topy: Representation of Numerical Results

209

a MATLAB implementation of the subfunction Plot_Data with two input parameters (Num and NURBS) is called in line 25 of the main function to construct some compulsory data for the latter representation, given as: [DenFied, Pos] = Plot_Data(Num, NURBS); Two output parameters are contained, namely DenFied and Pos. The first parameter contains the detailed information for the latter samples to plot the DDF by the MATLAB function surf, including the knot vectors, and the corresponding physical coordinates of all samples. The implementation of the DenFied is performed in lines 7–15, and the second parameter denotes the positions of figures, implemented in lines 2–6 of the subfunction Plot_Data and its MATLAB code is given as: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

function [DenFied, Pos] = Plot_Data(Num, NURBS) bdwidth = 5; topbdwidth = 30; scnsize = get(0,'ScreenSize'); Pos.p1 = [bdwidth, 3/5*scnsize(4)+bdwidth-50, scnsize(3)/2-2*bdwidth, 2*scnsize(4)/5-(topbdwidth+bdwidth)]; Pos.p2 = [Pos.p1(1)+scnsize(3)/2, Pos.p1(2), Pos.p1(3), Pos.p1(4)]; Pos.p3 = [bdwidth, 1/6*scnsize(4)+bdwidth-100, scnsize(3)/2-2*bdwidth, 2*scnsize(4)/5-(topbdwidth + bdwidth)]; Pos.p4 = [Pos.p1(1)+scnsize(3)/2, Pos.p3(2), Pos.p3(3), Pos.p3(4)]; Uknots = linspace(0,1,10*Num(1)); Vknots = linspace(0,1,10*Num(2)); [N_f, id_f] = nrbbasisfun({Uknots, Vknots}, NURBS); [PCor_U,PCor_W] = nrbeval(NURBS, {Uknots, Vknots}); PCor_U = PCor_U./PCor_W; PCor_Ux = reshape(PCor_U(1,:),numel(Uknots),numel(Vknots))'; PCor_Uy = reshape(PCor_U(2,:),numel(Uknots),numel(Vknots))'; DenFied.N = N_f; DenFied.id = id_f; DenFied.U = Uknots; DenFied.V = Vknots; DenFied.Ux = PCor_Ux; DenFied.Uy = PCor_Uy; end

The MATLAB command for the subfunction Plot_Topy with seven input parameters, including the DDF structural array X, control points and the Gauss quadrature points (GauPts and CtrPts), the detailed information for the samples of the DDF (DenFied), and the figure position Pos, is given as: [X] = Plot_Topy(X, GauPts, CtrPts, DenFied, Pos); The MATLAB code for the representation of control densities is called in lines 2–4 of the subfunction, and the representation of the densities at Gauss quadrature points is realized from lines 5–7. The MATLAB implementation of the representation of the DDF is called in lines 8–12. Lines 13–16 plot the representation of 2D view of the densities at Gauss quadrature points with the values larger than 0.5. In lines 17–19, the representation of the structural topology is implemented.

210 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

9 An In-House MATLAB Code of “IgaTop” for ITO function [X] = Plot_Topy(X, GauPts, CtrPts, DenFied, Pos) h1 = figure(1); clf; set(h1,'Position',Pos.p1, 'color',[1 1 1]); plot3(CtrPts.Cordis(1,:),CtrPts.Cordis(2,:),X.CtrPts,'.','color',[0.5 0 0.8]); axis equal; h2 = figure(2); clf; set(h2,'Position',Pos.p1,'color',[1 1 1]); plot3(GauPts.PCor(1,:),GauPts.PCor(2,:),X.GauPts,'.','color',[0.5 0 0.8]); axis equal; h3 = figure(3); clf; set(h3,'Position',Pos.p2,'color',[1 1 1]); X.DDF = sum(DenFied.N.*X.CtrPts(DenFied.id),2); X.DDF = reshape(X.DDF,numel(DenFied.U),numel(DenFied.V))'; surf(DenFied.Ux,DenFied.Uy,X.DDF); shading interp; colormap(jet(256)); alpha(0.5); axis equal; grid off; h4 = figure(4); clf; set(h4,'Position',Pos.p3,'color',[1 1 1]); GauPts_PCor = GauPts.PCor(1:2, X.GauPts>=0.5); plot(GauPts_PCor(1,:),GauPts_PCor(2,:),'.','color',[0.5 0 0.8]); axis equal; axis off; h5 = figure(5); clf; set(h5,'Position',Pos.p4,'color',[1 1 1]); contourf(DenFied.Ux, DenFied.Uy, X.DDF, [0.5 0.5], 'facecolor', [0.5 0 0.8], 'edgecolor', [1 1 1]); axis equal; axis off; end

9.10 Demos for Several Examples In this section, several numerical examples are tested to demonstrate the effectiveness and efficiency of the MATLAB code IgaTop2D for the ITO method. In all numerical examples, Poisson’s ratio is defined as 0.3, and Young’s moduli for solids and voids are equal to 1 and 1e − 9, respectively. A personal laptop with the MATLAB R2018b (9.5.0.944444) is used. Several benchmarks, including the cantilever beam, Michelltype structure, L beam and MBB beam, are all optimized by the current MATLAB implementation framework. The MATLAB implementations for L beam, cantilever beam, MBB beam and Michell-type structure are realized by calling the following MATLAB commands: IgaTop2D(10, IgaTop2D(10, IgaTop2D(18, IgaTop2D(10,

5, 5, 3, 4,

[1 [1 [1 [1

1], 1], 1], 1],

[101 [161 [241 [101

51], 81], 41], 41],

4, 1, 2, 3,

0.3, 0.2, 0.2, 0.2,

3, 3, 3, 3,

2) 2) 2) 2)

The optimized solutions for four numerical examples are shown in Fig. 9.10 for the L beam, Fig. 9.11 for the cantilever beam, Fig. 9.12 for the Michell-type structure and Fig. 9.13 for the MBB beam. The above-optimized numerical results of the L beam, cantilever beam, Michell-type structure and MBB beam can obviously demonstrate the effectiveness and efficiency of the MATLAB code IgaTop2D for the ITO method.

9.10 Demos for Several Examples

211

Fig. 9.10 Optimized designs of L beam: a control densities; b the densities at Gauss quadrature points; c the DDF in the design domain; d the 2D view of the densities with values higher than 0.5 at Gauss quadrature points; e the structural topology, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

Fig. 9.11 Optimized designs of cantilever beam: a control densities; b the densities at Gauss quadrature points; c the DDF in the design domain; d the 2D view of the densities with values higher than 0.5 at Gauss quadrature points; e the structural topology, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

Fig. 9.12 Optimized designs of Michell-type structure: a control densities; b the densities at Gauss quadrature points; c the DDF in the design domain; d the 2D view of the densities with values higher than 0.5 at Gauss quadrature points; e the structural topology, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

212

9 An In-House MATLAB Code of “IgaTop” for ITO

Fig. 9.13 Optimized designs of MBB beam: a control densities; b the densities at Gauss quadrature points; c the DDF in the design domain; d the 2D view of the densities with values higher than 0.5 at Gauss quadrature points; e the structural topology, reprinted from Ref. [68], copyright 2021, with permission from Elsevier

The codes of the IgaTop can be downloaded from https://link.springer.com/art icle/10.1007/s00158-021-02858-7, and all the results presented in the current paper can be reproduced using the MATLAB code. The data that support the MATLAB codes of the ITO method for 3D structures, namely IgaTop3D, are available from the author ([email protected]) upon reasonable request. The “nurbs” toolbox in the Octave and MATLAB can be obtained in https://octave.sourceforge.io/nurbs/.

9.11 Summary The main work of the current chapter is to present the detailed descriptions about the “IgaTop” code in MATLAB for the ITO method, and the details of the main function and all subfunctions are provided. In main function IgaTop2D, eleven MATLAB subfunctions (Geom_Mod, Pre_IGA, Guadrature, Shep_Fun, Boun_Cond, Stiff_Ele2D, Stiff_Ass2D, Solving, OC, Plot_Data and Plot_Topy) are developed for the ITO method. In the whole framework, it is mainly involved with construct the model of structural geometries using NURBS, the preparation for the implementation of IGA, the definition of two kinds of boundary conditions, the initialization of the DDF at control points (control densities) and Gauss quadrature points (discrete elements), the definition of smoothing mechanism, the related implementation of the IGA and solve unknown structural responses; calculate the objective function, constraint function and their sensitivity analysis, advance design variables and also the DDF, and the presentation of the optimized designs. Finally, several numerical examples are given to demonstrate the effectiveness and efficiency of the “IgaTop” framework.

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