Multiphysics and multiscale modeling 978-1-4822-4459-5

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MULTIPHYSICS and MULTISCALE MODELING

TECHNIQUES AND APPLICATIONS

Young W. Kwon

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

© 2016 by Taylor & Francis Group, LLC

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150810 International Standard Book Number-13: 978-1-4822-4459-5 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

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To my family, Sonja, Aric, and Elliot for sharing love; and to my graduate students for sharing knowledge and friendship.

© 2016 by Taylor & Francis Group, LLC

Contents Preface ........................................................................................................................................... xiii Acknowledgments ........................................................................................................................xv Author .......................................................................................................................................... xvii 1. Introduction .............................................................................................................................1 1.1 Overview ........................................................................................................................1 1.2 Computational Methods ..............................................................................................2 1.3 Organization of This Book ..........................................................................................2 2. Finite Element Method ..........................................................................................................5 2.1 Introduction ...................................................................................................................5 2.2 Method of Weighted Residual .....................................................................................5 2.3 Galerkin Finite Element Formulation ........................................................................9 2.4 Axial Bar and Beam .................................................................................................... 13 2.4.1 Axial Member ................................................................................................. 13 2.4.2 Beam ................................................................................................................ 17 2.4.3 Solution Techniques ...................................................................................... 21 2.5 Truss and Frame .......................................................................................................... 24 2.5.1 Truss................................................................................................................. 24 2.5.2 Frame ............................................................................................................... 26 2.6 Solid Element ............................................................................................................... 28 2.6.1 Theory of Elasticity ........................................................................................ 28 2.6.2 Finite Element Formulation ..........................................................................30 2.7 Isoparametric Formulation ........................................................................................ 36 2.8 Plate and Shell Structures .......................................................................................... 41 2.9 Acoustic Wave Equation ............................................................................................ 47 2.10 Interaction of Structure with Acoustic Domain ..................................................... 50 2.10.1 One-Dimensional Case ................................................................................. 50 2.10.2 Two-Dimensional Case ................................................................................. 52 3. Lattice Boltzmann Method ................................................................................................. 55 3.1 Introduction ................................................................................................................. 55 3.2 Standard Lattice Boltzmann Method ....................................................................... 55 3.3 Multiple Relaxation Lattice Boltzmann Formulation ............................................ 60 3.4 Finite Element–Based Lattice Boltzmann Method ................................................. 61 3.5 Element-Free-Based Lattice Boltzmann Method ....................................................64 3.6 Hybrid Lattice Boltzmann Formulation ..................................................................65 3.7 Multicomponent Flow ................................................................................................ 68 3.7.1 Color Fluid Model .......................................................................................... 68 3.7.2 Free-Energy Model ........................................................................................ 69 3.7.3 Mean Field Theory Model ............................................................................ 69 3.7.4 Interparticle Potential Model ....................................................................... 69 3.7.5 Immiscible Multicomponent Lattice Boltzmann Procedures.................. 70 vii © 2016 by Taylor & Francis Group, LLC

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3.8

Boundary Condition ................................................................................................... 71 3.8.1 Periodic Boundary ......................................................................................... 72 3.8.2 Fixed Rigid Boundary ................................................................................... 72 3.8.3 Pressure Boundary Condition ..................................................................... 73 3.8.4 Velocity Boundary Condition ...................................................................... 74 3.9 Turbulent Flow ............................................................................................................ 79 3.10 Wave Equation .............................................................................................................80 3.11 Scaling........................................................................................................................... 81 3.11.1 Viscosity Scaling ............................................................................................83 3.11.2 Velocity Boundary Condition Scaling ........................................................83 3.11.3 Pressure Boundary Condition Scaling .......................................................83 3.12 Example Problems ......................................................................................................85 3.12.1 Poiseuille Flow ...............................................................................................85 3.12.2 Backward-Facing Step ................................................................................... 87 3.12.3 Lid-Driven Cavity .......................................................................................... 88 3.12.4 Channel Flow over Cylinder ........................................................................ 92 3.13 Lattice Boltzmann Method Implementation on Graphics Processing Units ...... 96 3.13.1 Computational Requirements for the Lattice Boltzmann Method ......... 96 3.13.2 Basic Implementation on Graphics Processing Units ............................... 97 3.13.2.1 LBM Routine ................................................................................... 97 3.13.2.2 Data Layout ..................................................................................... 98 3.13.3 Performance Benchmark .............................................................................. 99 4. Cellular Automata .............................................................................................................. 101 4.1 Introduction ............................................................................................................... 101 4.2 Strengths and Weaknesses of Cellular Automata ................................................ 103 4.3 Modeling Moving Objects Using Cellular Automata .......................................... 103 4.3.1 Spring Constant ........................................................................................... 105 4.3.1.1 Left-End Particles ......................................................................... 105 4.3.1.2 Right-End Particles....................................................................... 106 4.3.2 Velocity, Mass, Momentum, and Energy.................................................. 107 4.4 Physical Examples Using Cellular Automata ....................................................... 108 4.4.1 Longitudinal Vibration of a Long Uniform Rod ..................................... 109 4.4.2 Transverse Vibration of String ................................................................... 111 4.4.2.1 String Plucked at the Midpoint .................................................. 111 4.4.2.2 String with a Force Applied on the Middle .............................. 112 4.4.3 One-Dimensional Wave Equation ............................................................. 114 4.5 Boundary Conditions ............................................................................................... 116 4.6 Discretization and Model Fidelity .......................................................................... 117 4.7 Convergence............................................................................................................... 118 4.8 Two-Dimensional Wave Problem ........................................................................... 119 4.8.1 Cellular Automata in Two Dimensions .................................................... 119 4.8.2 Example of Membrane Vibration............................................................... 120 4.9 Three-Dimensional Wave Problem ........................................................................ 122 4.10 Application to Underwater Acoustics .................................................................... 126 4.10.1 Transmission Loss Development ............................................................... 127 4.10.2 Various Boundary Conditions ................................................................... 129 4.10.3 Wave Propagation across a Flat-Bottom Ocean Floor............................. 129

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4.11

4.10.4 Wave Propagation over a Curved Hill ...................................................... 131 4.10.5 Wave Propagation over a Sloping Bottom ................................................ 132 Multigrid Technique of Cellular Automata .......................................................... 133 4.11.1 Global-Local Technique .............................................................................. 134 4.11.2 Example Problems ....................................................................................... 137

5. Molecular Dynamics .......................................................................................................... 145 5.1 Introduction ............................................................................................................... 145 5.2 Classical Molecular Dynamics Formulation ......................................................... 145 5.3 Time Integration Technique .................................................................................... 146 5.3.1 Prediction Step ............................................................................................. 148 5.3.2 Evaluation ..................................................................................................... 148 5.3.3 Correction ..................................................................................................... 148 5.4 Interatomic Potential Energy Function .................................................................. 149 5.4.1 Lennard-Jones Potential .............................................................................. 149 5.4.2 Morse Potential............................................................................................. 150 5.4.3 Abell-Tersoff-Brenner Potential.................................................................. 150 5.4.4 Embedded Atom Potential ......................................................................... 154 5.5 Molecular Mechanics Formulation ........................................................................ 154 5.6 Application to Carbon Nanotubes: Elastic Modulus ........................................... 156 5.6.1 Basic Structures of Carbon Nanotubes ..................................................... 157 5.6.2 Simulation Time Step .................................................................................. 158 5.6.3 Freestanding Thermal Vibration Method ................................................ 159 5.6.4 Equilibrium and Vibration Motion of CNTs ............................................ 159 5.6.5 Elastic Modulus of CNTs under Equilibrium .......................................... 162 5.6.6 Comparative Results of Equilibrium and Nonequilibrium Simulations ................................................................................................... 165 5.7 Application to Carbon Nanotubes: Vibrational Mode Shapes ........................... 168 5.7.1 Comparison of Radial Breathing Modes of Armchair SWCNTs .......... 168 5.7.2 Natural Frequencies and Mode Shapes for SWCNTs ............................. 168 5.7.3 Natural Frequencies and Mode Shapes for MWCNTs ........................... 171 5.7.4 Natural Frequencies and Mode Shapes for BSCNTs .............................. 172 5.8 Application to Polymers ........................................................................................... 173 5.8.1 Cross-Linking of Polymers ......................................................................... 173 5.8.2 Allocation of Polymers ................................................................................ 174 5.8.3 Potential Energy for Polymers ................................................................... 175 5.8.4 Autocorrelation ............................................................................................ 176 5.8.5 Examples ....................................................................................................... 176 5.9 Application to Nanoscale Fluid Flow..................................................................... 179 5.9.1 Fluid Flow Simulation ................................................................................. 179 5.9.2 Particle-Fluid Interaction ............................................................................ 180 5.10 Application to Fatigue of Metals ............................................................................. 183 5.10.1 Modeling ....................................................................................................... 183 5.10.2 Examples ....................................................................................................... 184 6. Coupling Techniques ......................................................................................................... 187 6.1 Introduction ............................................................................................................... 187 6.2 Coupling Technique between Finite Element and Atomic Models ................... 187

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6.4 6.5 6.6 6.7

Example Problems of Coupled Finite Element and Atomic Models ................. 188 6.3.1 Hexagonal Array of Atoms with Dislocation .......................................... 189 6.3.2 Atomic Array Embedded in the Finite Element Mesh with a Crack.... 191 6.3.3 Atomic Behavior at the Crack Tip ............................................................. 193 Homogenization of an Atomic Model into a Continuum Model....................... 194 Example Problems for a Smeared Atomic Model................................................. 196 6.5.1 Vibration of Atoms in a One-Dimensional Arrangement ..................... 196 6.5.2 Vibration of Atoms in 2-D Arrangement .................................................. 198 Coupling Technique between Finite Element and Lattice Boltzmann Models.... 200 6.6.1 Coupling Acoustic Domains ...................................................................... 200 6.6.2 Coupling Fluid and Structural Domains ................................................. 201 Coupling Technique between the Finite Element and Cellular Automata Models ........................................................................................................................ 203 6.7.1 Coupling Acoustic Domains ...................................................................... 203 6.7.2 Coupling the Acoustic and Structural Domains ..................................... 204

7. Multiscale Analysis of Composite Structures .............................................................. 207 7.1 Introduction ............................................................................................................... 207 7.2 Multiscale Modeling of Composite Structures ..................................................... 207 7.3 Fiber Model ................................................................................................................ 209 7.4 Fabric Model .............................................................................................................. 214 7.4.1 Fabric Model for Plain Weave Composite ................................................ 215 7.4.2 Fabric Model for 2/2 Twill Composite ...................................................... 219 7.5 Coordinate Transformation .....................................................................................222 7.6 Lamination Model .................................................................................................... 224 7.7 Examples of the Fiber Model ...................................................................................225 7.7.1 Particulate Composite Made of Al2O3 Particle/Aluminum Matrix ......225 7.7.2 Particulate Composite Made of SiC Particle/Aluminum Matrix .......... 226 7.7.3 Fibrous Composite Made of Boron Fibers and Aluminum Matrix ...... 226 7.7.4 Fibrous Composite Made of Graphite Fiber and Epoxy Matrix ............ 227 7.7.5 Short-Fiber Composite Made of SiC Whiskers and Aluminum Matrix ... 227 7.7.6 Porous Material Made of Al2O3.................................................................. 228 7.7.7 Thermal Stress of Glass Fiber/Epoxy Matrix Fibrous Composite ........ 229 7.7.8 Hierarchical Composite .............................................................................. 229 7.7.9 Glass Fiber/Epoxy Matrix Fibrous Composite under Tensile Load ..... 231 7.7.10 Filament Wound Cylinder .......................................................................... 233 7.7.11 Fibrous Composite Plate with a Hole ........................................................234 7.7.12 Particulate Composite Plate with a Hole .................................................. 236 7.7.13 Fibrous Composite with Statistical Consideration .................................. 237 7.7.14 Fibrous Composite for Strength................................................................. 239 7.8 Examples of Fabric Model ........................................................................................ 239 7.9 Elastoplastic Analysis of Composite Materials..................................................... 243 7.10 Examples of Elastoplastic Analysis of Composite Materials .............................. 246 7.10.1 Particulate Metal Matrix Composite Made of SiC and Aluminum ...... 246 7.10.2 Particulate Composite Made of SiC and Cu Alloy .................................. 248 7.10.3 Thermal Residual Stresses for a Whisker Composite............................. 249 7.10.4 Fibrous Metal Matrix Composite ............................................................... 250 7.10.5 Fibrous Composite Plate with Preexisting Crack.................................... 251 7.10.6 Laminated Composite Plate .......................................................................254

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8. Multiscale Analysis of Metallic Materials .................................................................... 255 8.1 Introduction ............................................................................................................... 255 8.2 Polycrystalline Materials.......................................................................................... 255 8.3 Previous Study of the Multiscale Analysis of Metals .......................................... 255 8.4 Procedure for Multiscale Analysis of Polycrystalline Metals ............................ 256 8.5 Macroscale Analysis ................................................................................................. 257 8.6 Mesoscale Analysis ................................................................................................... 258 8.7 Microscale Analysis .................................................................................................. 260 8.8 Nanoscale Analysis................................................................................................... 260 8.9 Example Problems .................................................................................................... 261 9. Multiscale Analysis of Biomaterials............................................................................... 267 9.1 Introduction ............................................................................................................... 267 9.2 Nanoscale Model ....................................................................................................... 267 9.2.1 Hydroxyapatite ............................................................................................. 267 9.2.2 Tropocollagen ............................................................................................... 269 9.2.3 Helical Spring ............................................................................................... 269 9.2.4 Results of the Nanoscale Model................................................................. 270 9.3 Microscale Model ...................................................................................................... 272 9.3.1 Two-Dimensional Fibril Structure ............................................................ 272 9.3.2 Three-Dimensional Fibril Structure.......................................................... 273 9.3.3 Micromechanics Fibril Model .................................................................... 274 9.3.3.1 Linear Fibril Subunit.................................................................... 275 9.3.3.2 Twisting Fibril Subunit ................................................................ 275 9.3.4 Fibril Results ................................................................................................. 276 9.3.5 Bone Fiber ..................................................................................................... 277 9.3.6 Micromechanical Fiber Model ................................................................... 278 9.3.7 Fiber Results ................................................................................................. 280 9.4 Macroscale Model ..................................................................................................... 281 9.4.1 Lamellar Bone .............................................................................................. 281 9.4.2 Lamellar Model ............................................................................................ 282 9.4.3 Lamellar Model Results .............................................................................. 283 9.4.4 Cortical Bone ................................................................................................284 9.4.5 Cortical Bone Model .................................................................................... 285 9.4.6 Cortical Bone Model Results ...................................................................... 286 9.4.7 Cancellous Bone ........................................................................................... 288 9.4.8 Trabecular Bone Model ............................................................................... 289 9.4.9 Result of Trabecular Bone ........................................................................... 289 9.4.10 Cancellous Bone Model............................................................................... 290 9.4.11 Cancellous Bone Results ............................................................................. 291 9.5 Further Adjustments in Models .............................................................................. 292 9.5.1 Modified Hierarchy ..................................................................................... 292 9.5.2 Adjustment Results...................................................................................... 293 9.5.3 Optimal Adjustment Results...................................................................... 294 9.5.4 Bone Loss Results ........................................................................................ 295 10. Multiphysics Analysis of Composite Structures ......................................................... 297 10.1 Introduction ............................................................................................................... 297 10.2 Fluid-Structure Interaction Modeling .................................................................... 297

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10.3 Low-Velocity Impact with FSI ................................................................................. 298 10.3.1 Single-Layer Plate Model ............................................................................ 299 10.3.2 Two-Layer Plate Model ...............................................................................300 10.3.3 Three-Layer Plate Model............................................................................. 302 10.3.4 Comparison to Experimental Data............................................................304 10.4 Vibration with FSI .....................................................................................................306 10.4.1 Numerical Modal Analysis ........................................................................ 307 10.4.2 Verification Study......................................................................................... 307 10.4.3 Example Problems .......................................................................................308 10.5 Fatigue Loading with FSI ......................................................................................... 318 10.5.1 Problem Description .................................................................................... 318 10.5.2 Example Problems ....................................................................................... 320 10.6 Hydrodynamic Loading .......................................................................................... 330 10.6.1 Model Description ....................................................................................... 330 10.6.2 Numerical Results with Constant Acceleration ...................................... 331 10.6.3 Numerical Results with Nonlinear Acceleration .................................... 333 10.6.4 Numerical Results with Free Surface........................................................ 335 10.6.5 Numerical Results with Different Material Properties .......................... 336 10.7 Hydrodynamic Ram ................................................................................................. 338 10.7.1 What Is Hydrodynamic Ram? ................................................................... 338 10.7.2 Numerical Models ....................................................................................... 338 10.7.3 Numerical Results........................................................................................ 339 11. Multiscale Analysis of Electromechanical System......................................................343 11.1 Introduction ...............................................................................................................343 11.2 Principle of Operation of a Rail Gun Launcher ....................................................343 11.3 Previous Study of Rail Gun Launchers .................................................................344 11.4 Mathematical Models for Multiphysics Analysis .................................................344 11.5 Electromagnetic Theory ...........................................................................................345 11.6 Thermal Stress Analysis ..........................................................................................348 11.7 Contact Theory .......................................................................................................... 349 11.8 Analysis Model.......................................................................................................... 352 11.9 Example Problems .................................................................................................... 353 12. Multiphysics Analysis of Biomechanics ........................................................................ 359 12.1 Introduction ............................................................................................................... 359 12.2 Review of Previous Work......................................................................................... 359 12.3 Description of Numerical Models .......................................................................... 360 12.4 Comparison of Models with and without FSI ...................................................... 362 12.5 Results of the Aneurysm Initiation Studies ..........................................................364 12.5.1 Blood Vessel Wall Modeling ......................................................................364 12.5.2 Blood Viscosity Effect.................................................................................. 373 12.6 Already-Developed Abdominal Aortic Aneurysm ............................................. 373 References ................................................................................................................................... 375 Index ............................................................................................................................................. 393

© 2016 by Taylor & Francis Group, LLC

Preface Multiscale and multiphysics analyses have become popular for better understanding of complex physical behaviors. However, to the best of my knowledge, books on these topics are limited. Most are focused on specific topics. As a result, this book is expected to serve a wider spectrum of engineers and scientists who work on multiscale and multiphysics analyses. This book is a collection of research in the subject areas of multiscale and multiphysics analysis that coworkers and I conducted during the past decade. The first several chapters present various computational techniques that are useful for multiscale and multiphysics analyses. Those include the finite element method, lattice Boltzmann method, cellular automata, molecular dynamics, and their coupling techniques. Some of those techniques are useful for multiscale analysis, while others are beneficial for multiphysics analysis. Then, the next three chapters present examples for multiscale analysis, which is used for composite materials, metallic materials, and biomaterials. The final three chapters present examples for multiphysics analysis. Those examples are fluidstructure analysis of composite structures, electromechanical rail guns, and blood vessel aneurysms. Young W. Kwon Monterey, California

xiii © 2016 by Taylor & Francis Group, LLC

Acknowledgments I am indebted to my many coworkers, especially former graduate students because they have made significant contributions to research programs. Especially, this book contains large portions of research undertaken with the following individuals (in alphabetical order): Ahmet Altekin, Jamal AlRowaijeh, Jermaine Bailey, Joe Berner, Stuart Blair, Brandon Clumpner, Ryan Conner, Linda Craugh, Jarema Didoszak, Anthony Harrell, Selcuk Hosoglu, Sunghoon Jung, Daniel Kidd, Scott Knutton, Chaitanya Manthena, Ryan McCrillis, Jung Joo Oh, Angela Owens, Moon Shik Park, Spyridon Plessas, Nikolaos Pratikakis, Eric Priest, Kevin Roach, Matthew Shellock, Fatma Gulden Simsek, Michael Violette, and Kangjie Yang. If there is anyone omitted here, I apologize for my mistake. Without their contribution, this book would not be possible. I also appreciate the guidance and help from the CRC staff, especially that of Jonathan Plant.

xv © 2016 by Taylor & Francis Group, LLC

Author Dr. Young W. Kwon is a distinguished professor in the Mechanical and Aerospace Engineering Department of the Naval Postgraduate School in Monterey, California. He was past chair of the department. He was also professor and chair of the Department of Mechanical Engineering and Energy Processes of Southern Illinois University Carbondale. He received his PhD degree from Rice University and a BS degree from Seoul National University, both in mechanical engineering. Before joining the Naval Postgraduate School, he was an assistant professor at the University of Missouri–Rolla. His research interests include multiscale and multiphysics computational techniques for material behaviors bridging the nanoscale to macroscale, fluid-structure interaction problems, ship shock modeling and simulation; composite materials; fracture and damage mechanics; nanotechnology; and biomechanics. He has authored or coauthored extensive technical publications, many of which appeared in archived, refereed publications. He wrote the textbook Finite Element Method Using MATLAB (CRC Press), which was translated into Greek; and contributed book chapters, including “Nanomechanics” in Nanoengineering of Structural, Functional and Smart Materials (CRC Press), “Multi-Scale Computational Modeling and Simulation” in Progress in Engineering Computational Technology (Saxe-Coburg Publishing), “Computational and Experimental Study of Composite Scarf Bonded Joints” in Structural Integrity and Durability of Advanced Composites: Innovative Modeling Methods and Intelligent Design (Woodhead Publishing), “Dynamic Loading on Composite Structures with Fluid-Structure Interaction” in Dynamic Deformation, Damage and Fracture in Composite Materials and Structures (Woodhead Publishing), and others. He edited a book, Multiscale Modeling and Simulation of Composite Materials and Structures, published by Springer. Dr. Kwon has received various awards, including the Cedric K. Ferguson Medal from the Society of Petroleum Engineers, Menneken Faculty Award, Excellent Research Award from the American Orthopedic Society of Sports Medicines, Outstanding Instruction and Research Awards, American Society of Mechanical Engineers PVPD (Pressure Vessels and Piping Division) Outstanding Service Award, National Dean’s List, Who’s Who in Science and Engineering, American Society of Mechanical Engineers Dedicated Service Award, American Society of Mechanical Engineers Board of Governors Award, and more. Dr. Kwon is a fellow of the American Society of Mechanical Engineers. He is the technical editor of the American Society of Mechanical Engineers Journal of Pressure Vessel Technology as well as the journal Materials Sciences and Applications, and he serves on the editorial boards for multiple journals. He is vice president for the International Society of Multiphysics. Dr. Kwon was the American Society of Mechanical Engineers PVPD chair and its senate president. He has served as a member of the executive committee of American Society of Mechanical Engineers PVPD, technical program chair for the 2009 American Society of Mechanical Engineers PVP Conference, and conference chair for the 2010 American Society of Mechanical Engineers PVP Conference. He is also an honorary theme editor of Pressure Vessels and Piping Systems of the Encyclopedia of Life Support Systems under the auspices of the United Nations Educational, Scientific, and Cultural Organization. Dr. Kwon has conducted extensive research projects sponsored by government agencies and private sectors and supervised more than 100 graduate students. Dr. Kwon provided numerous talks as an invited or keynote speaker at professional meetings and institutions worldwide. xvii © 2016 by Taylor & Francis Group, LLC

1 Introduction

1.1 Overview There are systems that exist in nature and there are systems developed by human beings. For example, the bodies of humans and animals are living systems in nature, and power plants are man-made systems. To understand or develop such systems, it is necessary to conduct multiphysics and multiscale analyses. Multiphysics analysis is the study of multiple physical behaviors as they interact with one another. For example, the human body is a good example of something that requires multiphysics analyses. Chemistry and biology are basic knowledge necessary to understand living cells, tissues, organs, and the like. However, those subjects are not considered in this book because they are outside of the scope of the study. Only the physical aspects are considered in multiphysics analysis. Considering the blood circulation system in the human body, blood vessels require fluid mechanics analysis to study blood flow and blood pressure; the system also requires structural analysis to investigate contraction and dilation as well as aneurysms and potential ruptures of the blood vessels. In this case, both analyses should be coupled because blood and vessels interact. As a result, multiphysics analysis is required. As an example for a man-made system, a power plant contains many heat pipes that carry hot fluid, which also requires fluid mechanics analysis and structural analysis, leading to multiphysics analysis. When there is an exchange of heat between the fluids inside and outside the heat pipes, heat transfer analysis should also be considered. In many cases, a single analysis is conducted, neglecting the interactive aspect of multiphysics because the single analysis is much simpler. However, if the interaction is strong and influencing each participant, a single analysis may not provide reliable results. In that case, multiphysics analysis must be undertaken even though it is more complex and requires more effort. With advances in computing power as well as computational techniques, multiphysics analysis has been more common in engineering design and analysis. On the other hand, almost every material and living organism has a complex hierarchical structure in different length scales. Human bones are good examples. They consist of simple elements at the nanoscale from a mechanics point of view. However, through the complex hierarchical structures in different length scales, bones are optimized and provide necessary strength and stiffness for the human skeleton. Likewise, man-made composite materials consist of fibers and matrix materials. Through aligning or weaving the fibers, the fiber composite structures become strong, stiff, and light for use for new technology. Metals also have many different characteristics at the different length scales. To understand and predict their behaviors as well as to develop new materials, it is necessary to undertake multiscale analysis. This analysis links the main characteristics 1 © 2016 by Taylor & Francis Group, LLC

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Multiscale and Multiphysics Modeling

at different length scales so that we can have more fundamental understanding of their behavior as well as knowledge of what are the most important hierarchical structures influencing the macroscale behavior.

1.2 Computational Methods Advances in computing capabilities as well as computational techniques contribute to the study of multiphysics and multiscale analysis. In this section, we discuss some computational methods used for multiscale and multiphysics analyses. There is no intention to list and discuss all the available computational methods here. Some computational methods are presented that are relevant to the contents of the book. The finite element method is arguably the most popular and powerful computational technique to analyze continuous media [1–3]. This technique can solve virtually any differential equation with proper boundary and initial conditions for complex shapes of domains. The finite element method was developed initially for structural analysis, and the technique has spread to other types of applications, including problems in fluid mechanics, electromagnetic waves, and so on. As a result, the finite element method is useful for multiphysics problems. The finite element technique is useful for solving continuous domain problems; the molecular dynamics technique is applicable to discrete domain problems such as atomistic and molecular modeling [4,5]. As a result, the length scale of the molecular dynamics problems is much smaller than the length scale of the finite element analysis problems. In order to combine the two techniques, a coupling technique is necessary for multiscale analyses. The lattice Boltzmann method as well as cellular automata can be applied to both continuous and discrete domain problems [6–9]. These are the advantages of the techniques. However, those methods are based on rules, and those rules are not straightforward for development for any differential equation. As a result, the lattice Boltzmann method and the cellular automata have been applied to a limited number of problems. Because those methods are based on rules, they are easy to program and are computationally efficient.

1.3 Organization of This Book This book is organized in the following manner: first, multiple computational techniques are presented in Chapters 2 through 5, respectively: the finite element method, lattice Boltzmann method, cellular automata, and molecular dynamics technique. Then, Chapter 6 presents coupling techniques to take advantage of these methods. Finally, several example problems are discussed for multiphysics and multiscale problems in Chapters 7 through 12. The first three of these chapters are for multiscale analyses, and the last three chapters are for multiphysics problems. Multiscale analysis for composite materials and structures is provided in Chapter 7; fibrous composite, particulate composite, and woven fabric composite materials and structures are presented as examples. Multiscale analysis for metals is given in Chapter 8. In this chapter, a simplified analysis is provided for crystalline structures using the finite

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Introduction

3

element method and molecular dynamics. Then, Chapter 9 discusses multiscale analysis of biomaterials using human bones as an example. Multiphysics problems are discussed for composite structures in Chapter 10. Fluid– structure interaction is presented for composite structures surrounded by water. Ship structures are examples of these structures. Chapter 11 provides an example of the multiphysics analysis of electromechanical problems. A rail gun is selected as an example problem; this couples electromagnetic waves, heat transfer, and rigid body dynamics. Finally, Chapter 12 presents a multiphysics analysis of biomechanics using the example of blood vessels, for which there is fluid–structure interaction.

© 2016 by Taylor & Francis Group, LLC

2 Finite Element Method

2.1 Introduction The finite element method (FEM) is one of the most widely used numerical solution techniques [1–3]. Especially, the FEM has been used almost exclusively for analyses of solids and structures. This chapter introduces the basic finite element concepts and presents formulations for various applications, in particular for solid and structural analyses. The FEM can solve differential equations whether they are boundary value problems, initial value problems, or eigenvalue problems. The method of weighted residual (MWR) is a good starting point for the finite element formulation [1]. Therefore, the next section presents the MWR to introduce the basic finite element concept and formulation. Then, the finite element technique is applied to various physical problems in the sections that follow.

2.2 Method of Weighted Residual To illustrate the MWR, let us begin with a simple example problem that is expressed as an ordinary differential equation: d2u − u = 0, 0 ≤ x ≤ 1 dx 2

(2.1)

and the differential equation should satisfy the following boundary conditions: u(0) = 0

and

u(1) = 1

(2.2)

This is a simple two-point boundary value problem. The exact solution is u( x) =

ex − e− x e − e −1

(2.3)

which satisfies both the governing equation, Equation 2.1, as well as the boundary conditions, Equation 2.2, simultaneously. Let us assume that we do not have the exact solution for the sake of introduction of the MWR. The purpose of the MWR is to find an approximate 5 © 2016 by Taylor & Francis Group, LLC

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Multiscale and Multiphysics Modeling

solution to the given problem. We want to make the approximate solution represent the exact solution as closely as possible even though we do not know the exact solution. To begin with the MWR, we assume a possible set of solutions. Because we do not know the exact solution, what would be the proper forms for the approximate solutions? One fact is that any complex function can be expressed in terms of a series of polynomials with an infinite number of terms. In a practical sense, we cannot assume an infinite order of polynomial function. Instead, a finite order of polynomial would be a reasonable choice. Therefore, let us assume a polynomial function. For the present example, it is easy to find an approximate solution that satisfies the boundary conditions. Because there are two boundary conditions, the lowest order of polynomial is a linear function. However, both coefficients of the linear function can be determined uniquely from the two boundary conditions. As a result, there is no room to optimize the approximate solution as closely as possible to the exact solution that also satisfies the differential equation. To this end, the lowest order of polynomial function is a quadratic function as given next: ũ(x) = a0 + a1x + a2x2

(2.4)

where ai (i = 0, 1, 2) is the coefficients to be determined, and ũ denotes an approximate solution. Such an assumed function is called the trial function. First, we want to satisfy the boundary conditions. Substitution of the boundary conditions as given in Equation 2.2 into the quadratic function of Equation 2.4 yields ũ(0) = a0 = 0

(2.5)

ũ(1) = a0 + a1 + a2 = 1

(2.6)

Because there are two equations for three unknowns, we can express any two unknowns in terms of the third unknown. In other words, we can obtain a0 = 0 and a1 = 1 − a2. Then, the approximate solution can be rewritten as ũ(x) = (1 – a2)x + a2x2

(2.7)

The mathematical expression in Equation 2.7 satisfies the boundary conditions regardless of the choice of the third coefficient a2. To check how the approximate solution of Equation 2.7 satisfies the differential equation, Equation 2.1, Equation 2.7 is plugged into Equation 2.1, leading to R(x) = 2a2 − (1 − a2)x − a2x2

(2.8)

in which R(x) is called the residual. The residual will be zero for any value of x within the problem domain if the approximate solution happens to coincide with the exact solution. However, because the approximate solution is different from the exact solution, the residual does not vanish for all x values within the problem domain. From the concept of the residual, it is reasonable to state that a smaller residual means the approximate solution is closer to the exact solution. As a result, we want to find the third coefficient a2 to minimize the residual. In particular, we want to minimize the residual throughout the whole problem domain. In that aspect, we plan to minimize the sum of the residual over the whole domain. However, we do not want to sum the residual as

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7

Finite Element Method

it is because the residual may be positive or negative from point to point in the domain. In that case, those residuals can cancel out one another, resulting in a negligible sum of residuals. In other words, the residuals, may be large at every point, but their sum may be small because of such cancellation. To avoid such a problem, the residual is multiplied by another function, which is called the test function or weighting function. Then, such a residual is called a weighted residual. Now, our goal is to make the sum of the weighted residual over the problem domain vanish. That is, 1

I=

1

∫ w(x)R(x) dx = ∫ w(x){2a − (1 − a )x − a x } dx = 0 2

0

2

2

2

(2.9)

0

The next question is what would be the best choice as the test function. There are some popular choices [1], and there is a specific name depending on the selection of the test function. In this study, the Galerkin method is presented. In that technique, the test function is selected based on the trial function, and their relationship to the present trial function is w( x) =

( x) du = − x + x2 da2

(2.10)

Inserting Equation 2.10 into Equation 2.9 gives 1

∫

{

}

I = (− x + x 2 ) 2 a2 − (1 − a2 )x − a2 x 2 dx = 0

(2.11)

0

5 Solving Equation 2.11 for the unknown a2 yields a2 = , and the approximate solution 22 becomes ( x) = u

17 x + 5 x 2 22

(2.12)

The approximate solution can be compared to the exact solution as shown in Figure 2.1. Both solutions agree well. To improve the approximate solution, we need to increase the order of the polynomials in the trial function. For example, we use a cubic function such as ũ(x) = b0 + b1x + b2x2 + b3x3

(2.13)

Applying the boundary conditions to this expression yields

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ũ(0) = b0 = 0

(2.14)

ũ(1) = b0 + b1 + b2 + b3 = 1

(2.15)

8

Multiscale and Multiphysics Modeling

1 Exact Approx. FEM

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

FIGURE 2.1 Comparison among the exact solution, approximate solution using quadratic polynomials, and finite element solution using two linear shape functions.

Elimination of two variables from the equations gives b0 = 0

(2.16)

b1 = 1 − b2 − b 3

(2.17)

Then, the trial function is expressed as ũ(x) = (1 – b2 – b3)x + b2x2 + b3x3

(2.18)

Substitution of the trial function into the differential equation results in the residual R(x) = 2b2 + (–1 + b2 + 7b3)x – b2x2 – b3x3

(2.19)

To apply the MWR, we need two test functions because there are two unknowns in Equation 2.19. Those are determined such that w1 ( x ) =

( x) du = − x + x2 db2

(2.20)

w2 ( x) =

( x) du = − x + x3 db3

(2.21)

The resulting two weighted residual expressions are 1

I1 =

∫ (− x + x ){2b + (−1 + b + 7b )x − b x 2

0

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2

2

3

2

2

}

− b3 x 3 dx = 0

(2.22)

9

Finite Element Method

1

I2 =

∫ (− x + x ){2b + (−1 + b + 7b )x − b x 3

2

2

3

2

2

}

− b3 x 3 dx = 0

(2.23)

0

Solving the two equations for b1 and b2 gives the approximate solution.

2.3 Galerkin Finite Element Formulation If a problem domain has a complex shape, it is not easy to assume a trial function over the domain while satisfying prescribed boundary conditions throughout the domain. In addition, the solution process should be conducted systematically using a computer algorithm. To that end, the problem domain is divided into a number of simple shapes of subdomains. For the one-dimensional (1-D) problem discussed previously, the problem domain is divided into two equal size subdomains as shown in Figure 2.2. Each subdomain is called a finite element. As a result, there are two finite elements in Figure 2.2, and each element has two nodes. However, because one node is shared between the first and the second elements, there are three nodes in the problem. The finite element formulation is going to determine solutions at the nodes, which is called the nodal variable. Let the nodal variable at the ith node be called ui. Then, the first element has nodal variables u1 and u2, and the second element has nodal variables u2 and u3. In the finite element formulation, a trial function is expressed in terms of so-called shape functions and nodal variables. The shape functions are used to interpolate the solution within an element. Because each element has two nodes, a linear interpolation would be suitable to express the solution inside the element with two nodal variables. For example, let an element have nodal variables ui and uj. The linear interpolation function within the element is expressed as ũ(x) = a + bx

(2.24)

We want to replace a and b by the nodal variables ui and uj. To achieve this, we evaluate the linear interpolation function at the nodal points as follows: ũ(xi) = a + bxi = ui

(2.25)

ũ(xj) = a + bxj = uj

(2.26)

Element #1

x=1 Node #1

Element #2

x = 1.5 Node #2

FIGURE 2.2 Finite element mesh with two linear elements and three nodes.

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x=2 Node #3

10

Multiscale and Multiphysics Modeling

Solving these two equations gives

a=

ui x j − u j xi x j − xi

(2.27)

u j − ui x j − xi

(2.28)

b=

Substitution of Equations 2.27 and 2.28 into Equation 2.24 yields

( x) = u

ui x j − u j xi u j − ui x + x j − xi x j − xi

(2.29)

Rearranging the expression with collecting terms for ui and uj, respectively, gives

( x) = u

xj − x x j − xi

ui +

 x −x x − xi j uj =  x j − xi  x j − xi 

x − xi   ui    x j − xi   u j   

(2.30)

The shape functions are expressed as H i ( x) =

xj − x x j − xi

(2.31)

H j ( x) =

x − xi x j − xi

(2.32)

and ũ(x) = Hi(x)ui + Hj(x)uj

(2.33)

The shape functions are called linear shape functions, and they are plotted in Figure 2.3. The functions have the following properties: Hi(xi) = 1, Hi(xj) = 0, Hj(xi) = 0, Hj(xj) = 1,

(2.34)

Hi(x) + Hj(x) = 1

(2.35)

and

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11

Finite Element Method

1 Hi(x)

Hj(x)

0 xi

xj

FIGURE 2.3 Plot of linear shape functions.

Now, the weighted residual formulation becomes 1

I=

∫

0.5

wR dx =

0

∫

1

wR dx +

0

∫

2

wR dx =

∑ ∫ wR dx

(2.36)

e = 1 Ωe

0.5

where Ω e denotes an element domain. For the present example,  d2u    dΩ = wR dΩ = w  2 − u  dx  e e

∫

Ω

∫

Ω

  dw du



 du

∫  − dx dx − wu dΩ + ∫ w dx dΓ

Ω

e

Γ

(2.37)

e

in which Γe denotes an element boundary. The first expression of Equation 2.37 is called the strong formulation, and the second expression is called the weak formulation. The Galerkin method results in the test function such that w1 =

  du du = H i ( x) and w2 = = H j ( x) du j dui

(2.38)

In addition, dH j  dH i du = ui + uj dx dx dx

(2.39)

Substitution of Equations 2.33–2.39 into Equation 2.37 gives     dw du     = + wu d Ω  dx dx   Ωe Ωe   

∫

∫

   =   Ωe   

∫

   dH i    dx dx 

dH i dx dH j

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dw1 dx dw2 dx

     du  w1 +   dx  w2 

dH j   H i  +  [H i dx   H j   

    dΩ u      u  i H j ]    uj  

(2.40)   dΩ 

12

Multiscale and Multiphysics Modeling

Let us plug the shape functions and evaluate the integral for an element whose left nodal coordinate is xi and its right nodal coordinate is xj. Then, the resultant matrix expression is  1 h  +  h 3  1 h − +  h 6

1 h + h 6 1 h + h 3

−

     

 u i  u  j

  

(2.41)

here, h = xj − xi. Let us apply this to each finite element as shown in Figure 2.2. Because each element has the same value of h = 0.5, the corresponding matrices become  13   6  23 −  12

23 12 13 6

     

 u 1   u2

  

(2.42)

23 12 13 6

     

 u 2   u3

  

(2.43)

−

for the first element and  13   6  23 −  12

−

for the second element. The two matrices must be added as shown in Equation 2.36. However, we cannot add 2 × 2 matrices on top of each other because the column vectors are different. To add them, each matrix expression is expanded into a 3 × 3 matrix as follows:  13   6  23 −  12  0

23 12 13 6 0

−

 0   0  0 

u  1  u2 u  3

    

      

u  1  u2 u  3

    

(2.44)

for the first element and 0  0   0 

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0 13 6 23 − 12

0 23 − 12 13 6

(2.45)

13

Finite Element Method

for the second element. Then, the two matrices can be summed together as  13   6  23 −  12   0 

23 12 13 3 23 − 12 −

 0   23  −  12  13  6 

u  1  u2 u  3

  =  

F  1  0 F  3

    

(2.46)

The right-hand side column vector comes from the boundary integral in Equation 2.37, which is explained further in the next section. From the boundary conditions, u(0) = u1 = 0 23 and u(1) = u3 = 1. Then, u2 = . The finite element solution is compared to the exact solu52 tion in Figure 2.1.

2.4 Axial Bar and Beam An axial bar carries a load along its longitudinal axis, while a beam carries a load through its bending. In this section, a 1-D bar and a beam are considered. Finite element formulations are presented for an axial member and then for a beam. 2.4.1 Axial Member The axial member is a 1-D structure deforming longitudinally. Let us consider an infinitesimal element as a free-body diagram, as indicated in Figure 2.4. For simplicity, let us assume a uniform cross-sectional area. The summation of force along the longitudinal direction is

∑

 ∂σ  ∂2 u F = σ + dx  A − σA = ρAdx 2  ∂x  ∂t

(2.47)

where A is the cross-sectional area, σ is the axial stress, ρ is the mass density, u is the axial displacement, and t denotes time. Simplifying, the equation becomes

σ σ + ——— dx x

σ dx

FIGURE 2.4 Free-body diagram of an infinitesimal axial member.

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14

Multiscale and Multiphysics Modeling

∂σ ∂2 u =ρ 2 ∂x ∂t

(2.48)

σ = Eε

(2.49)

Hooke’s law states

in which E is the elastic modulus, and ε denotes the normal strain. The strain-displacement relationship is ε=

∂u ∂x

(2.50)

Plugging Equation 2.50 into Equation 2.49 and the resultant expression into Equation 2.48 yields ∂  ∂u  ∂2 u  E  = ρ 2 ∂x ∂x ∂t

(2.51)

If the elastic modulus is constant along the bar, it is simplified as c2

∂2 u ∂2 u = ∂x 2 ∂t 2

(2.52)

where the speed of sound in the material is c=

E ρ

(2.53)

Let us apply the Galerkin finite element formulation to this equation. The weighted residual expression is

∫ Ω

 ∂2 u ∂2 u  w  2 − c 2 2  dΩ =  ∂t ∂x  n

=

Ω

∫ Γ

 ∂2 u 2 ∂w ∂u  ∂u 2  w 2 + c  dΩ − c w d Γ = 0 x x ∂n ∂ ∂ t ∂ e

∑∫ e =1 Ω

∫

 ∂2 u 2 ∂w ∂u  ∂u 2  dΩ − c w d Γ  w 2 + c ∂n ∂x ∂x  ∂t (2.54)

∫ Γ

For a typical element, let us use the linear shape function as follows: u( x , t) = H i ( x)ui (t) + H j ( x)u j (t) =  H i ( x) 

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 u (t)  i H j ( x)    u j (t)   

(2.55)

15

Finite Element Method

This expression shows that the shape functions are used to interpolate the solution in the spatial domain and the nodal variables vary as a function of time. As a result, their derivatives are computed as ∂H j   ui  ∂x   u j

∂u  ∂H i =  ∂x  ∂x

  

(2.56)

and ∂u =  Hi ∂t 

   H j    

     ∂t 

∂ui ∂t ∂u j

(2.57)

Substitution of Equations 2.56 and 2.57 into Equation 2.54 for each element level gives

  ∂2 u 2 ∂w ∂u  dΩ =  w 2 + c ∂ ∂ x x ∂t e

∫

Ω

   + c2   xi  xj

∫

xj

∫ xi

 H  i    H i H  j 

    H j  dx    

∂2 ui ∂t 2 ∂2 u j ∂t 2

       (2.58)

   ∂H i    ∂x ∂x 

∂H i ∂x ∂H j

∂H j   ui   dx   ∂x   u j   

Evaluation of the matrices in Equation 2.58 results in the following expression: } + [ k ]{u} = [m] {u

i  x j − xi  2 1   u c 2  1 −1   ui          + 6  1 2   u j  x j − xi  −1 1   u j     

(2.59)

in which the superimposed dot denotes a partial derivative in terms of time. Summing element-level matrices yields [M]{Ü} + [K]{U} = {F}

(2.60)

Let us discuss the column vector in Equation 2.60, which comes from the boundary integral in Equation 2.54. For the 1-D problem, the whole domain is from 0 to L, and the boundary integral is simplified as

∫ Γ

x= L

∂u ∂u ∂u(L, t) 2 ∂u(0, t) c w dΓ = c 2 w = c2w −c w ∂n ∂x ∂x x= 0 ∂x 2

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(2.61)

16

Multiscale and Multiphysics Modeling

The first term is for the last element, which can be written as

c2w

 H (L)  ∂u(L, t) i = c2   H ( L ) ∂ x j  

∂u(L, t) => c 2 ∂x

 0  ∂u(L, t)    1  ∂x

(2.62)

Therefore, the second component in Equation 2.62 goes to the last component of the column vector {F}. Likewise, the second term in Equation 2.61 is written as −c2 w

∂u(0, t) => − c 2 ∂x

 H (0)  ∂u(0, t) i = −c2   0 ( ) H ∂ x j  

 1  ∂u(0, t)    0  ∂x

(2.63)

The first term of Equation 2.63 contributes to the first term of {F}. As an example, let us consider an axial bar that is 3 m long. The bar has an elastic modulus of 12 GPa and mass density of 2000 kg/m. The bar is fixed at one end, and a constant force 2000 N is applied to the other end suddenly while the bar is at rest initially. The bar is divided into five equal-length elements. Each element has  } + [ k ]{u} =  0.2 [m]{u  0.1

0.1   0.2 

 u i    u j

  1 7  + 10   −1 

−1   1 

 u i  u  j

  

(2.64)

Summing all the matrices yields

 0.2   0.1  0  0   0  0 

0.1 0.4 0.1 0 0 0

 1   −1  + 107  0 0   0  0 

0 0.1 0.4 0.1 0 0

0 0 0.1 0.4 0.1 0

0 0 0 0.1 0.4 0.1

−1 2 −1 0 0 0

0 −1 2 −1 0 0

0 0 −1 2 −1 0

0   0  0  0   0.1  0.2  0 0 0 −1 2 −1

u   1 2 u u  3  4 u u   5 6  u 0   0  0  0   −1  1 

          u  1  u2 u  3   u4  u5   u6

     =    

 f   1   0   0     0   0     2000 

(2.65)

The boundary condition states u1 = 0, and the initial conditions are u1(0) = u2(0) = u3(0) = u4(0) = u5(0) = u6(0)

© 2016 by Taylor & Francis Group, LLC

(2.66)

17

Finite Element Method

and u 1 (0) = u 2 (0) = u 3 (0) = u 4 (0) = u 5 (0) = u 6 (0)

(2.67)

The technique to solve Equation 2.65 is discussed further in Section 2.4.3. 2.4.2 Beam A beam member supports transverse loading through bending like a bridge structure. An infinitesimal length of a beam is shown in Figure 2.5 for its free-body diagram. The force equilibrium in the transverse direction is (V + dV ) + qdx − V = ρAdx

∂2 v ∂t 2

(2.68)

Here, v is the transverse displacement, ρ is the mass density, and A is the cross-sectional area. This equation is simplified to dV ∂2 v + q = ρA 2 dx ∂t

(2.69)

M – (M + dM) – Vdx/2 – (V + dV)dx/2 = 0

(2.70)

dM +V = 0 dx

(2.71)

The moment equilibrium gives

This also simplifies to

Combining Equations 2.69 and 2.71 yields ρA

∂2 w ∂2 M + =q ∂t 2 ∂x 2

(2.72)

q

y

M V

FIGURE 2.5 Free-body diagram of an infinitesimal beam.

© 2016 by Taylor & Francis Group, LLC

M + dM

x

dx

V + dV

18

Multiscale and Multiphysics Modeling

Now, the bending moment is related to the curvature of the deformation, such as M = EI

∂2 v ∂x 2

(2.73)

where EI is the beam rigidity. Substitution of Equation 2.73 into Equation 2.72 results in

ρA

∂2 v ∂4v + EI 4 = q 2 ∂t ∂x

(2.74)

if EI is constant. This is the governing equation for classical beam theory. Applying the weighted residual formulation to the differential equation gives L

I=

∫ 0

  ∂2 v ∂4 v w  ρA 2 + EI 4 − q  dx = 0 ∂t ∂x  

(2.75)

This is the strong formulation. To develop the weak formulation, we take the integrations by parts twice for the fourth-order term as follows: L

I=

∫ 0

∂2 v wρA 2 dx + ∂t

L

∫ 0

L

L

L

∂2 w ∂2 v ∂3 v  ∂w ∂2 v  EI dx − wq dx + wEI EI 2  = 0  − ∂x 2 ∂x 2 ∂x 3  0 ∂x ∂x  0

∫

(2.76)

0

Discretizing the domain gives n

I=

∑∫ i=1 Ω

∂2 v wρA 2 dx + ∂t e

n

∑∫ i=1 Ω

∂2 w ∂2 v EI 2 dx + ∂x 2 ∂x e

n

∑∫ i=1 Ω

L

L ∂w  wq dx − wV  − M =0 0 ∂x  0 e

(2.77)

in which n is the number of elements. For the beam element, we need shape functions. However, the shape functions given in Equations 2.31 and 2.32 cannot be used for the beam bending. The shape functions in Equations 2.31 and 2.32 provide a continuous solution from one element to the next element, but their derivative is not continuous between two neighboring elements. As a result, these shape functions are called C0-type shape functions, that is, continuity up to the zeroth order of the shape functions. For beam-bending solutions, not only deflections but also slopes must be continuous from one element to the next element. Therefore, the beam element has both deflection and slope as nodal degrees of freedom, as illustrated in Figure 2.6. In addition, for the classical beam-bending theory, the deflection and slope are related to each other as follows: θ=

© 2016 by Taylor & Francis Group, LLC

∂v ∂x

(2.78)

19

Finite Element Method

v2

v1

θ2

I

FIGURE 2.6 Beam-bending element.

Therefore, the shape function should be the C1 type, that is, the functions and first-order derivatives must be continuous. To develop such shape functions, let us start with a cubic polynomial function such as the following: v = a0 + a1x + a1x2 + a1x3

(2.79)

The reason we use the cubic function is that it has four coefficients, and the number of the nodal variables for the beam element shown in Figure 2.6 is also four, so the cubic function can uniquely interpolate the solution within the element. Now, let us replace the coefficients ai by the nodal variables. To do that, we evaluate the cubic function and its derivative at the nodal points as follows: v  1  θ1   v2  θ2 

   =   

1  0 1  0

0 1 l 1

0 0 l2 2l

0 0 l3 3l 2

     

a  0  a1   a2  a3 

      

(2.80)

or, in a short notation, {v} = [C]{a}

(2.81)

Solving for {a} and substituting it into Equation 2.79 yields v = {X}T [C]–1 {v}

(2.82)

{X}T = {1 x x2 x3}

(2.83)

{H}T = {X}T [C]–1

(2.84)

where

The shape functions are

Explicit expressions for the shape functions are H 1 ( x) = 1 −

© 2016 by Taylor & Francis Group, LLC

3x 2 2 x 3 + 3 l2 l

(2.85)

20

Multiscale and Multiphysics Modeling

2x2 x3 + 2 l l

H 2 ( x) = x − H 3 ( x) =

(2.86)

3x 2 2 x 3 − 3 l2 l

(2.87)

x2 x3 + 2 l l

(2.88)

H 4 ( x) = −

Substitution of the shape functions into Equation 2.77 gives the following matrix expression, where the first term in the equation yields the elemental mass matrix:

l

[m]{ v} =

∫ 0

H  1 H ρA  2  H3  H4 

    H1   

{

 156  ρAl  22l = 420  54   −13l

}

H2

22l 4l 2 13l −3l 2

 v  1  θ 1 H3 H 4 dx   v 2  θ   2   −13l   v1    −3l 2   θ1    −22l   v2     4l 2   θ  2

54 13l 156 −22l

      

(2.89)

The second term in the equation yields the elemental stiffness matrix:

l

[ k ]{ v} =

∫ 0

H ,  1 xx H , EI  2 xx  H 3 , xx  H 4 , xx 

 12  EI 6l = 3  l  −12   6l

    H 1 , xx   

{

6l 4l 2 −6l 2l 2

H 2 , xx

−12 −6l 12 −6l

6l   2l 2  −6l   4l 2 

H 3 , xx

v  1  θ1   v2  θ2 

      

H 4 , xx

}

v  1 θ dx  1  v2  θ2 

       (2.90)

The third term yields the elemental force vector: l

{f} =

∫ 0

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H  1  H2   H3  H4 

    q( x) dx   

(2.91)

21

Finite Element Method

If the applied load intensity function q(x) is constant within the element and its value is qo, then the elemental force vector becomes  ql  o  2  qo l 2   { f } =  12  qo l  2  2  − qo l  12

           

(2.92)

If the load intensity function is linear over the element, such as q(x) = cx, then, the elemental load vector becomes  3cl   20  cl 2   { f } =  30  7 cl  20  2  − cl  20

           

(2.93)

Other loading cases can be found in Reference 1. Assembly of all the elemental matrices and vectors yields the following matrix equation:  } + [ K ]{V } = { F } [ M] {V

(2.94)

As an example, let us consider a cantilever beam with a tip load P as shown in Figure 2.6. We will use one element to represent the beam. Then, the system mass and stiffness matrices are equal to the elemental mass and stiffness matrices as given in Equations 2.89 and 2.90, respectively. The force vector becomes {F}T = {f1 m1 –P 0}

(2.95)

where f1 and m1 are the unknown reaction force and moment at the clamped node 1 of Figure 2.7. The boundary condition states v1 = 0 and θ1 = 0. The initial conditions state zero  0)} = {0}. nodal variables {V(0)} = {0} and zero nodal velocities {V( 2.4.3 Solution Techniques Let us consider the equations of motion as follows:  } + [C]{U } + [ K ]{U } = { F } [ M] {U

© 2016 by Taylor & Francis Group, LLC

(2.96)

22

Multiscale and Multiphysics Modeling

v2

v1 θ1

θ2

I P FIGURE 2.7 Cantilever beam with a tip load.

We discuss some techniques to solve the equations of motion. One way to solve the secondorder differential equation is to break it into two first-order differential equations. To this end, let us say { V} = {U }

(2.97)

[ M] {V } + [C]{V } + [ K ]{U } = { F }

(2.98)

Then, Equation 2.96 can be written as

Putting the two equations together gives  [ M]   [0]

[0]   [ I ] 

 {V }    +  {U } 

 [C]   −[ I ]

[K ]   [0] 

 {V }   =  {U } 

 { F }     {0} 

(2.99)

For convenience, we rewrite it in a simpler form at time t: t [ A] {X } + [B]{X }t = { P}t

(2.100)

Equation 2.100 can be solved using the finite difference method, such as using backward, forward, or Crank-Nicholson techniques [1]. If the forward difference technique is used, the temporal derivative is expressed as {X }t =

{X }t+∆t − {X }t ∆t

(2.101)

in which Δt is the time step size. Substitution of Equation 2.101 into Equation 2.100 and rearrangement of the resulting equation yields [A]{X}t+∆t = ∆t{P}t – ∆t[B]{X}t + [A]{X}t

(2.102)

The equation is solved as time progresses. For example, we set t = 0. Because {X}0 is given from the initial condition, {X}Δt is solved after applying the boundary condition. Then, we obtain {X}2Δt from the previous solution {X}Δt. This process continues until the time reaches

© 2016 by Taylor & Francis Group, LLC

23

Finite Element Method

the termination time set by the user. If the matrix [A] is a diagonal matrix, Equation 2.102 does not require inverting the matrix. It is called the explicit solver, and it is conditionally stable. In other words, if the time step size Δt is not so small, the solution diverges as time increases. To make [A] a diagonal matrix, the matrix [M] should be a diagonal matrix. The matrix [M] in Equation 2.59 is transformed into a diagonal matrix, such as [m] =

x j − xi  1  2 0

0  1

(2.103)

Likewise, the diagonal mass matrix for the beam element is expressed as 1  0 ρAl  [m] =  2 0  0 

0 l2 78 0

1

0

0

0   0    0  l2   78 

0 0

(2.104)

These are called diagonal mass matrices. On the other hand, the backward difference technique is an implicit solver and requires inversion of a matrix. The technique is unconditionally stable. That is, the solution is stable regardless of the time step size. The backward difference technique uses {X }t =

{X }t − {X }t−∆t ∆t

(2.105)

Plugging Equation 2.105 into Equation 2.100 and rearranging yields ([A] + ∆t[B]{X}t = ∆t{P}t + [A]{X}t–∆t)

(2.106)

This is solved in the same way as for the forward difference technique. The other way is to solve the second-order differential equation as it is. We use the central difference technique as follows: ∆t

∆t

{U }t+ 2 = {U }t− 2

 }t + ∆t {U

{U }t+∆t = {U }t + ∆t {U }

t+

∆t 2

(2.107)

(2.108)

To solve the problem, we first compute the initial acceleration from the initial displacement and velocity, such as the following:  } = { F } − [C]{U } − [ K ]{U }0 [ M] {U 0

© 2016 by Taylor & Francis Group, LLC

0

(2.109)

24

Multiscale and Multiphysics Modeling

Then, the velocity is computed from Equation 2.107 and the displacement is computed from Equation 2.108. This process repeats itself. One thing to note is that the velocity is ∆t computed at time t + . To compute the velocity at t, we use the average; for instance, 2

{U }t = {U }

t+

∆t 2

+ {U } 2

t−

∆t 2

(2.110)

The central difference technique is also an explicit solver and a conditionally stable scheme. The time step size should be smaller than the critical step size, which can be computed as ∆tc =

∆x cs

(2.111)

where Δx is the element length and cs is the wave speed in the material. An unconditionally stable technique is the Newmark method [1]. This technique can use any time step size without worrying about the instability of the solution. Of course, the solution accuracy is proportional to the time step size regardless of the stability.

2.5 Truss and Frame Truss structures consist of axial members, which are two-force members; the structures carry loads along the structural members’ axes. Every member of a truss structure is in either tension or compression. On the contrary, frame structures can carry loads transverse to structural members’ orientations. In this section, the finite element formulation is presented for both structures. 2.5.1 Truss A truss structure has axial members connected by pin joints as shown in Figure 2.8. Let us us consider a truss element in two dimensions (2-D) as shown in Figure 2.9. It shows two different coordinate systems; one is the global coordinate system, and the other is the local coordinate system, which rotates along with the truss member. All the variables associated with the former are denoted using capital letters; those for the local axis are symbolized using lowercase letters. Now, let us examine the relationship between the two coordinate systems. The displacements in those coordinates are related to each other as u  1  v1   u2  v2 

   =   

 cos β   sin β  0   0 

− sin β

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0

cos β

0

0

cos β

0

sin β

  0  − sin β   cos β  0

U  1  V1   U2  V2 

    or {d} = {T }{D}   

(2.112)

25

Finite Element Method

P

FIGURE 2.8 Truss structure. y, v x, u Y, V

β X, U FIGURE 2.9 A 2-D truss element.

where the angle β is the measure of the rotation of the local system from the global system. It is considered positive along the counter-clockwise direction. The stiffness matrix of the truss member is expressed as  1  AE  0 [ klocal ]{d} = l  −1  0 

0 0 0 0

−1 0 1 0

0  0 0 0 

u  1  v1   u2  v2 

      

(2.113)

Because the strain energy should be the same no matter how it is expressed in terms of any coordinate system, we can write 1 T 1 {d}  klocal  {d} = {D}T  k global  {D} 2 2

© 2016 by Taylor & Francis Group, LLC

(2.114)

26

Multiscale and Multiphysics Modeling

Substitution of Equation 2.113 into Equation 2.114 results in  c2  AE  cs T [ k global ] = [T ] [ klocal ][T ] = l  −c2   − cs

−c2 − cs c2 cs

cs s2 − cs − s2

− cs   − s2  cs   s2 

(2.115)

Here, c = cos β and s = sin β. Every truss element must be transformed as shown in Equation 2.115 and assembled together to make up the system stiffness matrix. On the other hand, the diagonal mass matrix is  c2  ρAl  0 [m] = 2  0   0

0 s2 0 0

0 0 c2 0

0 0 0 s2

     

(2.116)

A similar derivation can be conducted for a three-dimensional (3-D) truss, whose details are given in Reference 1. 2.5.2 Frame Let us us consider a 2-D frame element as shown in Figure 2.10. In terms of the local coordinate, the element stiffness matrix is a combination of those from the 1-D axial and beam members as expressed as follows:  Al 2   0 E  0 [ klocal ]{d} = 3  l  − Al 2  0   0

0 12 I 6 Il 0 −12 I 6 Il

0 6 Il 4 Il 2 0 − 6Il 2 Il 2

− Al 2 0 0 Al 2 0 0

0 −12 I − 6 Il 0 12 I − 6 Il

0   6 Il  2 Il 2   0  − 6 Il   4 Il 2 

u  1  v1 θ  1   u2  v2   θ2

         

(2.117)

The transformation matrix between the local and global axes is u  1  v1 θ  1   u2  v2   θ2

     =    

 c   −s  0  0   0  0 

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s c 0 0 0 0

0 0 1 0 0 0

0 0 0 c −s 0

0 0 0 s c 0

0  0 0 0  0 1 

U  1  V1 θ  1   U2  V2   θ2

      or {d} = [T ]{D}    

(2.118)

27

Finite Element Method

y, v x, u Y, V θ θ β X, U FIGURE 2.10 A 2-D frame element.

After the coordinate transformation, the element matrix in the global coordinate system becomes a  11  a21  a [ k global ] =  31  a41   a51 a  61

a12

a13

a14

a15

a22

a23

a24

a25

a32

a33

a34

a35

a42

a43

a44

a45

a52

a53

a 54

a55

a62

a63

a64

a65

a16   a16   a36  a46   a56  a66 

(2.119)

where  AE  2  12EI  2 a11 = − a14 = a44 =  c + 3  s  l   l 

(2.120)

 AE   12EI  a12 = − a15 = − a24 = a45 =  cs −  3  cs  l   l 

(2.121)

 6EI  a13 = a16 = − a34 = − a46 = −  2  s  l 

(2.122)

 AE  2  12EI  2 a22 = − a25 = a55 =  s + 3  c  l   l 

(2.123)

 6EI  a23 = a26 = − a35 = − a56 =  2  c  l 

(2.124)

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28

Multiscale and Multiphysics Modeling

a33 = a66 =

a36 =

4EI l

(2.125)

2EI l

(2.126)

aij = aji

(2.127)

The diagonal mass matrix for the 2-D frame is given as 1  0  0 ρAl  [m] = 2 0 0   0 

0 0

0 0

0

0

0 0

0 0 l2 78 0 0

1 0

0 1

0

0

0

0

0 1 0

0   0   0   0  0   l2  78 

(2.128)

The 3-D frame element matrices can be developed similarly (as given in [1]).

2.6 Solid Element In this section on the solid element, the theory of elasticity is briefly presented, and the finite element formulation is followed. Even though the 2-D theory and its formulation are given here, it is straightforward to extend them to 3-D problems [1]. 2.6.1 Theory of Elasticity Let us consider a 2-D infinitesimal element for a free-body diagram as shown in Figure 2.11. There are normal stresses, denoted by σ, and shear stresses, designated as τ. The equations of equilibrium can be developed as shown next. The summation of force in the x axis is 

∂σ x  dx  dy − σ x dy  ∂x  ∂τ yx  ∂2 u +  τ yx + dy  dx − τ yx dy + f x dxdy = ρdxdy 2 ∂y ∂t  

∑ F =  σ x

x

+

© 2016 by Taylor & Francis Group, LLC

(2.129)

29

Finite Element Method

∂σy

σy +

∂y

dy

τyx + τxy

∂τyx

dy

∂y

dy

σx

σx +

dx

σx + τyx

∂σx ∂x

∂σx ∂x

dx

dx

σy FIGURE 2.11 Free-body diagram for 2-D infinitesimal element.

in which fx is the body force per unit volume along the x axis and u is the displacement in the x axis. Simplifying the expression yields ∂σ x ∂τ yx ∂2 u + + fx = ρ 2 ∂x ∂y ∂t

(2.130)

Similarly, the force equilibrium in the y axis gives ∂τ xy ∂x

+

∂σ y ∂y

+ fy = ρ

∂2 v ∂t 2

(2.131)

The moment equilibrium about the z axis, which is in the direction perpendicular to the xy plane, states the stress tensor is symmetric, that is, τxy = τyx

(2.132)

The next set of equations presents the relationship between stresses and strains. These are called constitutive equations. For an isotropic material, their relationship is written as  1   ε   E x    ν  εy  =  − γ   E  xy    0 

© 2016 by Taylor & Francis Group, LLC

ν E 1 E

−

0

 0    0   1  G 

σ  x  σy τ  xy

   or {ε} = [C]{σ }  

(2.133)

30

Multiscale and Multiphysics Modeling

The inverse relationship is written as σ  x  σy τ  xy

  =  

 E  2  1− v  νE  2  1− v  0

νE 1 − v2 E 1 − v2 0

 0   0  G 

 ε  x  εy γ  xy

  σ } = [D]{ε}  or {σ  

(2.134)

The last set of equations is the kinematic equations, which relate strains to displacements as follows:  ∂u   ε   ∂x  x   ∂v  ε y  =  ∂y γ    xy   ∂u ∂v +   ∂y ∂x

        

(2.135)

Finally, there are two types of boundary conditions. One type is the given displacement, and the other is the prescribed traction, which can be expressed as p  x   py

  = 

σ n +τ n xy y  x x   τ xy nx + σ y ny

   

(2.136)

where px and py are the given traction, and nx and ny are the outward normal unit vector components at the given boundary. 2.6.2 Finite Element Formulation Let us apply the finite element formulation for the governing equations and the boundary conditions developed previously. To do that, we begin with the equations of motion and apply the weighted residual formulation to those equations:   2  w1  ρ ∂ u − 2   ∂t   ∂2 v Ω   w2  ρ ∂t 2 − 

∫

  ∂σ x ∂τ xy − − fx   ∂x ∂y    dΩ = 0   ∂τ xy ∂σ y − − fy   ∂y ∂x   

(2.137)

where w1 and w2 are two test functions. Now, we apply integrations by parts to this expression and substitute Equation 2.136 into the resultant expression to yield the following equations:

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31

Finite Element Method

∫ Ω

  ρw1    ρw  1  =

∫ Ω

∂2 u ∂t 2 ∂2 v ∂t 2

    dΩ +   

w f  1 x   w2 f y

∫ Ω

   dΩ + 

 ∂w 1   ∂x   0 

∫ Γ

∂w1   σ   x  ∂y   σ y  dΩ  ∂w2    τ  ∂y   xy  

0 ∂w2 ∂y

w p  1 x   w2 py

(2.138)

   dΓ 

The stress terms are replaced by the strain terms using Equation 2.134, and then the strain terms are replaced by the displacement terms using Equation 2.135. The resultant mathematical expression becomes

∫ Ω

  ρw1    ρw  1 

 ∂w 1  x ∂    0 

∂ u ∂t 2 ∂2 v ∂t 2

    dΩ +   

∫

∫

w f  1 x  w f  2 y

   dΩ + 

2

dΩ =

Ω

Ω

∫ Γ

0

∂w1 ∂y

∂w2 ∂y

∂w2 ∂y

w p  1 x  w p  2 y

 ∂u    ∂x   ∂v   [D]  ∂y    ∂u ∂v  +   ∂y ∂x

        

(2.139)

   dΓ 

Now, a problem domain is discretized into a number of finite element domains as sketched in Figure 2.12, and each finite element has a triangular shape with nodes at the vertices as seen in Figure 2.13. We will develop shape functions for the triangular element. The element has three nodes (nodes i, j, k), and each node has a nodal coordinate value as shown in Figure 2.13. Let node i have nodal variables (ui, vi), and the other two nodes have similar nodal displacements. Then, to interpolate the displacements inside the element in a unique way, the lowest order of polynomial function is a linear function in terms of x and y, such as the following:

FIGURE 2.12 Finite element mesh.

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32

Multiscale and Multiphysics Modeling

Node j (xj, yj) Node k (xk, yk)

Node i (xi, yi) FIGURE 2.13 Triangular element.

{1

u = a0 + a1 x + a2 y =

x

y

}

a  0  a1 a  2

    

(2.140)

As we evaluate the displacement at the three nodes, we obtain u  i  uj   uk

  =  

1  1   1

yi   yj   xk 

xi xj xk

a  0  a1 a  2

    

(2.141)

Inverting the matrix and substituting it into Equation 2.140 results in

u = {H i

Hj

u  i H k }  uj   uk

    

(2.142)

where

(

) (

) (

)

(

) (

) (

)

(2.144)

(

) (

) (

)

(2.145)

H1(x, y) =

1  x j y k − xk y j + y j − y k x + xk − x j y   2A 

H 2 (x, y) =

1  xk y i − xi y kj + y k − y i x + xi − xk y   2A 

H 3 (x, y) =

1  xi y j − x j y i + y i − y j x + x j − xi y   2A 

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(2.143)

33

Finite Element Method

and 1  1 A = det  1 2   1

yi   yj   xk 

xi xj xk

(2.146)

Because the two displacements u and v are independent, the same shape functions can be used for both displacements. That is,

v = {H i

v  i Hk }  vj   vk

Hj

    

(2.147)

Substitution of the displacements into the kinematic equation gives {ε} = [B]{d}

(2.148)

where {ε}T = {ε x

    [B] =     

εy

(2.149)

γ xy }

∂H 1 ∂x

0

∂H 2 ∂x

0

∂H 3 ∂x

0

∂H 1 ∂y

0

∂H 2 ∂y

0

∂H 1 ∂y

∂H 1 ∂x

∂H 2 ∂y

∂H 2 ∂x

∂H 3 ∂y

{d}T = {u1

v1

u2

v2

u3

 0   ∂H 3   ∂y  ∂H 3   ∂x  

v3 }

(2.150)

(2.151)

In addition, the same shape functions are used for the test functions w1 and w2. Then, we obtain the following expression: w  1  0

T

H 0   =  1 w2   0

© 2016 by Taylor & Francis Group, LLC

0

H2

0

H3

H1

0

H2

0

0   = [N ] H 3 

(2.152)

34

Multiscale and Multiphysics Modeling

Elemental mass matrix can then be computed as follows:          ρw1  [m] =   ρw Ωe  1  2  0 ρA  1 = 12  0  1 0 

∫

∂2 u ∂t 2 ∂2 v ∂t 2

∂2 u ∂t 2 ∂2 v ∂t 2

    = [ N ]{d}   

  w  1  dΩ = ρ   0  Ωe  

0   w2 

∫

0 2 0 1 0 1

1 0 2 0 1 0

0 1 0 2 0 1

      

(2.153)

∂2 u ∂t 2 ∂2 v ∂t 2

   T  dΩ = ρ[ N ] [ N ] dΩ  Ωe  

∫

0  1 0 1  0 2 

1 0 1 0 2 0

(2.154)

The diagonal mass matrix is [m] =

ρA [I ] 3

(2.155)

where [I] is a 6 × 6 identity matrix. The elemental stiffness matrix becomes

[k ] =

∫

Ω

e

 ∂w 1  ∂ x    0 

0

∂w1 ∂y

∂w2 ∂y

∂w2 ∂y

 ∂u     ∂x     ∂v   T B] dΩ  [D]  ∂y  dΩ = [B] [D][B   e  Ω  ∂u ∂v   +    ∂y ∂x 

∫

(2.156)

Evaluation of matrix [B] using the shape functions gives  (y − y ) j k 1   0 [B] = 2A   ( xk − x j ) 

0

(y k − yi )

0

(yi − y j )

( xk − x j )

0

( xi − x k )

0

(y j − y k )

( xi − x k )

(y k − yi )

( x j − xi )

© 2016 by Taylor & Francis Group, LLC

  ( x j − xi )  (2.157)  (yi − y j )   0

35

Finite Element Method

Because matrix [B] is constant (i.e., not a function of x and y), the element stiffness matrix can be evaluated easily over the element domain, and the resultant matrix becomes [k] = [B]T[D][B]A

(2.158)

Finally, the force column vector consists of two parts: body force and boundary traction. The column vector due to the body force is computed as w f  1 x   w2 f y Ωe    = [ N ]T   Ωe

{ fb } =

∫

∫

   dΩ =  Ωe f x   dΩ fy  

∫

w  1  0

0   f x     dΩ w2   f y    (2.159)

and the traction force vector is w p  w  1 x 1   dΓ =  w P 0   2 y  Γe  Γe  p   x = [ N ]T   d Γ  py  Γe

{ ft } =

∫

∫

0   px  w2   py 

   dΓ  (2.160)

∫

One of the main differences between the two force vectors is that the body force vector is an integral over the domain while the traction force vector is an integral over the boundary. As a result, the shape functions over the domain are simplified when they are evaluated along the boundary. In other words, 2-D shape functions for the 2-D domain become 1-D shape functions along the boundary. Let us say the element boundary ij is located along the boundary of the problem domain. Then, the shape function Hk disappears along the element boundary, and shape functions Hi and Hj become 1-D functions that have the axis along the boundary. For example, if s denotes the axis along the boundary, the shape functions used for the boundary integral are H i ( s) =

sj − s s j − si

(2.161)

H j ( s) =

s − si s j − si

(2.162)

Then, the traction force vector becomes

sj

{ ft } =

∫ si

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H  i  0   Hj  0 

0   Hi   0  H j  

p  x  p  y

   ds 

(2.163)

36

Multiscale and Multiphysics Modeling

If the traction is constant over an element boundary that is located on the problem domain boundary, the corresponding column vector becomes p  x s j − si  py { ft } =  2  px p  y

      

(2.164)

Assembly of element matrices and vectors results in the system matrix equation as follows:  } + [ K ]{D} = { F } [ M]{D

(2.165)

in which [ M] =

∑[m]

(2.166)

[K ] =

∑[k ]

(2.167)

{F} =

∑{ f } + ∑{ f } b

t

(2.168)

The set of equations can be solved using one of the techniques discussed previously using the finite difference method for the temporal derivative.

2.7 Isoparametric Formulation The linear triangular shape functions discussed previously are simple to use and do not require integration over the domain because the matrix [B] is a constant matrix. However, to obtain an accurate approximation, we have to use a large number of elements. Therefore, other types of finite elements and the corresponding shape functions were developed. Among them, isoparametric shape functions have been the most popular. For the isoparametric formulation, we consider mathematical mapping between two different coordinate systems. One coordinate system is called the natural coordinate domain, and the other coordinate system is called the physical coordinate domain. The problem domain is defined in terms of the physical coordinate system. As a result, the finite element mesh is made in the physical coordinate domain, and every finite element may have different shapes. However, shape functions are defined in terms of the natural coordinate system. As long as each finite element has the same kind of polygon, mostly the quadrilateral shape, with the same number of nodes per element, the same shape functions defined

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37

Finite Element Method

in the natural coordinate domain can be used for all those elements with proper mapping. The 2-D formulation is presented in the following material. The two coordinate systems are shown in Figure 2.14 along with their quadrilateral elements. In the natural coordinate denoted by (ξ, η), the square shape of the element remains the same all the time. On the other hand, the quadrilateral element in the physical coordinate system denoted by (x, y) can be in any location and of any shape depending on what element is chosen in the mesh of the physical problem. Each node is numbered sequentially in the counterclockwise direction in both coordinate systems. Shape functions are defined in terms of the natural coordinate system. They can be determined using the 1-D shape functions in both axes. In other words, the shape functions along the ξ axis are φ1 (ξ ) =

1− ξ 2

(2.169)

η 4

1.0

3

1.0

ξ

1.0

(a) 1

−1.0

2

y 3 (x3, y3) 2 (x2, y2)

4 (x4, y4)

1 (x1, y1) x (b) FIGURE 2.14 Isoparametric formulation: (a) natural coordinate and (b) physical coordinate.

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38

Multiscale and Multiphysics Modeling

φ 2 (ξ ) =

1+ ξ 2

(2.170)

ψ 1 (ξ ) =

1− η 2

(2.171)

ψ 2 (ξ ) =

1+ η 2

(2.172)

Similarly, along the η axis are

Then, the shape functions for the element in the natural coordinate system are H 1 (ξ , η) = φ1 (ξ)ψ 1 ( η) =

1 (1 − ξ)(1 − η) 4

(2.173)

H 2 (ξ , η) = φ2 (ξ)ψ 1 ( η) =

1 (1 + ξ)(1 − η) 4

(2.174)

H 3 (ξ , η) = φ2 (ξ)ψ 2 ( η) =

1 (1 + ξ)(1 + η) 4

(2.175)

H 4 (ξ , η) = φ1 (ξ)ψ 2 ( η) =

1 (1 − ξ)(1 + η) 4

(2.176)

To interpolate the physical variable in the physical domain, we use shape functions such as 4

u( x , y ) =

∑ H (ξ, η)u i

i

(2.177)

i=1

where ui is the nodal variable. The differential equation to be solved in terms of the physical coordinate system requires derivatives in terms of x and y. However, the shape functions are expressed in terms of ξ and η. As a result, we need to relate the two coordinate systems using mathematical mapping. To do that, we use the following geometrical mapping from the natural to physical coordinate systems: 4

x=

∑ H (ξ, η)x i

i

(2.178)

i=1

4

y=

∑ H (ξ, η)y i

i=1

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i

(2.179)

39

Finite Element Method

These expressions state that the ith node in the natural coordinate is mapped to the ith node in the physical coordinate shown in Figure 2.14. Likewise, the inside and boundary of the square element in the natural coordinate are mapped to the inside and boundary of the quadrilateral in the physical domain. Let us compute the Jacobian matrix associated with the mapping. It is defined as   [ J ] =   

∂x ∂ξ ∂x ∂η

∂y   ∂ξ  ∂y   ∂η 

(2.180)

The components of the Jacobian matrix are computed as follows: J11 =

J12 =

J 21 =

∂x = ∂ξ ∂y = ∂ξ ∂x = ∂η

∂y J 22 = = ∂η

4

∑ ∂H ∂(ξξ, η) x i

i

(2.181)

i

(2.182)

i

(2.183)

i=1 4

∑ ∂H ∂(ξξ, η) y i

i=1

4

∑ ∂H ∂(ηξ, η) x i

i=1

4

∑ ∂H ∂(ηξ, η) y i

(2.184)

i

i=1

Now, let us discuss how to take derivatives of the shape functions Hi(ξ, η) in terms of x and y. To this end, we consider functions φ(x, y) and (x, y) are functions of (ξ, η). Taking derivatives of φ(x, y) with respect to (ξ, η) results in the following expressions using the chain rule:       

∂ϕ   ∂ξ  = ∂ϕ  ∂η 

     

∂x ∂ξ ∂x ∂η

∂y   ∂ϕ  ∂ξ   ∂x  ∂y   ∂ϕ  ∂η   ∂y

      

(2.185)

Thus, premultiplying the inverse of the Jacobian matrix to the equation yields  ∂ϕ   ∂x   ∂ϕ  ∂y 

© 2016 by Taylor & Francis Group, LLC

   =   

     

∂x ∂ξ ∂x ∂η

∂y   ∂ξ  ∂y   ∂η 

−1

      

∂ϕ   ∂ξ   ∂ϕ  ∂η 

(2.186)

40

Multiscale and Multiphysics Modeling

These expressions are used for the shape functions. In other words, we have  ∂H i   ∂x   ∂H i  ∂y 

   =   

     

∂y   ∂ξ  ∂y   ∂η 

∂x ∂ξ ∂x ∂η

−1

      

∂H i   ∂ξ   ∂H i  ∂η 

(2.187)

As an example, we apply the isoparametric formulation to the 2-D solid elements. Then, we need to compute the matrix [B] for a quadrilateral shape of finite element in the physical domain. The matrix is expressed as     [B] =     

∂H 1 ∂x

0

∂H 2 ∂x

0

∂H 3 ∂x

0

∂H 4 ∂x

0

∂H 1 ∂y

0

∂H 2 ∂y

0

∂H 3 ∂y

0

∂H 1 ∂y

∂H 1 ∂x

∂H 2 ∂y

∂H 2 ∂x

∂H 3 ∂y

∂H 3 ∂x

∂H 4 ∂y

 0   ∂H 4   ∂y  ∂H 4   ∂x  

(2.188)

and it is a function of (ξ, η). The element stiffness matrix is written as 1 1

[k ] =

∫ [B] [D][B] dΩ = ∫ ∫ [B] [D][B] J dξ dη T

Ω

T

(2.189)

−1 −1

2

where |J| is the determinant of the Jacobian matrix and is equal to J11J22 − J12 J21. Evaluation of the double integrals in Equation 2.189 is undertaken numerically. The Gauss-Legendre quadrature rule is used for the numerical integration. Any numerical integration can be expressed in terms of sampling points and weight factors. For example, a function f(ξ) can be integrated numerically, and its result is written as 1

∫

n

f (ξ ) d ξ =

∑ W f (ξ ) i

(2.190)

i

i=1

−1

in which Wi is the weight factor for the sampling point ξi. In addition, n is the number of sampling points for numerical integration. Some sampling points and their corresponding weight factors are listed in Table 2.1 for the Gauss-Legendre quadrature, which is the most commonly used numerical integration technique for isoparametric finite elements. To extend this to double integrals, we apply the procedure one by one as follows: 1 1

∫∫

1

f (ξ , η) d ξ d η =

n

∫ ∑ W f (ξ , η)dη i

i

−1 i= 1

−1 −1

 Wj   j=1 m

=

n

∑ ∑

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i=1

 Wi f (ξ i , η j ) = 

m

(2.191)

n

∑∑ j=1 i=1

WjWi f (ξ i , η j )

41

Finite Element Method

TABLE 2.1 Gauss-Legendre Quadrature Sampling Points and Weight Factors No. of Sampling Points

Sampling Point Coordinate Value

n=1 n=2

Weight Factor

0.0 1 − 3

2.0 1.0

1

1.0

3 n=3

−

15 5 0

5 9 8 9

15 5

5 9

Therefore, the same sampling points and weight factors as in Table 2.1 can be used in both integrals. Because the shape functions have balanced expressions in terms of (ξ, η), the number of sampling points is equal in both directions. The Gauss-Legendre quadrature states that any polynomial function of order 2n – 1 can be numerically integrated exactly. In other words, the following integral 1

∫ (a + a x + a x 0

1

2

2

)

+  + a2 n−1x 2 n−1 dx

(2.192)

−1

requires n number of sampling points for exact integration.

2.8 Plate and Shell Structures Plates and shells are common structural components because they are efficient for carrying loading in the transverse direction. The plate is the 2-D extension of the beam. While the plate has a flat surface, the shell has a curved surface. As a result, the beam is a subset of the plate, and the plate is a subset of the shell. This section presents a shell formulation that can be easily degenerated into a plate or a beam element. The shell element developed here is like a 3-D solid element. It has four nodes at the bottom and top surfaces, respectively, leading to eight nodes total. In addition, the shell element has three displacements as nodal degrees of freedom but no rotational degrees of freedom. In that aspect, the shell element looks the same as the 3-D solid element. However, the constitutive equation is different between the two elements. Furthermore,

© 2016 by Taylor & Francis Group, LLC

42

Multiscale and Multiphysics Modeling

the shell element can model a geometry with a large aspect ratio (i.e., the ratio of the side length to the thickness). The advantage of the present shell element is that the shell element can be easily stacked on one another without special consideration at the interface between two neighboring shell elements. The shell element can be connected to the 3-D solid elements directly as it is, and it defines clear interfaces when in contact with other media, such as fluid. Let us consider three displacements in terms of the local coordinate system of the shell element, which is attached to the element. In-plane displacements u and v are parallel to the shell surface, and the transverse displacement w is normal to the shell surface as seen in Figure 2.15. The displacements are u = u(x, y, z)

(2.193)

v = v(x, y, z)

(2.194)

w = w(x, y, z)

(2.195)

where (x, y, z) is the local coordinate system. These displacements are interpolated using the following shape functions: n1

u( x , y , z) =

v( x , y , z) =

n2

∑ ∑ N (ξ, η)H (ζ)u i

i=1

j=1

n1

n2

i=1

j=1

j

∑ ∑ N (ξ, η)H (ζ)v i

y, v

j

z, w

Y, V Global coordinate X, U FIGURE 2.15 Shell element in terms of its local and global coordinate systems.

© 2016 by Taylor & Francis Group, LLC

(2.196)

ij

(2.197)

Local coordinate

x, u Z, W

ij

43

Finite Element Method

w( x , y , z) =

n1

n2

i=1

j=1

∑ ∑ N (ξ, η)H (ζ)w i

j

ij

(2.198)

in which Ni(ξ, η) is the 2-D shape functions for the in-plane interpolation, and Hj(ς) is the 1-D shape function for the transverse interpolation. Both shape functions are expressed in terms of the natural coordinate system (ξ, η, ς) because the isoparametric formulation is used. Because we use four nodes on the in-plane and two nodes along the thickness direction, n1 = 4 and n2 = 2. Of course, different numbers of nodes may be used for the in-plane and thickness directions, respectively. However, the total eight nodes are selected because they resemble the popular eight-node brick element. For the shell formulation, we consider in-plane strains, transverse shear strains, and the transverse normal strain independently. For the classical formulation, the transverse shear and normal strains are neglected. However, the present formulation includes those as a higher-order formulation. The in-plane strains are expressed as follows:    ε    x   {ε inpl } =  ε y  =  γ    xy    

∂ ∂x

0

0

∂ ∂y

∂ ∂y

∂ ∂x

 0   0   0 

u   v w  

(2.199)

The transverse shear strains are also expressed in the following manner:

 γ xz {ε trans } =  γ  yz

 = 

 ∂   ∂z   0 

0 ∂ ∂z

∂ ∂x ∂ ∂y

     

u   v w  

(2.200)

Finally, the transverse normal strain is given as εz =

∂w ∂z

(2.201)

The constitutive equations are

© 2016 by Taylor & Francis Group, LLC

{σinpl} = [Dinpl]{εinpl}

(2.201)

{σtrans} = [Dtrans]{εtrans}

(2.202)

σz = Dtransnεz

(2.203)

44

is

Multiscale and Multiphysics Modeling

If the material is orthotropic, the material property matrix for the in-plane components

D  11 [Db ] =  D12  0 

0   0  D33 

D12 D22 0

(2.204)

where D11 =

E1 1 − v12 v21

(2.205)

D12 =

E2 v21 1 − v12 v21

(2.206)

D22 =

E2 1 − v12 v21

(2.207)

D33 = G12

(2.208)

Here, Ei and νij are the elastic modulus and Poisson’s ratio, respectively, and Gij is the shear modulus of the in-plane directions. The material property matrix for the transverse shear components is expressed as G [Dtrans ] =  13  0

0   G23 

(2.209)

and the transverse normal component has Dtransn = E3

(2.210)

As the shape functions are plugged into the kinematic equation, we obtain {εinpl} = [Binpl]{d}

(2.211)

[Binpl] = [Binpl1 Binpl2]

(2.212)

where

© 2016 by Taylor & Francis Group, LLC

45

Finite Element Method

[Binpli ] =  H i      H i 

∂N 1 ∂x

0

0 Hi

∂N 2 ∂x

0

0 Hi

∂N 3 ∂x

0

0 Hi

∂N 4 ∂x

0

Hi

∂N 1 ∂y

0

0

Hi

∂N 2 ∂y

0

0

Hi

∂N 3 ∂y

0

0

∂N 1 ∂y

Hi

∂N 1 ∂x

0 Hi

∂N 2 ∂y

Hi

∂N 2 ∂x

0 Hi

∂N 3 ∂y

Hi

∂N 3 ∂x

0 Hi

∂N 4 ∂y

 0  ∂N 4  0 Hi ∂y  ∂N 4  0 Hi ∂x  (2.213) 0

{d}T = {d11 d21 d31 d41 d12 d22 d32 d42}

(2.214)

{dij} = {uij vij wij}

(2.215)

Furthermore, the transverse shear strains are expressed as follows: {εtrans} = [Btrans]{d}

(2.216)

where [Btrans ] = [Btrans1

Btrans 2 ]

(2.217)

[Btransi ] =  ∂N 3 ∂H 1 ∂N 1 ∂H 1 ∂N 2 ∂H 1 ∂H 1 ∂N 4   N1  H1 N2 H1 N3 0 0 0 H1 N4 0 H1 ∂ z ∂ x ∂ z ∂ x ∂ z ∂ x ∂ z ∂x    ∂N 3 ∂N 2 ∂H 1 ∂H 1 ∂N 4  ∂H 1 ∂N 1 ∂H 1 0 0 H1 0 N3 H1 0 N4 H1 N1 H1 N2   ∂y ∂z ∂y ∂z ∂y  ∂z ∂y ∂z  

(2.218) The transverse normal strain is εz = {Btrann}T{d}

(2.219)

in which {Btrann }T =

© 2016 by Taylor & Francis Group, LLC

{B

T trann1

T Btrann 2

}

(2.220)

46

Multiscale and Multiphysics Modeling

 ∂N i {Btranni }T = 0 0 H 1 ∂z 

0 0 H1

∂N i ∂z

0 0 H1

∂N i ∂z

0 0 H1

∂N i   ∂z 

(2.221)

The element stiffness matrix in terms of the local coordinate can be written as [ Klocal ] =

∫ [B

inpl

]T [Dinpl ][Binpl ]dΩ +

Ωe

+

∫

∫ [B

trans

]T [Dtrans ][Btrans ]dΩ

Ωe

(2.222)

{Btrann }T Dtrann {Btrann }dΩ

Ωe

The first term represents the bending energy, the second term represents the transverse shear energy, and the last term comes from the transverse normal energy. Finally, we need to transform the local displacements into the global displacements. To this end, we use the direction cosines between the x axis and the global axis denoted by (l1, m1, n1), the direction cosines between the y axis and the global axis denoted by (l2, m2, n2), and the direction cosines between the z axis and the global axis denoted by (l3, m3, n3). The compatibility of displacements between the local and global coordinate systems can be written as  ui 1 + ui 2  2   v i1 + v i 2  2   w i1 + w i 2  2 

    =    

l  1  l2 l  3

m1 m2 m3

n1   n2  n3 

 U i1 + U i 2  2   V i1 + V i 2  2   W i 1 + W i2  2 

        

(2.223)

and  ui 2 − ui 1  i2  v − v i1  i2 i1 w −w

  =  

 U i 2 − U i1  i2  V − V i1  i2 i1 W −W

    

(2.224)

where the superscript i denotes the node from 1 to 4. Solving the two sets of equations results in  ui 1  i1 v  w i 1  i2 u  vi2  i2  w

l +1   1   l2   1  l3 =   2  l1 − 1   l2   l   3

© 2016 by Taylor & Francis Group, LLC

m1

n1

l1 − 1

m1

m2 + 1

n2

l2

m2 − 1

m3

n3 + 1

l3

m3

m1

n1

l1 + 1

m1

m2 − 1

n2

l2

m2 + 1

m3

n3 − 1

l3

m3

  n2  n3 − 1  n1   n2  n3 + 1  n1

 U i1  i1 V  W i 1  i2 U  V i2  i2  W

        

(2.225)

47

Finite Element Method

Applying this transformation expression to all sets of nodal displacements results in the following equation: {dlocal} = [T]{dglobal}

(2.226)

Then, the element stiffness matrix in terms of the global coordinate system can be written as [Kglobal] = [T]T[Klocal][T]

(2.227)

This element stiffness matrix can be assembled into the system stiffness matrix.

2.9 Acoustic Wave Equation Sound wave propagates in a fluid medium. To develop the acoustic wave equation, fluid flow and its viscosity are neglected in the fluid medium. We will develop the velocity potential formulation for the acoustic wave equation. To do that, let us begin with the continuity equation: ∂ρ   + ∇ ⋅ ρv = 0 ∂t

(2.228)

  in which ρ is the fluid density, v is the velocity vector, ∇ is the gradient operator, and t denotes time. The change in density can be written as ρ = ρo(1 + s)

(2.229)

where s is called condensation, and ρo is an ambient fluid density. Plugging Equation 2.229 into Equation 2.228 with an assumption of s being so mall (i.e., s ≪ 1) results in the following equation: ∂s   +∇⋅v = 0 ∂t

(2.230)

  ∂v = −∇p ∂t

(2.231)

Applying Newton’s 2nd law yields vρo

Here, p is the pressure. The velocity potential is defined to express the velocity as follows:   v = −∇φ

© 2016 by Taylor & Francis Group, LLC

(2.232)

48

Multiscale and Multiphysics Modeling

Now, we substitute Equation 2.232 into Equation 2.231, which yields   ∂φ ∇  −ρo + p = 0 ∂t  

(2.233)

Pressure is proportional to condensation, and the proportional constant is called the bulk modulus; it is written as p = Bs

(2.234)

where B is the bulk modulus. Plugging Equations 2.232 and Equation 2.234 into Equation 2.230 results in 1 ∂p − ∇ 2φ = 0 B ∂t

(2.235)

Elimination of pressure from Equations 2.233 and 2.235 produces the final wave equation in terms of the velocity potential: ∂ 2φ 2 2 −c ∇ φ=0 ∂t 2

(2.236)

where c is the speed of sound, and c2 = B/ρo. The boundary conditions to the acoustic wave equation are either prescribed velocity or pressure. The velocity is computed from the velocity potential as defined in Equation 2.232. On the other hand, the pressure boundary condition is expressed from Equation 2.233, such as p = ρo

∂φ ∂t

(2.237)

The acoustic wave equation can also be expressed in terms of pressure instead of velocity potential: ∂2 p 2 2 −c ∇ p=0 ∂t 2

(2.238)

Either equation can be used depending on which expression is easier for a given application. For the fluid-structure interaction, the velocity potential formulation is selected, as will be discussed below. For a 1-D case, the acoustic wave equation is simplified to ∂2 p 2 ∂2 p −c =0 ∂t 2 ∂x 2 This is the same kind of equation as Equation 2.52.

© 2016 by Taylor & Francis Group, LLC

(2.239)

49

Finite Element Method

The weighted residual expression for Equation 2.236 yields  ∂ 2φ  w  2 − c 2 ∇ 2 φ dΩ =  ∂t  Ω

∫

∫

w

Ω

  ∂ 2φ ∂φ dΩ + ∇w ⋅∇udΩ − w d Γ = 0 2 ∂n ∂t

∫

∫

Ω

(2.240)

Γ

Discretizing the domain with a finite element mesh, this equation is rewritten as ne

∑∫

e = 1 Ωe

w

∂ 2φ dΩ + ∂t 2

ne

∑∫

  ∇w ⋅∇udΩ =

e = 1 Ωe

nb

∑∫ e =1

Γe

w

∂φ dΓ ∂n

(2.241)

If we use the linear triangular element for a 2-D domain problem, the first term in Equation 2.241 becomes

∂2 φ w 2 dΩ = ∂t e

∫

Ω

 A 2 [ H ] [ H ] dΩ{φ} = 1 12   1 Ωe

∫

T

1 2 1

1  1 2 

φ   1  φ 2   φ  3

    

(2.242)

 φ   1    φ2      φ3 

(2.243)

in which A is the area of the triangular element. The second term in Equation 2.241 becomes

∫

Ωe

  ∇w ⋅ ∇φ dΩ =

 a a a  11 12 13  ∂w ∂φ ∂w ∂φ  + = d Ω  a211 a22 a23  ∂x ∂x ∂y ∂y   Ωe  a31 a32 a33

∫

where a11 =

2 2 1  x3 − x2 ) + ( y 2 − y 3 )  (  4A

(2.244)

a12 = a21 =

1  ( x3 − x2 ) ( x1 − x3 ) + ( y2 − y3 ) ( y3 − y1 ) 4A 

(2.245)

a13 = a31 =

1  ( x3 − x2 ) ( x2 − x1 ) + ( y2 − y3 ) ( y1 − y2 ) 4A 

(2.246)

a22 =

a23 = a32 =

© 2016 by Taylor & Francis Group, LLC

1  ( x1 − x3 )2 + ( y3 − y1 )2  4A 

1  ( x1 − x3 ) ( x2 − x1 ) + ( y3 − y1 ) ( y1 − y2 ) 4A 

(2.247)

(2.248)

50

Multiscale and Multiphysics Modeling

a33 =

2 2 1  x2 − x1 ) + ( y1 − y 2 )  (  4A

(2.249)

The term on the right-hand side becomes ∂φ

∫ w ∂n dΓ = ∫ −wv dΓ

Γe

n

(2.250)

Γe

Here, vn is the normal velocity at the boundary. The outward direction is considered positive. Similarly, isoparametric quadralateral elements can be used for the finite element formulation.

2.10 Interaction of Structure with Acoustic Domain This section presents how to model coupling between a structure and an acoustic fluid medium. First, a 1-D problem is discussed for an axial member with an acoustic domain. Then, a 2-D problem is presented using a beam in a 2-D acoustic fluid medium. 2.10.1 One-Dimensional Case To model the effect of an acoustic fluid medium on the longitudinal vibration of a rod, the wave equations that follow are considered for the rod and the fluid, respectively, as presented previously. For the rod, the wave equation is expressed as ∂2 u ∂2 u = cr2 2 2 ∂t ∂x

(2.251)

where u, x, and t are the longitudinal displacement of the rod and the spatial and time variables, respectively. Furthermore, cr is the speed of sound in the rod; it is equal to Er /ρr , where Er and ρr are the elastic modulus and density of the rod material, respectively. The wave equation for the acoustic fluid is ∂2 p ∂2 p = c 2f 2 2 ∂t ∂x

(2.252)

where p is the acoustic pressure, and cf is the speed of sound in the fluid and is equal to B f /ρ f , where Bf and ρf are the bulk modulus and density of the fluid, respectively. The weighted residual finite element formulation for Equation 2.251 was developed previously, and the final matrix expression is [Mr]{ü} + [Kr]{u} = {Fr}+{Ff} = {Fr} + Arpint

© 2016 by Taylor & Francis Group, LLC

(2.253)

51

Finite Element Method

in which the superimposed dots denote the temporal derivative, and [Mr] and [Kr] are the mass and stiffness matrices, respectively, as given Equation 2.59. In addition, {Fr} and {Ff} are the force vectors applied to the rod by mechanical and fluid forces, respectively. The fluid force results from the pressure at the interface (i.e., pint), and Ar is the cross-sectional area of the rod. For simplicity, the cross section is assumed to be unity. The same formulation is applied to Equation 2.252, leading to Lf

∫ 0

 ∂2 p ∂2 p  w  2 − c 2f 2  dx = ∂x   ∂t

Lf

∫ 0

∂2 p w 2 dx + ∂t

Lf

∫

Lf

c 2f

0

 ∂p  ∂w ∂p dx −  c 2f w  = 0 ∂ n 0 ∂x ∂x 

(2.254)

The first and second terms on the right-hand side of Equation 2.254 yield [Mf] and [Kf], which are the same as those in Equation 2.59, like the rod, but the third term is for the boundary condition. For the fluid-solid interface, the boundary condition becomes ∂p ∂2 u = −ρ f 2 ∂n ∂t

(2.255)

The resulting matrix equation becomes int = 0 [ M f ]{ p} + [ K f ]{ p} − c 2f ρ f u

(2.256)

assuming the interface is located between the right end of the rod and the left end of the fluid domain. For the eigenvalue analysis of the rod in contact with fluid, the two equations, Equations 2.253 and 2.256, are solved together with an assumption of harmonic solutions for the rod displacement and the acoustic pressure. In the example provided in the following, the rod has an elastic modulus of 20 GPa and a density of 2000 kg/m3, resulting in the speed of sound, 3162 m/s. For the fluid, its speed of sound is varied from 1500 to 2500 and 4000 m/s. Some of this speed may not be realistic. However, as a demonstration of the effect of the fluid speed of sound, its value is varied. Figure 2.16 shows the first mode shape for each different case of the fluid speed of sound. The rod is fixed at the left end, while it has no constraint on the right end except for its contact with fluid. No fluid means there is no fluid on the right end of the rod. In this case, the first mode is a half-sine curve, as predicted by the vibration theory of the rod. When the fluid speed of sound is 1500 m/s, the mode shape is very close to the no-fluid case. When the fluid speed of sound is increased to 4000 m/s, the bulk modulus of the fluid becomes higher with a constant fluid density. This means the fluid becomes very incompressible. Then, the beam behaves like the case when both ends are constrained, resulting in a fullsine curve as shown in the figure. When cf equals 2500 m/s, the mode shape is between the two cases. This study also showed that the natural frequency and the mode shape of the wet structure did not change as the fluid density was varied while the ratio of the speeds of sound of the solid and fluid media remained constant. Thus, the ratio of the speeds of sound is the main parameter that can affect the frequency as well as the mode shape.

© 2016 by Taylor & Francis Group, LLC

52

Normalized longitudinal displacement

Multiscale and Multiphysics Modeling

0 0.1

No f luid Fluid speed = 1500 Fluid speed = 2500 Fluid speed = 4000

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Normalized distance

0.8

0.9

1

FIGURE 2.16 Mode shape of axial member coupled with 1-D acoustic domain.

2.10.2 Two-Dimensional Case To present a 2-D coupling example, a beam is considered. The dynamic equation for beam bending is written as previously: ρs As

∂2 u ∂2  ∂2 u  + EI = qbs ∂t 2 ∂x 2  ∂x 2 

(2.257)

As discussed previously, the finite element matrix equation becomes [Ms]{üs} + [Ks]{us} = {Fs} + bs = [Q fsi]{pfsi}

(2.258)

where the mass and stiffness matrices are given in Equations 2.89 and 2.90. The Helmholtz wave equation results in the following matrix equation as given previously:  fsi } [ M f ]{ p f } + [ K f ]{ p f } = ρ f [Q fsi ]T {u

(2.259)

with the following boundary condition: ∂p s = −ρ f u ∂n

(2.260)

Putting together the equations for the beam and acoustic problems and neglecting any external load gives  [M ] s   ρ f [Q fsi ]T 

 [0]   u  s [ M f ]   p f 

© 2016 by Taylor & Francis Group, LLC

 + 

 [K ]  s  [0] 

−bs [Q fsi ]   [K f ]  

 u s   p f

   0 =    0 

(2.261)

53

Finite Element Method

0.60

Frequency ratio

0.50 0.40 0.30 0.20 0.10 0.00

0

1000

2000

3000

4000

Beam density

5000

6000

FIGURE 2.17 Plot of the frequency ratio with and without acoustic domain versus beam density (kg/m3).

where [Q fsi] is only associated with the structural and fluid nodes at the fluid-structure interaction (FSI) interface. Therefore, the submatrix [Q fsi] contains many zero rows and columns for the nodal variables, which are not associated with the FSI interface. As an example, the fluid domain has a density of 1000 kg/m3, and the speed of sound is 1500 m/s. The beam has an elastic modulus of 20 GPa, and the density is 2000 kg/m3. Then, the first natural frequency is computed for the beam in the acoustic domain as well as without considering the acoustic domain. The ratio of the frequency with and without the acoustic domain interaction is 0.4. In other words, the acoustic domain reduces the frequency by 60%. As a parametric study, the ratio of the frequency is plotted as a function of the density of the beam as shown in Figure 2.17. The figure shows that the lighter the beam density is, the greater the effect of the acoustic domain is.

© 2016 by Taylor & Francis Group, LLC

3 Lattice Boltzmann Method

3.1 Introduction The Navier-Stokes equation has been solved using computational fluid dynamics techniques. The control volume technique is one of the most popular. Another common numerical method is the finite element method. Those formulations dealt with macroscopic variables, such as velocities and pressure, directly. As a result, those techniques are suitable for macroscale analysis of fluid flow. Another alternative computational technique is the lattice Boltzmann method (LBM). This technique is a bottom-up approach and is presented here. The LBM has been popular for modeling and simulating fluid flow [1,2]. In contrast to other conventional methods described previously, the LBM does not begin with the macroscale parameters as used in a continuous medium. Instead, the technique uses a collection of fictitious particles. The flow domain consists of regular shapes of lattices. Then, the particles move along the lattices to have local interactions with other particles in accordance with simple rules. The macroscale conservation laws, such as conservation of mass and linear momentum as described by the Navier-Stokes equations, are satisfied from the local collision of particles and their redistribution. Because of the simplicity of the concept using a collection of particles, the LBM has been applied to various flow problems, including those of porous media [3–5], two-phase flow [6–8], and magneto-hydrodynamics [9–11], among others. The LBM technique can also produce other partial differential equations of interest by altering the formulation. Those equations include the Burgers equation [12], the Korteweg-de Vries equation [13], the Brinkman equation [14], and the Schrodinger equation [15]. A review of the current state of the art in LBM can be found in Reference 1, with a more recent update in Reference 16. An analysis of LBM theory with a critique and comparison with traditional computational fluid dynamics techniques can be found [17]. This chapter presents various LBM formulations for the solution of the Navier-Stokes equations for single-component fluid flows as well as a limited number of multicomponent fluid flows. In addition, the LBM formulation is discussed for the wave equation.

3.2 Standard Lattice Boltzmann Method The LBM technique can be derived from the concepts of the cellular automaton (CA) [18,19]. The CA model underlying a fluid dynamics model incorporates movement of particles 55 © 2016 by Taylor & Francis Group, LLC

56

Multiscale and Multiphysics Modeling

from one lattice site to another along discrete lattice directions. The rule for the lattice site update is applied to all particles arriving at a given lattice site in a given time step and is represented formally in Equation 3.1:    f α ( x + δ x eα , t + δ t ) = f α ( x , t ) + Ω α ( f α )

(3.1)

 where fα is the distribution function of particle velocity along the α direction, t is time, eα is the discrete velocity vector along the α direction, and Ωα denotes the collision operator. The discrete velocity vector eα is provided for the specified directions of the lattice model used in the analysis. When the lattice space and the time increment are normalized as unity, Equation 3.1 is expressed as    fα ( x + eα , t + 1) = fα ( x , t) + Ωα ( fα )

(3.2)

The expression given in Equation 3.1 can also be recast into the following differential equation: ∂fα  + eα ⋅∇fα = Ωα ∂t

(3.3)

in which ∇ indicates the gradient vector. Equation 3.1 can be derived by applying the finite difference concept to Equation 3.3. Equations 3.1 through 3.3 are applied to a regular lattice of the same size in all directions. The lattice is defined by the problem dimension d and the number of lattice vectors q and is denoted as DdQq. One of the most common lattice structures is D2Q9, which suggests that the lattice is a two-dimensional (2-D) discretization and there are nine lattice velocity sets eα as shown in Figure 3.1. (0,1)

(−1,1) e6

(1,1)

e2

e5

e0 (−1,0)

e3

e7 (−1,−1) FIGURE 3.1 D2Q9 lattice with nine velocity vectors.

© 2016 by Taylor & Francis Group, LLC

e1 (1,0)

e4 (0,−1)

e8 (1,−1)

57

Lattice Boltzmann Method

The discrete velocity vectors are given next for the D2Q9 lattice structure of unit spacing as shown in Figure 3.1: 0  eα =  0

1 0

0 1

−1 0

0 −1

−1 1

1 1

−1 −1

1   −1 

(3.4)

   This means that the nine velocity    vectors for D2Q9 are  e0 = (0, 0), e1 = (1, 0), e2 = (0, 1), e3 = (−1, 0), e4 = (0, −1), e5 = (1, 1), e6 = (−1, 1), e7 = (−1, −1), e8 = (1, −1). For three-dimensional (3-D) analysis, the D3Q15 lattice model as illustrated in Figure 3.2 has the discrete velocity vectors expressed as 0   eα =  0  0

−1 0 0

1 0 0

0 1 0

0 −1 0

0 0 1

0 0 −1

−1 1 1

1 1 1

−1 −1 1

1 −1 1

1 1 −1

−1 1 −1

−1 −1 −1

1   −1  (3.5) −1 

Other common 3-D lattice structures are D3Q19 and D3Q27. Figure 3.3 compares the three lattice structures. The discrete velocity vectors for D3Q19 and D3Q27 are 0 1 −1 0 0 0 0 1 −1 1 −1 1 −1 1 −1 0 0 0 0   eα = 0 0 0 1 −1 0 0 1 1 −1 −1 0 0 0 0 1 −1 1 −1 0 0 0 0 0 1 −1 0 0 0 0 1 1 −1 −1 1 1 −1 −1

(−1,1,1)

e7 (1,1,1)

e8 e5 e3

(−1,−1,1) z

e10 (1, −1,1)

e9 e0

e2 y

(−1,1,−1) e12 x

(−1,−1,−1)

e13

FIGURE 3.2 D3Q15 lattice with 15 velocity vectors.

© 2016 by Taylor & Francis Group, LLC

e4

e1 e11 (1,1,−1)

e6 e14 (1,−1,−1)

(3.6)

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Multiscale and Multiphysics Modeling

D3Q15 lattice

Z

Y

D3Q19 lattice

Y

X

Z

(a)

X

(b) D3Q27 lattice

Z

Y

X

(c) FIGURE 3.3 Comparison of (a) D3Q15, (b) D3Q19, and (c) D3Q27 lattice structures.

for D3Q19 and

0   eα =  0  0

−1 0 0 0 1 0

0 0 1

0 −1 0

0 0 −1

−1 −1 0

1 1 0

0 −1 0

1 0 1

−1 1 0 1 0 −1

−1 0 −1 0 1 1

−1 0 1 0 1 −1

0 −1 −1 1 1 1

1 1 −1

0 −1 1 1 −1 1

−1 −1 −1

−1 −1 1

1   −1  −1 

−1 1 −1

−1 1 1

1 0 0

…

(3.7)

for D3Q27. In general, more lattice vectors improve the accuracy of the solution at the expense of the computational cost.

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The collision operator in Equations 3.1 through 3.3 can be expressed in different ways. When a single relaxation time technique is used for the collision operator, such as the BGK technique [20], the collision operator is expressed as 1 Ωα = − ( fα − fα ) τ

(3.8)

where fα denotes the local equilibrium distribution of fα. The relaxation parameter τ can be related to the fluid kinematic viscosity ν using the following expression: τ=

ν 1 + c s2 2

(3.9)

where cs is the lattice speed and is equal to 1/ 3 for the unit spacing lattice structures. The local equilibrium distribution is written as     2   f = ρω 1 + 3 ν ⋅ eα + 9( ν ⋅ eα ) − 3 ν ⋅ ν   α α c s2 2 c s4 2 c s2  

(3.10)

 in which ρ is the macroscopic fluid density, and ν is the fluid velocity. The pressure and velocity vector can be expressed, respectively, as ρ=

∑f

(3.11)

α

α

and  1 ν= ρ



∑f e

α α

(3.12)

α

In addition, ωα is the weighting parameter for each velocity direction as follows:  4/9  ω α =  1/9  1/36

 α=0  α = 1, 2 , 3, 4  α = 5, 6, 7 , 8 

(3.13)

 2/9  ω α =  1/9  1/72

α=0   α = 1 to 6  α = 7 to 14 

(3.14)

for D2Q9;

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for D3Q15;  1/3  ω α =  1/18  1/36

α=0   α = 1 to 6  α = 7 to 18 

(3.15)

  α = 1 − 3, 14 − 16  α = 10 − 13, 23 − 26   α = 4 − 9, 17 − 22 

(3.16)

for D3Q19; and    ωα =   

α=0

8 27 2 27 1 54 1 216

for D3Q27. The computational procedure for the LBM consists of repeats of the two processes called streaming and collision (or called redistribution) processes. For the streaming process, particles at each lattice point move to its neighboring lattice points as specified by DdQq. The distribution of the particles along those directions is determined by ωα as expressed in Equations 3.13 through 3.16, depending on the lattice structure used in the analysis. This streaming process is completed at all the lattice points. Then, Equation 3.8 is used for the collision process, after which the particles at every lattice point are updated using Equation 3.1 or 3.2. Then, the two processes repeat themselves.

3.3 Multiple Relaxation Lattice Boltzmann Formulation The single relaxation collision operator is simple and computationally efficient because the update is based on a single parameter. However, the collision operator has a problem of stability in some applications. To improve stability, the multiple relaxation collision operator, also referred to as the generalized lattice Boltzmann equation, was presented [21]. The objectives of the multiple relaxation collision operators were to resolve the fixed Prandtl number associated with the single-parameter collision operator. The technique allows for varying kinematic and bulk viscosities and introduces a mechanism for improving simulation stability. The multiple-relation collision technique utilizes the projection of the density distribution functions fα onto an orthogonal vector space of momenta using the operator M. The particular momenta depend on the lattice structure chosen, but all include a combination of the mass density, kinetic energy, momentum flux, energy flux, and viscous stress tensor. They are expressed in the vector R. Relaxation occurs over the momentum space using the relaxation times given in S, and the result is transformed back to the density space fα using the inverse of M. For the D2Q9 lattice, the momentum space and transformation matrix are given in Equations 3.17 and 3.18, respectively. RD 2Q 9 =  ρ 

© 2016 by Taylor & Francis Group, LLC

e

ε

jx

qx

jy

qy

pxx

qxy  

T

(3.17)

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Lattice Boltzmann Method

M D 2Q 9

 1   −4  4  0  =  0  0   0  0  0 

1 −1 −2 1 −2 0 0 1 0

1 −1 −2 0 0 1 −2 −1 0

1 −1 −2 −1 2 0 0 1 0

1 −1 −2 0 0 −1 2 −1 0

1 2 1 1 1 1 1 0 1

1 2 1 −1 −1 1 1 0 −1

1 2 1 −1 −1 −1 −1 0 1

1   2  1  1   −1  −1   −1  0  −1 

(3.18)

where R = Mfα, and ρ is fluid density, e is the energy, ε is related to the square of the energy, jx and jy are mass fluxes, qx and qy correspond to energy flux, and pxx and pxy correspond to the diagonal and off-diagonal components of the viscous stress tensor. The coefficients for relaxation over this momentum space are given in a diagonal matrix as shown in Equation 3.19: S = diag(0, s2, s3, 0, s5, s7, s8, s9)

(3.19)

In Reference 22, it was shown that the same fluid viscosity is given in the fluid flow when s8 = s9 = 1/τ. The other parameters in Equation 3.19 can be set as desired to promote solution stability or as required to further tailor fluid behavior. If all nonzero coefficients of S are set to 1/τ, the single-relaxation technique is recovered. Once the coefficients of S are provided, the LBM collision operator is given as shown in Equation 3.20: Ω MRT = −M−1SM(f − f eq)

(3.20)

where all values of fα are relaxed with a single matrix collision operator. While the multiple relaxation technique requires more computations per time step, it has been found that simulations are able to be conducted with much lower lattice density with improved stability. Furthermore, fewer time steps are typically required for flows to overcome the noise of nonequilibrium initial conditions and to reach accurate flow configurations.

3.4 Finite Element–Based Lattice Boltzmann Method One of the requirements of the standard LBM is the use of regular lattice structures of square or cubic shapes of equal spacing. If there is a curved boundary, such a regular lattice structure requires a very fine mesh to represent the curved boundary accurately. This is directly related to computational cost. One way to overcome this limitation is to use the finite element–based LBM, called FELBM [23]. Because more general shapes of finite elements, such as triangular or quadrilateral shapes for a two-dimensional (2-D) domain, can be used with FELBM, any complex boundary shape can be modeled using a reasonable size of lattice structure of a nonregular shape like a finite element mesh.

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Multiscale and Multiphysics Modeling

The formulation of the FELBM starts with Equations 3.3 and 3.8 for the single-relaxation time operator. The derivation shown next is also applicable to the multiple-relaxation time operator with a minor change. The weighted residual technique is applied to the equation:  ∂fα



∫ w  ∂t + e Ω

α

 1 ⋅ ∇fα + ( fα − fα ) dΩ = 0  τ

(3.21)

where w is a test or weighting function. To develop the FELBM, the problem domain is discretized into a number of finite elements called a finite element mesh. Then, the unknown variable fα is expressed in terms of the interpolation (or called the shape) functions and nodal variables for every finite element as follows: n

fα =

∑H f i

i α

= [ H ]{ fα }

(3.22)

i=1

where Hi is the spatial shape function associated with the ith node of the finite element, fαi is the ith nodal variable of the finite element, and n is the number of nodes per element. In addition, [H] is a row vector consisting of the shape function Hi, and {fα} is a column vector containing unknown solutions at the nodes. Substitution of Equation 3.22 into Equation 3.21 yields the weighted residual equation:

∑∫

Ω

e

   1 {w}  [ H ]{ fα } + eα ⋅ [∇H ]{ fα } + [ H ] { fα } − { fα }  dS = 0 τ  

(

)

(3.23)

where the integration is conducted over each finite element domain Ω e, and the summation is performed over the total number of elements. Furthermore, {w} is a column vector consisting of weighting functions. Calculation of each element integral from Equation 3.23 and summation of the resulting matrices and vectors results in the following matrix equation: [ M]{ Fα } + [ K ]{ Fα } + [C]{ Fα } − [C]{ Fα } = 0

(3.24)

where the matrices and vectors in Equation 3.24 are expressed as follows: [ M] =

[K ] =

{w}[ H ] dS

(3.25)

 {w} ( eα ⋅ [∇H ]) dS

(3.26)

Se

∑ [k ] = ∑ ∫

[C] =

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∑ [m] = ∑ ∫ Se

∑ [c ] = ∑ ∫

Se

1 {w}[ H ] dS τ

(3.27)

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Lattice Boltzmann Method

{ Fα } =

∑{ f }

(3.28)

{ Fα } =

∑ { f }

(3.29)

α

α

Depending on the choice of the weighting function, there are many choices in the weighted residual method. Some of them are the Galerkin method, collocation method, method of moments, least-square method, or subdomain method. In this section, the Galerkin method is presented because it is one of the most popular choices. For the Galerkin method, the weighting function is selected to be the same as the shape functions used in Equation 3.22, that is, {w} = [H]T. Then, Equations 3.25 through 3.27 are rewritten, respectively, as [ M] =

[K ] =

∑ [m] = ∑ ∫

∑ [k ] = ∑ ∫

[C] =

[ H ]T [ H ] dS

(3.30)

 [ H ]T ( eα ⋅ [∇H ]) dS

(3.31)

Se

Se

∑ [c ] = ∑ ∫

Se

1 [ H ]T [ H ] dS τ

(3.32)

For the collocation method, the weighting functions are selected to be Dirac delta functions. The Dirac delta functions may be defined at the nodal points of each element for convenience. Therefore, for a 2-D case, w(x, y) = δ(x − xi) δ(y − yj), where xi and yj are the nodal coordinate values. For the 3-D case, w(x, y, z) = δ(x − xi) δ(y − yj) δ(z − zk). On the other hand, for the method of moment the weighting functions are chosen to be monomial terms, such as xpyqzr (where p, q, and r are nonnegative integers), starting from the lowest order. Once the matrix equation of the first-order temporal derivative, as given in Equation 3.24, is developed from the weighted residual finite element formulation, a numerical time integration scheme is applied to the equation. There are many different numerical techniques for time integration. Those include, but are not limited to, the forward difference technique, backward difference technique, Crank-Nicolson technique, Runge-Kutta technique, and predictor-corrector technique. The forward difference technique is the simplest and easiest. If the matrix [M] becomes a diagonal matrix, the forward difference technique becomes the explicit method. As a result, even though a small time step size Δt is used because of the conditional stability, the overall computation is efficient. If an unconditionally stable method is preferred, the Crank-Nicolson technique may be selected. Using the forward difference scheme for time integration to Equation 3.24 yields the following equation:

(

{ Fα }t+ ∆t = { Fα }t + ∆t[ M]−1 [C]{ Fα }t − [C]{ Fα }t − [ K ]{ Fα }t

)

(3.33)

Equation 3.33 is solved for the given initial and boundary conditions. How to apply the boundary conditions using the LBM is described in Section 3.8.

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Multiscale and Multiphysics Modeling

3.5 Element-Free-Based Lattice Boltzmann Method Consider a lattice point x and its neighborhood as a subdomain Ωx. Inside the subdomain, there are n numbers of randomly distributed lattice points. To represent the solution u(x) inside the subdomain, a polynomial function is assumed: u(x) = {p(x)}T{a(x)}

(3.34)

where {p(x)} is a vector containing a complete monomial basis of order m as expressed as follows: {p(x)T} = {1 x y}

for 2-D with m = 3

{p(x)T} = {1 x y x2 xy y2} for 2-D with m = 6 {p(x)T} = {1 x y z}

for 3-D with m = 4

(3.35) (3.36) (3.37)

and {a(x)} is a vector consisting of coefficients of the monomial terms. One thing to be mentioned here is that the coefficient vector {a(x)} is not a constant vector but a function of x, which will be determined subsequently. The coefficient vector is determined to best fit the solutions at the lattice points inside the subdomain. To achieve the goal, the weighted least square technique is utilized. The sum of the weighted square is expressed as n

J( x) =

∑ w (x)[{p(x )} {a(x)} − uˆ ] k

k

T

k

2

(3.38)

k =1

where wk(x) is the weighting function associated with the lattice point k, and ûk is the solution at the same node. Minimization of this equation with respect to {a(x)} results in [A]{a} = [B]{û}

(3.39)

in which n

[ A( x )] =

∑ w (x){p(x )}{p(x )} k

k

k

T

(3.40)

k =1

[B(x)] = [w1(x){p(x 1)} w2(x){p(x 2)} … wn(x){p(x n)}]

(3.41)

Solving for the coefficient vector and substituting the resulting expression into Equation 3.34 yields u(x) = {Φ}T{û}

(3.42)

where the interpolation function vector {Φ} is given as {Φ(x)}T = {p(x)}T [A(x)]−1[B(x)]

© 2016 by Taylor & Francis Group, LLC

(3.43)

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Lattice Boltzmann Method

The interpolation function, Equation 3.43, is applied to the weighted residual formulation as described in the previous section to develop the element-free-based LBM, called EFLBM [23]. This interpolation function vector is different from that used in the finite element method. The interpolation function vector for the finite element method satisfies the following relationship: u(x k) = ûk

(3.44)

However, the interpolation vector developed for the element-free technique does not satisfy the same relationship. For the matrix [A] to be invertible, n ≥ m should be satisfied, and the weighting function is selected to be a nonnegative function, such as a spline function, defined as follows: 2 3 4   1 − 6  dk  + 8  dk  − 3  dk  0 ≤ d ≤ r k k  r   r   r  wk ( x) =  k k k  0 dk ≥ rk 

(3.45)

where dk is the distance between the lattice points x and x k, and rk is the size of the support for the weighting function wk. The derivative of the interpolation function requires the derivatives of {p(x)}, [A(x)]−1, and [B(x)], respectively. For the derivative of [A(x)]−1, the following expression is used: ([A]−1)l = − [A]−1([A])l [A]−1

(3.46)

3.6 Hybrid Lattice Boltzmann Formulation To mitigate the computational demands of the FELBM while retaining the ability to model a domain with complex and irregular shapes without an unnecessarily dense lattice, the hybrid LBM (HLBM) was developed. The HLBM couples classical LBM (CLBM) described in Section 3.2 or 3.3 and FELBM [24]. The CLBM requires a regular lattice structure, but FELBM does not. Therefore, any domain with curved boundaries can be constructed as a sum of two types of subdomains. One subdomain has a regular lattice structure; the other subdomain consists of a random lattice structure. It is also acceptable to have multiple regular lattice subdomains and irregular lattice subdomains. The CLBM is applied to the subdomains made of regular lattices, and FELBM is applied to the other subdomains made of irregular lattices. The two techniques are properly coupled at the interfaces of the two different types of subdomains. Either the single-relaxation or multiple-relaxation technique can be used for CLBM as well as FELBM. All of the theoretical development from the CLBM and FELBM formulations is preserved, and the logical sequence of computations is maintained on each individual subdomain. A typical HLBM time step is portrayed schematically in Figure 3.4.

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Multiscale and Multiphysics Modeling

Collision

CLBM subdomain:

FELBM subdomain:

classical streaming

advective time step

FIGURE 3.4 Schematic of the hybrid LBM time step. Methodology differs only in implementation of the particle streaming phase.

To couple two neighboring subdomains, an interface layer is provided. Computationally, the streaming process of the CLBM domain and the advection process of the FELBM domain can be executed concurrently, with each subdomain retaining a “halo” of depth 1 into the adjoining subdomain [24]. Within each subdomain, the outermost layer of lattice points represents the halo as shown in Figures 3.5 and 3.6. While this coupling scheme is conceptually simple, great care must be taken with the time and space integration methods used for advecting particle density data on the FELBM subdomain. First-order time integration schemes tend to have too much dissipation error, while second-order schemes suffer from dispersion errors. Both effects propagate onto the CLBM subdomain and have an impact on solution quality and stability everywhere. For the results presented here, simple bilinear elements are used for the spatial discretization, and a four-stage third-order explicit time integration method shown in Equation 3.47 is used for temporal integration [24].

(a)

(b)

FIGURE 3.5 Schematic hybrid lattice on regular domain. Assignment following streaming in the CLBM domain and advection in the FELBM domain is only made to the interior of each respective subdomain. Data drawn from the lattice points on the halo facilitate communication between each subdomain. (a) Solid circles: CLBM interior; open circles: CLBM halo. (b) Solid squares: FELBM interior; open squares: FELBM halo.

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67

Lattice Boltzmann Method

Hybrid test patch 32 30

Lattice Y position

28 26 24 22 20 18 35

40

45

50

Lattice X position FIGURE 3.6 Interface region for CLBM and FELBM domains on a uniform mesh. Lattice points with both an asterisk and circle belong to the interface.

F(U ) = FEM advection operator on FELBM sub-domain n elements U n ← fOUT n U (1) = U n +

1 ∆t F(U n ) 2

U ( 2 ) = U (1) + U ( 3) =

FELBM sub-domain

1 ∆t F(U (1) ) 2

(3.47)

2 n 1 (2) 1 U + U + ∆t F(U ( 2 ) ) 6 3 3

U n+ 1 = U ( 3 ) +

1 ∆t F(U ( 3) ) 2

U n+1 → fIN n+1

FELBM interior

This method effectively controls both dissipation and dispersion errors during the advection process, and it allows a single FELBM advection time step for every CLBM streaming step. The CLBM subdomain undergoes the classical LBM streaming to adjacent neighbors,

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Multiscale and Multiphysics Modeling

restricted to only destination lattice points that lay within the CLBM interior as indicated in Equation 3.48. fOUT n

CLBM sub-domain

streaming  → fIN n+1

CLBM interior

(3.48)

Geometrically simple portions of the domain are discretized with a uniform, regular lattice for the CLBM. Subdomains containing complex or irregular shapes are identified and discretized with the FELBM. For example, in simulations such as those requiring fluid-structure interaction (FSI), in the case of flow past a heat exchanger tube bundle, it would be sufficient to employ the FELBM only in a region around the actual tubes. This area could employ a mesh with isoparametric elements to efficiently describe the shape of the tube and be used with the FELBM, while the remainder of the domain could use a uniform, regular lattice and the CLBM. The resulting HLBM could accurately capture the flow properties while reducing the total number of lattice points required significantly. To make the most of the computational benefits, the size of the FELBM subdomains should be much smaller than the size of the CLBM subdomains.

3.7 Multicomponent Flow One of the advantages of the LBM lies in its natural amenability to multicomponent fluids. There are four main multicomponent fluid models using the LBM theory: the color fluid model [25], the interparticle-potential model [26], the free-energy model [27], and the mean-field theory model [28]. The different methods are classified based on the way in which the surface tension of the component interface is taken into account in the evolution of the particle distribution functions as well as how the location of this interface is determined. References 2 and 16 provide good surveys of multicomponent fluid flows. In this section, the interparticle potential model is discussed in greater detail and the other methods are briefly reviewed. 3.7.1 Color Fluid Model The color fluid model [25] allows the simulation of immiscible binary fluids in two dimensions. The method is based on the two-component CA model introduced in Reference 29 and is modified for use with LBM. In this technique, each component of the binary fluid mixture is denoted by a color. That is the reason the method is referred to as the color fluid model. For example, one component is called “red” particles and the other “blue” particles. The LBM is carried out for each fluid species, and the effect of surface tension on particle distributions is considered with an additional perturbation term appended to the collision operator. Furthermore, a “recoloring” step is applied to make a correction based on the local color gradient that forces a shift to a direction leading to other like-colored particles. This method is frequently criticized in the literature for the “artificial” recoloring process [30], although each of the multicomponent models has some heuristic process that can be subjected to the same criticism. Because of that, spurious resultant velocities are commonly exhibited in the vicinity of fluid interfaces. More importantly, the recoloring step

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Lattice Boltzmann Method

is executed based on a costly and time-consuming calculation of local color gradients that requires considerably more information sharing between neighboring particles compared to other methods. 3.7.2 Free-Energy Model The free-energy model proposed in Reference 27 takes a different approach. Instead of maintaining density distributions for each phase, a single-density function ρ is used along with a density difference Δρ. Despite the terminology, which leads one to believe that the method is intended mainly for single-component multiphase flow, the method was originally introduced to model phase separation in nonideal one- and two-component fluids. The free-energy model gets its name through the use of the so-called Cahn-Hilliard’s approach for nonequilibrium thermodynamics [31]. In this approach, the form of the pressure tensor is defined based on a nonlocal pressure and a parameterized van der Waals equation of state. This pressure tensor is added to an expanded equilibrium distribution function that produces the desired interfacial effect. This approach has been used to simulate Rayleigh-Taylor instability [32], bubble motion [33], and simulation of spontaneous emulsification of liquid droplets in oil-water-surfactant systems [34]. 3.7.3 Mean Field Theory Model The mean-field theory was introduced in Reference 28 for nonideal gas flow. In this method, two distribution functions are used. The first distribution function is used to calculate the pressure and velocity fields of an incompressible liquid. The second is an index function that is used to locate the interface. The model is so named because the interparticle interactions are treated using a mean-field approximation in the same way that the Coulomb interaction among charged particles of a plasma is treated in the Vlasov equation [20,35]. As several authors have pointed out, this approach is similar to the traditional computational fluid dynamics methods for interface capturing and is the LBM analogy to the level set [36] and volume of fluid methods [37]. This model has been successfully used to model Rayleigh-Taylor and Kelvin-Helmholtz instabilities [38,39] with nonideal dense fluids, among other applications. 3.7.4 Interparticle Potential Model The interparticle potential model [26] simulates multiphase and multicomponent fluids as a simple means. The fundamental idea is that the surface tension effect is microscopic, and the same effect could be incorporated into the LBM via these same interparticle potential forces. In this model, only the nearest-neighbor particle densities are considered, and they are introduced as follows: q−1

∑ w ψ(x + e ∆t, t)e

F ( x , t) = −Gψ ( x , t)

α

α

α

(3.49)

α =1

where x is particle position, G is a parameter indicating interaction strength, ψ(x, t) is a function describing the interaction potential, wα is the lattice weight for direction α, and eα is the corresponding lattice velocity. The form of the potential function can be varied

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Multiscale and Multiphysics Modeling

to obtain the desired interparticle potential behavior. For multiphase flow, ψ is commonly expressed as in Equation 3.50:  ρ  ψ (ρ) = ψ o exp  − o   ρ

(3.50)

where ψo and ρo are arbitrary parameters selected to achieve appropriate dynamics for a selected fluid system. The multicomponent fluid systems used in this work are only qualitatively correlated with real fluid systems in that only density and viscosity are set and scaled consistently with the LBM. The parameter G is set to generate a desired level of immiscibility while maintaining simulation stability. The potential function is set as ψ(x, t) = ρ(x, t). Notice that Equation 3.49 specifies a weighted summation of the value of ψ for each lattice position in the neighborhood of a given lattice particle. 3.7.5 Immiscible Multicomponent Lattice Boltzmann Procedures The basic LBM process with multiple components is similar in most ways to that used for single-component systems. The obvious difference is that there are now two sets of distribution functions: as before fα and for a second component that will conventionally be referred to as gα. A second difference is that, as discussed in the preceding paragraphs, the interparticle force must be calculated in accordance with Equation 3.49 and incorporated into the calculation of macroscopic velocity used for computing f eq and g eq. The biggest and most fundamental difference is the need to structure the computations to account for the fact that, due to the desired macroscopic dynamic evolution of the system, some lattice sites will have zero density for one or the other component and only the interfaces will have a significant mixture. To be as explicit as possible, the immiscible multicomponent LBM time step for fluid lattice points carried out for this work is implemented as follows: 1. Compute the macroscopic density of each fluid: q−1

ρf =

∑f

(3.51)

α

α =1

q−1

ρg =

∑g

(3.52)

α

α =1

2. Compute the macroscopic momentum of each fluid: q−1

ρf uf =

∑e

f

α α

(3.53)

α =1

q−1

ρg ug =

∑e g α

α =1

© 2016 by Taylor & Francis Group, LLC

α

(3.54)

71

Lattice Boltzmann Method

3. Compute a weighted combined macroscopic density and velocity: ρω = ρfωf + ρgωg u=

(3.55)

ρ f u f ω f + ρg ug ω g ρω

(3.56)

1 where we recall ω = . τ 4. Compute the interparticle force using Equation 3.49, setting G to a constant and ψ(x, t) = ρ: q−1

F f = −Gρ g

∑ w ρ (x + e ∆t, t) α

g

α

(3.57)

α =1

q−1

Fg = −Gρ f

∑ w ρ (x + e ∆t, t) α

f

α

(3.58)

α =1

5. Apply these interparticle forces as momentum inputs to each respective lattice population: ueqf = u f − τ f Ff

(3.59)

ueq g = u g − τ g Fg

(3.60)

6. Complete the usual Lattice Bhatnagar-Gross-Krook (LBGK) collision and streameq ing steps using u f , ueq g , and the corresponding macroscopic density for computaeq eq tion of f and g accordingly.

3.8 Boundary Condition Boundary conditions must be applied to the boundary of the fluid domain. Because the LBM has different variables from conventional fluid mechanics, applying the boundary condition to the LBM model is not straightforward like the conventional computational fluid dynamics application. The LBM has the particle velocity distributions as the nodal variable, while conventional fluid mechanics has fluid velocity and pressure as the nodal variables. As a result, fluid velocity and pressure boundary conditions must be translated into the particle density distribution conditions. Some of common boundary conditions are discussed in this section.

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72

Multiscale and Multiphysics Modeling

Before streaming

After streaming with periodic boundary FIGURE 3.7 Streaming from north to south as a periodic boundary.

3.8.1 Periodic Boundary Periodic boundary conditions are common and easy to implement into the LBM. For nodes along a periodic boundary, the nodes along the counterpart periodic boundary are assigned as the nearest neighbor for streaming purposes. For example, the density distribution along the north direction at the north boundary node is streamed into the north direction at the south boundary node, which is supposed to be overlapped with the north boundary node with the periodic boundary condition, as shown in Figure 3.7. 3.8.2 Fixed Rigid Boundary Fixed rigid boundaries appear in a wide variety of applications. A flat rigid boundary is modeled using the so-called bounce-back technique in which all unknown values of fα are replaced with the values that are known, but from the opposite direction. In addition, directions parallel to the solid boundary are also reversed, resulting in the exchange of density distribution values for all opposing directions. This is illustrated in Figure 3.8. Before bounceback

Boundary FIGURE 3.8 Application of on-grid bounce-back boundary condition.

© 2016 by Taylor & Francis Group, LLC

After bounceback

Boundary

73

Lattice Boltzmann Method

Fluid lattice point

Solid boundary

Solid lattice point

Bounced-back directions FIGURE 3.9 Halfway bounce-back solid boundary condition schematic.

Solid boundaries implemented in this fashion are often referred to as “dry nodes” because they do not undergo the collision process. This simplifies implementation and computational efficiency considerably because macroscopic values need not be computed at these nodes and the equilibrium distribution does not need to be evaluated. This so-called ongrid version of the bounce-back boundary conditions has been shown to be first-order accurate. An alternate scheme was introduced where the lattice points are arranged so that the physical wall is actually located exactly halfway between the first fluid point inside the domain and the corresponding solid node representing the wall. This scheme is illustrated in Figure 3.9 and has been shown to exhibit second-order convergence. 3.8.3 Pressure Boundary Condition The pressure boundary condition at the inlet is described first for the D2Q9 lattice. The inlet is assumed to be located at the left side (i.e., west boundary) of the domain, which is normal to the x axis direction. First, the inlet pressure pin is converted into the density as ρin. Then, the distribution density function is written as f1 + f5 + f8 = ρin − (f0 + f2 + f3 + f4 + f6 + f 7)

(3.61)

f1 + f5 + f8 = ρin (uin)x + (f3 +f6 + f 7)

(3.62)

f5 − f8 = − f2 + f4 − f6 + f 7

(3.63)

Because f1, f2, and f8 contribute to the flow domain, they are obtained from these equations along with the inlet velocity (uin)x.

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Multiscale and Multiphysics Modeling

(uin)x = 1 − ( f0 + f2 + f4 + 2f3 + 2f6 + 2f 7)/ρin

(3.64)

2 ρin (uin )x 3

(3.65)

f 5 = f7 −

1 1 ( f2 − f 4 ) + ρin (uin )x 2 6

(3.66)

f 8 = f6 +

1 1 ( f2 − f 4 ) + ρin (uin )x 2 6

(3.67)

f1 = f 3 +

The pressure boundary condition at the outlet is described as follows: The outlet is located at the right side (i.e., east boundary) of the domain, which is normal to the x axis direction. First, the outlet pressure pout is converted into the density as ρout. Then, the distribution density function is written as f3 + f6 + f 7 = ρout − (f0 + f1 + f2 + f4 + f5 + f8)

(3.68)

f3 + f6 + f 7 = − ρout (uout)x + (f1 + f5 + f8)

(3.69)

f6 − f 7 = − f2 + f4 − f5 + f8

(3.70)

Because f3, f6, and f 7 contribute to the flow domain, they are obtained from these equations along with the outlet velocity (uout)x. (uout)x = [(f0 + f2 + f4 + 2f1 + 2f5 + 2f8)/ρout] − 1

(3.71)

2 ρout (uout )x 3

(3.72)

f6 = f 8 −

1 1 ( f2 − f 4 ) − ρout (uout )x 2 6

(3.73)

f7 = f 5 +

1 1 ( f2 − f 4 ) − ρout (uout )x 2 6

(3.74)

f 3 = f1 −

3.8.4 Velocity Boundary Condition The velocity inlet for the west side of the domain of the D2Q9 lattice, which is normal to the x axis, has the following density distribution function: f1 + f5 + f8 = ρ − ( f0 + f2 + f3 + f4 + f6 + f 7)

© 2016 by Taylor & Francis Group, LLC

(3.75)

75

Lattice Boltzmann Method

f1 + f5 + f8 = ρ(uin)x + ( f3 + f6 + f 7)

(3.76)

f5 − f8 = ρ(uin)y− f2 + f4 − f6 + f 7

(3.77)

From these equations, ρ=

1 ( f 0 + f 2 + f 4 + 2 f 3 + 2 f6 + 2 f7 ) 1 − (uin )x

(3.78)

2 ρ(uin )x 3

(3.79)

f 5 = f7 −

1 1 1 ( f2 − f 4 ) + ρ(uin )x + ρ(uin )y 2 6 2

(3.80)

f 8 = f6 +

1 1 1 ( f2 − f 4 ) + ρ(uin )x − ρ(uin )y 2 6 2

(3.81)

f1 = f 3 +

The velocity outlet for the east side of the domain, which is normal to the x axis, has the following density distribution function: f3 + f6 + f 7 = ρ − ( f0 + f1 + f2 + f4 + f5 + f8)

(3.82)

f3 + f6 + f 7 = −ρ(uout)x + (f1 + f5 + f8)

(3.83)

f6 − f 7 = ρ(uout)y− f2 + f4 − f5 + f8

(3.84)

From these equations, ρ=

1 ( f 0 + f 2 + f 4 + 2 f1 + 2 f 5 + 2 f 8 ) 1 + (uin )x

(3.85)

2 ρ(uout )x 3

(3.86)

f6 = f 8 −

1 1 1 ( f2 − f 4 ) − ρ(uout )x + ρ(uout )y 2 6 2

(3.87)

f7 = f 5 +

1 1 1 ( f2 − f 4 ) − ρ(uout )x − ρ(uout )y 2 6 2

(3.88)

f 3 = f1 −

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76

Multiscale and Multiphysics Modeling

For the D3Q15 model, the velocity boundary conditions may be described as follows: If the flow goes into the negative x plane, the density distribution functions satisfy the following equations: f1 + f 7 + f10 + f11 + f14 = ρ − ( f0 + f2 + f3 + f4 + f5 + f6 + f8 + f9 + f12 + f13)

(3.89)

f1 + f 7 + f10 + f11 + f14 = ρux + f2 + f8 + f9 + f12 + f13

(3.90)

f 7 − f10 + f11 − f14 = ρuy − f3 + f4 − f8 + f9 − f12 + f13

(3.91)

f 7 + f10 − f11 − f14 = ρuz − f5 + f6 − f8 − f9 + f12 + f13

(3.92)

From these equations, we obtain the following equations: ρ=

1 ( f0 + 2 f2 + f3 + f 4 + f5 + f6 + 2 f8 + 2 f9 + 2 f12 + 2 f13 ) 1 − ux

(3.93)

2 ρux 3

(3.94)

1 1 1 1 ( f 3 − f 4 + f 5 − f6 ) + ρux + ρuy + ρuz 4 12 4 4

(3.95)

f10 = f12 −

1 1 1 1 (− f 3 + f 4 + f 5 − f6 ) + ρux − ρuy + ρuz 4 12 4 4

(3.96)

f11 = f9 −

1 1 1 1 ( f 3 − f 4 − f 5 + f6 ) + ρux + ρuy − ρuz 4 12 4 4

(3.97)

f14 = f8 −

1 1 1 1 (− f 3 + f 4 − f 5 + f6 ) + ρux − ρuy − ρuz 4 12 4 4

(3.98)

f1 = f 2 +

f7 = f13 −

If there is a flow out of the positive x plane, the density distribution functions satisfy the following expressions: f2 + f8 + f9 + f12 + f13 = ρ − ( f0 + f1 + f3 + f4 + f5 + f6 + f 7 + f10 + f11 + f14)

(3.99)

f2 + f8 + f9 + f12 + f13 = −ρux + f1 + f 7 + f10 + f11 + f14

(3.100)

f8 − f9 + f12 − f13 = ρuy − f3 + f4 − f 7 + f10 − f11 + f14

(3.101)

f8 + f9 − f12 − f13 = ρuz − f5 + f6 − f 7 − f10 + f11 + f14

(3.102)

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Lattice Boltzmann Method

From the equations, we obtain the following expressions for the density and distributions: ρ=

1 ( f0 + 2 f1 + f3 + f 4 + f5 + f6 + 2 f7 + 2 f10 + 2 f11 + 2 f14 ) 1 + ux

(3.103)

2 ρux 3

(3.104)

1 1 1 1 ( f 3 − f 4 + f 5 − f6 ) − ρux − ρuy − ρuz 4 12 4 4

(3.105)

f10 = f12 +

1 1 1 1 (− f 3 + f 4 + f 5 − f6 ) − ρux + ρuy − ρuz 4 12 4 4

(3.106)

f11 = f9 +

1 1 1 1 ( f 3 − f 4 − f 5 + f6 ) − ρux − ρuy + ρuz 4 12 4 4

(3.107)

f14 = f8 +

1 1 1 1 (− f 3 + f 4 − f 5 + f6 ) − ρux + ρuy + ρuz 4 12 4 4

(3.108)

f 2 = f1 −

f13 = f7 +

When there is a flow into the negative y plane, the density distributions have the following relationships: f3 + f 7 + f8 + f11 + f12 = ρ − ( f0 + f1 + f2 + f4 + f5 + f6 + f9 + f10 + f13 + f14)

(3.109)

f 7 − f8 + f11 − f12 = ρux − f1 + f2 + f9 − f10 + f13 − f14

(3.110)

f3 + f 7 + f8 + f11 + f12 = ρuy + f4 + f9 + f10 + f13 + f14

(3.111)

f 7 + f8 − f11 − f12 = ρuz − f5 + f6 − f9 − f10 + f13 + f14

(3.112)

From these equations, ρ=

1 ( f0 + f1 + f2 + 2 f 4 + f5 + f6 + 2 f9 + 2 f10 + 2 f13 + 2 f14 ) 1 + uy 2 ρuy 3

(3.114)

1 1 1 1 ( f 3 − f 4 + f 5 − f6 ) + ρux + ρuy + ρuz 4 12 4 4

(3.115)

f3 = f 4 +

f7 = f13 −

(3.113)

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78

Multiscale and Multiphysics Modeling

f10 = f12 −

1 1 1 1 (− f 3 + f 4 + f 5 − f6 ) + ρux − ρuy + ρuz 4 12 4 4

(3.116)

f11 = f9 −

1 1 1 1 ( f 3 − f 4 − f 5 + f6 ) + ρux + ρuy − ρuz 4 12 4 4

(3.117)

f14 = f8 −

1 1 1 1 (− f 3 + f 4 − f 5 + f6 ) + ρux − ρuy − ρuz 4 12 4 4

(3.118)

A similar expression can also be derived for the positive y plane as well as z planes. Those expressions derived previously require intensive bookkeeping in terms of computer programming because the identity of each boundary plane must be provided for each boundary lattice point. For instance, it is necessary to know the normal vector to each boundary node as well as the direction of flow, such as inflow or outflow. To avoid such cumbersome data, another approach was proposed to describe the velocity boundary conditions. This approach does not require bookkeeping because it can be applied to any lattice point regardless of its location and the flow direction. In this approach, the present velocities are computed from the present density distributions at the lattice point where velocities are prescribed as follows for D3Q15:  ux = ( f1 + f7 + f10 + f11 + f14 − f2 − f8 − f9 + f12 − f13 )/ρ

(3.119)

 uy = ( f3 + f7 + f8 + f11 + f12 − f 4 − f9 − f10 + f13 − f14 )/ρ

(3.120)

 uz = ( f5 + f7 + f8 + f9 + f10 − f6 − f11 − f12 + f13 − f14 )/ρ

(3.121)

Then, the present velocities are subtracted from the described velocities as follows:  ∆ux = uxp − ux

(3.122)

 ∆uy = uyp − uy

(3.123)

 ∆uz = uzp − uz

(3.124)

where the superscript p denotes the prescribed value as a boundary condition. The final density distributions are given as

© 2016 by Taylor & Francis Group, LLC

f1 = f1 + ρΔux/3

(3.125)

f2 = f2 − ρΔux/3

(3.126)

f3 = f3 + ρΔuy/3

(3.127)

79

Lattice Boltzmann Method

f4 = f4 − ρΔuy/3

(3.128)

f5 = f5 + ρΔuz/3

(3.129)

f6 = f6 − ρΔzz/3

(3.130)

f 7 = f 7 + ρΔux/24 +ρΔuy/24 + ρΔuz/24

(3.131)

f8 = f8 − ρΔux/24 + ρΔuy/24 + ρΔuz/24

(3.132)

f9 = f9 − ρΔux/24 − ρΔuy/24 + ρΔuz/24

(3.133)

f10 = f10 + ρΔux/24 − ρΔuy/24 + ρΔuz/24

(3.134)

f11 = f11 + ρΔux/24 + ρΔuy/24 − ρΔuz/24

(3.135)

f12 = f12 − ρΔux/24 + ρΔuy/24 − ρΔuz/24

(3.136)

f13 = f13 − ρΔux/24 − ρΔuy/24 − ρΔuz/24

(3.137)

f14 = f14 + ρΔux/24 − ρΔuy/24 − ρΔuz/24

(3.138)

As stated, these expressions can be used for any boundary lattice point with a prescribed velocity condition regardless of whether the point is located on an inlet or outlet or x plane, y plane, or z plane.

3.9 Turbulent Flow A turbulent model is incorporated into the LBM by modifying the viscosity. The total viscosity of a fluid is the sum of the molecular viscosity and the eddy viscosity resulting from turbulence [40]: ν t = νo + νe

(3.139)

where νt is the total viscosity, and νo and νe are the molecular and eddy viscosities, respectively. The viscosity is related to the relaxation constant τ as follows:  1 ν = τ −  2 

(3.140)

As a result, the relaxation constant can be written as τt = τo + τe

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(3.141)

80

Multiscale and Multiphysics Modeling

The eddy viscosity is computed using the Prandtl mixing length approach [37] as follows: νe = (κlmix)2 |ε|

(3.142)

where κ is the von Karman constant and is equal to 0.41, and |ε| is the magnitude of the strain rate tensor, which can be computed from ε≈−

1 2pτ

∑e e ( f ji ji

ji

− f ji )

(3.143)

ji

in which p is the pressure. The mixing length is determined based on the two-layer model. If the flow is close to a boundary wall, the distance from the wall is assumed equal to the mixing length. On the other hand, if the flow is far from the boundary wall, the mixing length is considered a constant value.

3.10 Wave Equation The previous sections presented various LBM formulations to solve fluid flows. This section discusses the LBM formulation to analyze the wave equation. The LBM equation based on the single parameter is used as before and expressed as

(

    1 eq  fi (r + ei ∆t , t + ∆t) − fi (r ,t) = fi (r ,t) − fi (r ,t) τ

)

(3.144)

  where fi (r , t) denotes the probability of finding a particle at lattice site r and  timet that moves along the ith lattice direction with the local discrete particle velocity ei . fieq (r , t) is the local equilibrium solution. Furthermore, Δt is the time increment, and τ is the relaxation time. For the wave equation using the 2-D square lattice, the D2Q5 lattice structure is used [41]. The five lattice points include the points at the center, east, west, north, and south. The local equilibrium of particle distribution is expressed as

fieq =

 2 c s2  − 1   ϕ if i = 0 e2    c s2 1 ei ⋅ ζ ϕ+ if i ≠ 0 2 e2 2e 2

(3.145)

where ϕ=

∑f i

© 2016 by Taylor & Francis Group, LLC

i

(3.146)

81

Lattice Boltzmann Method

and  ζ=



∑fe

(3.147)

i i

i

and cs is the wave propagation speed. The local velocity vectors are given as follows: 0  eα =  0

1 0

0 1

−1 0

0   −1 

(3.148)

With the choice of τ = 1/2, Equation 3.144 is rewritten as

  fi (r + ∆tei , t + ∆t) =

 n2 − 1 ϕ − fo (r , t) if i = 0 2 n  µ ϕ − fi+ 2 (r , t) if i ≠ 0 2 2n

2µ

(3.149)

and the wave equation is recovered. For these equations, 0 ≤ μ ≤ 1 is the wave attenuation factor. If μ = 0, it denotes perfect reflection of the wave. On the other hand, μ = 1 indicates perfect transmission. Any in-between value for μ represents that the wave is partially absorbed. Furthermore, n ≥ 1 is the refraction index, which is defined as the ratio of the maximum propagation speed of the model to cs [41].

3.11 Scaling Most of the LBM literature is cast in “lattice units,” with the lattice spacing and the time increment unity. To solve a problem in a physical domain, it is necessary to convert the physical units into the lattice units. As an intermediate step, it is sometimes customary to rescale physical units to nondimensional units. This is particularly useful if some knowledge of the system state in terms of some nondimensional parameters such as the Reynolds number is needed. Figure 3.10 illustrates the process to change from the physical units to the lattice units by way of the nondimensional units.

TP, LP

TD

TP LP LD T0,P L0,P

Physical units

T0, L0

TLBM

Dimensionless units

FIGURE 3.10 Scaling from physical units to dimensionless units to LBM units.

© 2016 by Taylor & Francis Group, LLC

T0

Nt

LLBM

L0 Nx

TLBM, LLBM

Lattice units

82

Multiscale and Multiphysics Modeling

To illustrate the unit conversion, an example is presented as shown in Figure 3.11. The problem involves flow within a 2-D channel around a circular object. Flow enters from the left boundary with a prescribed parabolic velocity and exits the right boundary with a prescribed constant pressure. The top and bottom boundary are modeled as no-slip walls. The process of scaling for this example problem is completed in two steps as described previously. First, a characteristic time scale T0 and length scale L0 are identified. For this problem of a circular obstruction in 2-D channel flow, the natural choice is to use the conventions for Reynolds scaling where the characteristic length is the diameter of the circle. Therefore, the characteristic length in physical units L0,p = 0.2 m. The characteristic time is assigned to be the time required for an average fluid particle to traverse the diameter of the cylinder. For the assigned inlet boundary condition, the average fluid velocity is twothirds of the maximum inlet velocity of 0.5 m/s for the parabolic profile. Consequently, T0,p = L0,p/U0,p = 0.2/0.5 = 0.4 s. All of this corresponds to a Reynolds number of 100, which is convenient to know when comparing the output of the LBM simulation against experimental data or benchmark values. The second step is to decide how finely the reference time and space scales are to be subdivided. For this example, the reference length L 0 will be represented with 25 lattice points. In terms of dimensionless units, L 0 = 1. The conversion between dimensionless units and the LBM units is LLBM = δx = 1/(25 – 1) = 0.0417. To convert between a distance in terms of lattice units and a distance in physical units, one would multiply by both the conversion factors. Therefore, the physical spacing of the lattice points is δx × L0,p = 0.0083 m. Similarly, the time domain is discretized by deciding how many time steps will be used to traverse a single unit of the reference time T0. For this problem, the reference time will be divided by 250 time steps, so δt = T0/Nt = 1/250 = 0.004 s. As with the spatial scaling, to convert a single lattice time step to physical elapsed time, one must multiply by the scaling parameters between the dimensionless units and lattice units in addition to the conversion between physical and dimensionless units. For this problem, those conversions are TP = TLBM × δt × T0 = 0.0016. Once the spatial and temporal scaling factors are determined, the properly scaled LBM parameters must be determined from the given physical data.

Y

X, Y = (0,1) 1 u (0, y)= 1 2

y 0.5 0.5

2

No-slip boundaries top/bottom L=1m

P = 0 Pa

D = 0.2 m

X, Y = (0,0) v = le

3

m2

FIGURE 3.11 Schematic diagram of the channel flow example problem.

© 2016 by Taylor & Francis Group, LLC

X

s

ρ = 1000

kg

m3

X, Y = (4,0)

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Lattice Boltzmann Method

3.11.1 Viscosity Scaling To obtain dynamic LBM behavior that corresponds to the desired physical fluid under the prescribed conditions, the temporal and spatial scaling factors need to be applied to the specified fluid kinematic viscosity ν. Because ν has units of square meters per second, the necessary conversion can be accomplished using Equation 3.150. νLBM = ν physical ×

T0δ t (L0δ x )2

(3.150)

Carrying out this conversion for the specified fluid with the chosen discretization results in νLBM = 0.023. This is the value that is applied to Equation 3.9 to determine the LBM relaxation parameter, τ = 0.57. 3.11.2 Velocity Boundary Condition Scaling The problem shown in Figure 3.11 has a prescribed inlet velocity profile. The velocity is expressed in terms of meters per second, which is not compatible with the unit system assumed when the LBM boundary conditions were developed. The velocity is simply scaled in accordance with Equation 3.151. uLBM = uphysical ×

T0δ t L0δ x

(3.151)

For this problem, this conversion reads: u(0, y ) =

 ( y − 0.5)  ( y − 0.5)  0.0016 3 = 0.144 1 − 1 − ×  0.5  4 0.5  0.0083 

(3.152)

This is the velocity that will be passed to the LBM time-stepping routine to set the prescribed velocity at the inlet lattice points. 3.11.3 Pressure Boundary Condition Scaling For the problem shown in Figure 3.11, the prescribed outlet pressure is set to 0 Pa. This is a relative pressure, of course. Otherwise, the density for lattice points at the outlet would be set to zero. The pressure in the physical units is expressed as δ L  c  x 0  δ tT0 

2 LBM s

Pphysical = ρ

2

(3.153)

The solution procedure is provided in Figure 3.12; the solutions after 50,000 time steps are illustrated in Figure 3.13 [42].

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84

Multiscale and Multiphysics Modeling

Start

Increment time step

Apply bounce-back boundary condition

Solid

Type of lattice point

BC

Apply microscopic and macroscopic boundary conditions

Fluid Compute equilibrium density distribution No

Collide

Stream

Last time step?

Yes End FIGURE 3.12 LBM time step flowchart.

© 2016 by Taylor & Francis Group, LLC

85

Lattice Boltzmann Method

FIGURE 3.13 Velocity magnitude (top), pressure (middle), and vorticity magnitude (bottom) for example flow case after 50,000 time steps.

3.12 Example Problems 3.12.1 Poiseuille Flow The first test case is the Poiseuille flow which is a 2-D flow between parallel plates. The flow condition is depicted in Figure 3.14. In the figure, the channel width 2b is 1 m wide, fluid density ρ is 1000 kg/m3, and fluid viscosity μ is 1 N s/m2. The maximum inlet velocity Umax is 0.015 m/s. The solution to this problem is known to be a function of y only and is given as u( y ) = − where

1 dp 2 (b − y 2 ) 2µ dx

(3.154)

dp is given as a function of maximum velocity and fluid viscosity as follows: dx dp 2µ = − 2 U max dx b Y

(3.155)

No slip wall

b X

b

u(0, y, t) = Umax 1

y b b

FIGURE 3.14 Poiseuille flow configuration.

© 2016 by Taylor & Francis Group, LLC

2

No slip wall

p(L, y, t) = pout = constant

86

Multiscale and Multiphysics Modeling

For the LBM model of this problem, the D2Q9 lattice with the single-parameter collision operator is used along with the on-grid boundary conditions for the prescribed velocity on the west boundary and constant prescribed pressure on the east boundary. The initial lattice discretization is set so that 30 lattice points can span the channel entrance. The time step is set to achieve a relaxation parameter ω of 1.30. While refining the grid to test for convergence, the time step is adjusted to maintain a constant relaxation parameter for all tests. The results are shown in Figure 3.15 [42]. As expected, first-order convergence is obtained for this boundary condition. The halfway bounce-back boundary condition is also implemented and used in an identical set of tests. The goal of this step, in addition to showing the convergence properties of the boundary condition, is to illustrate the second-order convergence properties of the LBM. Results for double precision are shown in Figure 3.16 [42]. In this study, it may seem arbitrary to have selected a constant relaxation parameter ω = 1/τ = 1.25. This conclusion is partially correct insofar as there is considerable flexibility On-grid double precision

1

Relative error –1 slope –2 slope

Relative error

0.1 0.01 0.001 0.0001 0.00001

1.48

1.78 2.08 2.38 Log grid resolution

2.68

FIGURE 3.15 Poiseuille flow convergence with on-grid bounce-back boundary conditions. Halfway double precision

1

Relative error –1 slope –2 slope

Relative error

0.1 0.01 0.001 0.0001 0.00001 0.000001

1.48

1.78

2.08 2.38 Grid resolution

FIGURE 3.16 Poiseuille flow convergence with halfway bounce-back boundary conditions.

© 2016 by Taylor & Francis Group, LLC

2.68

87

Lattice Boltzmann Method

Horizontal velocity fluctuation x/Lx = 0.75 vs. time step, Re = 10 0.06 0.05

Umax (m/sec)

0.04 0.03 0.02 0.01 0 0.01 (a)

0

1

2

3

4 5 6 Time step

7

8

9

10 ×104

Horizontal velocity fluctuation x/Lx = 0.75 vs. time step, Re = 10 0.06 0.05

Umax (m/sec)

0.04 0.03 0.02 0.01 0 0.01 (b)

0

1

2

3

4 5 Time step

6

7

8 ×105

FIGURE 3.17 1 Stabilization time for Poiseuille flow, Re = 10, ω = = 1.3: (a) with Ny = 30 and (b) with Ny = 480. τ

regarding how this value is picked. Recall that the fluid viscosity in lattice units is scaled δt by 2 in accordance with Equation 3.150. Consequently, if δx is reduced by a factor of 2, δt δx must be reduced by a factor of 4. With this refined time step, the number of time steps is increased by a factor of 4 for the fluid simulation, including the time required for the LBM simulation to arrive at the equilibrium from nonequilibrium initial conditions. For a given value of ω, this initial instability can last for many time steps. Even for this simple problem geometry, the LBM system does not reach a stable answer for many time steps. An illustration of this is given in Figure 3.17. This figure shows variation in the horizontal velocity at the center of the channel geometry for lattice density of Ny = 30 and 480, respectively. Note the change in time scales for the time step axis. 3.12.2 Backward-Facing Step The occurrence of flow separation of internal flows by sudden geometric changes is well known and is important to engineering applications. While the Poiseuille flow test case is

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convenient for code validation insofar as analytic solutions are known, it does not fully test the ability of the LBM to reproduce correct fluid behavior. It will be shown that for modest Reynolds numbers, the LBM captures flow separation typified by the backward-facing step problem accurately. For this benchmark study, the experimental results presented in Reference 43 are used. The problem setup is as depicted in Figure 3.18. For these computations, the inlet boundary conditions are prescribed velocity and for the outlet is prescribed pressure. The multiple relaxation scheme for D2Q9 is used for bulk dynamics. Measurements were taken based on streamlines computed from the velocity data. An example for a Reynolds number of 100 is provided in Figure 3.19 [42]. To conveniently compare with measured benchmark values, representative data points are pulled from Reference 43 and plotted separately along with the values taken from the computed data. For each successively increased Reynolds number, the lattice density and time step were both adjusted to maintain a constant relaxation factor ω = 1.25. Results are given in Figure 3.20 [42]. Good agreement can be seen in this case with experimental results. 3.12.3 Lid-Driven Cavity Another validation example of the LBM implementation is the lid-driven flow in two dimensions. A commonly used benchmark for this flow condition is given in Reference 44. A schematic illustration of the 2-D lid-driven cavity problem is given in Figure 3.21. No-slip wall b

Y

b p(L, y, t) = pout = constant X

No-slip wall

FIGURE 3.18 Schematic of domain and boundary conditions for backward-step benchmark in two dimensions. Backward step, Re = 100

0.731 FIGURE 3.19 Backward-step simulation. Step height = 0.25 m, outlet width = 0.5 m, Re = 100.

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Primary vortex length vs. Reynolds number 10 9

Vortex length (×1/s)

8 7 6 ×1 Armaly et al. 1983

5

×1 LBM

4 3 2 1 0

100

400 200 300 Reynolds number

500

FIGURE 3.20 Comparison of primary vortex reattachment length normalized by step height with results reported in Reference 45.

u(x,t) = constant = ulid Moving boundary Y No-slip boundary X

FIGURE 3.21 Schematic of the 2-D lid-driven cavity problem.

The on-grid bounce-back technique is applied to model no-slip stationary walls. The moving-wall boundary condition is used to model the lid. The lid velocity is set to a constant value in the x direction with the desired speed. The standard benchmark stipulates a Reynolds number of 1000, although other authors have published results at higher Reynolds numbers. In two dimensions, the D2Q9 lattice is used with either the single- or multiple-relaxation LBM techniques.

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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

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1

FIGURE 3.22 Lid-driven cavity in two dimensions with 1600 × 1600 lattice showing, from left to right, streamlines, vorticity contours, and pressure contours for Re = 1000. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

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FIGURE 3.23 Lid-driven cavity in two dimensions with 1600 × 1600 lattice showing, from left to right, streamlines, vorticity contours, and pressure contours for Re = 5000. X-velocity comparison to benchmark

1

LBM simulation Ghia et al.

0.9 0.8

Y-position

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 –1

–0.8

–0.6

–0.4

–0.2

X-velocity

FIGURE 3.24 Comparison of X velocity for lid-driven cavity flow for Re = 1000.

© 2016 by Taylor & Francis Group, LLC

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Figures 3.22 and 3.23 show the streamlines, pressure contours, and vorticity contours for Reynolds numbers of 1000 and 5000, respectively [42]. The results compare well with the results in References 45 and 46. In Figures 3.24 through 3.29, two samples of velocity are taken in the computed problem domain. The first is the x velocity component sampled along the vertical centerline. As can be seen, good agreement with benchmark values is obtained for all macroscopic fluid parameters. Y-velocity comparison to benchmark

0.4 0.3 0.2

LBM simulation Ghia et al.

Velocity

0.1 0

–0.1 –0.2 –0.3 –0.4 –0.5 –0.6

0

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0.3

0.4 0.5 0.6 X-position

0.7

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1

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1

FIGURE 3.25 Comparison of Y velocity for lid-driven cavity flow for Re = 1000.

Pressure profile horizontal slice

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LBM simulation Ghia et al.

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Pressure (Pa)

0.06 0.04 0.02 0 –0.02 0

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X-position FIGURE 3.26 Comparison of pressure along the central horizontal line for lid-driven cavity flow for Re = 1000.

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Pressure profile vertical slice

1

LBM simulation Ghia et al.

0.9 0.8

Pressure (Pa)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 –0.02

0

0.02

0.04 0.06 Y-position

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0.12

0.1

FIGURE 3.27 Comparison of pressure along the central vertical line for lid-driven cavity flow for Re = 1000. Vorticity comparison horizontal slice

4

LBM simulation Ghia et al.

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0 –2 –4 –6 –8 –10

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X-position FIGURE 3.28 Comparison of vorticity along the central horizontal line for lid-driven cavity flow for Re = 1000.

3.12.4 Channel Flow over Cylinder The flow configuration for the example of channel flow over a cylinder is illustrated in Figure 3.30. Constant velocity is specified on the inlet, and constant pressure is specified on the outlet. The top and bottom of the domain are simulated as periodic boundaries with the result that the effective domain is a linear array of cylinders in uniform flow. There is similar work for the benchmarks [47–53]. The trailing vortex region for this benchmark was measured visually following flow simulation. Numeric results are given in Table 3.1

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Vorticity comparison vertical slice

1 0.9

LBM simulation Ghia et al.

0.8

Y-position

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 –5

0

5 10 Vorticity

15

20

FIGURE 3.29 Comparison of vorticity along the central vertical line for lid-driven cavity flow for Re = 1000.

Y

X, Y = (0,1)

Periodic top/bottom

X

L=1m

u(0, y) = U0

P = 0 Pa

D = 0.1 meter

X, Y = (0,0)

m2 v = le – 3 sec

ρ = 1000

Kg

m3

FIGURE 3.30 Channel with cylindrical obstacle for the 2-D problem.

TABLE 3.1 Comparison of Trailing Vortex Length to Benchmark Values Reference 49 Reference 51 Reference 53 Reference 55 Present solution

© 2016 by Taylor & Francis Group, LLC

Re = 20

Re = 40

0.92 0.91 0.94 1.04 0.95

2.20 2.18 2.35 2.55 2.05

X, Y = (4,0)

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and graphically in Figure 3.31 [42]. It can be seen from the table that the computed values using LBM are comparable to those reported in the literature. Above a Reynolds number of approximately 45, the trailing vortex detaches in an alternating pattern referred to as a Von Karman street. As a last measure, the rate of vortex shedding was measured and the nondimensional Strouhal number was evaluated, with the result compared to the literature. A visualization of the vorticity in the wake of a circular cylinder in channel flow at a Reynolds number of 100 is given in Figure 3.32. Notice the alternating regions of positive and negative vorticity resulting from the vortex shedding alternately from the top and bottom of the cylinder. The equation for the Strouhal number is given in Equation 3.156: St =

f L0 U0

(3.156)

where St is the nondimensional Strouhal number, f is the frequency of vortex shedding, L0 is the characteristic length, and U0 is the characteristic velocity. For this problem, the characteristic length is the diameter of the cylinder, and the characteristic velocity is the average flow velocity. During flow simulation, the drag and lift forces were computed; results are presented in Figure 3.33. The Strouhal number for this simulation is determined by taking the discrete Fourier transform of the computed coefficient of lift data and applying this along with U0 and L0 in Equation 3.156. The resulting spectrum is presented in Figure 3.34 [42]. The result was compared with others reported in the literature and showed good agreement (Table 3.2).

0.950

(a)

2.05

(b) FIGURE 3.31 Streamline visualization of trailing vortex at (a) Re = 20 and (b) Re = 40.

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Vorticity Z 0

100 151.979

100 146.2506

FIGURE 3.32 Vorticity plot for cylinder in 2-D flow at Re = 100. Drag/lift coef f icient versus time, Re = 100

2

CL CD

Drag/lift coef f icient

1.5 1 0.5 0 0.5 0

1

2

3 4 5 6 Simulation time (sec)

7

8

9

FIGURE 3.33 Drag and lift coefficient for cylinder in uniform flow, Re = 100. Strouhal number from CL energy spectrum

Energy magnitude

0.12 0.1 0.08 0.06 0.04 0.02 0

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Strouhal number

0.8

0.9

FIGURE 3.34 Strouhal number computed from the energy spectra of the lift coefficient at Re = 100.

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TABLE 3.2 Comparison of Strouhal Number to Benchmark Values Strouhal Number Reference 55 Reference 52 Reference 49 Present study

0.16 0.171 0.172 0.169

3.13 Lattice Boltzmann Method Implementation on Graphics Processing Units With recent advances in modern graphics processing units (GPUs), the interest in using these devices for scientific calculations has been growing. In particular, the memory bandwidth and computing capability of GPUs compared to contemporary central processing units (CPUs) has made their use for LBM applications particularly appealing. Implementations of the LBM using CUDA® (NVIDIA) and the C programming language have been published and the viability of the GPU as an effective platform for executing the LBM has been demonstrated extensively [54–56]. In this section, the computational requirements for the LBM are reviewed. Next, the NVIDIA CUDA GPU computing platform is introduced, and its use for the LBM simulations presented in this work is outlined. Comparisons are made to recently published performance benchmarks for systems with a single GPU. Last, multi-GPU implementations of the LBM in hybrid parallel schemes employing CUDA with Open Multi-Processing (OpenMP) as well as CUDA with Message Passing Interface (MPI) are presented. Performance of these codes is compared with a more conventional parallel implementation with MPI. 3.13.1 Computational Requirements for the Lattice Boltzmann Method Although the operations to be executed for each lattice point during each time step are conceptually straightforward and easy to implement on a computer, it has been well recognized that the LBM is particularly computationally intensive and memory demanding [57]. Considerations for precision and stability combine to dictate a requirement for a large number of lattice points to effectively discretize a problem domain. For a large number of lattice points, each lattice point in the domain requires storage for each value of fα: 9 values for the popular D2Q9 lattice and 13–27 values for most commonly used 3-D lattices. This is roughly double the requirement for a more traditional solver for the incompressible Navier-Stokes equation. In addition to ample memory and computational capability, the LBM requires great memory bandwidth so that data can be streamed into the computing cores. For each time step, each value of fα must be loaded from memory and stored again at least once. The number of floating point operations needed depends on the choice of boundary conditions and collision operator, but as a rule of thumb, roughly 20 floating point operations are required for every value of fα. This implies that to achieve a computational performance of

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97

a high-end CPU, say 500 billion floating point operations per second (500 GFLOPS), with single-precision arithmetic, the computing device would require a memory bandwidth of at least 200 gigabytes (GB) per second. 3.13.2 Basic Implementation on Graphics Processing Units The basic implementation of the LBM on the GPU is discussed here. The discussion has two parts. The essential calculations required for the LBM routine are considered followed by the question of how to best arrange the main LBM variables—fα for all of the lattice points—in memory. 3.13.2.1 LBM Routine Every LBM routine must provide for certain identifiable milestones; briefly, these are the following: • Problem initialization • Computation of macroscopic flow properties such as ρ and u • Enforcement of boundary conditions to force the proper flow and solve the correct problem • Collision to relax toward equilibrium • Streaming to propagate information across the LBM grid • Exportation of data to allow postprocessing Several methods for initializing the values of fα at each lattice point have been analyzed in the literature [56]. For this work, all lattice points are initialized by setting u = 0 and ρ equal to the nominal density of the fluid to be used in the simulation. Then, fαeq is computed using Equation 3.10. These tasks either can be done with the CPU prior to copying the lattice data to the GPU or can be implemented in a separate kernel prior to commencing time stepping. Computation of macroscopic properties and enforcement of boundary conditions are frequently done in conjunction with the collision step. This is done because the macroscopic properties are often required, within the context of the LBM simulation, for calculation of fαeq, which is required for collision and, for some schemes, boundary condition enforcement. To compute macroscopic properties separately from either boundary condition enforcement or collision would require storing the values in global GPU memory. For this reason, in light of the CUDA programming guidance to minimize global memory transactions, the steps of computing macroscopic flow properties, boundary condition enforcement, and collision are always done in the same kernel. The streaming step for the classical LBM is simply a data copy operation. While this is simple to implement, it is the subject of much research regarding how to best execute the streaming step in a way that memory accesses are coalesced. Last, any simulation is pointless if there is no way to evaluate the results. For this work, intermediate values for the fluid velocity and pressure field were periodically transferred from the GPU to the CPU and written to disk using Visualization Toolkit (VTK) file formats. For FSI computations, displacement, velocity, and acceleration data were similarly stored for later postprocessing.

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3.13.2.2 Data Layout The two principal alternative data layouts for the LBM computations are the so-called array of structures (AoS) or structure of arrays (SoA). The two alternatives are illustrated schematically in Figure 3.35 [42]. In AoS, the density distribution values fα for a given lattice point are assigned in consecutive memory locations. In SoA, the density distribution for all of the lattice points for a given lattice speed are assigned consecutive memory locations; these are followed by the density distribution function for the next velocity and so on until all of the data have a location. For the LBM calculations conducted on the CPU, it is most appealing to use the AoS because this will allow the CPU to access sequential memory locations while accessing the data for a particular lattice point. This allows for efficient memory transfers as well as effective use of the memory cache hierarchy. In contrast, most LBM implementations on the GPU use the SoA approach. With the SoA, when data are loaded from memory within a kernel, each thread in a given warp reads from consecutive memory locations, as illustrated in Figure 3.36. When loads are coalesced in this fashion, the data are transferred

Consecutive memory locations 1

f0

1

f1

1

f2

i f0

i f1

i f2

N

N

N

N–2

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f α–3 fα–2 fα–1

(a) 1

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fi

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fi

(b)

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rea

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rea d3 Th rea d4

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rea d

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rea

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rea d Th

F loat f1 = fEven(1*nnodes+tid);

d1

f0

d1

Th

rea

Th

1

f0

0

Float f0 = fEven(0*nnodes+tid);

Th

rea

d0

FIGURE 3.35 Schematic of data layout schemes: (a) the AoS and (b) the SoA. Superscripts indicate lattice node number; subscripts indicate the lattice velocity.

4

5

f1

Consecutive memory locations FIGURE 3.36 When using SoA, load instructions executed by consecutive threads read from consecutive locations in memory.

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from memory in a single transaction. A similar condition exists during store operations as well. This is in conformance with the guidance to ensure coalesced memory access. As a rule of thumb, using the AoS approach on the GPU penalizes achievable performance by a factor of approximately two. 3.13.3 Performance Benchmark To compare the effectiveness of the implementation strategies adopted for this work, a 3-D lid-driven cavity problem was selected as a benchmark [42]. Three comparable works recently published in the literature were used as comparisons [54–56]. To make a fairer comparison between all of the results, the reported performance figures were normalized for memory bandwidth capability for the GPU device on which each comparable result was computed. The relevant characteristics of these devices are listed in Table 3.3. The normalized performance is shown in Figure 3.37 [42]. It can be seen from Figure 3.37 that the result of this work is comparable to or better than other recently reported implementations. To achieve the best performance for each problem size, the number of threads per block must be adjusted accordingly. The dependence of execution performance on the thread block size is illustrated in Figure 3.38. Using 96 threads per block performs well for all problem sizes, while the use of 256 threads in a block performs very poorly for all problem sizes. TABLE 3.3 Properties of GPU Devices Used in Benchmark Computations in Figure 3.38 Device

Million of lattice point updates per second

Number of CUDA cores Global memory (MB) Memory bandwidth (GB/s) Estimated peak performance (GFLOPS)

GTX-260

GTX-295

GTX-480

GTX-580

192 896 111.9 805

240 × 2 896 × 2 111.9 × 2 805 × 2

480 1536 177.4 1345

512 1536 192.2 1581

Lid driven cavity D3Q19 (scaled for memory bandwidth)

800 700 600 500

This work Rinaldi et al. (2012) Obrecht et al. (2010) Astorino et al. (2011)

400 300 200 100 0

32 3

64 3

96 3 128 3 144 3 160 3 N

FIGURE 3.37 Performance benchmark for LBM on a 3-D lid-driven cavity scaled for device memory bandwidth.

© 2016 by Taylor & Francis Group, LLC

100

Millions of lattice point updates per second

Multiscale and Multiphysics Modeling

Lid driven cavity D3Q19

800 700 600 500

64 threads per block 96 threads per block 128 threads per block 256 threads per block

400 300 200 100 0

32 3

64 3

96 3

N

128 3 144 3 160 3

FIGURE 3.38 LBM on a 3-D lid-driven cavity with various numbers of threads per block.

© 2016 by Taylor & Francis Group, LLC

4 Cellular Automata

4.1 Introduction Instead of providing a formal definition of a cellular automaton (plural: cellular automata, CA) directly, a simple example is used to explain CA. A square board is divided into equal spacing in both directions, and each small square is called a cell. Then, a finite number of colors are selected, and each cell is assigned to one of the colors. However, each cell cannot have the transition of any colors. For example, one cell can be red, another may be blue, but no cell can be half red and half blue. In this board, time is also as discrete as the colors of the cells are. In every time step, the colors of the cells can change according to a specified rule. As an example, the rule dictates the change in the color of any cell depending on the colors of neighboring cells and the cell’s own color. This rule applies to every cell at each time step, and the process repeats itself with every time step. The whole board with the given rule is called the CA. The objective of the CA technique is to represent or explain a complex physical or social phenomenon using a simple, automated rule. These cells do not have to be colored, but they are assigned to one of a finite number of states at any given time step. These states may be represented by colors or may be represented by integer numbers (0, 1, 2, …). Furthermore, a finite number of alphabets will do. Usually, the number of states is small, but in principle any finite number is acceptable. The way that the neighboring cells are defined may be different. One can only use the four cells on the east, west, north, and south (von Neumann neighborhood), but another can use eight cells, such as east, west, north, south, northeast, southeast, southwest, and northwest (Moore neighborhood). One can even use a hexagonal lattice instead of a square lattice. These are for the two-dimensional (2-D) cases. The concept can be extended to three-dimensional (3-D) problems. For example, the equivalents to von Neumann neighbors in 3-D are east, west, north, south, front, and back. In the beginning, it may be assumed that every cell is in the same state, except some finite number of cells that are in a different state. This is called the configuration. Let us give an example that was initially proposed by Edward Fredkin [1]. The rule is defined on a 2-D plane that consists of regular lattices (i.e., cells). Each cell is labeled by its position (i, j), where i and j are the row and column indices, respectively. Furthermore, each cell positioned at (i, j) has the state defined at iteration t. The state can be either 1 or 0. The CA rule determines the state of each cell at iteration t + 1 by using the states at iteration t. The rule starts from an initial condition at time t = 0 with a given configuration of the

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values ϕ 0 (i, j) on the lattice. The state at time t = 1 at a cell is determined based on the following rule: 1. For each cell, compute the sum of states (1 or 0) on the four nearest neighboring cells (north, south, east, and west). The domain has periodic boundaries on both i and j directions so that this calculation applies to all cells in the domain. 2. If the sum of the four neighbors is an even number, the new state ϕ1 (i, j) becomes 0 (or is called white) and 1 (or is called black) otherwise. 3. Repeat the process for the next time step. This CA rule can be expressed by the following mathematical expression: ϕi+1 (i, j) = ϕi (i + 1, j) + ϕi (i − 1, j) + ϕi (i, j + 1) + ϕi (i, j − 1)

(4.1)

ϕi+1 (i, j) = remainder of (ϕi+1 (i, j)/2)

(4.2)

Applying the CA rule starting with a small black square at the center of the domain develops a complex shape as the number of iterations continues. Figure 4.1 shows the shape after 119 iterations. This example shows that, despite the simplicity of the local rule, the behavior of a CA model can be complex [1]. In the previous example, the rule is applied to every cell homogeneously, leading to a synchronous dynamics. One way to introduce spatial inhomogeneity is using nonperiodic boundary conditions. The cells at the boundary do not have as many neighboring cells as those inside the domain. Therefore, if the boundary condition is not periodic, a new rule must be applied to the boundary cells. In other words, the boundary cells do not follow the same rule as the cells inside the domain. Furthermore, multiple rules can be applied selectively in tandem. For example, one rule is applied to one set of cells, while another rule is applied to the other set of cells. This application can also be in tandem in terms of the temporal domain. t = 119

50

100

150

200

250

50

100

150

200

250

FIGURE 4.1 The rule expressed in Equations 4.1 and 4.2 on a 256 × 256 periodic lattice after 119 iterations.

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The history of CA goes back to the 1940s when von Neumann introduced the concept of extracting the abstract mechanisms leading to self-reproduction of biological organisms [2]. In 1970, Conway developed the “game of life,” the well-known example of a CA [3]. The game of life has attracted much interest because the example showed that simple rules could generate complex patterns. In the 1980s, Wolfram showed that a CA can be applied to many behaviors of a continuous system, as summarized in References 4–6. Other areas that received attention regarding the CA technique include, but are not limited to, fluid dynamics problems such as porous media, granular flows, spreading of a liquid droplet and wetting phenomena, microemulsion and physical situations such as pattern formation, reaction-diffusion processes, nucleation-aggregation growth phenomena, traffic process, and so on.

4.2 Strengths and Weaknesses of Cellular Automata The power of the CA approach comes from its simplicity. In searching for a way to model a physical system, the traditional methodology has been to solve a set of equations (e.g., differential equations) that satisfies the complex behavior of the system and whose solution gives the desired answers to the system. With the increasing processing power of computers, a new way became feasible. Instead of trying to model the system as a whole, modeling it as a sum of parts becomes possible. By using the CA approach, one can model the system by means of simple local rules governing the behavior of the whole system. The CA model must use some simple (and intuitive or experimental at some level) local rules at the microscopic level but at the same time it must reflect the macroscopic behavior of the physical system under consideration. Numerically, an advantage of the CA approach is its simplicity and its ease of implementation on computers and parallel machines. In addition, working with Boolean quantities prevents the problem of instability. The weaknesses of the CA approach mostly result from its discrete nature. Some of them are the statistical noise requiring a systematic averaging process and a less flexible rule to describe a wider range of physical situations [1]. According to its definition and its nature, the CA approach seems not to be suitable for modeling large-scale moving objects because in the CA approach the only things changing are the states of the cells, not the positions of the cells. However, there are some suggested models [7] to overcome this problem. In the 1980s, some researchers [8,9] showed that directly working with real numbers representing the state of the cells has some advantages instead of working with Boolean cellular states. This approach is called the lattice Boltzmann method and is numerically more efficient than Boolean dynamics.

4.3 Modeling Moving Objects Using Cellular Automata The interest here is to model vibration of a string using the CA approach. In its current form, the main elements that constitute a CA domain are cells, which can be in a finite number of states, namely, 0, 1, 2, … ; white/black or dead/alive; and so on. The example in

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Section 4.1 demonstrated a CA model that consists of cells with the state of either white or black (1 or 0). On an automaton, the particles do not move from one site to another. Instead, the states of the cells change during iterations without any transport of matter. To model moving objects, Chopard proposed a simple model [7] with features to deal with large-scale objects that can move and interact with their surroundings and to allow these objects to have adjustable mass, energy, and momentum. In addition, in the mode, these objects maintain their sizes and their integrity during the evolution but the particles composing them do not spread out in the entire space. In the proposed model, the CA space is composed of particles on the lattice and the springs that connect and hold the particles together on the lattice. Only one particle can be linked to each end of a spring. The particles can be of two kinds: white and black. The end points of a spring can have either color, and the consecutive particles should have different colors. That is, both neighbors of a particle have the same color (e.g., both of the neighbors of a white particle must be black). A spring has an orientation and length. According to this definition, a particle initially on the arbitrary positive side of a spring should not pass to the other side during the evolution process. The particles are allowed to alternate in a 3-D cubic lattice, but no two particles are allowed to occupy the same lattice position in the same time step (this means no spring can be zero length or fold onto itself). An example of a one-dimensional (1-D) lattice (called a string) is illustrated in Figure 4.2. In this example, the positive x direction is arbitrarily selected to the right. The rule for time evolution of the internal particles (particles with two neighbors) is given as [7] u(t + 1) = u+ + u− − u(t)

(4.3)

where u(t) represents the position of a particle at time t, and u+ and u− represent the positions of two neighboring particles. This equation implies that the position of a particle at the next time step is a reflection of the particle with respect to the center of mass of its two neighbors. In Figure 4.2, u(t) represents the position of the second black particle, u(t) = 5. Therefore, u− and u+ denote the positions of the first and second white particles, respectively. It is apparent that Equation 4.3 is valid only for particles with two neighbors. For the particles at both end points, because they only have one neighbor, the reflection is performed with respect to u± a, where a is a constant that represents the unstretched length and orientation of the spring that links the particles. For convenience with the sign convention, a should be positive. Furthermore, to prevent the particles from moving off the lattice points, a should be an integer or half integer (assuming that the lattice coordinates are given in integers). We discuss a in detail further in the chapter.

1.5 1 0.5 0 –0.5

1

2

FIGURE 4.2 One-dimensional CA lattice.

© 2016 by Taylor & Francis Group, LLC

3

4

5

6 t=0

7

8

9

10

11

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(a)

(b)

FIGURE 4.3 Time evolution of one-dimensional string particles: (a) internal particle and (b) right-end particle.

The evolution rule for a particle at the left end of a string, whose current position of the particle is u(t), is given by u(t + 1) = 2(u+ − a) − u(t)

(4.4)

Similarly, the new position of a particle at the right end becomes u(t + 1) = 2(u− + a) − u(t)

(4.5)

The time evolution has two phases. At each time step, only one kind of particle can move. For instance, in the first time step, all the white particles are held fixed and the equations described previously are applied to black particles only. In the first time step, only black particles change their positions, and they move simultaneously. In the second time step, black particles are held fixed, and only white particles move on the lattice simultaneously, and so on [7]. Figure 4.3 demonstrates the time evolution of an internal particle and the right-end particle on a 1-D lattice with a = 3/2. The new position of the internal particle is calculated by reflecting the particle with respect to the center of mass of its two neighboring black particles using Equation 4.3. The new position of the particle at the right end of the string is calculated by reflecting the particle with respect to (u− + a) using Equation 4.5. 4.3.1 Spring Constant As mentioned previously, there are some constraints for defining the spring constant a. The constant should be positive and be an integer or half integer to prevent the particles from moving off the lattice points when assuming the lattice coordinates are given as integers. In addition, there are two other requirements. A spring should not fold over itself. For instance, a particle at the left end of the string should not pass to the right of its right neighbor in any of the time evolution steps. Similarly, a particle at the right end of the string should not pass to the left of its left neighbor in any of the time evolution steps. Here, right and left are defined assuming the positive x coordinate is increasing to the right. Last, the end particles should not go out of the lattice (not the same as moving off the lattice points). This last requirement can be relaxed according to the physical system modeled. If this is the case, one can ignore the constraint on a that comes from the last requirement. These additional two requirements are studied separately for the left-end and right-end particles next. 4.3.1.1 Left-End Particles The third requirement is that a spring should not fold over itself, which implies r(t + 1) < r+. Applying Equation 4.4 to this expression yields

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u(t + 1) < u+ 2(u+ − a) − u(t) < u+ 2 u+ − 2 a − u(t) < u+ a>

(4.6)

u+ − u(t) 2

The last requirement implies u(t + 1) ≥ xo, where xo represents the left-end coordinate of the lattice. Again, by substituting Equation 4.4 into the left-hand side of this equation, we obtain u(t + 1) ≥ xo 2(u+ − a) − u(t) ≥ xo a≤

(4.7)

xo + u(t) − 2 u+ 2

By combining these two expressions, the allowed interval of a for the left-end particle on the lattice becomes u+ − u(t) x + u(t) − 2 u+ r−. Substitution of Equation 4.5 into this expression yields u(t + 1) > u− 2(u− + a) − u(t) > u− 2 u− + 2 a − u(t) > u− a>

(4.9)

u(t) − u− 2

The last requirement implies u(t + 1) ≤ xL, where xL represents the right-end coordinate of the lattice. By substituting Equation 4.5 into the left-hand side of this equation, we obtain u(t + 1) ≤ xL 2(u− + a) − u(t) ≤ xL a≤

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xL + u(t) − 2 u− 2

(4.10)

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By combining these two, the allowed interval of a for the right-end particle on the lattice is r(t) − r− x + r(t) − 2 r− Rij(1)

1,   π(r − R(1) )   1 + cos  ( 2 ) ij(1)  /2 ,   Rij − Rij   0,

((

∑

Rij(1) < r < Rij( 2 )

(5.35)

r > Rij( 2 )

))

) (

Bij = 1 + Gi (θijk ) fik (rik ) ⋅ exp α ijk ⋅ rij − Rij( e ) − rik − Rik( e )      k (≠i , j)   c2 c02 G(θijk ) = a0 1 + 02 − 2 d0 d0 + (1 + cos θ)2 

{

}

− δi

(5.36)

  

(5.37)

where θ is the angle between atoms i − j and i − k bonds, r is the distance between atoms, and other constant variables are constant. These equations require a total of 11 fitting parameters for carbon atoms. The values for these parameters are given in Table 5.1, where the subscripts cc means that those parameters are used just for the carbon-carbon bonding network without other elements, such as hydrogen. The A-T-B potential energy function for two carbon atoms is plotted as a continuous function in Figure 5.1 using Equations 5.31 through 5.37. In the equilibrium state, two carbon atoms are separated by the equilibrium separation distance r0 ~ 1.42 Å, which is called the effective carbon bond length. At equilibrium, the potential energy of the system becomes minimal. Consequently, any equilibrated carbon nanotube (CNT) structure will be formed only if the potential energy per each carbon atom can attain a minimum-energy value as they approach each other. This minimum energy is defined as the bond energy. The bond energy for the CNT is known to be ε ≈ 2.5 eV, which is consistent with the carboncarbon tight bonding overlap energy accepted from numerous prior experiments [11,12].

TABLE 5.1 Optimized Parameters for the A-T-B Type Potential Parameter

Value

Parameter

Value

Parameter

Value

Rcc( e )

1.42 Å

αccc

0.0

d02

3.52

6.0 eV

( 1) cc

1.7 Å

(2) cc

2.0 Å 0.00020813 3302

(e) cc

D

βcc Scc δcc

2.1 Å 1.22 0.5

R

R a0 c02

Source: From D. W. Brenner, Physical Review B, vol. 42, 1990, pp. 9458–9471.

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3 2.5 2 1.5 1 0.5 0

–0.5 –1 –1.5 –2 –2.5 –3

Bond length ~ 1.42 Å

Bond energy ~ 2.5 eV

1

1.25

1.5

1.75

2

2.25

2.5

Interatomic separation, r (Å) FIGURE 5.1 Potential energy versus interatomic separation of A-T-B potential from MD simulation.

The simulated A-T-B potential energy as a function of interatomic separation distance can be seen in Figure 5.1. Although these arguments are considered for only two carbon atoms, similar arguments can be applied to the entire bonding networks consisting of many carbon atoms, as in CNTs. Because any system tends toward the minimum potential energy, carbon atoms exert forces on one another so that they can adjust themselves to the bottom of the potential energy as shown in Figure 5.1. Figure 5.1 shows that the A-T-B potential curve is asymmetric. The A-T-B potential curve is broader in the region r > r0. Thus, when two paired atoms vibrate about their equilibrium positions at a given temperature (i.e., stretching and compressing the bond according to kinetic molecular theory); the atoms will spend more time in the region r > r0, that is, more time in stretching the bond than in compressing the bond with respect to the equilibrium bond length r0. However, when the temperature is approximately room temperature 3 (~300 K), the mean vibration kinetic energy is 0.039 eV, which is computed from kT, and 2 the atoms will vibrate near the bottom of the potential well. In this case, the A-T-B curve can be taken as approximately symmetric; consequently, the thermal vibrational motion of two atoms can be approximated as a simple harmonic oscillator motion [13,14]. It is interesting to discuss the attractive force region because it gives us a hint for studying the deformation behaviors and Young’s modulus of CNTs under an external tensile load. For the range of 1.42 < r < 2.0 Å, the paired carbon atoms experience a significant force change when their separation reaches around 1.72 Å; namely, the interaction force derived from the potential energy function suddenly changes from the gentle positive slope to the steep negative slope at around 1.72 Å when it goes far away from the equilibrium position, as shown in Figure 5.2. Figure 5.2 shows the magnified parts of the attractive force region from Figure 5.1, and ∂U . the force function is evaluated as the negative gradient of the potential energy, F = − ∂r In the figure, the positive y axis represents the attractive force region, while negative y axis represents the repulsive force region. At the equilibrium position r0, the interaction force between two carbon atoms becomes zero, and then the attractive force increases

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5

Potential energy and force

4

Force function

3 2 dF

1 0

dr

1 2

Potential function

3 4 5 1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

Separation distance, r (Å) FIGURE 5.2 Idealized interaction force and potential function of one pair of carbon atoms.

until the separation distance reaches around 1.72 Å. However, as it goes away from that point, the attractive force decreases until the separation distance reaches 2.0 Å. For separations larger than about 2.0 Å, the interaction force between two atoms becomes essentially zero; this is a reason that the A-T-B potential is called a short-range interaction potential, and the interacting force becomes a strong repulsive force for r ≤ 1.42 Å. Thus, from this perspective, it can be anticipated that CNTs will show strong resistance against external compressive loads. If a CNT is subjected to an external tensile load, the system is in a nonequilibrium state. The displacement of neighboring atoms due to the tensile force results in a net attractive force; subsequently, this net force will be balanced by a portion of the applied force acting on these atoms. The elastic modulus can be evaluated by the measurement of the balance mentioned [14]. Thus, the elastic modulus E depends on the gradient of the interaction force F versus separation distance r at the equilibrium position r0 as shown in Figure 5.2 and can be written E=

1 r0

 ∂F  ⋅   ∂r  r = r0

(5.38)

Based on Equation 5.38, the computer-generated elastic modulus is 16.74 TPa when r0 = 1.42 Å. This elastic modulus is unrealistically high due to the consideration of the idealized one pair of carbon atoms. That is, there is no special effect of many-body interaction forces. Therefore, the realistic elastic modulus of a CNT system can be found only when the many-particle interaction forces are considered, including the bonding angle effect represented in Equation 5.37 within the previously mentioned A-T-B potential function. Meanwhile, it is geometrically impossible for every pair of carbon atoms in a CNT system to be separated corresponding to the minimum energy. Moreover, the thermal energy, which is manifested as interactions among atoms, constantly knocks atom pairs away from the minimum-energy separation, and the overall behavior of the collection of atoms, therefore, may be complex. Thus, it is expected that the actual elastic modulus of CNTs

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will converge to a realistically accepted value, the order of approximately 1 TPa, when all the facts mentioned are considered. 5.4.4 Embedded Atom Potential The embedded atom method describes the total energy of a system of metallic atoms: Etot =

∑ E (ρ ee

h ,i

)+

i

1 2

∑ φ (r ) ij

ij

(5.39)

i, j

Equation 5.39 states that the total energy of the system of atoms is the sum of the embedding energy and the core-core repulsions. Specifically, the embedding energy term treats each atom i as an impurity and is a calculation of the amount of energy required to embed atom i into the electron density of its actual location. The electron density of its actual location is determined by ρh , i =

∑ ρ (r ) a j

ij

(5.40)

j≠ i

Equation 5.40 states that the electron density at the position of atom i is a sum of the densities contributed by all other atoms in the sample within a given distance rij. Atoms beyond rij have no effect on atom i. The core-core repulsion term in Equation 5.39 is a calculation of the core-core repulsions ϕij of each atom i paired with all other atoms in the sample within a given distance rij. Similar to the electron density, atoms beyond rij have no core-core repulsion with atom i. The use of one-half in Equation 5.39 is necessary to avoid counting the same repulsion two times during summation over the entire sample.

5.5 Molecular Mechanics Formulation The molecular mechanics used for discrete atoms are described next: The force acting on an atom is computed from  ∂Φ  F(rij ) = − nr ∂rij

(5.41)

 in which nr is the unit position vector between two atoms. All resultant forces among N atoms along with any external load in a given system must be in equilibrium. Consider any two atoms located at positions i and j, called atom i and atom j, as shown in Figure 5.3. In that figure, solid circles indicate present positions of the two atoms under equilibrium before applying external loads. As external loads are applied to the system, the two atoms move to the positions denoted by circles, called atoms i* and j*, as seen in Figure 5.3. Then, displacement vectors from initial positions to final positions of the atoms

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Atom #j*

uj

Rij

Atom #i* ui

Atom #j

z

rij y

Atom #i

x FIGURE 5.3 Relative positions of two atoms before and after movement.

  are expressed as ui and u j , respectively. The position vectors of the two atoms at the initial   and final positions, respectively, are denoted by rij and Rij , as shown in the figure. The displacement vectors and position vectors are related as       Rij = rij + u j − ui = rij + ∆uij

(5.42)

   where uij = u j − ui is the relative displacement vector of the two atoms. The force between the two atoms i and j at their new equilibrium positions is expressed as Equation 5.43:   Fij (Rij ) = Fij (Rij )nR

(5.43)

 between the two displaced atoms. nR is the directional unit vecin which Rij is the distance  tor along the vector Rij, and it is expressed as follows:   rij + ∆uij  nR = Rij

(5.44)

Equation 5.44 is substituted into Equation 5.43, resulting in the following expression:    ∆uij rij Fij (Rij ) = Fij (Rij ) + Fij (Rij ) Rij Rij

(5.45)

Applying Equation 5.45 to atoms i and j yields a matrix expression as shown in Equation 5.46:

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        

k 0 0 −k 0 0

0 k 0 0 −k 0

0 0 k 0 0 −k

−k 0 0 k 0 0

0 −k 0 0 k 0

0 0 −k 0 0 k

          

uix   Fix     uiy   Fiy     uiz   Fiz  = u jx   Fjx     u jy   Fjy     u jz   Fjz 

(5.46)

where k=

Fix = − Fjx =

Fiy = − Fjy =

Fiz = − Fjz =

Fij (Rij )

(5.47)

Rij Fij (Rij ) Rij Fij (Rij ) Rij Fij (Rij ) Rij

( xi − x j )

(5.48)

(yi − y j )

(5.49)

( zi − z j )

(5.50)

Here, (xi, yi, zi) and (xj, yj, zj) are the coordinates of the atoms in Figure 5.3. The matrix expression, Equation 5.46, is computed for all atoms that interact with one another and is assembled into the system matrix [K] consisting of all atoms’ displacements {u}. The resultant system matrix equation is [M]{ü} + [K]{u} = {F}

(5.51)

where matrix [M] is the diagonal system mass matrix constructed from atomic masses, vector {F} is the system force vector resulting from atomic interacting forces and additional external loads, and the superimposed dot indicates a temporal derivative. In the  of  system equations, the stiffness matrix [K] and force vector {F} are nonlinear because Rij and Fij (Rij ) are not known a priori.

5.6 Application to Carbon Nanotubes: Elastic Modulus Understanding the basic geometrical structures of CNTs is a preliminary essential step to start MD simulations with the initial geometrical configurations of the given CNT systems. For example, it should be a structure that is properly energy minimized. This section describes the basic structural properties of CNTs, explains the synthesis possibility of the heterojunction-type nanotubes such as the bamboo shape nanotube (BSNT), and then

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presents the resultant initial geometrical models of single wall nanotubes (SWNTs) and bamboo shape single wall nanotubes (BSWNTs) [15,16]. 5.6.1 Basic Structures of Carbon Nanotubes Many experiments have confirmed that CNTs are cylindrical tubular shell structures based on a benzene-type hexagonal lattice of carbon atoms. The cylindrical tube can be tens of microns long, and the ends of nanotubes are “capped” with half-dome-shaped fullerene molecules. The ideal single-wall carbon nanotube (SWCNT) can be thought of as the fundamental structure that forms the building blocks of both multiwall carbon nanotubes (MWCNTs) and an ordered array of SWCNTs called nanoropes. It is known that Young’s modulus of MWCNTs is not significantly different from SWCNTs because the high modulus mainly results from the carbon-carbon interaction bonds within the individual layers. In addition, there is only a weak van der Waals interaction force between two graphite layers because the average interlayer spacing distance of 3.4 Å is larger than the maximum carbon-carbon bond length of about 2.0 Å. Thus, once understanding of the SWCNT structures is obtained, the consideration of other CNT structures is much easier. Consequently, the outline for the basic structural characteristics of CNTs focuses on the SWCNT. It has been found that three types of CNTs are possible: armchair, zigzag, and chiral SWNTs. These depend on how the two-dimensional graphene layer is rolled up. To understand the structural features of SWCNTs, Figure 5.4 shows the two-dimensional graphene layer with atoms labeled using (n, m) notation, where n and m are integers and typically  In Figure 5.4, the  n ≥ m; this notation was suggested by Dresselhaus et al. [17]. vector C is called the chiral vector of the nanotube and is defined as C = (m, n) in the twodimensional hexagonal graphene sheet. The magnitude of these unit vectors is known to be 0.246 nm. The physical properties of CNTs are significantly determined by their diameter and chiral angle, both of which depend on n and m. When the graphene layer is rolled up to form the cylindrical part of the nanotubes, the ends of the chiral vector meet each other. Thus, the chiral vector forms the circumference of the nanotube’s circular cross section, and different values of n and m therefore lead to different nanotube structures. The diameter of nanotubes dt is simply the length of the chiral vector divided by π; thus, it can be expressed as a function of n and m:

(0,2) (0,1)

(0,0)

(1,2) (1,1)

(1,0)

(2,2) (2,1)

(2,0)

(3,2) (3,1)

(3,0)

FIGURE 5.4  Graphene layer with carbon atoms identified by the notation C = (m, n).

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C = (3,2)

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Multiscale and Multiphysics Modeling

dt =

0.246 2 (n + n ⋅ m + m2 )1/2 (nm) π

(5.52)

The chiral angle can be calculated by   31/2 ⋅ m θ = sin −1  2 2 1/2   2(n + n ⋅ m + m ) 

(5.53)

According to prior research, armchair nanotubes are formed when n = m and the chiral angle is 30°. Zigzag nanotubes are formed when either n or m is zero with the chiral angle 0°. All other nanotubes are known as chiral nanotubes, with the chiral angle between 0° and 30° [18]. In particular, the chiral vector (n, m) can determine whether a SWCNT can be either a metal or a semiconductor. For example, a metallic nanotube can be obtained when the difference n – m is a multiple of three. If the difference is not a multiple of three, a semiconductor nanotube can be obtained. These characteristics are important for studying the electronic properties of SWCNTs. 5.6.2 Simulation Time Step The simulation is propagated through time at intervals of Δt by iteration using Gear’s method at every time step. Based on the energy conservation principles, typically Gear’s algorithm is about one order higher in accuracy than Verlet’s. A reasonable simulation time step Δt is estimated as 0.01 ps using the relation between the bond length r0 and the bond energy ε. The method to find the reasonable time step size is presented next. For the isolated system, the kinetic energy of an atom should be about the same as the potential energy of the atom. Let us assume that one atom can move within the bond length r0 during the time step Δt. Then, the kinetic energy Ek and the velocity v of the atom are written, respectively, by Ek =

1 mv 2 = ε 2

(5.54)

r0 ∆t

(5.55)

v=

From Equations 5.54 and 5.55, the maximum possible Δt value is computed as ∆t = r0 m/2 ε ≤ 0.02 ps

(5.56)

where m is the mass of the carbon atom, and r0 and ε are the carbon bond length and C–C tight bonding overlap energy, respectively. Based on Equation 5.56, the time step 0.01 ps is considered to have a reasonable value.

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5.6.3 Freestanding Thermal Vibration Method This section describes the method to calculate Young’s modulus of CNTs by introducing the freestanding room temperature vibration method presented in Reference 19. According to the kinetic molecular theory, carbon atoms vibrate about their equilibrium positions with a 3 mean vibration kinetic energy that increases with the temperature as kT. From this fun2 damental principle, Young’s modulus of SWCNTs is estimated as Y = 1.25 − 0.35 / + 0.45 (TPa) by observing SWCNTs’ freestanding room temperature vibrations in a transmission electron microscope (TEM). The formula to calculate Young’s modulus was derived from the relationship between the motion of a vibrating clamped cylindrical cantilever rod governed by the classical fourth-order wave equation and the quantum mechanics–statistical probability theory given by the Boltzmann factor. The resultant relationship between Young’s modulus Y; length L; inner and outer tube radii b and a, respectively, that form a cylindrical CNT wall; and the root mean square (rms) displacement σ, which is represented by the vibration amplitude at the tip of a CNT at a temperature T, can be expressed by σ 2 = 0.8486

L3 kT YWG(W 2 + G 2 )

(5.57)

where W is the SWCNT width (diameter), G = (a − b) is the graphite interlayer spacing of 0.34 nm, and k is the Boltzmann constant. Equation 5.57 is used to predict the Young’s modulus as follows: Y = 0.8486

L3 kT σ 2WG(W 2 + G 2 )

(5.58)

From this equation, notice that Young’s modulus depends strongly on the rms displacement σ at the tip of the SWNT for a given length and width W of the CNT and the given ambient temperature T. 5.6.4 Equilibrium and Vibration Motion of CNTs The system with atoms randomly assigned initial velocities taken from a Gaussian distribution must be allowed to reach equilibrium before any reliable measurements can be taken. Equilibrium is monitored using the Boltzmann H function that represents the instantaneous velocity distribution: the distribution at one moment during the simulation. The H function for the x component can be written as ∞

H x (t) =

∫ f (v ) ln f (v ) dv x

x

x

(5.59)

−∞

where f(vx)dvx is Maxwell’s velocity distribution, which is the fraction of atoms having velocities between vx and vx + dvx, and can be evaluated analytically, giving

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f ( vx )dvx =

 − mvx2  m exp  dv  2 kT  x 2 πkT

(5.60)

Based on Boltzmann’s H theorem, the H function decreases from an initial value to its value given by the Maxwell velocity distribution Equation 5.60 until the system reaches equilibrium. Once equilibrium is attained, the H function shows steady-stable fluctuations with respect to an average value. Figure 5.5a and b show the simulated H function for the SWCNT and bamboo-structured single-wall carbon nanotube (BSCNT) models under study from the initial to the equilibrium states, respectively [15]. Both H functions converge to some values, and they stay close to the values. It is an indication that the equilibrium of both systems is achieved. In Figure 5.5, the H function of the BSCNT model displays more fluctuations than that of the SWCNT model within the same range of time steps between 200 and 400. This indicates that the fluctuation of carbon atom velocities in the BSCNT model is larger than in the SWCNT model. As a result, it may be expected that Young’s modulus of BSCNT should be lower than for the SWCNT under the same temperatures and constraints because the atomic rms displacement σ of the BSCNT model is expected to be larger than SWCNT by the relation between Y and σ in Equation 5.58. Figure 5.6 shows the equilibrated top configuration of the armchair (4, 4) SWCNT model made of 185 carbon atoms after the equilibrium MD simulation. The configuration is captured when the potential energy per carbon atom of the SWCNT model reaches the minimum value and atoms are vibrating about the equilibrium state. In the center of Figure 5.6, the local topology of the cap region of the SWCNT model shows a pentagon-shape defects ring pole. This pentagon topology becomes a seed for the stable cap formation mechanism 0 H function

–1

0

200

400

600

800

1000

800

1000

–2 –3 –4 –5 –6 Time steps

(a) 0 –1

0

200

400

600

H function

–2 –3 –4 –5 –6 –7 (b)

Time steps

FIGURE 5.5 H function of SWNT and BSNT model from MD simulation: (a) SWCNT model’s H function with time steps and (b) BSCNT model’s H function with time steps.

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Observed atomic vibration

FIGURE 5.6 Top view of SWCNT model under equilibrium.

Knots

FIGURE 5.7 The top view of BSCNT model under equilibrium.

of the SWCNT model [20]. We calculate the Young’s modulus of the SWCNT model using the thermal vibration method because it consists of the tip of the SWNT model. A few atomic vibration motions along the circumference of the SWCNT model can be seen in Figure 5.6. The equilibrated configurations for the BSCNT, which consists of two armchair (5, 5) SWCNTs and a total of 210 carbon atoms, is shown in Figure 5.7. The equilibrium process for the BSCNT model is similar to the previous SWCNT model. Figure 5.7 shows the equilibrated top view. The equilibrated BSCNT model shows somewhat different features with respect to the SWCNT model. The remarkable local topology changes (e.g., knot-shaped defects that look like protuberant lumps at a point from which a stem or branch grows) are created at the heterojunction region along the circumference of the BSCNT. Atomic

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vibrational motions are also observed, but the magnitudes of those are more conspicuous than for the SWCNT model. In both cases, the central pentagon ring of the end tip of both models shows visible vibration motions because it is difficult to catch the rms displacements of those pentagon rings’ atoms by eye on the atomic scale. Thus, to find the atomic scale’s displacements of end tips of both models, computer algorithms were developed for calculating Young’s modulus. 5.6.5 Elastic Modulus of CNTs under Equilibrium The two models considered previously were tested under equilibrium MD simulation at room temperature (300 K), and then the elastic moduli of SWCNT and BSCNT were evaluated based on Equation 5.58. The numerical values were compared with the theoretical and experimental results presented from other studies. Figure 5.8a and b show histograms that demonstrate the spread in the evaluated Young’s modulus values for the SWCNT and BSCNT by running 100 simulations each, and measured objects were the carbon atoms of the pentagon ring pole at the tip in both models. The MD code was designed to calculate the rms displacements of the tip carbon atoms  3  when the instantaneous kinetic energy per atom of both models reached 0.039 eV  ~ kT   2  at the given room temperature of 300 K. Then, Young’s modulus was calculated using Equation 5.58. In both cases, the use of 100 simulations was considered sufficient to reduce statistical errors due to the random initial velocities for both models. 30 25 Occurence %

20 15 10

(a)

1.

0– 1

1.

0.

75

–1 .

0

0

.2 5 15 –1 .5 1. 5– 1. 75 1. 75 –2 .0 2. 0– 2. 25 2. 25 –2 .5 2. 5– 2. 75 2. 75 –3 .0 3. 0– 3. 25

5

Elastic modulus (GPa) 30 25

Occurence %

20 15 10

45

0. 4

0. 4

−0 .

0

5− 0. 5 0. 5− 0. 55 0. 55 −0 0. .6 6− 0. 65 0. 65 − 0. 07 7− 0. 75 0. 75 −0 0. .8 8− 0. 85 0. 85 −0 .9

5

(b)

Elastic modulus (GPa)

FIGURE 5.8 Histograms of Young’s modulus from MD for (a) SWCNT and (b) BSCNT.

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The observed distributions of Figure 5.8a and b are fairly asymmetrical about the mean values, displaying a tail extending to higher values. These asymmetrical distributions may affect the ability to interpret the reliability of a mean elastic modulus without a normal distribution assumption. Thus, under the equilibrium MD simulation, there are essential embedded random errors during the simulation process due to the time evolution method and different initial conditions on each independent simulation, such as random initial velocities. Although embedded time evolution errors in MD code cannot be completely removed, a normal distribution assumption about the randomly distributed elastic moduli evaluating from MD results is feasible due to the fact that the assigned initial velocities are normally distributed by the Maxwell-Boltzman velocity distribution. According to the assumption that the measured Young’s modulus shows a normal distribution about a mean value, normal probability density functions for the SWCNT and BSCNT models can be computed. The normal probability density function for a random variable x, which represents the Young’s modulus value, with mean μ and standard deviation σ is given by  −( x − µ)2   1  f ( x) =  exp   , where − ∞ ≤ x ≤ +∞  2  σ 2π   (2 σ ) 

(5.61)

The parameters used for this normal probability density function for the computed Young’s moduli are shown in Table 5.2. The data were collected from the equilibrium MD simulation results of the SWCNT and BSCNT models. Figure 5.9 shows the normal probability density functions of both SWCNT and BSCNT models plotted using Equation 5.61 with the parameters of Table 5.2. The mean value of Young’s modulus for the SWCNT model was 1.42 TPa with a standard deviation of 0.34 TPa. Based on these data, the expected mean Young’s modulus for the real population of SWCNTs can be predicted as 1.42 ± 0.23 TPa with 50% confidence intervals. These values are in good agreement with the theoretical and experimental Young’s modulus values reported by others, as shown in Tables 5.3 and 5.4 [19,21–27]. In particular, considering about 48% of the elastic values for the SWCNT model, which is distributed within the range 1.0 to 1.5 TPa as shown in Figure 5.8a, and comparing with the previously mentioned mean Young’s modulus of SWCNT 1.25 TPa as evaluated by the same thermal vibration method presented by Krishnan et al. [19] and obtained from 27 SWCNTs, the consistency of both results are comparable. In their method, the tube’s length and tip vibration amplitudes were estimated directly from the digital micrographs (TEM images). From these consistent results, the MD simulation is reasonable for use for evaluating Young’s modulus of the SWCNT model, and it may be feasible to simulate the BSCNT model. TABLE 5.2 Statistical Data Measured from MDSS Simulations of SWCNT/ BSCNT Models Parameter Number of simulations Mean Young’s modulus (TPa) Standard deviation (TPa) Maximum Y value (TPa) Minimum Y value (TPa)

© 2016 by Taylor & Francis Group, LLC

SWCNT

BSCNT

100 1.4243 0.342 3.4202 0.7402

100 0.6041 0.1002 0.9241 0.4046

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4

×10−12

Normal probability density

3.5

BSNT

3 2.5 2 1.5

SWNT

1 0.5 0 0

0.5

1

1.5 Υ (Pa)

2

2.5

3 ×10−12

FIGURE 5.9 Normal probability density functions of elastic moduli (Y value) for SWCNT and BSCNT models.

TABLE 5.3 Various Theoretical Calculations for Young’s Modulus Method

MD Using TersoffEmpirical Force Nonorthogonal Brenner Potential [21] Constant Method [22] Tight Binding [23]

Wall thickness Elastic modulus

0.06 nm SWCNT: 5.5 TPa

0.34 nm SWCNT: 0.97 TPa

0.34 nm SWCNT: 1.2 TPa

Ab Initio DFT [24] 0.34 nm SWCNT rope: 0.8 TPa MWCNT: 0.95 TPa

TABLE 5.4 Various Experimental Studies for Young’s Modulus Method Wall thickness Elastic modulus

Thermal Vibration [25]

Restoring Force of Bending [26]

Thermal Vibration [19]

0.06 nm SWCNT: 1.8 ± 1.4 TPa

0.34 nm SWCNT: 1.28 ± 0.59 TPa

0.34 nm SWCNT: 1.3 to 0.4/+0.6 TPa

Deflection Forces [27] 0.34 nm SWCNT: 1.0 TPa

The equilibrium MD simulation results of the SWCNT and BSCNT models show two important features. One is that although the SWCNT MD results and prior studies were consistent with each other within the mean error, the elastic modulus of the mean value 1.42 TPa of the SWCNT model calculated from MD simulation shows a somewhat higher value than the elastic modulus of the mean 1.25 TPa used with the same thermal vibration method reported by Krishnan et al. [19]. A second is that the calculated elastic modulus of the BSCNT model shows an obvious difference with respect to the SWCNT model. The mean elastic modulus of the BSCNT model is calculated as 0.604 TPa with a standard deviation of 0.1 TPa. Consequently, if BSCNTs can be synthesized in practice, the expected  mean elastic modulus for a real BSCNT population would be predicted to be 0.604 ± 0.07 TPa with a 50% confidence level. Contrary to the general belief that expects

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165

high elastic modulus values due to the fact that the BSCNT has an internal support structure, the results show a significantly lower elastic modulus than that of the SWCNT model, but still show a strong elastic modulus compared with other materials, such as hardened steel, on the order of 210 GPa. Unfortunately, so far there are no available data for the heterogeneous CNTs’ elastic moduli to be compared with that of the BSCNT model. One reason for a somewhat higher mean elastic value of the SWCNT model calculated from MD simulation can be understood from the parameter relationships of Equation 5.58. In this equation, the elastic modulus is inversely proportional to the width of the CNTs W when the other parameters are held the same. This means that the elastic modulus of CNTs has dependence on the diameter of the tube. Krishnan et al. used nanotube widths as a range of 1.0–1.5 nm (10–15 Å) to calculate Young’s modulus using Equation 5.58, while the present SWCNT and BSCNT model diameter used in MD simulation was just 6.78 Å. Therefore, it is natural to expect that the measured Young’s modulus of the SWCNT model is higher than Krishnan et al.’s Young’s modulus from the fundamental relations mentioned. There is another possible factor in explaining the difference in the elastic moduli between our MD results and the prior experimental results reported by others. In most of the prior experiments, the CNTs not only were made of carbon atoms but also had impurities due to the limit of the SWCNT synthesis process in practice. Typically, it is known that impurities contained in CNTs lower their mechanical strength. Thus, it is a reasonable inference that the Young’s modulus evaluated from the MD simulation using the SWCNT model made of pure carbon atoms may have higher strength than real CNTs. On the other hand, why is the elastic modulus of the SWCNT model roughly twice as high than BSCNT model? Earlier studies reported that the Young’s modulus of CNTs probably depends on the presence of structural imperfections, such as the nesting of tubular cylinders, which can create a joint or “knuckle,” thereby weakening the tube, and structural defects with the pentagon (5)–heptagon (7) rings. The defects that appeared along the circumference of the BSCNT model can prove the arguments presented in Reference 28: When heterogeneous CNTs are formed, the 5–7 defect rings are induced, and the induced defects absorb partial energy from the total energy of the system because it requires defect formation energy. Thus, the defects at the junction regions of the BSCNT model may absorb energy, with the other region’s energy per atom increasing for balance so that the total energy of an isolated system is constant. As a result, the energy needed for the atomic vibration in the BSCNT tips would increase, and then the elastic modulus of the BSCNT model would be lower than that of the SWCNT model based on the relationship of Equation 5.58. This analytical interpretation is also consistent with the results of Figure 5.5b, as mentioned previously. However, it is not clear which energy (i.e., kinetic or potential energy) has an important role for that mechanism. 5.6.6 Comparative Results of Equilibrium and Nonequilibrium Simulations In this section, a relative Young’s modulus ratio concept for the SWCNT and BSCNT models is introduced with the force-strain diagram that is extracted from the nonequilibrium MD simulation, which is then compared with the Young’s modulus values calculated from the previous equilibrium MD simulation. By conventional continuum mechanics, the ultimate tensile strength and Young’s modulus for a bulk material can be determined from the force-displacement data under the external tensile loading test. The ultimate tensile strength is measured as the maximum stress prior to fracture. To obtain the Young’s modulus from the collected data, a second-degree

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polynomial function can be fitted to the applied force-displacement diagram, with the modulus measured as the slope within the small strain limit range [29]. However, because it is difficult to estimate a definite cross-sectional area of CNTs due to the nanoscale model configuration, the previous conventional method is not appropriate for calculating a realistic Young’s modulus directly from the applied force-strain diagram of nonequilibrium MD simulation. For that reason, a relative Young’s modulus ratio, in which two materials are assumed to have the same cross-sectional area, is introduced to remove the effect of cross-sectional area in computing the Young’s modulus. To collect data for the analysis of atomic-scale tensile deformations of our SWCNT and BSCNT models, the tensile forces were applied to both models to move the end atoms with constant strain rates (~10 –2/ps), and the strain per atom at every force was measured. Once the force-strain per atom data are collected from the simulation, a relationship between the applied force and strain can be plotted as the force-strain diagram. The tensile force-strain diagram for the SWCNT and BSCNT models obtained from the nonequilibrium MD simulation is plotted in Figure 5.10. The applied tensile forces per atom of the SWCNT and BSCNT models were recorded on the y axis, and the average strains of carbon atoms in the location right below the forced atoms were recorded on the x axis, respectively. Overall, the observed force-strain diagrams showed a significant nonlinear relationship. In particular, observe that the strain of the BSCNT model was more linearly increased than that of the SWCNT model as the applied force increased. From this observation, it is possible to expect that the Young’s modulus of the BSCNT model would be less than the modulus of the SWCNT model.

Force per atom (nM)

60 50 40 30 20 10 0 (a)

0

2

4

6 Milli-strain

8

10

12

Force per atom (nM)

50 40 30 20 10 0 (b)

0

2

4

6

8 10 Milli-strain

12

14

16

FIGURE 5.10 Tensile force-strain per atom for the SWCNT and BSCNT models: (a) SWCNT and (b) BSCNT.

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To calculate the relative Young’s modulus ratio value between two materials and to satisfy an elastic limit criteria, an applied tensile force range of 15–30 nN/atom was selected, and the linear regression analysis was applied to this selected data as shown Figure 5.11. Based on the linear regression analysis under the assumption that there is an elastic limit on a force-strain diagram, the ratio of force to strain of a specific material can be represented as the slope of the regression line. In Figure 5.11, although the data set of the SWCNT model shows a characteristic that was more nonlinear than those of the BSCNT model within the same applied force range, the linear regression analysis could be used for both data sets due to a good-fitting statistical coefficient R 2, which quantifies goodness of fit and is a fraction between 0.0 and 1.0, with no units. Higher R 2 values indicate that the fitting line or curve comes closer to the data. In this case, the computed R 2 using the statistical software package was 0.91 for the SWCNT and 0.99 for the BSCNT models. These numerical values indicate that the linear assumption to calculate the slope of a regression line for SWCNTs and BSCNTs is appropriate. The resultant slope of the regression line was 7416.3 for the SWNT and 3691.3 for the BSNT models, as seen in Figure 5.11. Consequently, the ratio value for the BSCNT model with respect to the SWCNT model was calculated as 0.498. This means that the average Young’s modulus of the BSCNT model was 49.8% of that of SWCNT. From the results of the previous equilibrium MD simulations, the evaluated mean Young’s modulus of the SWCNT and BSCNT models were 1.424 TPa and 0.604 TPa, respectively. Based on these two values, the calculated ratio is 0.424 (42.4%). This is consistent

Force per atom (nM)

40 30 20 10 0

0

0.5

1

1.5

2

2.5

3

3.5

2.5

3

3.5

Milli-strain

(a)

Force per atom (nM)

40 30 20 10 0

0

0.5

(b)

1

1.5

2

Milli-strain

FIGURE 5.11 The linear regression fitting for the SWCNT/BSCNT within the elastic limit: (a) SWCNT and (b) BSCNT.

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with the ratio value of the nonequilibrium MD simulation results, 0.498 (49.8%), within an error of ±15%. Consequently, although we could not find an exact numerical value for the Young’s modulus of the SWCNT and BSCNT models from the nonequilibrium MD simulations, the Young’s modulus of the BSCNT model was observed to be lower than that of the SWNT model; that is, a heterogeneous CNT might be found to have a lower Young’s modulus than pure CNTs.

5.7 Application to Carbon Nanotubes: Vibrational Mode Shapes In this section, radial breathing modes of armchair nanotubes of different diameters are first discussed, and a comparison is provided for vibrational modes of the same tubes computed using different methods and interatomic interactions based on tight-binding and ab initio discrete Fourier transform (DFT) approaches. The other low-frequency modes are described in material that follows; these modes include axial breathing, nonaxial bending, and torsional shear for single-wall, multiwall, and bamboo-like CNTs. 5.7.1 Comparison of Radial Breathing Modes of Armchair SWCNTs The natural frequencies of radial breathing modes in the low-frequency region were calculated for armchair-type nanotubes with different diameters (Table 5.5). The results obtained with the present technique were summarized along with the results obtained using other techniques [30–33]. Overall, the results obtained using the present technique were comparable to results using other methods. In particular, these results agreed well with the ab initio results. On the other hand, these were slightly higher than the experimental data and the constant force results and slightly lower than the results using the tight-binding methods. The computational results from the constant force model agreed well with experiments because the constant force models were generally fitted directly to the experimental data. 5.7.2 Natural Frequencies and Mode Shapes for SWCNTs Natural frequencies and mode shapes were computed for an (8, 8) armchair-type SWCNT with both ends fully constrained to their equilibrium position. There were various vibrational mode shapes for the SWCNT, such as the axial vibrational mode, lateral bending mode, radial breathing mode, twisting mode, axial shear mode, and so on. Each vibration type had different mode shapes associated with natural frequencies, which ranged from the lowest to the highest. However, only the first few lowest-mode shapes for some TABLE 5.5 Natural Frequencies (in THz) Associated with the Radial Breathing Mode in the Low-Frequency Region Chirality (7, 7) (8, 8) (9, 9) (10, 10)

Present Technique

Constant Force [30]

Tight Binding [31]

Ab Initio [32]

Experiments [33]

7.39 6.86 5.83 5.57

– 6.18 5.49 4.95

8.04 7.14 6.42 5.85

– 6.57 5.85 5.25

6.90 6.18 5.46 4.86

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vibration types are discussed here. This is because the lowest natural frequencies and mode shapes are more important in most thermomechanical and vibrational resonatortype engineering applications. In the following discussion, all the mode shapes are presented in the ascending order of the natural frequency, starting from the lowest. The first two accordion-like axial vibrational mode shapes, with natural frequencies 4.683 and 9.328 THz, are shown in Figure 5.12. The first mode is the movement of all atoms in one direction along the tube axis, and the second mode shows the atoms moving away from the center line toward the top and bottom ends of the nanotube’s longitudinal direction. A set of lateral bending vibrational modes is shown in Figure 5.13. The first mode is a half-sine shape, and the second mode is a full-sine shape. The exact frequency and shape 16

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2

2

0

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–2 –10

0

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–10

–2 –10

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–10

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FIGURE 5.12 First two axial vibration modes of the SWCNT.

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8

8

6

6

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4

2

2

0

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–2 –10

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10

0

–10

FIGURE 5.13 First two bending vibration modes of the SWCNT.

© 2016 by Taylor & Francis Group, LLC

–2 –10

–5

0

5

–10 0 10

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of the modes depend on the length of the nanotube under consideration. The longer the tube is, the lower the natural frequency is for lateral bending vibration. Figure 5.14 shows the first two radial breathing modes with frequencies of 8.690 and 9.664 THz. Because of the fixed end boundary constraints, the atoms at the center of the tube have the largest displacement for the first radial mode. On the other hand, the second mode has a vase shape. Figure 5.15 illustrates a twisting (or torsional) mode shape with a frequency of 5.331 THz. In other words, the atoms move along the circumferential direction except for the constrained top and bottom atoms. Another vibrational mode was the axial shear mode with a frequency of 5.331 THz, as seen in Figure 5.16. In this case, atoms at one side of the tube diameter move in one axial direction while rest of the atoms move in the other direction of the axis. 16

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10

8

8

6

6

4

4

2

2

0

0

–2 –10

0

0

–10

–2

–10

0

10

FIGURE 5.14 First two radial vibration modes of the SWCNT.

–6 –4 20

–2

10

0

0

2

–10

4

–20

–10

FIGURE 5.15 Twisting mode of the SWCNT.

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6 10

0 –10

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16 14 12 10 8 6 4 2 0 –2 –10

0

10

0

–10

FIGURE 5.16 Axial shear mode of the SWCNT.

5.7.3 Natural Frequencies and Mode Shapes for MWCNTs For MWCNTs, the interaction among atoms in the same nanotube was modeled using the Tersoff-Brenner potential; the interaction of atoms located in two different concentric nanotubes was modeled using the van der Waals interactions of the LJ 6-12 type. The parameters for the LJ potential were selected such that ε = 0.0778 kcal/mol and σ = 3.4 Å for Equation 5.27. The MWCNTs consisting of two concentric SWCNTs can have symmetric and asymmetric modes. For the symmetric modes, both the inner and outer tubes have deformations in the same directions for the same mode, while for the asymmetric modes the inner and outer tubes have deformations in different directions for the same mode shape. The analysis of MWCNTs did not show any vibrational mode, which consisted of two different types of modes for the inner and outer nanotubes, such as a bending mode for one tube and a radial mode for the other. This is probably because van der Waals interactions are strong enough to disallow any partial or full mixing of different vibrational modes of the inner and outer tubes. This may also be because both the inner and outer tubes in the multiwall tube model are of same chirality (i.e., both are armchair nanotubes). The mode mixing may be possible in tubes of different chiralities, and it needs to be further investigated. The results also showed that, for the same chirality, the symmetric and asymmetric mixing of modes in the nanotubes was possible through weak van der Waals forces in a multiwall nanotube. The natural frequencies of the MWCNTs were compared to those of the two individual (5, 5) and (10, 10) SWCNTs of the same length under similar conditions. The computed natural frequencies of the MWCNTs were sensitive to the strength of the van der Waals forces acting between two concentric tubes. The present results were obtained using the LJ parameters given previously. As far as the axial accordion and lateral bending modes are concerned, all three cases had almost similar natural frequencies. Similar natural frequencies in the (10, 10) and (5, 5) SWCNTs for these modes were probably because bending and axial modes do not vary with the radii of the nanotubes. This behavior was similar to the one seen in continuum models of axial vibration of a rod and bending vibration of a hollow beam. The natural frequency depends only on length and not radius in the case of axial vibration. In the case of bending of a hollow beam, length has a more dominant effect on the natural frequency than the cross-sectional area.

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TABLE 5.6 Comparison of Natural Frequency of the First Radial Mode between MWCNT and SWCNT MWCNT

Frequency (THz)

Symmetric

Asymmetric

(5, 5) SWCNT

(10, 10) SWCNT

4.684

5.667

5.665

4.682

For the symmetric and asymmetric mixing, however, the MWCNT has two natural frequencies, for the first accordion and bending modes, respectively. The frequencies for the asymmetric modes are slightly higher than those for symmetric modes. For the radial mode of vibration, the (10, 10) SWCNT had a 17% lower frequency than the (5, 5) SWCNT. The natural frequency associated with the symmetric radial mode of MWCNT was slightly higher than the (10, 10) SWCNT, while the frequency related to the asymmetric radial mode was slightly higher than the (5, 5) SWCNT, as shown in Table 5.6. 5.7.4 Natural Frequencies and Mode Shapes for BSCNTs As one of the possible heterogeneous CNTs, the BSCNT was studied for its natural frequencies and mode shapes and compared to a conventional SWCNT. The simulated stable structure was investigated for low-frequency vibrational characteristics. Radial breathing modes of a BSCNT are also shown in Figure 5.17. Those mode shapes for the SWCNT and BSCNT look similar in a global sense. However, the local behaviors were different between the two CNTs depending on the mode shape. Natural frequencies of SWCNT and BSCNT are compared in Table 5.7 for different vibrational modes for the CNTs with the same diameter. The table shows the ratios of the BSCNT natural frequencies in relation to the SWCNT natural frequencies. As shown in Table 5.7, 16

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8

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6

4

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2

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–2 –5

0

–5

0

FIGURE 5.17 First two radial vibration modes of the BSCNT.

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TABLE 5.7 First Natural Frequency Ratios between the SWCNT and BSCNT Vibrational Mode

Accordion Mode

Bending Mode

Radial Mode

BSCNT frequency/ SWCNT frequency

0.917

0.917

1.160

the natural frequency of BSCNT associated with the first axial accordion or lateral bending mode shape was lower than that of the SWCNT. On the other hand, the first natural frequency of the radial mode of the BSCNT was higher than that of the SWCNT. These results can be explained as follows: The additional number of the atoms in the cross-sectional layer at the junction in a BSCNT can contribute to both stiffness and mass of the overall CNT structure. For the first accordion-like axial and lateral bending modes, the effect of the additional mass was more critical than the change of stiffness because the internal atomic membrane structure of a BSCNT does not contribute much to the axial and bending stiffness. However, the case was opposite for the radial vibration. The cross-sectional layer structure at the junction increased the radial stiffness significantly so that the radial frequency of the BSCNT increased compared to the SWCNT.

5.8 Application to Polymers 5.8.1 Cross-Linking of Polymers Molecular dynamics simulations are in heavy demand for the study of randomly crosslinked polymers. Barsky and Plischke previously reported on simulations that involved short chains and small system sizes [34]. These simulations showed that there was a universal function that stated the distribution of localization lengths for a wide range of crosslink density. The relationship between the shear modulus and density were never taken into account. During later studies, Barsky and Plischke extended their simulations to longer polymer chains and systems [34]. Their focus was on the shear modulus as a function of the density of cross-links. The purpose of the cross-links is to tie all the polymer molecules together. This prevents molecules from flowing past or around each other once the temperature increases and increases their resistance to melting. The advantage of being tied together is that the molecules are not easily broken apart from each other. The difference between uncross-linked polymer chains and a cross-linked network is described in Figure 5.18. However, in this study the temperature was held constant as was the cross-linking density. This section describes the construction of a molecular model. The goal is to randomly generate polymer chains within a cube and determine how many molecules per given chain are necessary to produce homogeneous behavior. Computer simulations were used to focus on randomly dispersed particles in a three-dimensional (3-D) space. These simulations contained different volumes as well as polymers with different lengths, N = 5, …, 45, with unit multiples of five molecules in a chain. The 3-D models of polymers were used to create structures that properly represented real molecules, thus providing various

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When polymers become cross-linked, this becomes this FIGURE 5.18 How polymers appear once they are cross-linked. 35 30 25 20 15 10 5 0 –5 –10 150

100

50

0

–50 –20

0

20

40

60

80

100

FIGURE 5.19 Composite material with particles randomly dispersed in a 90 × 90 × 15 space volume.

construction and analysis strategies. Figure 5.19 shows a simulation with 12 polymer chains extending in both the x and y directions and 2 chains in the z direction. Using cross-linking potential systems, MD simulations were conducted in the 3-D polymeric matrix state. The simulations emphasized the relation between the correlation volume of a space and the concentration of the random particles. 5.8.2 Allocation of Polymers Molecules formed in this study were first confined to a cube with a set of dimensions in the x, y, and z directions. For the construction of the polymer chain, an allowable distance between molecules within the polymer was first established. All measurements during the construction were dimensionless. To ensure atoms were not too close or too far apart, an allowable tolerance input was set. After creating the foundation, the actual construction of the polymer chain began. For optimum results that are not too computationally expensive, the number of polymer chains in the x and y directions were the same, and the z direction was one-third of the x direction’s value. The number of molecules contained in each chain was chosen to be constant.

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Molecular Dynamics

To determine the spacing in each direction, the lengths of the cube were divided by the number of polymer chains in the same direction. The total number of atoms contained in the cube was the product of the number of molecules in a chain and the number of polymer chains in the x, y, and z directions. As far as the atomic structure was concerned, the bead-spring model was used. In essence, the bead-spring model simulates the hydrodynamic properties of a chain macromolecule. These chains consist of a sequence of beads, each of which offers hydrodynamic resistance to the flow of the surrounding medium. The beads are connected to each other by a spring, which does not contribute to the frictional interaction. However, the spring is responsible for the elastic and deformational properties of the chain. The mutual orientation of the springs is random. For this project, each bead, or mapping point, represented a specific monomer. This required a fixed distance of the beads along the backbone, bond angles around the backbone, and torsion states around the bead-bead connection to be held constant. A loop was generated to account for the individual atoms and their physical location. To initiate the chain creation, the first atom’s position was generated. The remainder of the atoms along the chain were randomly generated. During the random generation, the atoms’ positions were checked to verify whether they were within the tolerance. 5.8.3 Potential Energy for Polymers The interactions between the polymers and atoms led to prediction of the large-scale bulk properties of material. This MD program used the LJ potential method to evaluate molecule interactions between the polymers and atomic atoms. The LJ potential is an effective potential that describes the interaction between two uncharged molecules or atoms. The LJ potential was mildly attractive as two uncharged molecules or atoms approached each other from a distance, but strongly repulsive when they approached too closely. At equilibrium, the pair of atoms or molecules tended to go toward a separation corresponding to the minimum of the LJ potential. The strong close-in repulsion between atoms or molecules is understandable, resulting from mutual deformation of their structures (one atom cannot diffuse through another). The mild attraction at larger distances was due to the induced dipole-dipole moment interaction of the particles. As stated, the LJ potential is used to ensure self-avoidance of polymers. The following formula was used to compensate for the added attractive potential between neighboring atoms:  2   1 2   rij   − kR ln 1 −  U nn (rij ) =  2 o  Ro      ∞

   r 0.02). Yielding started at subcell 5 in Figure 7.2 if the loading was applied along the 3 direction. The composite strain at the start of yielding was about 0.0015 by both the present and the FEM models. At the composite strain of 0.01, the von Mises stress and the equivalent plastic strain of subcell 5 were 267 MPa and 0.01565, respectively, for the micromechanics model. Using the FEM, those values were 264 MPa and 0.01562, which were calculated by the average over the volume equivalent to subcell 5. Several strengthening mechanisms, including the size effect [15,30], may contribute at the initial yielding range such that the strength of the composite may have higher strength than the classical theory. Several failure mechanisms, including particle-matrix interface decohesion [30,31], may occur at the postyielding range so that hardening of the composite

400 PMMC 15% SiCp/A356 T6 d = 16 µm

Stress (MPa)

300

Matrix only A356 T6

200

Experimental Ref. 15 Ref. 13 FEM Present

100

0

0

0.01

Strain

0.02

0.03

FIGURE 7.28 Stress-strain curves of 15% SiC–A356 T6 particulate composite (particle diameter: 16 μm).

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TABLE 7.13 Comparison of Computational Statistics between the Present Micromechanics Method and the Conventional Finite Element Method 100-Increment Scheme Maximum Strain Level: 0.05 Total number of elements Total number of increments and iterations Average iterations per increment Total CPU time(s) CPU time per iteration(s)

20-Increment Scheme

Present

FEM

Present

FEM

1 101 1.01 9.2 0.091

8000 111 1.11 1947 17.54

1 38 1.9 4.7 0.124

8000 44 2.2 675.9 15.36

may be lower than for the present model. The experiments in Reference 15 showed the size effect on strengthening; that is, the smaller particle diameter yielded higher composite strength. Because the present model does not consider such strengthening and failure mechanisms, the stress-strain curve in Figure 7.28 is reasonable in terms of both strength and deformation. Computational aspects of the present method along with the FEM are summarized in Table 7.13. The FEM used 8000 (20 × 20 × 20) hexahedral elements, but the present method used only 1 element. The analysis used 100 increments or 20 increments up to the global maximum strain level of 0.05, respectively. A laptop computer with an Intel Core 2 Duo at 1.86 GHz was used for comparison. Not only a small number of iterations per increment but also dramatically less central processing unit (CPU) time per iteration were observed by the present model. 7.10.2 Particulate Composite Made of SiC and Cu Alloy The next example considers a particulate composite consisting of 20% SiC and an Al-3.5 wt% Cu alloy as studied in Reference 32, which shows the stress-strain curves for the composite and the matrix alloy. Material properties are given in the following, and the stress-strain equation of the matrix alloy was fitted using the modified Hollomon equation, σ = K(ε0 + εp)n: SiC: E = 427 GPa, ν = 0.22 Al-3.5 wt% Cu: E = 72 GPa, ν = 0.33, σ0 = 172.4 MPa, K = 618.5 MPa, n = 0.2117 The particle diameter varied from 1 to 7 μm, and its median value was between 3 and 4  μm. As a result, 3.5 μm was selected for the present model. Because of the extensive aging process after the fabrication for the composite [32], the size effect by quenching was considered negligible. Figure 7.29 compares the present result to other experimental, FEM, and analytical results. The result by the present model was close to the experimental result up to a strain level of 0.015. The FEM overestimated and the method in Reference 13 underestimated the experimental result.

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400 PMMC 20% SiCp/Al-Cu d = 1–7 µm

Stress (MPa)

300

200

Matrix only Al-3.5 pct Cu

100

Experimental Ref. 32 Ref. 13 FEM Present

0

0

0.01

0.02

0.03

Strain FIGURE 7.29 Stress-strain curves of 20% SiC–Al-3.5 wt% Cu particulate composite (particle diameter: 1–7 μm).

7.10.3 Thermal Residual Stresses for a Whisker Composite Most composite materials have a cooling period during the fabrication process, which results in residual stresses in the composite because of mismatch in CTE between the reinforcing inclusions and the soft matrix materials. Such thermal residual stresses in the metal matrix composite were studied in Reference 33 using the Eshelby method. The composite was the whisker-shaped SiC with the 6061 aluminum matrix. The aspect ratio of the short fiber was 1.8. Material properties were as follows: SiC: E = 427 GPa, ν = 0.17, CTE = 4.3 × 10−6/°K Al 6061: E = 72 GPa, ν = 0.33, CTE = 2.36 × 10−5/°K, σ0 = 47.5 MPa, ET = 2.3 GPa The aluminum matrix was assumed to be an isotropic, linear-hardening material of initial yield stress σ0 and plastic modulus ET. The cooling temperature was ΔT = 200° K. The theoretical value of the residual stress in the matrix material was derived in a matrix volume average form [33]. Figure 7.30 shows the longitudinal stress 〈σzz〉M and tangential stress 〈σθθ〉M obtained from Reference 33. The tangential and radial stresses were in tension and compression, respectively, with an identical magnitude, while the longitudinal stress was in compression. As a result, the absolute stress values are plotted in Figure 7.30. The effective von Mises stress 〈σvm〉M can be calculated from component stresses, and the result is also indicated in Figure 7.30. The effective von Mises stress was also calculated by the present model as shown by the solid line straightforwardly in a similar fashion by the

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70 SiC–Al 6061

Residual stress (MPa)

60

T = 200˚K

vm M

50 40

zz M

30 20

θθ M

Ref. 33

10 0

Present 0

0.05

0.15 0.1 SiC volume fraction

0.2

0.25

FIGURE 7.30 Predicted residual stresses for SiC–Al 6061 whisker composite by cooling.

matrix average. Both predictions were close for a dilute system (less than 5% volume fraction). For a larger volume fraction, the result given by Reference 33 begins to overestimate the residual stress. Physically, the von Mises residual stress must be bounded by a certain value because of the yield criterion. It is evident that the present method predicted the bounded value as seen in Figure 7.30. 7.10.4 Fibrous Metal Matrix Composite Studies of fibrous metal matrix composites (FMMCs), including those for matrix material plasticity, can be found in References 34 and 35. Although extensive experimental and analytical investigations were presented in Reference 34, basic constituents’ material properties were not explicitly provided. Therefore, the composite material used in Reference 35 was selected for this study. The composite was made of 34% boron fibers with 2024 aluminum matrix. The stress-strain curve for the 2024 aluminum was fitted by the Ludwik equation, σ = σ 0 + K ε np. Constituents’ material properties were as follows: Boron: E = 379 GPa, ν = 0.20 Al 2024: E = 60.5 GPa, ν = 0.33, σ0 = 83.5 MPa, K = 562 MPa, n = 0.447 Figure 7.31 shows the stress-strain predictions for transverse and longitudinal loading by the present model and FEM. Both results agreed well with each other. The figure also

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250 Long.

Stress (MPa)

200

FMMC 34% B–Al 2024

150 Trans. 100 Matrix only Al 2024

50

Ref. 35 FEM Present

0

0

0.001

0.002

Strain

0.003

0.004

0.005

FIGURE 7.31 Stress-strain curves of 34% B–Al 2024 unidirectional FMMC.

compares the experimental results for the transverse loading. Three experiments for the composite and a uniaxial test of the 2024 aluminum are included in the figure. The experimental data by the transverse loading were a little bit higher than the present analysis for the elastic range of the stress-strain behavior. The elastic modulus could be affected by several factors, such as the packing arrangement of the fibers and boundary conditions such as plane strain or plane stress. The present model considered the plane stress condition, while the experiments may be between plane stress and plane strain conditions. Therefore, the experimental modulus was higher than for the present model. The plane strain condition caused higher hydrostatic stress, which led to higher yield stress by the von Mises yield criterion. Postyielding behaviors (ε > 0.2%) by the two analyses deviated from the experiments. The experimental results showed severe failure or damage after initial yielding, while the analyses exhibited apparent hardening after initial yielding. This can be explained by the damage observed during the experiments. According to the detailed description in Reference 35 through photomicrographic observations, the test specimen failure was associated with boron filament failure along its diametrical plane normal to the direction of the applied stress. It was investigated that boron fibers may fail at an applied stress of about 170 MPa. In addition, other damages from microflaws, such as interfacial defects and matrix voids, may lower the strength. This is possible because the diameter of the boron filament is normally large (over 100 μm). Because the present model does not account for any failure other than matrix plasticity, the prediction resulted in higher hardening, as seen in Figure 7.31. 7.10.5 Fibrous Composite Plate with Preexisting Crack A composite plate with an initial crack was studied as a structural example using the present model. The plate was 100 × 100 mm, and its thickness was 1 mm. The crack was

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15 mm long. The material for the plate was a unidirectional composite made of 34% boron fibers and 2024 aluminum, as in the previous example. Adopting ideas from the study in Reference 34, the crack orientation as well as fiber orientation were the parameters for this study. Figure 7.32a shows a crack perpendicular to the loading direction with an arbitrary fiber orientation angle α. Figure 7.32b shows a slanted crack at an angle θ, with fibers always parallel to the crack. This example focused on a local-global effect due to matrix yielding and plastic deformation. The present study can predict many physical aspects around a crack tip, such as the crack propagation direction, matrix failure area, and residual stresses for the composite material from a micromechanics viewpoint. Here, an extensive qualitative study is presented, and results are compared with experimental observations described in Reference 34. Figure 7.33 compares propagation of the plastic yielding zone in the matrix material around the perpendicular crack tip of Figure 7.32a for the composites with fiber angles α = 0°, α = 45°, and α = 90°. The applied stress level is indicated as λ = σ/σmax (σmax = 100 MPa) in

α

θ

(a)

(b)

FIGURE 7.32 A 34% B–Al 2024 fibrous composite cracked plate problem: (a) perpendicular crack and (b) oblique crack.

λ = 3/4

λ = 3/4

λ = 3/4

λ=1

λ=1

λ=1

λ = 1/2

λ = 1/2 λ = 1/4

λ = 1/2 λ = 1/4

λ = 1/4 1 mm

(a) FIGURE 7.33

1 mm

1 mm

(b)

(c)

(

)

Perpendicular crack: propagation of plastic failure region εp > 0 : (a) fiber angle α = 0°; (b) fiber angle α = 45° and (c) fiber angle α = 90°.

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(

)

the plots. Contour lines in the figure represent frontal lines of plastic failure εp > 0 . The plastic yielding propagates along the fiber’s longitudinal direction. This is clear for fiber orientation of α = 45° or 90° as shown in Figure 7.33b and c, where the yielding emanates from the notch tip and eventually propagates toward the fiber orientation. This prediction coincides with the experimental results for the fatigue crack propagation with α = 90° for up to 20,000 cycles [34]. The slanted crack results are shown in Figure 7.34 for various crack orientation angles θ in Figure 7.32b. The plot was reoriented to make the slanted cracks at different angles be aligned to one another for easy comparison. At applied stress level of λ = 1/2, the plastic yielding zone is shown in Figure 7.34. It agrees well with the results from Reference 34. Finally, residual stress analysis was undertaken. A complete cycle of loading up to 100 MPa followed by unloading was applied to the perpendicular crack plate with α = 0° in Figure 7.32a, and the results are shown in Figure 7.35. At the maximum applied stress, the entire plate is in plastic deformation. After complete unloading, at a remote material

λ = 1/2

θ = 0° θ = 30°

θ = 60⁰

1 mm

FIGURE 7.34

(

)

Oblique crack: Plastic failure region εp > 0 .

S, Mises (Avg: 75%)

B A

210 190 170 150 130 110 90 70 50 30 10

(a)

(b)

(c)

FIGURE 7.35 Perpendicular crack: Residual von Mises stress: (a) composite stress; (b) fiber stress and (c) matrix average stress.

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point, the fibers remain in a slight compression (4 MPa) longitudinally, while the matrix remains in slight tension (1 MPa) along the fiber direction. The net stress for the composite remains stress free. Near the crack tip, the stress field is severe and complicated, as shown in Figure 7.35. Those residual stresses occurred at two distinct levels. One was the global structural level due to nonuniform near- and far-field stresses, and the other was the micromechanics level resulting from the CTE mismatch in the fiber and matrix materials. Fibers were in biaxial compression with about 200 MPa. Along the fiber direction, however, the matrix was in compression (35 MPa) at point A while in tension (35 MPa) at point B in Figure 7.35c near the crack tip. The matrix was in compression in the loading direction (50–190 MPa). Therefore, the maximum residual von Mises stress was approximately 230 MPa, which is a high stress. Note that the initial yield stress of the matrix material was about 80 MPa. These detailed aspects, including both structural and material behaviors, can be predicted directly by the present model. 7.10.6 Laminated Composite Plate The last example demonstrates the capability of the present model for a laminated composite structure. The same 34% boron–aluminum fibrous composite was used for each ply with various fiber orientations. The plate had 10 plies of [45°/90°/–45°/90°/0°]s with equal thickness (0.1 mm). At the structural level, the plate was quasi-isotropic because of the layer orientations. However, each ply should have behaved differently. Figure 7.36 shows the results of plastic deformation and stresses at the respective layers when applied stress was 100 MPa. Figure 7.36 shows larger plastic deformation in plies 1 and 3 (i.e., ±45° layers) and smaller plastic deformation in ply 2 and 4 (i.e., 90° layer). This result was consistent with analyses carried out previously for the single ply. Therefore, it can be predicted that the structural plastic deformation may initiate in ply 1 and ply 3.

Ply −1 (45°) 3

Ply −3 (–45°)

1

Ply −5 (0°) Ply −2 and 4 (90°)

Ply −2 and 4 (90°)

Ply −5 (0°) Ply −3 (–45°) Ply −1 (45°)

1 mm

3

1

FIGURE 7.36 Plastic deformation contours in each ply of [45°/90°/–45°/90°/0°]s laminated composite when λ = 1 (applied stress: 100 MPa).

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8 Multiscale Analysis of Metallic Materials

8.1 Introduction Most metallic materials have grain structures in the microscale. With the advancements of nanotechnology, new materials are constructed with nanoscale grain sizes because the material strength is inversely proportional to the grain size. Strength of the material greatly depends on the grain size and characteristics. This chapter presents a simple multiscale analysis technique for a polycrystalline material. The nature of such a material is still too complex to fully understand in all length scales. More refined multiscale techniques will be developed as more knowledge is available to bridge different length scales.

8.2 Polycrystalline Materials Polycrystalline metallic materials are commonly used for engineering applications, especially for load-carrying structural members. The stiffness and strength of those materials are influenced by various factors associated with various length scales. For instance, atomic defects such as vacancies, impurities, and dislocations play important roles at the atomic or molecular level, which is at the nanoscale. On the other hand, grain boundaries at the microscale also affect material strength and deformation. Furthermore, macroscale defects such as notches, holes, and cracks can also influence the strength of the bulk material. Most of them in various length scales cannot be avoided, but some of them may be altered through some engineering processes. Therefore, it is necessary to include all those characteristics in different length scales to understand and predict the mechanical behaviors of engineering structural materials.

8.3 Previous Study of the Multiscale Analysis of Metals Multiscale models and analyses have been developed for metallic materials [1–10] in which microstructural characteristics, including voids, dislocations, and the like, were examined through multiscale analysis. The multiscale models considered two neighboring length scales; some researchers undertook coupling of a discrete model, such as an atomistic

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model and a continuum model [11,12]. Some of the previous multiscale modeling works on metallic materials are discussed next. Chen and Mehraeen [3] used the principle of virtual power for an asymptotic expansion of field variables into multiscale Euler equations at different scales. The approach was applied to grain deformation and evolution under stress. Takano et al. [4] applied both the enhanced mesh superposition method and the asymptotic homogenization theory for multiscale finite element analyses to study porous materials. Cuitino et al. [5] identified controlling unit processes at the microscale (e.g., dislocation mobility, interactions and evolution for plastic deformation). Then, atomistic modeling was conducted for the controlling unit process assuming independency of those processes. The atomistic parameters were used to correlate the macroscopic driving force to the macroscopic response. Tadmor et al. [9] developed the quasi-continuum method to link the atomic and continuum models. Zhu et al. [10] incorporated an interatomic potential into a continuum finite element analysis using the Cauchy-Born hypothesis [13]. They studied dislocation nucleation induced by nanoindentation. The finite element result incorporating the interatomic potential energy compared well with the pure molecular dynamics result. One of the important aspects of multiscale modeling is linking one scale to the next scale. For example, it is necessary to link an atomic model to a continuum model. Liu et al. [14] provided a recent survey on multiscale modeling and simulation. They summarized and compared various coupling techniques between the atomistic model and the continuum model. Some of those techniques were hierarchical [10], concurrent [9], and bridging scale methods [15]. This chapter presents a systematic approach for multiscale modeling to implement characteristics of different length scales of a polycrystalline material into the structural behavior, from the nanoscale (i.e., the atomic dimension) to the macroscale (i.e., the engineering structure dimension).

8.4 Procedure for Multiscale Analysis of Polycrystalline Metals The proposed multiscale model consists of four different length scales: macroscale, mesoscale, microscale, and nanoscale. The macroscale is an engineering structural level whose dimension can range generally from centimeters to meters. This scale is a continuum level, so finite element analysis is conducted for the macroscale analysis. At this scale, the structure may have holes, notches, cracks, and the like, and it is subjected to external loading, possibly with geometric constraints. The mesoscale consists of a multigrain whose size is usually on the order of millimeters or less. This level is also a continuum level so that another finite element analysis is conducted at this scale. Intergrain boundary characteristics can be employed at this level. Therefore, special care is provided for the grain boundaries, as discussed further in the chapter. The microscale model is for a single-grain boundary, which is generally in the dimension of micrometers. The single-grain domain is divided into finite elements whose material properties are obtained from smeared atomic behaviors. In this model, material defects inside grains, such as dislocation, vacancies, impurities, and the like, can be considered collectively.

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Macroscale:

Mesoscale:

Microscale:

Nanoscale:

Test coupon level

Multigrain level

Single grain level

Atomic level

FIGURE 8.1 Schematic diagram of the multiscale analysis of polycrystalline metal.

Finally, the nanoscale is for a local section inside a single grain where dislocations, vacancies, impurities, and so on at the atomic level can be considered explicitly. For example, interaction among different atomic defects can be examined at this level. Molecular mechanics is suitable for this length scale. Coupling of different length scale models is depicted in Figure 8.1. The figure illustrates how each length scale analysis is connected the others. The following sections describe each scale analysis in more detail and how the neighboring length scales are connected to each other.

8.5 Macroscale Analysis Macroscale analysis is undertaken for load-carrying engineering structures such as test coupons or structural components subjected to loading. Many different numerical analysis techniques can be applied at this level. For simple problems, analytical solutions may be available, although those are limited cases. Among all different solution techniques, the finite element method is the technique used most often because of its flexibility and generality. The structure to be investigated is discretized into a finite element mesh as illustrated in Figure 8.2. To obtain a more accurate solution, an adaptive mesh technique may be utilized for a local zone where multiscale analysis will be undertaken. At the selected local points, the following set of information is obtained from the finite element analysis: {∂u/∂x

∂u/∂y ∂u/∂z

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∂v/∂x

∂v/∂y ∂v/∂z

∂w/∂x ∂w/∂y

∂w/∂z}

(8.1)

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FIGURE 8.2 Macroscale finite element mesh of a structure with a concentrated force and supports at both ends.

for a three-dimensional (3-D) domain analysis, and {∂u/∂x

∂u/∂y

∂v/∂x

∂v/∂y}

(8.2)

is used for a two-dimensional (2-D) domain analysis, where u, v, and w are the displacements along the x, y, and z coordinate axes, respectively. The solutions given in Equation 8.1 or 8.2 are utilized for the mesoscale analysis. Because 2-D and 3-D multiscale analyses use the same techniques, 2-D analysis is presented in the descriptions that follow.

8.6 Mesoscale Analysis A representative 2-D unit cell of a rectangular shape was selected for the mesoscale analysis. The size of the unit cell is generally in millimeters or micrometers, depending on the average grain sizes in the metallic material under study. The unit cell consists of a number of grains. In most cases, because grain sizes and shapes vary from location to location inside the material, it may be necessary to construct generic grain sizes and shapes properly representing the actual grains that can be measured using a scanning electron microscope. In 2-D analysis, the Voronoi diagram may be utilized to generate grain boundaries inside the unit cell as shown in Figure 8.3. The average size of Voronoi

FIGURE 8.3 Finite element mesh of Voronoi polygons in the mesoscale unit cell (solid bold lines indicate the boundary of Voronoi polygons, and the broken lines are the finite element boundaries).

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polygons is determined based on the average size of the measured grains. If the average grain size is longer in one axis due to a prior unidirectional deformation process, Voronoi polygons are constructed to have a longer dimension in the same axis. The seed points to generate Voronoi polygons are determined using a random number generation process. The number of grains is determined by dividing the unit cell length by the average grain size. Each Voronoi cell, which represents a grain, is further divided into a finite element mesh for the mesolevel finite element analysis (Figure 8.3). Then, the deformation gradient solution obtained from the macroscale analysis as given in Equation 8.2 is used as boundary conditions of the mesoscale model as indicated in Figure 8.4. In the figure, line segments AB and DC are parallel to each other, while line segments BC and AD are also equal and parallel, too. Because grain boundaries have different crystallographic orientations with dislocations, the interface stiffness of a grain boundary is different from the internal stiffness of the mostly regularly positioned grains. As a result, the grain boundary is modeled using different finite elements. A simple way is to use spring elements as illustrated in Figure 8.5. Two neighboring finite elements at a grain boundary have independent nodes that are overlapped at the grain boundary as shown in the figure. Then, the two independent nodes are coupled using spring elements in both directions. The spring constant should be properly determined to represent the stiffness at the interface. A low value of the spring constant means low stiffness of the interface along the spring orientation. A molecular dynamics simulation may be undertaken to determine the proper stiffness of the interface for various conditions, such as different crystallographic orientations and dislocations. Depending on the arrangement of the Voronoi polygons, a point at multigrain boundaries may have multiple nodes, so each grain sharing the point can have its own node. The finite element solution of the mesoscale unit cell provides the nodal displacements of a grain, which is analyzed in the subsequent microscale analysis. The selected grain with known nodal displacements is further analyzed as described in the next section. ∂u L ∂y y

C D

∂v L ∂y y

B

Ly

∂v L ∂x x A

Lx ∂u L ∂x x

FIGURE 8.4 Sketch of deformation of the mesoscale unit cell.

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Grain boundary

Elem. A

Elem. A Double Elem. B

Nodes Elem. B

Spring element FIGURE 8.5 Two finite elements meeting at a grain boundary and connected by spring elements.

8.7 Microscale Analysis Microscale analysis is typically at the length scale of micrometers or less. The unit cell is a single grain selected from the previous mesoscale analysis. Independent multiple grains can be analyzed if necessary. Each single grain is also analyzed using the finite element method because there are still too many atoms in a single grain so a continuum model is still applicable. As a result, finite element analysis is undertaken for the single-grain unit cell with the boundary displacements obtained from the mesoscale analysis. The grain may have defects, such as dislocations or vacancies. The stiffness of each finite element in the grain may be adjusted based on the respective state of defects. If there is no specific information available for the locations and states of defects inside the grain, a random assignment of defects for each element may be assumed in terms of statistical data. From the microscale analysis, the deformation gradients as in Equation 8.2 are obtained at a selected location for the next-level nanoscale analysis. If desirable, the microscale analysis and nanoscale analysis can be conducted simultaneously without two subsequent analyses.

8.8 Nanoscale Analysis Eventually, nanoscale analysis is performed at the atomic scale, which can represent all the defects as they are. The dimension of the nanoscale model is nanometers. Because there are still too many atoms in the nanoscale model in general, we cannot model all atoms in the nanoscale model. Instead, the nanoscale model consists of three subdomains. Individual atoms are modeled in the inner subdomain, while a smeared continuum model is used for the outer subdomain. Then, two subdomains are coupled through the interface subdomain. Figure 8.6 shows a nanoscale model. The inner subdomain contains a detailed atomic arrangement, such as atomic vacancies, dislocations, impurities, and so on. Molecular mechanics is applied to the inner subdomain. The outer subdomain is modeled as a continuum with smeared material properties of the atoms, and it is modeled using the finite

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Continuum region with smeared atoms

Discrete atomic region

Interface region

FIGURE 8.6 Nanoscale analysis model containing three subdomains. The broken lines indicate a finite element mesh.

element technique [11]. Eventually, the interface subdomain contains both finite elements and discrete atoms as denoted by bold lines in Figure 8.6. This subdomain provides coupling between the discrete atoms and the smeared continuous subdomain as discussed in Chapter 6 [16]. Boundary conditions are applied to the outer subdomain using the deformation gradients obtained from the microscale analysis as expressed in Equation 8.2.

8.9 Example Problems As an example for multiscale analysis, a large plate with a hole at the center is considered [17]. The plate is subjected to a uniform applied stress σo as shown in Figure 8.7. At the edge of the hole, a nanoscale crack is located, denoted by A in the figure. Multiscale analysis is applied to point A. Because the nanoscale crack is too small to be included in the macroscale analysis, the macroscale model neglects the nanoscale crack. In this example case, an analytical solution is available so it is used for the macroscale level instead of finite element analysis. The deformation state at point A in Figure 8.7 is given as ∂u 3µσ o =− , ∂x E

∂u = 0, ∂y

∂v = 0, ∂x

∂v 3σ o = ∂y E

(8.3)

where u and v are the displacements along the x and y axes, respectively, and E and μ are the elastic modulus and Poisson ratio of the material, respectively. Pure aluminum is chosen as the material, so the elastic modulus is 70 MPa and the Poisson ratio is 0.3. Figure 8.8 shows the mesoscale analysis model at point A in Figure 8.7. The average grain size is assumed to be 10 μm. The multigrains are generated using the Voronoi diagram with random point generation. Each grain in Figure 8.8 is discretized into a number of finite

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σo Plate

y

Nanoscale crack x A

Hole

σo FIGURE 8.7 A very large perforated plate under remote uniform stress with a nanoscale crack.

Selected grain for microscale analysis FIGURE 8.8 Multigrains for mesoscale analysis in its local coordinate system with the unit in micrometers. The grain in the bold-line boundaries is for the next-level microscale analysis.

elements. Approximately 1000 finite elements are used in the mesh. If the grain boundary information (e.g., grain orientations and defects in grain boundaries) is known at the specific location, it should be used in the mesoscale model. Otherwise, statistical data for the grain boundary can be utilized. For example, the stiffness of grain boundaries (i.e., the interface spring constants in Figure 8.5) of the mesoscale model is assigned randomly based on the selected statistical model to represent the random grain orientations within grains and grain boundary defects. For the present example, a uniform random distribution is considered for the grain boundary stiffness, which is assumed to vary between the

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prescribed ranges. Four different cases are considered in this study. It is assumed that the grain boundary stiffness varies randomly between (a) 90% and 100%, (b) 50% and 100%, (c) 10% and 100%, and (d) 1% and 100% of the value that represents no defect in the grain boundary. The major stress component that occurs in the y axis of Figure 8.7 is calculated for the grain, denoted by bold boundary lines in Figure 8.8 and normalized with respect to the applied tensile stress. The stresses are calculated 200 times for each case of the four scenarios for statistical analysis. Then, the probability of events at different stress states is plotted in Figure 8.9 for the four cases. As expected, the actual state of stress depends on the grain boundary state. Microscale analysis is conducted for the bottom left grain as selected in Figure 8.8 using the finite element method. The selected grain is shown in Figure 8.10 with a finite element mesh. Approximately 2000 elements are used in the microscale model. However, for visual clarity, a much coarser mesh is shown in Figure 8.10. The boundary conditions of the microscale model are obtained from the deformation of the single-grain boundary from the mesoscale analysis. The elastic modulus of each finite element inside the grain is randomly varied to represent possible defects inside the grain, such as dislocations or vacancies. The first case assumes a uniform random variation of the elastic modulus between 90% and 100% of the modulus of the no-defect case. A total of 100 computer simulations are performed with the randomly generated modulus. The second case assumes a Gaussian distribution of the elastic modulus between 90% and 100% of the modulus of the no-defect case, with 95% for the mean elastic modulus. Figure 8.11 plots the histogram of

0.2 Probability of event

Probability of event

0.2 0.15 0.1 0.05 0 (a)

1

2 3 Normalized stress

4

0.15 0.1 0.05 0

(b)

Probability of event

Probability of event

0.15 0.1 0.05 0

2 3 Normalized stress

4

1

2 3 Normalized stress

4

0.2

0.2

(c)

1

1

2 3 Normalized stress

4

0.15 0.1 0.05 0

(d)

FIGURE 8.9 Probability of stress states at the selected grain location in Figure 8.8. The grain boundary stiffness varies randomly within (a) 90% to 100%; (b) 50% to 100%; (c) 10% to 100%; and (d) 1% to 100% of the stiffness of the grain boundary without any defect.

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Selected element for nanoscale model

0.16

0.16

0.14

0.14

0.12

0.12

0.1

Probability of event

Probability of event

FIGURE 8.10 Finite element mesh for the microscale model of the selected grain in the mesoscale model.

0.08 0.06

0.1 0.08 0.06

0.04

0.04

0.02

0.02

(a)

0 2.7

2.8

2.9 3 3.1 Normalized stress

3.2

(b)

0 2.7

2.8

2.9 3 3.1 Normalized stress

3.2

FIGURE 8.11 Probability of stress states at the selected location in the grain as shown in Figure 8.10. The material defects in the finite elements of the grain vary in (a) uniform random distribution or (b) Gaussian random distribution between 90% and 100% of the stiffness of the grain boundary without any defect.

the probability of events at the stress state at the bottom left corner of the selected grain in Figure 8.10. Finally, Figure 8.12 shows the nanoscale analysis model. The discrete atomic region is surrounded by the smeared continuum region. Figure 8.12 shows the coupled atoms and the smeared continuum. Even though more than 1500 atoms are used, only a small number of atoms are shown in the figure for visual clarity.

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Discrete atomic subdomain

Smeared continuum subdomain FIGURE 8.12 Nanoscale model with the discrete atomic subdomain surrounded by a smeared continuum subdomain that was divided into a finite element mesh.

In the nanoscale analysis, the nanoscale crack is modeled discretely. The nanoscale model is subjected to the stress field at the boundary as determined in the microscale analysis. Because the finite element model is used for the smeared continuum region, the boundary conditions are applied to the finite element mesh. The displaced atomic positions near the crack tip due to the load are shown in Figure 8.13. The figure illustrates the atomic separation around the crack tip indicating crack propagation at the bottom side of the crack tip.

Crack growth

FIGURE 8.13 Crack growth shown with separation of atoms.

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9 Multiscale Analysis of Biomaterials

9.1 Introduction Biomaterials are complex living tissues that have evolved for many years. Those materials have distinct hierarchical structures constructed of simpler materials [1]. The human body has a limited number of basic materials to make the desired biomaterials in the macroscale [2]. Bones, tendons, ligaments, muscles, and skin are examples of such biomaterials. This chapter studies the bone because it is the main part of the body skeleton that can support the load in both tension and compression [3–5]. Complex multiscale hierarchies of the bones can carry various large loading with an optimal weight and serve as the storage site for bone marrow, calcium, and phosphate. To understand and predict the structural properties of the bones, a multiscale model was developed and its analysis was conducted. The multiscale model began with the nanoscale model and continued up the hierarchies to the macroscale level. Figure 9.1 shows a sketch of multiscale hierarchical structures of the bones. In the following sections, some biological descriptions in different length scales are provided. Then, multiscale biomechanics models and analyses are presented [6].

9.2 Nanoscale Model The major nanoscale components of the bone are hydroxyapatite (HA) and tropocollagen. These two components become the building blocks for the next scale model: the microscale model. The bone contains microscopic inclusions of noncollagenous proteins and proteoglycans other than hydroxyapatite and tropocollagen. These two materials act as a binding matrix, while noncollagenous proteins also act for metabolic functions [7]. Because the noncollagenous proteins and proteoglycans do not serve for mechanical functions and their volume fractions are low, they are not considered in the nanoscale model discussed here. As a result, only hydroxyapatite and tropocollagen are further discussed and modeled. 9.2.1 Hydroxyapatite Hydroxyapatite is a hexagonal, close-packed crystal lattice crystalline solid material consisting of calcium phosphate, and it is the only ceramic-type material created naturally inside the human body [8,9]. Hydroxyapatite is sometimes called bone mineral, and it is 267 © 2016 by Taylor & Francis Group, LLC

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Nanoscale

(a)

(b)

(c)

(d)

Microscale

(e)

Macroscale

(f )

(g)

FIGURE 9.1 Diagram of bone hierarchies from nanoscale to macroscale: (a) Tropocollagen; (b) Hydroxopatite; (c) Fibril; (d) Fiber; (e) Lamellar bone; (f) Trabecular bone; and (g) Cortical bone.

the major component providing bone stiffness. The material has a polycrystalline form as small thin plates within the body [4,10]. Hydroxyapatite has a chemical formula of Ca10(PO4)6(OH)2. The growth of hydroxyapatite is regulated by heterogeneous nucleation factors that promote the growth of the mineral phase and contribute to the highly ordered structure of the microlevel hierarchy of the bone. The growth is considered concentrated within the gap zones. Because hydroxyapatite exhibits a hexagonal close-packed crystal lattice, a single crystal can be tested to determine material properties [10].

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TABLE 9.1 Properties of Hydroxyapatite Materials Source

Elastic Modulus, E (GPa)

Poisson Ratio, ν

Measuring Technique

150 144 114 165

– – 0.27 –

Nanoindentation along [0001] Nanoindentation along [1010] Ultrasonic –

[10] [11] [8]

Nanoindentation testing on single hydroxyapatite crystals showed that the stiffness in the [0001] direction of the hexagonal close-packed crystal lattice was greater than that in the [1010] direction [10]. Some of the previous test results are listed in Table 9.1. The data showed some variations. Because the nanoindentation test data were the median, those values were used in this study as the reference values. Although the size of hydroxyapatite crystals varies depending on locations, average crystal size is considered 50 × 25 × 3 nm [1,3,4,8,10,12], and the crystals are organized into small plates. Hydroxyapatites also store ions, and they are responsible for 99%, 90%, 90%, and 50% of the body’s store of calcium, phosphorus, sodium, and magnesium, respectively. Hydroxyapatites have a large ratio of surface area to volume, which allows for rapid absorption and dissolution of ions as needed [4,10]. This is important in determining effects of bone diseases and disorders that affect bone density and calcium absorption. 9.2.2 Tropocollagen Collagen is constructed of tropocollagen molecules as the basic unit. Collagen has 28 different variations to serve different physiological purposes as needed in the body [13]. Collagens in human bones consist mostly of collagen I type, about 95% of the total collagen [14]. Collagen I is composed of three helical protein strands. Two are alpha-1 type 1 (COL1A1) strands, and the third one is the alpha-1 type 2 (COL1A2) strand [13,15–19]. Each strand is different in terms of its amino acid sequence, but each strand maintains a similar helical structure. Both COL1A1 and COL1A2 are left-handed polyproline II (PPII) helices, and they are 300 nm long. They also contain approximately 1000 amino acids [19–25]. Those helices have a common repeating subunit, called Gly-X-Y [13,16–18,20,21,26]. Gly is the amino acid glycine, which is the smallest of the amino acids. For each repeated strand, an amino acid glycine is along the central axis of the tropocollagen molecule [27,28]. This arrangement makes the three strands orient themselves to form stable hydrogen bonds. There is a pattern between the X and Y positions of the three-amino-acid subunit. Two commonly repeated patterns are Gly-Pro-Y and Gly-X-Hyp, where Pro and Hyp represent the amino acids proline and hydroxyproline, respectively [16–18,23]. A common model for the tropocollagen molecule utilizes the repeating sequence Gly-Pro-Hyp. 9.2.3 Helical Spring A helical spring model was selected to estimate the stiffness of a single molecule of the tropocollagen. This model was selected because each polyproline helix of the tropocollagen molecule can be independently represented as a spring. The stabilizing hydrogen bonds as well as the van der Waals force among the three helices keep the tropocollagen from buckling and make sure that all three helices deform symmetrically. The close packing

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of the tropocollagen molecules within collagen fibrils also prevents buckling, which is discussed further in this chapter. The stiffness of the tropocollagen molecule is the sum of the stiffness of the COL1A1 and COL1A2 strands. Equation 9.1 was used to determine the spring stiffness of each helix. k=

Gd 4 8D3 N A

(9.1)

where k represents the spring constant, G represents the shear modulus, d is the wire diameter, D denotes the mean coil diameter, and NA represents the number of coils. To compute the quantitative values for each protein helix, we assumed that each repeating subunit is consists of the Gly-Pro-Hyp sequence. This representative subunit provides a relatively stable position for each amino acid. It also allows for proper crosslinking and hydrogen bonds to form between strands [18,20,23,29]. The presence of the amino acids proline and hydroxyproline greatly influences the stiffness of collagen [30]. Modeling every subunit with this sequence provides good agreement with the correct bond angles and axial repeat, 10/3, for the majority of the polyproline helices [18]. Some recent studies discussed the predominance of an axial repeat of 7/2 over portions of the PPII. However, this variation is more important for molecular dynamics simulations in which strain energies and steric effects of the amino acid interactions are computed and analyzed [27,28]. The wire diameter for the helical spring model as expressed in Equation 9.1 was taken to be the average spacing between residues, 0.286 nm [19,27,28]. The coil diameter of the helix was determined from the diameter of a tropocollagen molecule. A tropocollagen molecule has a diameter of 1.5 nm. Hence, a single PPII helix was assumed to have a mean coil diameter of 0.5 nm [18,20,21]. The number of coils was determined from a constant axial repeat of 10/3 [16,23]. In other words, the number of coils was 300 for 1000 amino acids. The shear modulus was the most difficult parameter to derive because no data are available for the shear modulus of a single-chain amino acid helix. The bond energy of the backbone of a single subunit was computed to estimate the shear modulus. There are six C–N bonds and three C–C bonds along the backbone inside the Gly-Pro-Hyp triplet. Bond energies were 5.06 × 10 –19 J/bond and 5.75 × 10 –19 J/bond for C–N and C–C bonds, respectively; therefore, the total bond energy of the backbone was then 4.76 × 10–18 J/subunit. As a subunit had a volume of 7.02 × 10 –29 m3, the shear stress of the backbone was computed as energy over volume, which was equal to 67.9 GPa. This is close to the typical values of aluminum (70 GPa) and thus not unreasonable. This calculation did not include all strengthening effects of crosslinking, nonbackbone atoms, and other atomistic considerations. However, this value provides a basic starting point to calculate the overall stiffness of collagen. 9.2.4 Results of the Nanoscale Model Table 9.2 is a summary of the parameters and their values used for Equation 9.1. The spring constant k for a helix is 0.0015 N/m. The elastic modulus of a helix was computed from the spring constant using the following equation:  L E= k  A

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(9.2)

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TABLE 9.2 Parameter Values Used for Equation 9.1 Parameter

Value

Units

G d D NA k

67.9 0.286 0.5 300 0.0015

GPa nm nm – N m

where L is the length and A is the cross-sectional area of the helix. The length and diameter of a PPII helix were 300 nm and 0.5 nm, respectively. However, due to the helical twist, the cross-sectional diameter of the helix was defined as 0.7 nm. Using the helix length of 300 nm and cross-sectional area of 3.85 × 10 –19 m2, Equation 9.2 yielded the elastic modulus of 1.18 GPa for each protein helix. Because each tropocollagen molecule had three helices, the elastic modulus of a tropocollagen molecule was 3.54 GPa. The present estimated modulus is compared to other experimental data and theoretical analysis results in Table 9.3. All results showed quite a variation in the elastic modulus of the tropocollagen, from less than 1 GPa to more than 10 GPa. When compared, the present nanoscale model using the equivalent helical spring presented a reasonable elastic modulus of the tropocollagen, confirming the validity of this model.

TABLE 9.3 Comparison of Elastic Modulus for Tropocollagen Elastic Modulus (GPa) 3.54 1.2 2.8 6–16 (average 6) 7 2.4 3 9 5.1 3 0.35–12 4.8 ± 1 1.4 (PHG)a 11.37 (PPG)b 13.43 a b

Method

Source

Spring model Property used in finite element model Property used in finite element model Molecular dynamics Atomistic modeling Atomistic modeling X-ray diffraction Brillouin light scattering Brillouin light scattering Estimate from persistence length Estimate from persistence length Molecular dynamics Debye-Waller factor Molecular dynamics Molecular dynamics

Present study [1] [31] [29] [32] [33] [34] [35] [36] [37] [33,38] [39] [23] [29] [29]

PHG is a Gly-Pro-Hyp model. PPG is a Gly-Pro-Pro model.

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9.3 Microscale Model The previous section considered the material properties of the elemental composite materials of bone at the nanoscale, that is, hydroxyapatite and tropocollagen. These two simple constituents produce a more complex hierarchy at the next length scale, called the microscale. The structures at the microscale have organized and regular arrangements of collagen and hydroxyapatite. A micromechanics model can be developed and used at this length scale. 9.3.1 Two-Dimensional Fibril Structure The fibril is the smallest size of the microstructures at the microscale. It consists of a staggered two-dimensional (2-D) array of hydroxyapatite and tropocollagen molecules. Those arrays depend on the binding and cross-linking tendencies of tropocollagen molecules. The regular fibrils are organized during a process called fibrillogenesis, which involves the expansion and regrowth of fibrils. A collagen molecule has a C terminal and an N terminal, so named for the peptides cleaved during the transition from procollagen to collagen [19,25]. As each end of a tropocollagen exhibits different cross-linking tendencies, the amino acid sequence at each terminal of the tropocollagen defines a polar reference for each molecule. When the collagen molecules are aligned near each other, the C and N ends join to produce an organized structure. Not only do the ends of the tropocollagen molecules have preferential alignment, but also the entire length of the tropocollagen molecule contains segments of amino acids preferential to cross-linking with adjacent tropocollagen molecules [26]. These preferential segments form a collagen network with a 40-nm gap and a regular periodicity of 67 nm [22,25,26,40–44]. Hydroxyapatites grow preferentially in these gap regions, allowing for the inclusion of the reinforcing particles within the fibril composite. Figure 9.2 shows the staggered alignment of hydroxyapatite in the fibril composite. This figure is not to scale. The height of the crystals and tropocollagen matrix is 3 nm, and the hydroxyapatite crystals are 50 nm in length. Figure 9.2 is used to portray the periodic arrangement known to occur. This shows a regular structure driven by the molecular sequence of the tropocollagen molecules. Furthermore, such cross-linking results in stability on the tropocollagen phase of the fibril [22,30]. The cross-linking keeps the individual molecules from shearing. This in turn increases the strength and brittleness of the structure [22,31]. The 67 nm

67 nm

67 nm

67 nm

67 nm

67 nm 1 3

~N

C~ ~N

C~ ~N

C~

FIGURE 9.2 Visualization of 67-nm periodicity in bone fibril. Gray regions denote hydroxyapatite crystals, and white regions depict tropocollagen.

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presence of cross-linking also allows the collagen phase to be represented as a single, continuous, material in a simplified model. An important observation of the tropocollagen and hydroxyapatite microstructures shows that the hydroxyapatite crystals are aligned in the parallel direction to the microfibril orientation. This results from the growth behavior of the mineral phase within the gap zones. The initial deposition of hydroxyapatite at a molecular level is influenced by heterogeneous nucleation factors. One of the factors is bone sialoprotein, which binds directly to the ends of collagen I molecules. In addition, bone sialoprotein is found within the gap zones of microfibrils in a high concentration and has been found to be a nucleator of hydroxyapatite formation, creating a hydroxyapatite crystal within a supersaturated medium [45]. Hydroxyapatite crystals grow in a preferential direction, depending on the substrate chemistry on which they are grown [46]. As a result, mineral components supersaturate the gap zone, followed by crystal growth on the ends of the collagen I molecules initiated by the bone sialoproteins. Subsequently, crystals grow in an ordered manner that aligns all crystals parallel with the microfibril axis [45]. Because all crystals have grown in the same orientation, they can be considered as a transversely isotropic material using the material values given in Table 9.1. There is one conflict in the physical models based on tropocollagen cross-linking. The collagen network has 40-nm gaps, which are smaller than the 50 × 25 × 3 nm hydroxyapatite crystals. It is believed that the ends of the tropocollagen molecules must be surrounded by hydroxyapatites, which grow between adjacent molecules [44,47]. This helps to prevent shearing of the hydroxyapatite–tropocollagen-bonded surfaces. The width of the crystals represents another challenge in modeling and determining the three-dimensional (3-D) structure of collagen fibrils. 9.3.2 Three-Dimensional Fibril Structure Even though the 2-D regular staggered array of a collagen fibril can be easily visualized, 3-D fibrillogenesis in vivo is still a debated topic [25,26]. When fibrils are examined using electron microscopy, they have shown different 3-D structures. One is a linear fibril structure, and the other is a twisting fibril structure. A linear fibril model has been accepted for many years [25,27,47,48]. The linear model was considered as a long, thin filament with alternating bands of mineral-rich phase and mineral-deficient phase. The former is sometimes referred to as the gap, while the latter is referred to as the overlap. The mineral content in rich phases is approximately twice that of the deficient phase [48]. Figure 9.3 shows the 3-D packing of hydroxyapatite crystals in the linear fibril model.

2 1 3

FIGURE 9.3 Simplified linear fibril model.

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67 nm

2

67 nm

67 nm

1 3

FIGURE 9.4 Twisting fibril model.

Even though the linear fibril structure is a widely accepted model for the proposed theory of 3-D fibrillogenesis, there has not been any evidence that the fibril grows purely linearly in the lateral direction. A recent study with the in vitro organization of fibril growth showed that fibrils grow laterally in 4-nm steps [41]. Each lateral step was defined by electrostatic attraction during the formation of the microscale fibril, but the lateral growth was not related to longitudinal growth [41]. More recently, a twisting crystalline structure was considered [7]. A spiral orientation resulted in a more even distribution of hydroxyapatite crystals. In the twisting model, periodicity was assumed in the lateral direction with 67 nm, and the 2-D stacking also directed the 3-D pattern. A simplified twisting fibril model is shown in Figure 9.4. Figures 9.3 and 9.4 show the two proposed fibril models and illustrate a simplified visualization of hydroxyapatite crystal distributions. The organization of bone fibril in vivo is much more complex because it includes noncollagenous proteins; errors in collagen stacking, amino acid sequencing, and continuous absorption and regrowth of fibrils. Furthermore, biological materials are never made of pure substances. The collagen used in bone contains only 95% collagen I [14]. It is postulated that the other collagen species assist in regulating the fibrillogenesis process, but they may also introduce irregularities into the structure [19]. Fibrils are generally considered to have a circular cross section whose diameter varies from 50 to 300 nm [49,50]. It is reasonable to state that only one crystal width is present across the structure in those small-diameter fibrils. This would validate the use of the linear fibril model. However, either linear or twisting structures may be possible for fibrils with larger diameters. To explore the two models discussed previously, a micromechanics model was modified to characterize both structures, and their results were compared to other experimental and theoretical data to determine their validity. 9.3.3 Micromechanics Fibril Model Kwon and his coworkers [51–59] developed a micromechanics model for the analysis of composite structures. This model is presented in Chapter 7, and the representative unit cell is shown in Figure 7.2. The micromechanics model was used for the present analysis. For the present work, the collagen network was assumed to have a staggered array with at least a 25-nm width to allow for the generally accepted dimensions of the hydroxyapatite crystals. Then, the micromechanics model would provide relevant results for both the linear packing model and the twisted packing model. Table 9.4 lists the material properties used to model the fibril with the micromechanics model.

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TABLE 9.4 Material Properties for Fibril Components Material Hydroxyapatite Tropocollagen (compression) Tropocollagen (tension)

E1 (GPa)

E 2 (GPa)

ν12

G 12 (GPa)

150.38 3.428 3.256

143.56 3.428 3.256

0.23 0.35 0.35

59.744 1.270 1.206

Water is present inside fibrils. The presence of water makes the fibril a bimodulus composite material because water can only support compressive load. Bone is known to contain approximately 10% to 25% water [31], and some of this is thought to be contained inside the nanoscale tropocollagen and hydroxyapatite. Water serves as a binding and stabilizing agent within the tropocollagen [26,30], and small amounts of water are tightly bound within the hydroxyapatite crystal [31]. In addition, the rest of the water is assumed to be held within the various hierarchies. For the present fibril model, each unit cell was postulated to have 8% water by volume. In other words, the stiffness of collagen included 8% water by volume in compression and an 8% void space volume in tension. Tropocollagen in compression and tension exhibits the different properties shown in Table 9.4. 9.3.3.1 Linear Fibril Subunit The linear packing model was a lateral repeat of the 2-D fibril array shown in Figure 9.2. Each crystal had a length of 50 nm, width of 25 nm, and height of 3 nm. Figure 9.3 shows the cross section used to create a repeating subunit of the array. This subunit section repeats throughout the array in 67-nm increments. The dimensions used for the micromechanics unit cell are listed in Table 9.5. Subcell 1 was assigned hydroxyapatite properties; the remaining subcells were assigned tropocollagen properties. 9.3.3.2 Twisting Fibril Subunit The twisting fibril takes into account periodicity in the 2 direction as well as the 3 direction, as shown in Figure 9.5. Table 9.6 lists the dimensions used for the twisting model. The dimensions for the a1, a2, b1, and c1 lengths were determined with the 67-nm periodicity in the 1 direction and the known size of the hydroxyapatite crystals. For the twisting model, subcell 1 was assigned hydroxyapatite properties, and the remaining subcells were assigned tropocollagen properties. TABLE 9.5 Unit Cell Dimension for Linear Fibril Model Dimension a1 a2 b1 b2 c1 c2

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Value (nm) 50 17 25 3 3 9

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2 1 3

FIGURE 9.5 Subunit employed in the twisting fibril model.

TABLE 9.6 Unit Cell Dimension for Twisting Fibril Model Dimension

Value (nm) 50 17 25 75 3 6

a1 a2 b1 b2 c1 c2

9.3.4 Fibril Results The micromechanics model calculated the effective properties of the fibril unit cell. Fibrils are expected to have transverse isotropic properties because of the random dispersion in the radial orientations. The 1 axis in the micromechanics model lies along the fiber axis, but the 23-plane orientation will be different for each fibril. To consider the random dispersion in the radial direction to compute transverse isotropic material properties, the elastic and shear moduli and Poisson ratios were averaged in the 2 and 3 axis. The resulting values are shown in Table 9.7, and the results are compared to other experimental and theoretical values in Table 9.8. All the results except for the experimental nanoindentation and dynamic mechanical analysis evaluated the longitudinal elastic modulus in tension. The others determined the transverse elastic modulus of a fibril in compression. The micromechanics model resulted in the elastic moduli, which compared well with other data for bone fibrils. The micromechanics model provided a simple solution to a complex problem. The relatively elementary approach yielded realistic results for both the TABLE 9.7 Transverse Isotropic Fibril Results Model Linear Twisting

Compression Tension Compression Tension

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E1 (GPa)

E 23 (GPa)

G 23 (GPa)

G 12 (GPa)

ν12

ν32

6.050 5.760 4.264 4.054

6.601 6.298 3.812 3.621

1.534 1.458 1.354 1.286

1.677 1.593 1.368 1.299

0.299 0.299 0.309 0.309

0.295 0.295 0.366 0.366

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TABLE 9.8 Comparison of Values for Fibril Results Elastic Modulus (GPa) 7.65 ± 3.85 1.00 ± 0.75 7.5 ± 5.5 4.96 ± 0.57 3.07 ± 0.23 4.36 38 10 ± 2.0 5 4.81 2.05 ± 0.75 4.75 ± 3.06 a

Method

Source

Nanoindentation of rat tail tendon (transverse) Finite element model, tension (longitudinal) Finite element model, compression (longitudinal) Dynamic mechanical analysis using AFM, peak (transverse)a Dynamic mechanical analysis using AFM, trough (transverse)a Molecular multiscale modeling (small strain) Molecular multiscale modeling (large strain) 3-D molecular dynamics (longitudinal) Molecular modeling (longitudinal) Molecular dynamics and finite element modeling (longitudinal) Full atomistic model (longitudinal) Finite element model (longitudinal)

[60] [31] [31] [50] [50] [22] [22] [42] [44] [47] [61] [48]

Peak and trough refer to mineralized and unmineralized sections, respectively. AFM, atomic force microscopy.

TABLE 9.9 Fibril Model Mineral Volume Fraction Fibril Model Linear Twisting

Mineral Volume Fraction (%) 16.66 6.22

linear fibril model and the twisting fibril model. In addition, the micromechanics model calculated the Poisson ratios for the fibril rather than assuming a value, as did all results in Table 9.8. The micromechanics model resulted in accurate results for the elastic modulus of bone fibrils in both tension and compression. The complete data set for the fibril calculated using the micromechanics model can be used to increase the accuracy of subsequent hierarchies, as the micromechanics model was used for the bone fiber. The bone mineral content was used as a metric for comparing the validity of theoretical models. However, the mineral content at the microstructure level is not well known. Therefore, the mineral content was tracked at each length scale and was compared at the macrolevel in further study. Table 9.9 shows the mineral content of the two fibril models. 9.3.5 Bone Fiber In the hierarchical structure, the bone fibril makes up the bone fiber. Fibers consist of fibrils and hydroxyapatite. The fiber contains hydroxyapatite in the form of extrafibrillar mineral, deposited between densely packed fibrils [2,7,62,63]. Bone fibers are sometimes called fibril bundles. Minerals are deposited on the outside of fibrils, which are arranged in parallel. The extrafibrillar minerals provide a stiffener for the bone fiber. These mineralized fibrils are tightly packed in a uniform direction [7]. Each closely packed fibril is surrounded by a crust of hydroxyapatite minerals, which grow as small plates with an approximate thickness of 20–30 nm to coat the fibril [7,63]. As a result, there is an ordered arrangement of minerals such that the 1 axis of minerals and

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3 2

1

FIGURE 9.6 Three-dimensional fiber array. Dark rectangles represent hydroxyapatite crystals; the rod structures represent fibrils.

the longitudinal axis of fibrils are parallel to each other [63]. This allows each fiber to be modeled as a transversely isotropic material. The thickness of the crust was considered to be 26 nm, and the diameter of a single uniform fibril was chosen to be 150 nm [7]. Figure 9.6 illustrates the 3-D structure with the surface mineral removed for visual clarity. The fibrils in Figure 9.6 show a parallel arrangement, and both linear and twisting mineral fibril models were analyzed. There was little information regarding the degree of mineralization along the lengths of the fibrils [7]. Different variations of mineralization were considered to analyze the effects of extrafibrillar mineralization (EFM). Mineral packing at the fiber level is not a close-packed assembly because there are void spaces among the minerals. Water is contained in those voids along with dissolved nonstructural proteins and macromolecules [7]. This solution is often called the extrafibrillar matrix. This liquid phase at the fiber level represents the remaining component of bone water, and these spaces are treated as water, resulting in a bimodulus property for fibers as considered for the fibril model. 9.3.6 Micromechanical Fiber Model The fibers are made of single and multiple bundles of fibrils. The bundles have organized and repeating arrangements of fibrils and hydroxyapatite. Therefore, the micromechanics model discussed in Chapter 7 was also applicable to model the bone fiber. The micromechanics model relies on volume fractions of each material. To ensure a volume of unity, Equation 9.3 was used: a1 + a2 = b1 + b2 = c1 + c2 = 1

(9.3)

Furthermore, a circular cross section of the fibrils was modeled as an equivalent square shape and minerals surrounding the fibrils. In the micromechanics model, the 1-axis cross section remained constant like a continuous fiber composite model. This area percentage was based on a total crust thickness of 26 nm and a fibril diameter of 150 nm. To apply this to the micromechanics model, Table 9.10 shows the dimensions applied to the 1-axis face. This resulted in a fibril area of 72.25% when converted into the micromechanics model. This is shown in Figure 9.7, where subcell 1 represents the fibril. To test the effect of fibril mineralization, the degree of mineralization was varied. Percentage EFM (%EFM) was defined as the total percentage of fibril covered by mineral and was equal to the volume of subcells 3, 5, and 7, shown in

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TABLE 9.10 Fiber Unit Cell 1-Axis Face Dimension Dimension Parameter

Fraction 0.85 0.15 0.85 0.15

b1 b2 c1 c2 3 c2

7

5

c1

3

1

b2

b1

2

(a)

(b)

FIGURE 9.7 Longitudinal fiber cross section as compared to micromechanics model cross section. (a) Fiber cross section: dark area represents mineral, light section represents fibril. (b) Micromechanics model longitudinal cross section.

Figure 9.8. The %EFM was altered by changing the length of a1. It was postulated that the fiber consisted of no more than 95% EFM. The different levels of mineralization analyzed are listed in Table 9.11. The effect of water was also studied under compression. Subcells 4, 6, and 8 were assigned the properties of water in Figure 9.8. In tension, these subcells were assumed to be void spaces. 3

6 8

15

85

8

4

5 7

3

7

7

3

1

1 2

85 a2

FIGURE 9.8 Unit cell for fiber micromechanics model.

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a1

15

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TABLE 9.11 Fiber Unit Cell Extrafibrillar Mineralization (EFM) Dimensions %EFM

a1

a2

50 70 90 95

50 70 90 95

50 30 10 5

9.3.7 Fiber Results The material properties were computed using the fiber micromechanics model for compression and tension. As expected, the fiber exhibited transverse isotropic material properties because the radial arrangement of the fiber arrays had a random distribution. Tables 9.12 and 9.13 show the results for compression and tension, respectively. The compressive results showed that the linear fibril model produced a stiffer fiber in elastic and shear moduli than the twisting fibril model. For all %EFM, the fiber shear modulus in the plane 12 direction was much less than for the plane 23 directions. As %EFM increased from 50% to 95%, the plane 12 shear modulus increased by an order of magnitude, while the plane 23 modulus approximately doubled. This was exhibited for both the linear and the twisting models.

TABLE 9.12 Fiber Results in Compression 50% EFM

E1 (GPa) E2 = E3 (GPa) G23 (GPa) G12 (GPa) ν21 ν32

70% EFM

90% EFM

95% EFM

Linear

Twisting

Linear

Twisting

Linear

Twisting

Linear

Twisting

6.511 16.66 1.053 0.012 0.369 0.140

5.446 14.62 0.931 0.012 0.413 0.132

7.567 21.41 1.473 0.019 0.364 0.128

6.509 19.16 1.302 0.019 0.397 0.119

11.60 26.35 1.893 0.056 0.295 0.126

10.47 23.91 1.672 0.056 0.307 0.118

16.17 27.72 1.998 0.109 0.245 0.127

14.97 25.23 1.765 0.108 0.250 0.120

TABLE 9.13 Fiber Results in Tension 50% EFM

E1 (GPa) E2 = E3 (GPa) G23 (GPa) G12 (GPa) ν21 ν32

70% EFM

90% EFM

95% EFM

Linear

Twisting

Linear

Twisting

Linear

Twisting

Linear

Twisting

4.527 14.10 1.001 0.012 0.227 0.113

3.374 12.80 0.885 0.012 0.266 0.099

4.691 19.74 1.401 0.019 0.318 0.113

3.592 17.95 1.237 0.019 0.372 0.099

4.870 25.38 1.800 0.056 0.408 0.113

3.841 23.03 1.589 0.056 0.478 0.099

4.919 26.79 1.900 0.109 0.431 0.113

3.911 24.31 1.677 0.107 0.505 0.099

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TABLE 9.14 Fiber Model Mineral Volume Fraction Mineral Volume Fraction (%) %EFM 50 70 90 95

Linear

Twisting

25.91 31.46 37.01 38.40

18.37 23.92 29.47 30.86

The tensile data also showed similar results to those in compression. The linear model exhibited greater moduli than the twisting model. In addition, the values of the transverse elastic modulus were much greater than the longitudinal elastic modulus. The shear moduli in the plane 12 direction were very low for a low %EFM. This was due to a relatively large amount of void space present within the unit cell. However, this is discussed further with the structure of lamellar bone. The inclusion of fibers within the disordered fibrillar matrix can increase the shear modulus in the plane 12 direction. Because fibers exist within the macrostructures of bone and are not easily isolated for testing, there are no data available for the fiber model. Therefore, the present data cannot be compared to assess the fiber model. The mineral content completely surrounds the fibril. The mineral content of the different fiber models was calculated. These calculations included both intrafibrillar and extrafibrillar hydroxyapatite. The resulting values are shown in Table 9.14.

9.4 Macroscale Model So far, nanoscale fibril and fiber models were developed from the nanoscale hydroxyapatite and tropocollagen. These microscale components served as the base unit for the macroscale biomaterials. To investigate the macroscale characteristics of bones, three different models were examined. The first was a unit cell model of lamellar bone; the second was a layered fiber-reinforced composite model of cortical and trabecular bone; and the third was a finite element model of a tetrakaidecahedron modeling cancellous bone. Although the lamellar layer was still measured in microns, it was quantified in the macroscale section because it is the main constituent of cortical and cancellous bone. 9.4.1 Lamellar Bone Lamellar bone is the next hierarchical structure of bones. It is equivalent to a fiberreinforced composite made of bone fibers and bone fibrils. More recently, the presence of a disordered fibril matrix was discovered in the lamellar bone. As a result, the lamellar bone comprises bone fibers surrounded by a matrix of disordered fibrils [7]. This is a new finding that many previous models have not taken into account. Previous studies have postulated that lamellar layers are constructed merely of an ordered array of fibers [3,64]. The discovery of the disordered fibril matrix required the previous micromechanics unit cell model for the present hierarchical level. It was found that the lamellar bone exhibited

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3 2 1 FIGURE 9.9 Single lamellar bone layer.

an ordered motif of 2–3 μm, which represents the layer thickness, and a disordered matrix thickness of 0.25 to 1 μm, which was associated with the fibrillary matrix [7]. Small pores with a diameter of 50 nm were also found along with the disordered matrix [7]. These pores were represented as voids in the lamellar matrix. The voids could be included in the model, but their volume seemed small. As a result, voids were neglected in the model. The simplified view of a single lamellar layer is shown in Figure 9.9. 9.4.2 Lamellar Model The model used for lamellar bone to predict the material properties was similar to that utilized for bone fibers (i.e., the micromechanics unit cell model for continuous fiber composites). Because fibers within a lamellar layer are unidirectional and the disordered matrix has a relatively random thickness, the lamellar properties are considered as transversely isotropic material. The dimensions used for the lamellar unit cell model were as follows: layer thickness of 2.5 μm and matrix thickness of 0.375 μm. These were the averages of the ranges obtained in Reference 7. The fiber diameter was then computed to be 1.75 μm. Figure 9.10 shows the representation of the longitudinal cross section used for the lamellar model. The longitudinal cross section of the unit cell model had 38.4% fiber volume fraction for the lamellar unit cell. The presence of the small pores was neglected as they accounted for approximately 0.1% of the current model’s volume. The unit cell model used the volume fractions of each constituent. To model the lamellar layer as a continuous fiber, the dimensions of a1 and a2 were unimportant; other dimensions are listed in Table 9.15. 3

(a)

c2

7

5

c1

3

1

b2

b1

2 (b)

FIGURE 9.10 Longitudinal lamellar cross section as compared to micromechanics model cross section. (a) Lamellar cross section with shaded area representing disordered matrix and light section representing fiber. (b) Micromechanics model longitudinal cross section.

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TABLE 9.15 Unit Cell Dimensions for Lamellar Model Dimension

Value 0.50 0.50 0.62 0.38 0.62 0.38

a1 a2 b1 b2 c1 c2

The lamellar composite matrix contained a random distribution of fibrils, resulting in isotropic material behavior. To determine the isotropic material properties, the averaging technique was used to compute the equivalent elastic modulus and the Poisson ratio. The values listed in Table 9.7 were used. In addition, fiber material properties of varying %EFM were considered. Because the lamellar model contained the bimodulus materials, the lamellar layer was expected to show transverse isotropic, bimodulus, material properties. 9.4.3 Lamellar Model Results Material properties were computed using the lamellar model for both tensile and compressive loading, respectively, as well as four different values of %EFM. The results are shown in Tables 9.16 and 9.17. TABLE 9.16 Lamellar Results in Compression 50% EFM

E1 (GPa) E2 = E3 (GPa) G23 (GPa) G12 (GPa) ν21 ν32

70% EFM

90% EFM

95% EFM

Linear

Twisting

Linear

Twisting

Linear

Twisting

Linear

Twisting

6.428 9.001 1.336 0.622 0.328 0.237

4.609 6.425 1.156 0.529 0.366 0.266

6.841 9.717 1.552 0.629 0.322 0.235

5.024 6.928 1.338 0.536 0.350 0.269

8.391 10.35 1.705 0.665 0.280 0.243

6.547 7.409 1.466 0.572 0.286 0.288

10.14 10.55 1.737 0.715 0.242 0.252

8.272 7.576 1.492 0.620 0.237 0.302

TABLE 9.17 Lamellar Results in Tension 50% EFM

E1 (GPa) E2 = E3 (GPa) G23 (GPa) G12 (GPa) ν21 ν32

70% EFM

90% EFM

95% EFM

Linear

Twisting

Linear

Twisting

Linear

Twisting

Linear

Twisting

5.514 8.305 1.270 0.592 0.312 0.235

3.718 5.992 1.098 0.503 0.370 0.262

5.578 9.174 1.475 0.599 0.338 0.230

3.801 6.538 1.271 0.510 0.394 0.257

5.646 9.781 1.621 0.635 0.355 0.226

3.895 6.902 1.393 0.546 0.407 0.254

5.665 9.905 1.651 0.684 0.359 0.225

3.922 6.976 1.418 0.593 0.410 0.253

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TABLE 9.18 Lamellar Model Mineral Volume Fraction Mineral Volume Fraction (%) %EFM 50 70 90 95

Linear

Twisting

20.22 22.35 24.48 25.02

10.89 13.02 15.16 15.69

Table 9.16 shows that the linear model produced a greater elastic modulus than the twisting model. Furthermore, the elastic modulus in the 1 direction was smaller than that in the 2 or 3 direction for a low value of %EFM. As the %EFM value increased, the two values became closer. For the twisting model, the elastic modulus in the 1 direction was greater than that in other directions. The tensile property also showed that the linear model yielded a higher elastic modulus than the twisting model. In addition, the modulus in the 2 or 3 direction was larger than that in the 1 direction for all considered values of %EFM. The mineral content of the lamellar layers was analyzed as before, and it is tabulated in Table 9.18. The mineral volume fraction within lamellar layers is a representative parameter for the macroscale bone mineral content. As a result, the predicted mineral content could be compared to previous theoretical and experimental values. There was a wide variation in previous data, which showed the mineral content from around 30% to 70% [65–68]. Among them, the data with better understanding of the hierarchical structure of the bone gave approximately 30% to 40% mineral contents. Comparing the present bone mineral content results to those data, the linear crystal pattern was considered the more viable model. The mineral content of the twisting model was much lower than those values. Even though some perturbations were considered in the fibril and fiber models, the mineral content of the twisting model would not match the current estimates. As a result, the calculations that follow used the linear model. 9.4.4 Cortical Bone Cortical bone is the part of the bone with high density, and it is the outer shell of the bone surrounding the spongy center of cancellous bone. Cortical bone makes up a large percentage of the weight fraction of the skeletal system and provides major stiffness of and strength to the bone. Cortical bone has concentric layers of lamellar bone, called osteons, which can be made of either primary or secondary bone. Primary bone is found where bone has been grown de novo. Secondary bone is created when the primary bone is broken down and regrown in place [7]. Secondary osteons are called Haversian systems, which were modeled in this study. The concentric layers composing Haversian systems form a cylindrical structure whose diameter is approximately 200 μm [7,64,69]. Haversian canals are located at the center of these cylinders, and they contain the blood supply and nerve endings for the surrounding bone. The diameter of the canals is approximately 30–50 μm [7,64,69]. Lamellar bone constitutes each concentric layer inside the osteon. As a result, each layer has unidirectional fibers. The Haversian canal is represented as unbound water [7]. Multiple Haversian systems are packed together inside cortical bone. Due to their circular shape, there are incomplete layers at the interface of each Haversian system where the

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boundaries intersect. These boundaries are defined by a cement line, which is an identifiable region where osteon the growth direction has transitioned. The properties of cement lines are similar to those of the surrounding bone, despite the misnomer cement [70]. The longitudinal axis of a Haversian system is preferentially aligned with the longitudinal axis of the bone for a majority of cortical bone [64]. Transversely oriented Haversian canals do exist that interconnect each longitudinal Haversian canal, but they are short compared to the longitudinal canals. This study assumed all Haversian systems were aligned parallel to the long axis of the bone. The fiber directions of the concentric lamellar layers vary for each layer like a filament winding composite cylinder. Two distinct patterns of fiber orientation have been found in lamellar bone. One is a periodic alternating pattern, and the other is a continuous fiber twist. The distinct alternating pattern has high- and low-angle fibers. The high-angle fibers are at 65°–80°, and low-angle fibers are at 15°–30°. Because of the imperfect nature of biomaterials, there are some fibers with intermediate angles. The average orientations of three samples were calculated [71]. The results showed 45% high-angle fibers, 35% lowangle fibers, and 20% intermediate fibers [71]. The findings in Reference 64 also showed a section of femur exhibiting a continuous twist of fiber angles from 0° to 180° between lamellar layers. The continuous twist fibers were varied with an approximately 10° increment. Both fiber orientation patterns (i.e., the periodic alternating pattern and the continuous twist) can be modeled as a laminated fibrous composite. 9.4.5 Cortical Bone Model The lamellar layers of the cortical bone behave like a laminated composite. Each layer is a fibrous composite with a transverse isotropic material. The material properties of each layer can be determined from the fiber orientation. Figure 9.11 shows the respective layers when the cylindrical shape of an osteon is cut and arranged flat on top of one another. The material property of the cortical bone was computed using the volume average of all the layers. The material properties of a single lamellar layer can be determined by the coordinate transformation technique, which is discussed in Chapter 7 for a laminated composite as given in Equations 7.82 through 7.84. In addition to considering the fiber orientation of the individual lamellar layer, the Haversian canal and canaliculi were addressed in the macroscale model. Taking into consideration the dimensions of the Haversian system, the Haversian canal, and the microscopic canaliculi, the macroscale cortical bone had 75% densely packed bone and 25% void space. As done with models at a smaller length scale, the void space was addressed as a liquid in compression and a void in tension.

FIGURE 9.11 Osteon cross-sectional view.

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TABLE 9.19 Preferential Orientation Layered Composite Model Parameters Fiber Direction 72.5° 22.5° 0° 40° 55° 90°

Volume Percentage 0.45 0.35 0.05 0.05 0.05 0.05

TABLE 9.20 Smooth Orientation Layered Composite Model Parameters Fiber Direction 0° 10° 20° 30° 40° 50° 60° 70° 80° 90°

Volume Percentage 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10 0.10

The periodic alternating pattern and the continuous fiber twist are referred to as preferential orientation and smooth orientation, respectively, from this point. Their effects were examined. The two models relied on volume percentage of a defined fiber orientation. Tables 9.19 and 9.20 list the volume percentages present for each fiber orientation of the preferential model as well as for the smooth model. The stiffness matrix of each layer was calculated using Equations 7.82 through 7.84 and the fiber orientations provided in Tables 9.19 and 9.20. Because of the random radial distribution of osteons, the material properties in the radial direction were assumed to be constant. 9.4.6 Cortical Bone Model Results The results of the layered cortical bone model were computed using only the linear fibril model because it was more viable than the twisting fibril model. Both the preferential and smooth orientation models were used for tension and compression. Table 9.21 shows the compressive material properties; Table 9.22 shows the tensile properties. Comparing the results for both tensile and compressive properties shows that the preferential fiber orientation model yielded greater stiffness than the smooth orientation model for all %EFM values considered here. However, the difference between the two models was not large. The elastic and shear moduli increased as %EFM increased, as expected. The results of the present cortical model are compared to other experimental and theoretical data in Table 9.23. Comparing the results of the present model to the experimental

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TABLE 9.21 Cortical Bone in Compression 50% EFM

E1 (GPa) E2 = E3 (GPa) G23 (GPa) G12 (GPa) ν21 ν32

70% EFM

90% EFM

95% EFM

Pref.

Smooth

Pref.

Smooth

Pref.

Smooth

Pref.

Smooth

5.074 5.836 0.700 0.944 0.319 0.232

4.642 5.679 0.735 0.957 0.357 0.226

5.390 6.217 0.773 1.037 0.316 0.227

4.914 6.045 0.818 1.050 0.355 0.221

5.944 6.818 0.839 1.144 0.311 0.222

5.453 6.586 0.889 1.157 0.350 0.219

6.424 7.282 0.870 1.213 0.306 0.218

5.923 6.990 0.920 1.226 0.344 0.218

TABLE 9.22 Cortical Bone in Tension 50% EFM

E1 (GPa) E2 = E3 (GPa) G23 (GPa) G12 (GPa) ν21 ν32

70% EFM

90% EFM

95% EFM

Pref.

Smooth

Pref.

Smooth

Pref.

Smooth

Pref.

Smooth

4.110 4.779 0.666 0.887 0.375 0.257

3.704 4.649 0.699 0.900 0.427 0.250

4.396 5.138 0.735 0.970 0.378 0.256

3.930 5.009 0.778 0.983 0.432 0.247

4.622 5.408 0.798 1.039 0.398 0.268

4.120 5.286 0.846 1.054 0.455 0.257

4.696 5.485 0.829 1.069 0.423 0.284

4.193 5.371 0.876 1.085 0.482 0.272

Note: Pref., preferential.

TABLE 9.23 Comparison of Values for Cortical Results Elastic Modulus (GPa) 6 9 15 21 21.4–22.1 17.5 ± 1.9 17.8 ± 2.1 19.1 ± 5.4 17.1 ± 3.15 17.0 14.91 ± 0.52 20.55 ± 0.21 16.58 ± 0.32 23.45 ± 0.21 22.5 ± 1.3 a

Method

Source

FEM, 20% mineral volume fraction FEM, 30% mineral volume fraction FEM, 40% mineral volume fraction FEM, 50% mineral volume fraction Uniaxial tension Uniaxial tension Uniaxial tension Nanoindentation (transverse) Uniaxial compression and tensiona Uniaxial compression and tension Acoustic microscopy (transverse) Acoustic microscopy (longitudinal) Nanoindentation (transverse) Nanoindentation (longitudinal) Nanoindentation (longitudinal)

[1] [1] [1] [1] [66] [72] [73] [74] [75] [76] [77] [77] [77] [77] [78]

Only two femurs sampled had statistically significant differences in compression and tension.

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TABLE 9.24 Elastic Moduli of Compact Bone (GPa) Location

Femur

Tibia

Femur

Femur

Method

Ultrasound

Mechanical Testing

Ultrasound

Ultrasound

Preferential Model 95% EFM, Compression

[76] 11.17 11.17 17.21 3.3 3.3 3.6 0.595 0.298 0.298

[79] 6.94 8.56 18.45 4.91 3.56 2.41 0.495 0.142 0.119

[80] 13.18 14.56 21.67 6.56 5.85 4.74 0.377 0.237 0.225

[81] 18.57 19.43 28.29 8.71 8.71 8.71 0.323 0.209 0.126

This study 6.42 6.39 8.46 0.87 0.97 1.62 0.419 0.191 0.193

Source E1 E2 E3 G23 G13 G12 ν12 ν13 ν23

data showed that the present model resulted in a lower longitudinal modulus. There were many factors affecting the result. First, the mineral content assumed in our model was generally lower than those obtained in other studies. Increasing the mineral content would increase the stiffness, which is shown for the solutions obtained using the finite element method in Table 9.23. At higher mineral contents, the finite element results approached the experimental data. The present study utilized a maximum of 25% mineral volume fraction, which resulted in a longitudinal modulus of approximately 7 GPa. From the finite element solutions in Table 9.23, the results would predict a longitudinal modulus of approximately 7.5 GPa. This shows that the present simplified multiscale models presented results that were comparable to other solutions and the difference results from the assumptions made at various hierarchical levels. While most results showed either a longitudinal or a transverse modulus, some studies determined complete material properties, which are also compared to the present data in Table 9.24. The values depicted previously were the orthotropic material properties. The present model exhibited similar trends as other experimental tests. The modulus in the 3 direction was greater than both those in the 1 and 2 directions. As discussed previously, the present model underestimated the properties compared to others. 9.4.7 Cancellous Bone Cancellous bone is a porous macrostructure located inside the cortical bone layer, and it is composed of small plates and beams. The porous nature of cancellous bone stores bone marrow and reduces the weight of the skeletal system. The individual beams of the cancellous bone are called trabeculae. While cortical bone provides more strength and stiffness, cancellous bone is more metabolically active, resulting in more frequent absorption and deposition of bone. The deposition of new bone is related to the applied stresses on the cancellous bone [82], and the directed deposition makes the outermost layers of trabecular bone oriented parallel to the longitudinal axes of the trabeculae [7]. The orientations of adjacent layers are slightly different because they are formed under a different state of stress. The trabecula has a resulting structure of lamellar layers with a relatively uniform fiber direction. Because the layers have low fiber angles with respect to the longitudinal direction of the trabecula, a

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single trabecula is considered to have laminated composite layers at angles of –10°, –5°, 0°, 5°, and 10°. While a laminated fibrous composite material model is a good representation of a single trabecula, it cannot predict the properties of the cancellous bone. The structure of cancellous bone is truly 3-D, and it has much heterogeneity at different anatomical locations [83]. Early models for cancellous bone used beams or plates. The early models are useful as simple models of cancellous bone, but these models result in asymmetrical properties for cancellous bone and consider Euler buckling a failure mechanism [82]. Later experimental studies showed that the cancellous bone failed mostly due to microscopic cracking. Therefore, buckling was not considered as a failure mode for trabeculae [84]. Three-dimensional finite element models were developed using microcomputed tomography to replicate small sections of bone, and two unit cells were proposed to accurately model cancellous bone [85]. One was a prismatic unit cell, and the other was a tetrakaidecahedral unit cell. It was found that both unit cells accurately represented the mechanical properties. Furthermore, a study in Reference 86 showed that a complex finite element model of tetrakaidecahedral cells could accurately represent different levels of bone loss resulting from aging. This study used a tetrakaidecahedral unit cell to compute the macroscale properties of cancellous bone. Several values of trabecular bone properties were used for the tetrakaidecahedral unit cell along with different bone densities. The results of the macroscale properties were then compared to previous experimental data. 9.4.8 Trabecular Bone Model Trabecular bone was modeled in the same way as the cortical bone, but the fiber orientations and volume percentage of each layer were different for the two. The parameters used for trabecular bone are given in Table 9.25. Due to the lack of Haversian canals in trabecular bone, the void space results from microscopic canaliculi. This void space was assessed to be 5%, and it was treated as a liquid only responsible for compressive properties. 9.4.9 Result of Trabecular Bone Table 9.26 shows both compressive and tensile material properties predicted using the trabecular model. The modulus was higher in compression than in tension because of the liquid contained in the void. The difference between tensile and compressive moduli became larger as %EFM increased. The results of the present trabecular bone model were also compared to previous experimental and theoretical results, as shown in Table 9.27. Early methods tested macroscale cancellous bone to back calculate the material properties TABLE 9.25 Trabecular Bone Layered Composite Model Parameters Fiber Direction –10° –5° 0° 5° 10°

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Volume Fraction 0.2 0.2 0.2 0.2 0.2

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TABLE 9.26 Trabecular Bone in Compression (Comp.) and Tension (Tens.) 50% EFM

E1 (GPa) E2 = E3 (GPa) G23 (GPa) G12 (GPa) ν21 ν32

70% EFM

90% EFM

95% EFM

Comp.

Tens.

Comp.

Tens.

Comp.

Tens.

Comp.

Tens.

6.047 8.519 1.259 0.658 0.242 0.234

5.099 7.762 1.197 0.623 0.254 0.253

6.427 9.186 1.461 0.672 0.235 0.232

5.158 8.569 1.389 0.636 0.264 0.241

7.848 9.782 1.605 0.717 0.236 0.240

5.223 9.134 1.526 0.673 0.318 0.237

9.451 9.975 1.636 0.773 0.241 0.248

5.245 9.252 1.555 0.720 0.382 0.237

TABLE 9.27 Comparison of Values for Trabecular Results Elastic Modulus (GPa) 11.38 12.7 ± 2.0 15 3.81 5.35 ± 1.36 10.4 ± 3.5 14.8 ± 1.4 13.4 ± 2.0 18.0 ± 2.8 11.4 ± 5.6 17.5 ± 1.12 18.14 ± 1.7

Method

Source

Inelastic buckling Ultrasound (isotropic) Microhardness (transverse) Three-point bending Four-point bending Tensile test Ultrasound (isotropic) Nanoindentation (transverse) Tension and compression tests Nanoindentation (transverse) Acoustic microscopy (transverse) Nanoindentation (transverse)

[87] [88] [89] [90] [91] [92] [92] [78] [73] [74] [77] [77]

of trabecular bone. On the other hand, current methods use microelectronic-mechanical systems and nanoindentation to directly measure trabecular material properties. The results for the trabecular bone model were similar to those for the cortical bone model, resulting in underestimation of the stiffness values. However, the trabecular result was closer to the mean values of accepted longitudinal and transverse results. The same discussion for the cortical bone model also applies to the trabecular bone model. 9.4.10 Cancellous Bone Model A tetrakaidecahedral unit cell was used as the cancellous bone model. We developed the unit cell to analyze cellular materials [93]. The tetrakaidecahedron is a regular truncated octahedron and was developed by Lord Kelvin in 1887 to model the soap bubble formation in foam. It is a compact and optimal geometry for space filling by repeating itself. It provides minimum surface area for a given volume. The tetrakaidecahedron has 8 hexagonal faces, 6 square faces, 36 edges, and 24 vertices. For the open-cell structures present in cancellous bone, the faces were treated as voids, and the edges were treated as trabeculae. A simple finite element model was developed by considering the symmetry of the unit cell [93]. Because the unit cell had open faces,

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TABLE 9.28 Anatomical Cancellous Bone Indices BV/TV

Tb.Th (mm)

0.2654 0.2067 0.27 0.27 0.105 0.111 0.34 0.41 0.27

a

0.172 0.2 0.19

0.36 0.33 0.47 0.186 0.174

Anatomical Location

Source

Femoral neck Femoral head Distal femur Proximal femur Greater trochanter Tibia Radius Distal tibia Distal tibia Average (young age)a Average (medium age)a

[94] [95] [96] [96] [94] [94] [97] [98] [98] [99] [99]

Young age was defined as 16–39 years old, and medium age was defined as 40–59 years old.

the edges were modeled as beams with the properties of trabecular bone. As a result, the tetrakaidecahedral unit cell was modeled as a 3-D frame structure consisting of beams. The cross section of the beam elements depended on the volume fraction provided for the unit cell [93]. Each beam element had one node at each end with six degrees of freedom per node. Loads were applied to the model at vertices. No buckling was considered. The finite element model for the tetrakaidecahedral unit cell yielded isotropic material properties. In addition, because each beam element was treated as an isotropic material, two variations were considered: one using compressive material properties and the other using tensile material properties. This created a bound on the material properties of cancellous bone for each %EFM used. The beam elements of the tetrakaidecahedral model were represented by the material properties of the trabecular long axis. The properties for the different %EFM, in tension and in compression, were considered, and the effect of bone marrow was neglected. The radius of the beam elements and the volume density of the bone were determined as follows: A universally accepted set of indices for trabecular structure are bone volume per tissue volume (BV/TV), bone surface per tissue volume (BS/TV), and bone surface per bone volume (BS/BV) [83]. These indices helped derive the structure of the trabeculae: trabecular thickness (Tb.Th), trabecular separation (Tb.Sp), and trabecular number (Tb.N) [83]. Table 9.28 shows the parameters discussed previously at various anatomical locations. The diameter of the beams was set to the average thickness of the femoral trabeculae, which was 0.19 mm. 9.4.11 Cancellous Bone Results The results of the present model were determined using the trabecular results obtained previously, and their values are tabulated in Table 9.29. As expected, the modulus of the cancellous bone was greater in compression than in tension. The distal and proximal femur were stiffer than both the femoral neck and the head. This was expected due to its greater density. The present predictions are compared to experimental and theoretical results in Table 9.30. The comparison was good overall.

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TABLE 9.29 Cancellous Model Elastic Modulus (MPa) 50% EFM

Femoral neck Femoral head Distal/proximal

70% EFM

90% EFM

95% EFM

Comp.

Tens.

Comp.

Tens.

Comp.

Tens.

Comp.

Tens.

292 196 300

246 165 253

311 209 319

249 167 256

379 255 390

252 170 259

457 307 469

253 170 260

Note: Comp., compression; Tens., tension.

TABLE 9.30 Comparison of Values for Cancellous Results Elastic Modulus (MPa) 47 ± 2 300 ± 200 110 ± 90 550 ± 450

Method

Source

Finite element method Microcompression testing Compressive testing Mechanical testing

[100] [101] [102] [103]

9.5 Further Adjustments in Models In this section, we further discuss each hierarchical model for the multiscale analysis to improve the results and capability. In addition, it determined which hierarchical model had the greatest influence on macroscale properties. Minerals are the main element to result in necessary stiffness and strength of the biomaterial. The fiber model showed that there could be a high level of EFM within living bone. Comparing the linear and twisting fibril models based on the assessed mineral volume contents, the linear model seemed to be the proper one for fibril crystal packing. Therefore, the selected model for further analysis comprised the linear fibril hydroxyapatite packing and 95% EFM. Moreover, the model was studied for compressive properties. We used the preferential fiber model for the macroscale hierarchical structure. We examined further the macroscale longitudinal stiffness, transverse stiffness, and mineral volume fraction. The results from the present modification were assessed for the cortical and trabecular bones. Another independent analysis was conducted to assess bone loss. To examine the effects of bone loss on the material properties of cancellous bone, different bone densities were considered for the macroscale cancellous bone. 9.5.1 Modified Hierarchy Hierarchies in both microscales and macroscales were altered to find out their effects on the cortical and trabecular properties. The fibrillar subunit was changed at the microscale level. The diameter of the fibrils and the width of the EFM were varied. On the other hand, the dimensions of the fiber and the disordered matrix were changed at the macroscale level. Finally, to define the upper and lower bounds of the composite stiffness, both the Voigt and the Reuss averages of cortical and trabecular bone are presented [104]. The modification to the hierarchy was conducted one by one, and the results from each modification are presented and compared. The first change was for the unit cell of the

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TABLE 9.31 Unit Cell Alteration to Fibrillar Model Value (nm) Dimension a1 a2 b1 b2 c1 c2

Model 1

Model 2

Model 3

50 17 25 3 3 6

50 17 25 1 3 6

50 17 25 1 3 3

fibrillar model. The hydroxyapatite crystal had the same dimensions as previously, but the volume of the surrounding tropocollagen matrix was decreased as shown in Table 9.31. These changes increased the mineral volume percentage of the fibrillar model. Each change was called model 1, model 2, and model 3. The next modification was made for the bone fiber and depended on the volume percentage of both the bone fibril and the extrafibrillar mineral. Two variations were tested by changing the volume percentage of the fibril. The original model assumed the fibril volume percentage of 72.25%, which was changed to either 62.25% or 82.25%. These variations are referred to as model 4 and model 5, respectively. The only adjustment at the macroscale hierarchies was that of the lamellar bone. The original model had a 38.44% fiber volume percentage. The fiber volume percentage was increased such that model 6 had 50% fibers, model 7 had 70% fibers, and model 8 had 90% fibers. 9.5.2 Adjustment Results Table 9.32 summarizes the results obtained using models 1 through 8. These models altered only the hierarchy discussed without other changes. From the results, the transverse elastic modulus was greater than the longitudinal modulus for all models. As expected, the TABLE 9.32 Elastic Moduli of Adjusted Hierarchies (GPa) Cortical Bone Voigt Model 1 2 3 4 5 6 7 8 Original

Trabecular Bone

Reuss Model

Voigt

Reuss

E1

E2

E1

E2

E1

E2

E1

E2

Mineral Volume Fraction (%)

7.05 8.27 10.26 6.95 5.84 7.21 8.93 11.25 6.42

8.03 9.53 11.87 7.87 6.61 8.30 10.49 13.39 7.28

3.75 3.92 4.33 3.58 3.61 3.35 2.75 1.81 3.58

4.48 4.72 5.15 4.33 4.23 4.21 3.81 2.84 4.29

10.28 11.63 14.18 10.79 8.15 10.49 12.29 14.08 9.45

11.19 13.87 17.73 10.64 9.07 11.84 16.07 22.04 9.98

8.04 8.76 10.20 8.15 6.84 7.77 7.68 6.38 7.52

9.00 10.48 12.49 8.60 7.72 9.12 10.34 10.05 8.24

29.97 31.50 42.18 28.03 22.00 27.53 31.89 36.22 25.01

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increasing mineral volume fraction increased the macroscale stiffness. Reduction in the tropocollagen matrix within the fibrillar subunit increased the macroscale properties. On the other hand, an increase in the volume percentage of the fibril within the fiber model showed an inverse effect on its stiffness because it was due to the extra fibrillar mineral. In other words, because the fibril had less volume, the mineral occupied a larger volume. An increase in the size of the bone fibers within the lamellar layers increased the macroscale modulus. As far as the Voigt and Reuss models are concerned, the Voigt model assumed isostrain conditions, whereas the Reuss model assumed isostress conditions. Therefore, the Voigt model resulted in the upper bound of the material properties, while the Reuss model yielded the lower bound. This was derived from averaging the stiffness matrices or the compliance matrices, respectively. The results of the cortical models showed a large difference between the Voigt and Reuss models. This was due to the large theta values of the rotated layers. The trabecular models showed less variation because the layers were constrained to small thetas. Finally, the fibrillar model had the greatest influential hierarchy affecting the macroscale properties. Small changes in this hierarchy were compounded throughout the upper scales. Changes to the fibrillar model could be physically embodied by reducing the vertical spacing between hydroxyapatite crystals. The lamellar model was the second-most influential hierarchy. The relatively small volume percentage of fibers present in this model produced a large increase in the fiber diameter. This increase in the fiber diameter decreased the relative proportion of the fibrillar matrix. 9.5.3 Optimal Adjustment Results The multiple hierarchies were adjusted simultaneously to optimize the macroscale results. Both the fibrillar subunit and the fiber model were modified at the microscale. Although the unit cell dimensions in model 2 were selected for the fibrillar subunit, the fiber model was assumed to have a fibril volume percentage of 60%. The lamellar model had a fiber volume percentage of 70%. The results are summarized in Table 9.33. The optimized models of the multiscale analysis showed a mineral volume percentage of 43.82% for the macroscale bone. The values of E1 and E2 for both models were within the bounds of the tested results. An observation is that the accepted moduli of the cortical bone were greater than those of the trabecular bone. The previous tested data showed that the cortical values average was greater than the trabecular results average. This study showed that the trabecular material was stiffer than the cortical material. This difference resulted from some variations at the hierarchies. TABLE 9.33 Optimized Macroscale Bone Properties Cortical

E1 (GPa) E2 (GPa) G23 (GPa) G12 (GPa) ν21 ν32

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Trabecular

Voigt

Reuss

Voigt

Reuss

12.71 15.03 0.914 1.62 0.272 0.153

2.82 3.98 0.004 0.004 0.413 0.176

16.72 24.17 2.18 0.52 0.165 0.164

8.91 12.73 0.020 0.019 0.304 0.176

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One of the reasons was the simplification of the void space within the macrostructures. While the cortical bone models assumed 25% void space due to the combination of Haversian canals and canaliculi, the trabecular model assumed only 5% void space due to the lack of Haversian canals. More information is needed for correct void space representation. Another factor relates to the lamellar layers, which should physically be portrayed as layers that are similar in orientation to the surface layers. However, this assumption will be unique for each individual strut based on its historical loading. It can be hypothesized that each trabecular strut exhibited different mechanical properties independent of its dimensions. 9.5.4 Bone Loss Results Bone loss occurs with aging and disease. Some studies identified quantitative differences between healthy individuals and individuals suffering from bone loss. The differences are shown with respect to the cancellous indices of BV/TV and Tb.Th. The effects of bone loss are shown in terms of the parameter in Table 9.34 that shows both a reduction in BV/TV and thinning of trabeculae. To study the effects of reductions in bone density, both modes of bone loss were modeled. The cancellous model assumed baseline values of 0.27  and 0.19  mm for BV/TV and Tb.Th, respectively. The models used the optimized material properties of trabecular bone, as found through Voigt averaging. The baseline cancellous results were computed for the optimized trabecular properties. The finite element model for the cancellous bone used a ligament radius and length. Either the ligament length or radius was varied to reflect the bone loss. In solving for the cancellous modulus, the BV/TV index was the major parameter. Table 9.35 tabulates the results, which showed a steady decrease in macroscale stiffness resulting from the decreasing density. The effect of a 10% reduction in density yielded a 50% reduction in modulus. Furthermore, the effects of the trabecular thinning may apply more to the failure of trabecular bone than to the material stiffness. TABLE 9.34 Cancellous Bone Indices for Individuals with Bone Loss BV/TV 0.26

Tb.Th (mm)

Anatomical Location

Group Identifier

Source

0.33 0.157

Distal Radius Proximal tibia Iliac crest

Osteoporotic postmenopausal females 60–79 years old 70–90 years old

[98] [99] [105]

0.16

TABLE 9.35 Cancellous Bone Material Properties after Bone Loss BV/TV 0.27 0.24 0.20 0.16 0.27 0.27 0.27

© 2016 by Taylor & Francis Group, LLC

Tb.Th (mm)

Stiffness (MPa)

0.19 0.19 0.19 0.19 0.17 0.15 0.13

830 689 514 357 830 830 830

10 Multiphysics Analysis of Composite Structures

10.1 Introduction Recently, composite materials have been used increasingly for maritime, aerospace, and automotive structures. Early use of composites was limited to secondary structures; however, as knowledge and understanding of mechanical characteristics of composites have grown, more primary load-bearing structures have been fabricated. In recent years, large composite structures have been incorporated into ships and aircraft to increase operational performance while lowering ownership costs. For example, carbon fiber composite material provides high strength and stiffness with low weight, which in turn translates to increased fuel efficiency and increased payload. Further advantages of composites over metals are lower maintenance costs and resistance to corrosion, which make composites desirable for maritime applications. While composites provide advantages over metals, they also come with complex and challenging engineering problems for analysts and designers to overcome [1]. This chapter focuses specifically on the implications of utilizing composite structures in maritime applications below the waterline because the structural behavior is affected by fluid-structure interaction (FSI).

10.2 Fluid-Structure Interaction Modeling The FSI generally has two-way interactions between fluids and composite structures. Fluids apply pressure loading to structures, while the structures affect the fluid velocity at the FSI interface. If the effect on the fluid velocity is negligible, the FSI becomes a one-way interaction. The one-way interaction is easier to model because it does not require mutually interactive solutions of both media. In this case, the fluid medium is solved first, and then the fluid pressure loading is applied to the structures to solve the structural responses. On the other hand, two-way interactions require simultaneous solutions of both media because the solutions are influenced by each other. If not solved simultaneously, one medium is solved first, followed by the solution of the other medium, and the process iterates until the solutions are converged. We may consider two types of fluid media. One type of medium includes fluid flow, and the other type neglects fluid flow. The former problem can be solved using the NavierStokes equation, while the latter is solved using the wave equation. The Navier-Stokes equation can be modeled using the lattice Boltzmann method, while the wave equation can be analyzed using the cellular automata (CA) as discussed in previous chapters. 297 © 2016 by Taylor & Francis Group, LLC

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In modeling FSI, the structure is modeled using the finite element method (FEM). Especially, the three-dimensional (3-D) solidlike plate/shell elements are useful for modeling plate/shell structures with FSI because the FSI interfaces can be easily defined on the bottom or the top of the thin structures. If 3-D solid elements are used for a thin shell structure, the mesh requirement becomes prohibitive in terms of computation costs. On the other hand, if the fluid is modeled using the FEM or control volume method, the compatibility conditions at the FSI interface can be easily enforced for those techniques use physical variables such as fluid velocities and pressure as the unknown variables at the nodal points. The lattice Boltzmann method uses fictitious particle densities as the nodal variables, which should be translated into fluid velocities and pressures. Such a translational process may not be unique and requires special attention. The CA method has some difficulties in taking derivatives of their own variables, which are needed to satisfy compatibility conditions. However, both techniques are computationally efficient. Coupling between the finite element structural model and the lattice Boltzmann or CA fluid model is discussed in Chapter 6, so their details are omitted here.

10.3 Low-Velocity Impact with FSI This section discusses low-velocity impact on composite structures that are in contact with water. The structures may be in contact with water on only one side or on both sides. For the one-side contact case, water can touch the structures inside or outside. For a composite plate considered in this study, Figure 10.1 illustrates four possible cases. The first case had no water around the plate; the second case had water on the bottom side of the plate that was impacted on the top side; the third case had water only on the top side of the plate; and finally the fourth case had water on both sides of the plate. Because the fluid motion during the dynamic response of the impacted plate may be neglected as a simplified model, the fluid domain is considered an acoustic medium represented by the wave equation. The impactor was modeled as a rigid material with a given initial velocity. The impact/contact condition was applied between the impactor and the plate. The plate was modeled using the 3-D solidlike plate/shell element, which Impact side of plate

(a)

Air

Impact side of plate

(b)

Water

Air

Impact side of plate

(c)

Air

Water

Impact side of plate

(d)

Water

FIGURE 10.1 Four different cases as a composite plate is in contact with water: (a) no water contact; (b) water on the bottom of the plate; (c) water on the top of the plate; and (d) water on both sides of the plate.

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had displacement only as nodal degrees of freedom but did not have rotational degrees of freedom as presented in Section 2.8. The acoustic domain was modeled using the CA technique, while the composite plate was analyzed using the FEM. However, as discussed in Chapter 6, a finite element structural model has some difficulty with direct coupling with the CA. As a result, the immediate neighbor of the structure was modeled using the finite element technique, which was enclosed by the CA as sketched in Figure 6.20. The overall analysis was conducted as follows: At a given time, the impact/contact analysis was conducted for the structure with the fluid pressure loading determined previously. Then, the structural analysis yielded the velocity at the FSI interface, which was applied to the fluid analysis. The pressure loading at the FSI interface was computed from the fluid analysis. The computed pressure was compared to the previous pressure loading. If these were within a given tolerance, the iteration stopped. Otherwise, the process iterated until convergence. 10.3.1 Single-Layer Plate Model A clamped square plate was studied. The plate was 0.3048 × 0.3048 m and was 6.35 mm thick. Because the plate was made of quasi-isotropic layers of composites, it was modeled as a single layer of the equivalent isotropic material. Both continuous Galerkin (CG) and discontinuous Galerkin (DG) techniques were used for modeling the composite plate. For fluid, freshwater was considered. Figure 10.2 compares the transverse displacement of the center of the plate subjected to a constant concentrated force applied at its center. In the figure, dry means no FSI, while wet means inclusion of the FSI. The dry structure oscillated about its predicted static deflection, while the wet structure showed an altered frequency and varying magnitude as a result of FSI. As studied in Reference 2, the effect of the elastic modulus and density of the plate was investigated along with FSI as shown in Figures 10.3 and 10.4. The baseline model shown in Figure 10.2 had a structural density that was 2.7 times larger than that of the fluid, and it had a frequency ratio of the dry-to-wet response of 1.84 and an amplitude ratio between the first peak and the first trough of the dry-to-wet response of 2.49. As seen in Figure 10.3, doubling the elastic modulus of the base model 0

×10–4

Displacement (m)

–1 –2 –3 –4 –5 –6 –7

Dry Wet 0

0.002

0.004 0.006 Time (s)

0.008

0.01

FIGURE 10.2 Displacement of a clamped plate with and without fluid-structure interaction.

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0

×10–4 Dry Wet

Displacement (m)

–0.5 –1

–1.5 –2

–2.5 –3 0

0.002

0.004 0.006 Time (s)

0.008

0.01

FIGURE 10.3 Displacement of a clamped plate of doubled elastic modulus of the base model with and without fluid-structure interaction.

0

–4

×10

Displacement (m)

–1 –2 –3 –4 –5 –6 –7

Dry Wet 0

0.002

0.004 0.006 Time (s)

0.008

0.01

FIGURE 10.4 Displacement of a clamped plate of doubled density of the base model with and without fluid-structure interaction.

showed a slightly higher-frequency oscillation about a smaller static deflection for the dry structure, as expected. In this case, the dry-to-wet frequency ratio was 1.86, and its amplitude ratio was 2.15, which was smaller than that of the base case. As seen in Figure 10.4, doubling the density of the base model resulted in the expected lower-frequency oscillation for the dry structure. Its dry-to-wet frequency ratio was 1.49, and the dry-to-wet amplitude ratio for the double density case was 2.00. 10.3.2 Two-Layer Plate Model The clamped plates discussed in this section were each 0.3048 × 0.3048 m and 3.5 mm thick; they had two thickness layers and 16 elements in each planar direction. The material

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was a quasi-isotropic E-glass. Initial conditions were zero displacement and zero velocity, and a constant concentrated force of 1000 N was applied at the center of the plate at the first time step. The damaged plates had debonding of four elements by four elements between the two layers at the center of the plate. The debonded area was modeled using the contact element without friction. Figures 10.5 and 10.6 display time histories of the displacement and normal strain in the plane on the bottom of each plate. The dry plate had larger displacement and strain than the wet plate under this loading condition. Furthermore, the strains at the center of the plate for both dry and wet cases had greater magnitudes than those at the edge of the damage zone. This is not realistic and suggests that an interface layer is necessary to properly model debonding in laminated composites and other layered structures. 0

Displacement (m)

–0.002 –0.004

Dry Wet

–0.006 –0.008 –0.01 –0.012 –0.014

0

1

2 Time (s)

3

4 ×10–3

FIGURE 10.5 Displacement of a damaged clamped two-layer E-glass plate with and without fluid-structure interaction.

7

–3

×10

6

Strain

5 4 3 2 Center dry Center wet

1 0

0

1

2 Time (s)

3

4

×10

–3

FIGURE 10.6 Strain at the center of a clamped two-layer E-glass plate with and without fluid-structure interaction.

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10.3.3 Three-Layer Plate Model The two-layer model of the previous section did a poor job of reflecting debonding damage within a laminated plate. A three-layer model included a thin interface layer with properties representing the resin used for the composite between two layers of E-glass, and it was subjected to the same loading and boundary conditions as used previously. Responses were calculated for 500 time steps. Figure 10.7 shows that the displacement of the center of the plate did not reflect the presence or absence of a debonding zone but did demonstrate the FSI effect in a fashion similar to that of the two-layer model. Figure 10.8 shows that the strain calculated in the E-glass element reflected the presence or absence of damage mildly. Figure 10.9, on the other hand, shows clearly that the interface layer was profoundly affected by the presence of a damage zone. To be clear, the strain values at the centers of the interface layers of the damaged plates were not identically zero, but they were four orders of magnitude lower than their undamaged counterparts. Figure 10.10 compares the strains in the interface layer at the center and the edge of the potential debonding zone of the dry structure which was assumed to have no damage or damage. Likewise, Figure 10.11 compares the strains in the interface layer at the center and the edge 0 Displacement (m)

–0.005 –0.01 Dry (damaged) Wet (damaged) Dry (undamaged) Wet (undamaged)

–0.015 –0.02

0

0.5

1

1.5 Time (s)

2

2.5

3 ×10–3

FIGURE 10.7 Displacement of a clamped three-layer E-glass plate with and without fluid-structure interaction and with and without damage. 0.01

Strain

0.008 0.006

Dry (damaged) Wet (damaged) Dry (undamaged) Wet (undamaged)

0.004 0.002 0 0

0.5

1

1.5 Time (s)

2

2.5

3

×10–3

FIGURE 10.8 Strain of a clamped three-layer E-glass plate with and without fluid-structure interaction and with and without damage at center of lower E-glass layer.

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5

×10–4 Dry (damaged) Wet (damaged) Dry (undamaged) Wet (undamaged)

Strain

4 3 2 1 0 0

0.5

1

1.5 Time (s)

2

2.5

3 ×10–3

FIGURE 10.9 Strain at the center of the interface layer of a clamped three-layer E-glass plate with and without fluid-structure interaction and with and without damage.

Strain

4

×10–5 Center Zone edge

8

Center Zone edge

Strain

×10

10

–4

2

6 4 2

0

0

–2 0

1

(a)

2

Time (s)

0

3

×10–3

0.5

1

(b)

1.5 Time (s)

2

2.5

3 ×10–3

FIGURE 10.10 Strains at the interface of a dry clamped three-layer E-glass plate (a) without and (b) with damage.

4

×10–4 Center Zone edge

3 2 1

(a)

6

Center Zone edge

4 2

0 –1 0

×10–5

8 Strain

Strain

10

0 0.5

1

1.5 Time (s)

2

2.5

3

×10–3

0.5 (b)

1

1.5 Time (s)

2

FIGURE 10.11 Strains at the interface of a wet clamped three-layer E-glass plate (a) without and (b) with damage.

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3

×10–3

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of the debonding zone of the wet structure with and without damage. Once debonding occurred at the center of the plates, the strain in the interface layer became nearly zero at the plate center. 10.3.4 Comparison to Experimental Data Some experimental work was conducted to examine the response of composite plates to low-velocity impact with and without FSI [3–5]. In general, it was found that, for a given impact weight dropped from the same height, structures with FSI experienced higher resultant forces and consequently greater damage than the same structure in dry conditions. The experimental data are compared to the numerical results here. The vacuum-assisted resin transfer molding technique was used to construct a series of 0.3048 × 0.3048 m composite plates consisting of 16 layers of E-glass (approximately 3.5 mm thick in total). The plates were subjected to low-velocity impact forces that resulted from dropping a 10.8-kg weight from various heights to the center of the plates using the impact equipment shown in Figure 10.12. The plates were instrumented with strain rosettes at multiple positions. Numerical comparison with these experimental data was conducted using a DG structural model consisting of a single layer of plate elements with a discretization of 12 elements in each planar direction. The overall structure had a length-to-thickness ratio of 87:1, and each element had a length-to-thickness ratio of 7.3:1. In this model, the material properties used were those of E-glass, but the material was treated as quasi-isotropic. Those nodes closest to the positions of the strain gauges in the experimental work were compared to the experimental strains. Figures 10.13 through 10.15 are for the dry plate and Figures 10.16 through 10.18 are for the wet plate, including FSI [6]. The locations for strain gauges in the plate are shown in Figure 10.19. All plots show reasonable qualitative agreement between experimental and numerical data. There were many reasons for discrepancies between the numerical and experimental results. For example, the experiment had neither exact clamped boundary conditions nor homogeneous material properties. In addition, for FSI, there was some flow motion, which was neglected in the computational model.

FIGURE 10.12 Impact equipment.

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0.2

Calculated Measured

1000 µ strain

0

–0.2 –0.4 –0.6 –0.8 0

0.005

0.01 0.015 Time (s)

0.02

FIGURE 10.13 Measured versus calculated strain along the x axis in a dry plate at gauge 1.

1000 µ strain

0.5

Calculated Measured

0

–0.5 –1 0

0.005

0.01 Time (s)

0.015

0.02

FIGURE 10.14 Measured versus calculated strain along the x axis in a dry plate at gauge 2. 0.2

Calculated Measured

1000 µ strain

0.1 0

–0.1 –0.2

0

0.005

0.015 0.01 Time (s)

0.02

FIGURE 10.15 Measured versus calculated strain along the x axis in a dry plate at gauge 3.

1000 µ strain

0.5

Calculated Measured

0

–0.5 –1

0

0.002 0.004 0.006 0.008 0.01 0.012 Time (s)

FIGURE 10.16 Measured versus calculated strain along the x axis in a wet plate at gauge 1.

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1000 µ strain

0.5

Calculated Measured

0 –0.5 –1 –1.5 0

0.002

0.004

0.006 0.008 Time (s)

0.01

0.012

FIGURE 10.17 Measured versus calculated strain along the x axis in a wet plate at gauge 2.

1000 µ strain

0.8 0.6 0.4 0.2

Calculated Measured

0

–0.2 0

0.002 0.004 0.006 0.008 Time (s)

0.01

0.012

FIGURE 10.18 Measured versus calculated strain along the x axis in a wet plate at gauge 3.

y

Gauge #1

Gauge #2

x Gauge #3 Gauge #4

FIGURE 10.19 Strain gauge locations.

10.4 Vibration with FSI Vibrational characteristics of composite structures were analyzed while they were submerged in water. The fluid domain was modeled as an acoustic domain by neglecting fluid motions. The wave equation was solved using CA. On the other hand, the structures were modeled using the beam and plate/shell elements, which had displacements only as nodal degrees of freedom.

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10.4.1 Numerical Modal Analysis Modal analysis extracts modal parameters like natural frequencies, mode shapes, and damping ratios of a structure. In the experimental modal analysis, multiple sensors were attached to a structure to be tested. Then, an impulse loading was applied to the structure. The sensors mostly used were accelerometers because they are sensitive, small, and light so that they would not alter the mass and stiffness of the original structure. Velocimeters are bulkier and heavier than accelerometers, so the former sensors were not suitable for a small structure used in a laboratory. To measure a mode shape accurately, many sensors must be attached to a structure with small spacing. On the other hand, the numerical modal analysis can obtain dynamic responses at every nodal point of the finite element mesh. Therefore, as long as the mesh is fine enough, an accurate mode shape can be determined from the numerical modal analysis, for which any dynamic response (e.g., displacement, velocity, acceleration, etc.) can also be selected. Once the time domain responses of an excited dynamic structure were obtained, the fast Fourier transform (FFT) was used to convert the time domain data into the frequency domain data. Then, the modal parameters were obtained from the frequency domain data at each nodal point. Reference 7 explained the modal analysis technique to extract the mode shapes. Therefore, the explanation is omitted here. 10.4.2 Verification Study The results obtained using the numerical modal analysis were compared to some known solutions. The example cases were vibrations of beams in air without FSI. The first example case was a cantilever beam. The beam was 1.0 m long, 0.05 m thick, and 0.1 m wide. The modulus of elasticity was 20 GPa, and its density was 2000 kg/m3. The analytical solutions for natural frequencies and mode shapes were given in Reference 8. The mode shapes of a cantilever beam are given in the following equation: ϕ(x) = A(cos βx − cosh βx) + (sin βx − sinh βx)

(10.1)

where A=−

sin βl + sinh βl cos βl + cosh βl

(10.2)

mω 2 EI

(10.3)

β=

4

Here, l is the length of the beam, m is the mass per unit length, EI is the beam rigidity, and ω is the frequency in radians/second. The coordinate x was measured from the fixed end of the beam. The first three mode shapes had βl = 1.875104 for mode 1, βl = 4.694091 for mode 2, and βl = 7.854757 for mode 3, from which the natural frequency could be determined. The natural frequencies and mode shapes of the cantilever beam were computed using numerical modal analysis. In addition, finite element eigenvalue analysis was also conducted to determine the natural frequencies and mode shapes using the mass and stiffness matrices obtained from the finite element models. Forty beam elements were used for the analysis. Table 10.1 compares the first three natural frequencies among three different

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TABLE 10.1 Comparison of Natural Frequencies of Cantilever Beam in Air Analytical Solution (Hz)

FEM Eigenvalue Analysis (Hz)

Error of Eigenvalue Analysis (%)

Numerical Modal Analysis (Hz)

Error of Modal Analysis (%)

25.54 160.06 448.19

25.54 160.06 448.19

0 0 0

26.20 160.00 460.00

2.58 0.03 2.63

Mode 1 Mode 2 Mode 3

solutions. Percentage errors are provided in the table compared to the analytical frequencies. The numerical modal analysis resulted in natural frequencies with an error of approximately 2%. The mode shapes obtained from the numerical modal analysis compared well with the other results from the analytical solution as well as the FEM eigenvalue solution. They were not distinguishable in the plots. The second example was a clamped beam at both ends. The geometric data and material properties were the same as previously measured. The mode shapes could be obtained from Equation 10.1 except for the constant A, which was

A=−

sin βl − sinh βl cos βl − cosh βl

(10.4)

The first three mode shapes had βl = 4.730041 for mode 1, βl = 7.853205 for mode 2, and βl = 10.995608 for mode 3, from which the natural frequencies could be determined. The first three natural frequencies are listed in Table 10.2. The results from the numerical modal analysis agreed well with the other solutions. 10.4.3 Example Problems As an example for a vibrational study including FSI, a composite beam clamped at both ends was considered. The geometric and material properties of the beam were the same as those used in the previous section. Water had a density of 1000 kg/m3, and the speed of sound was 1500 m/s. The nonreflected boundary condition was applied to the fluid domain. The numerical modal analysis was conducted using the multiphysics-based computational techniques described previously. As the first case, a short impulse load was applied to the clamped composite beam. Then, the natural frequency and the mode shapes were computed from the numerical modal analysis. To obtain the second rotationalsymmetric mode shape, the impulse load was applied to a finite element nodal point other than the center for the center was the node of vibration for the second mode. TABLE 10.2 Comparison of Natural Frequencies of Clamped Beam at Both Ends in Air

Mode 1 Mode 2 Mode 3

Analytical Solution (Hz)

FEM Eigenvalues Analysis (Hz)

Error of Eigenvalues Analysis (%)

Numerical Modal Analysis

Error of Modal Analysis (%)

162.52 448.01 878.29

162.52 448.01 878.29

0 0 0

170.00 460.00 910.00

4.6 2.67 3.61

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The first natural frequency obtained with FSI was 63.64 Hz, which was much smaller than the first frequency without FSI, as shown in Table 10.2. The effect of FSI reduced the frequency by more than 60%. Such reduction was comparable to what was measured in the physical experiment for a cantilever beam. The experimental study in Reference 9 showed an approximately 70% reduction in the first natural frequency. Because the cantilever beam was less constrained than the beam clamped at both ends, the effect of FSI was slightly larger. The first two mode shapes were compared in Figures 10.20 and 10.21 between the two solutions with and without FSI, respectively. The mode shapes look similar between the two cases. Then, the modal curvatures were compared; these were the second derivatives of the mode shapes and were proportional to bending strains. As shown in Figures 10.22 and 10.23, there was greater difference in the modal curvatures than in the mode shapes resulting from the FSI effect, especially for the second mode. Modal curvatures were directly proportional to the bending strains of the plates. The next study investigated the free vibration of a beam clamped at both ends. As the initial deformation, the static deflection of the beam subjected to a central force was used,

Normalized modal deflection

0 In air In water

–0.2 –0.4 –0.6 –0.8 –1

0

0.2

0.4 0.6 Normalized beam length

0.8

1

FIGURE 10.20 Comparison of the first mode shape of a beam clamped at both ends with and without FSI effect.

Normalized modal deflection

1

In air In water

0.5

0

–0.5

–1

0

0.2

0.4 0.6 Normalized beam length

0.8

1

FIGURE 10.21 Comparison of the second mode shape of a beam clamped at both ends with and without FSI effect.

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Modal curvature

5

0

–5

In air In water

0

0.2

0.4

0.6

Normalized beam length

0.8

1

FIGURE 10.22 Comparison of the first modal curvatures of a beam clamped at both ends with and without FSI effect. 80 60

In air In water

Modal curvature

40 20 0

–20 –40 –60 –80

0

0.2

0.4 0.6 Normalized beam length

0.8

1

FIGURE 10.23 Comparison of the second modal curvatures of a beam clamped at both ends with and without FSI effect.

and the initial velocity was set to zero. The beam was freed to vibrate without any external load. Figures 10.24 and 10.25 show snapshots of deformed beams during free vibration of the beam at some selected times. Figure 10.24 is for vibration in air, while Figure 10.25 is for vibration in water. The initial static deflection of the beam is close to its first mode shape. As a result, the first mode shape was the major vibrational mode. The free vibration in air looked like more or less the first mode shape of vibration. However, the free vibration in water contained visible high-frequency modes on top of the first mode. In other words, the FSI resulted in higher mode shapes to store the vibrational energy. This finding was also observed in the previous experiment of a composite cantilever beam [9]. Figures 10.26 and 10.27 show the experimentally measured vibrations of a cantilever beam in air and water, respectively, using the digital image correlation technique. The figures are also snapshots at selected times. The experimental data also confirmed the high-frequency motion in water resulting from FSI because the effect of FSI was not uniform over the beam.

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0

×10–3

–0.2

Beam deflection

–0.4 –0.6 –0.8 –1 –1.2 –1.4 –1.6 –1.8

0

0.2

0.4 0.6 Normalized beam length

0.8

1

FIGURE 10.24 A snapshot of the deformed shape of a beam clamped at both ends with initial static deflection without FSI.

3

×10–7

2

Beam deflection

1 0 –1 –2 –3 –4 –5 –6

0

0.2

0.4 0.6 Normalized beam length

0.8

1

FIGURE 10.25 A snapshot of the deformed shape of a beam clamped at both ends with initial static deflection with FSI.

The next set of studies was conducted for a composite plate whose material properties were the same as the previous ones. The plate was square, 1 × 1 m, and plate thickness was 0.02 m. A short-duration impulse force was applied to the composite plate while it was clamped all around the boundary. The plate had zero initial displacement and velocity. The deformed shapes of the vibrating composite plate in air and water, respectively, are plotted in Figures 10.28 and 10.29 at arbitrary selected times. The deformed shape of the vibrating composite plate in air resembled that of the static deformation under the central load as shown in Figure 10.28. However, the vibrating composite plate in water showed a different transverse displacement, as shown in Figure 10.29,

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Amplitude (mm)

0 –0.05 –0.1 –0.15 –0.2 0

50 100 150 200 250 Distance from fixed end of beam

300

Amplitude (mm)

FIGURE 10.26 A snapshot of the deformed shape of a cantilever beam with initial static deflection without FSI, measured using the digital image correlation technique.

t= 8 s

0

–0.05

–0.1 0

50

100

150

200

250

300

Distance from fixed end of beam FIGURE 10.27 A snapshot of the deformed shape of a cantilever beam with initial static deflection with FSI, measured using the digital image correlation technique.

×10–6

Plate deflection

0 –1 –2 –3 –4 0

0.2

0.4

0.6

0.8

1 0

0.2

0.4

0.6

FIGURE 10.28 Vibrational shape of a clamped square plate without FSI under impulse load.

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0.8

1

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×10–7 1

Plate deflection

0.5 0 –0.5 –1 –1.5 –2 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

FIGURE 10.29 Vibrational shape of a clamped square plate with FSI under impulse load.

where there are local peaks near the four corners of the clamped plate that were not observed for the vibration in air. Furthermore, the excited mode shapes were compared between the two vibrations in air and water, respectively. Because the force was applied at the center, only symmetric modes were excited by the impulse load. The first two mode shapes in air are shown in Figures 10.30 and 10.31. The first mode shapes in air and water were close to each other. As a result, the first mode shape in water is not provided separately. However, the next symmetric modes excited by the central impulse force were very different between the vibrations in air and water. Figure 10.31 shows the second symmetric mode excited in air, while Figure 10.32 shows the counterpart mode in water. The mode shape shown in Figure 10.32 explains the local peak deformation near the corners as observed in the experimental results [4,5]. When a clamped composite plate was impacted

Modal deflection

0 –0.2 –0.4 –0.6 –0.8 –1 1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

FIGURE 10.30 First mode shape of a clamped square plate under a central impulse force in air.

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0.8

1

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Modal deflection

1 0.5 0 –0.5 –1 1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

FIGURE 10.31 Second symmetric mode shape of a clamped square plate under a central impulse force in air.

Modal deflection

1 0.5 0 –0.5 –1 0

0.2

0.4

0.6

0.8

1

1

0.8

0.6

0.4

0.2

0

FIGURE 10.32 Second symmetric mode shape of a clamped square plate under a central impulse force in water.

at the center while the plate was in air and water, respectively, the measured strains were very different near the corners of the plate for the two situations compared to near the center location [4,5]. That is because the excited mode shapes were different between the vibrations in air and water. The next examples examined a clamped plate with a given initial condition without any external load. Two cases were considered. The first case used the static deformation of the square plate under central force as the initial displacement along with zero initial velocity. The second case applied the static deformation under a concentrated bending moment at the center as the initial displacement with zero initial velocity. Those initial displacements were obtained from the static bending finite element analysis of the same composite plate clamped along the boundary. The same mesh was used for both static and dynamic analyses. The first case corresponded to the initial deformation close to the first mode of vibration, which was a symmetric mode. On the other hand, the second case corresponded to the initial deformation close to the second mode of vibration, which was rotationally symmetric about one axis.

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Figures 10.33 and 10.34 compare the snapshots of deformed plates during free vibrations in air and water, respectively, at arbitrary times for the symmetric initial deflection. Likewise, Figures 10.35 and 10.36 compare the plate vibrations in air and water, respectively, with the rotationally symmetric initial deflection. As shown in the beam study, the free vibration of the composite plate in water stored the vibrational energy into modes of higher frequencies, as shown in the figures. All deformed plots did not consider slopes for smooth deformed shapes for simplicity. As a result, the deformed plates showed facets.

×10–5

Plate deflection

0 –1 –2 –3 –4 1

0.8

0.6

0.4

0.2

0

0

0.2

0.4

0.6

0.8

1

FIGURE 10.33 Snapshot of the deformed shape of the free vibration of a clamped square plate in air with symmetric initial displacement.

×10–8 Plate deflection

5 0 –5

–10 –15 –20 1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

FIGURE 10.34 Snapshot of the deformed shape of the free vibration of a clamped square plate in water with symmetric initial displacement.

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×10–5

3 2 Plate deflection

1 0

–1 –2 –3 –4 –5 1 0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6

0.8

1

FIGURE 10.35 Snapshot of the deformed shape of the free vibration of a clamped square plate in air with antisymmetric initial displacement.

1.5

×10–5

1

Plate deflection

0.5 0

–0.5 –1

–1.5 –2 0 0.2 0.4 0.6 0.8 1

1

0.8

0.6

0.4

0.2

0

FIGURE 10.36 Snapshot of the deformed shape of the free vibration of a clamped square plate in water with antisymmetric initial displacement.

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The frequency spectra are plotted in Figures 10.37 and 10.38 for the symmetric initial displacement, which was obtained from the time history of the central displacement of the plate. The plot for vibration in air shows the first natural frequency as a sharp peak, while the frequency spectrum for vibration in water shows the downward shift of the first natural frequency with a much smoother peak because of contributions from various frequency components. Furthermore, Figures 10.39 and 10.40 compare frequency spectra of the plate motion with the rotationally symmetric initial displacement. Almost the same kind of observation can be made for the two plots except the frequency in Figure 10.39 represents the second mode shape. 10 9 8

Magnitude

7 6 5 4 3 2 1 0

0

50

100 150 Frequency (Hz)

200

250

FIGURE 10.37 Frequency spectrum of the center displacement of a clamped square plate in air with symmetric initial displacement. 0.3

Magnitude

0.25 0.2 0.15 0.1 0.05 0

0

50

100 150 Frequency (Hz)

200

250

FIGURE 10.38 Frequency spectrum of the center displacement of a clamped square plate in water with symmetric initial displacement.

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100 90 80 Magnitude

70 60 50 40 30 20 10 0

0

50

100

150 200 Frequency (Hz)

250

300

FIGURE 10.39 Frequency spectrum of the center displacement of a clamped square plate in air with rotationally symmetric initial displacement. 1 0.9 0.8 Magnitude

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

50

100

150 200 Frequency (Hz)

250

300

FIGURE 10.40 Frequency spectrum of the center displacement of a clamped square plate in water with rotationally symmetric initial displacement.

10.5 Fatigue Loading with FSI 10.5.1 Problem Description Let us consider a composite structure that supports vibrating equipment on its top side and is in contact with water on the bottom side, as shown in Figure 10.41. The water domain is considered semi-infinite. The corresponding simplified engineering model is presented in Figure 10.42, with detailed explanations presented next [10].

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Vibrating equipment Flexible composite structure

Support of structure

Support of structure Water domain

Semi-infinite water boundary FIGURE 10.41 Structure supporting vibrating equipment while in contact with water. F = Fo sin(ωt)

K

Composite beam or plate

Water subdomain modeled using FEM Water subdomain modeled using CA

FIGURE 10.42 Engineering model to represent a structure supporting vibrating equipment while in contact with water.

The vibrating equipment was simplified as a single mass and a linear spring under a harmonic motion. Damping was neglected in the model. Neglecting damping would not affect the objective of the present study because the structural response was compared with and without the FSI effect under the same condition otherwise. The vibration of the equipment was modeled as a harmonic force applied to the mass. The structure was modeled as a beam or a plate/shell depending on the structural dimensions. Both beam and plate structures are considered here. The FEM was used to analyze the dynamic response of the beam or plate. Because the beam or plate must be coupled with the fluid (i.e., water), the beam or plate finite elements were formulated to have only displacement degrees of freedom at each node like a 3-D solid element, as discussed in Chapter 2. This makes the compatibility at the fluid-structural interface easy. The boundary of the structure was either simply supported or clamped. The water medium was assumed to be an acoustic medium represented by the wave equation. Then, the water domain was broken into two subdomains as shown by broken lines in Figure 10.42 to achieve the computational efficiency as well as easy compatibility at the fluid-structure interface, as described in more detail in this chapter. Water subdomain 1 was analyzed using FEM, while water subdomain 2 was solved using the CA technique. The top boundary of the water domain, represented by thick solid lines, was free while the

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other boundary, denoted by double solid lines, were nonreflective, like an infinite boundary. Both boundary conditions could be easily represented by CA. In addition, both water subdomains were properly coupled to have continuity of the solution. As a harmonic force was applied to the single mass, the dynamic response of the structure was analyzed. The structure may or may not have been in contact with water during the harmonic excitation. The results from the two cases were compared to study the effect of FSI. 10.5.2 Example Problems First, a beam structure was analyzed. The beam was made of a composite, aluminum, or steel. Their material properties are given in Table 10.3. To avoid the resonance effect, the vibration equipment had a frequency that was much lower than the natural frequencies of the structures. The beam was 1.0 m long, 0.1 m wide, and 0.02 m thick. Table 10.4 lists the first natural frequency of the beam in air for all three different materials. The composite beam clamped at both ends had the lowest frequency of 65 Hz. This was the frequency in air, that is, without the effect of water. In addition, the natural frequency of the clamped composite beam, including the additional single mass and spring as shown in Figure 10.42, was calculated using the FEM. The mesh sensitivity study showed that using 20 or more beam elements resulted in the first several natural frequencies being consistent. The frequencies depended on the spring-to-mass ratio K/M . Figure 10.43 shows the first natural frequency of the clamped composite beam with the centrally attached mass and spring (called a beam system from this point) as a function of the spring-to-mass ratio K/M . The graph is almost linear up to the ratio of K/M = 1000. This suggests the first natural frequency of the beam system was almost linearly proportional to the ratio of K/M . When the ratio K/M was approximately less than 500, the first natural frequency of the clamped composite beam system was smaller than that of the beam only without the spring and mass. In other words, the attached spring and mass decreased the first natural frequency of the beam. The vibrational frequency in water is much lower than that in air because of the added mass effect. A previous experimental study [9] showed the FSI reduced the natural frequency of an E-glass composite cantilever beam by 70%. In other words, the frequency in water was 30% of that in air. Metallic beams have smaller reductions in natural frequency TABLE 10.3 Material Properties E-glass composite Aluminum Steel

Elastic Modulus (GPa)

Density (kg/m3)

20 70 200

2000 2700 8000

TABLE 10.4 First Natural Frequencies of Beams in Air (Hz) E-glass composite Aluminum Steel

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Clamped

Simply Supported

65.09 104.8 102.9

28.68 47.18 45.35

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150

Frequency (Hz)

100

50

0

0

200

400 600 √2 (K/M) (1/s)

800

1000

FIGURE 10.43 Plot of first natural frequency of a clamped composite beam in air with centrally attached mass and spring as a function of the spring-to-mass ratio.

because of a smaller FSI effect. A numerical modal analysis was conducted for the composite beam as well as the composite system as described in the following so as to determine the natural frequencies. A short impulse duration was applied to a location of the beam structure, and the dynamic response was obtained at many locations as a function of time. For example, displacements, velocities, or accelerations can be computed from FEM. Among them, time histories of nodal displacements were used for the present study. Then, the FFT was conducted to convert the time domain response to the frequency domain data to determine the natural frequencies. For the present clamped composite beam without the spring and mass, an impulse was applied to near the center of the beam, and the center displacement was obtained for FFT. Figure 10.44 shows the FFT of the displacement. From the graph, the clamped composite beam submerged in water had the first natural frequency of 16 Hz, which is approximately one-fourth of the frequency in air. The same numerical modal analysis was conducted for the clamped composite beam in water with centrally attached mass and spring to find the natural frequency. Figure 10.45 shows the FFT plot with the first three natural frequencies of 25 Hz, 51 Hz, and 76 Hz, respectively, for K/M = 100. This ratio of K/M was selected because it resulted in the first natural frequency of the dry composite beam system to be 16 Hz, like the wet composite beam. The first natural frequency of the wet composite beam increased with the attached spring and mass, while the second natural frequency was reduced from 187 Hz. When selecting values for the singles mass M and spring K, the force transmissibility was considered. Assuming the rigid base structure, the force transmissibility TR was expressed as follows without damping: TR =

1 1 − r2

(10.5)

where r = ω/ωn, and ω is the applied frequency, while ω n = K/M . To consider the largest force transmission to the structure as the worst case, the frequency ratio r should be

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0.1 0.09

Absolute value

0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

10

20

30 40 50 Frequency (Hz)

60

70

80

60

70

80

FIGURE 10.44 The FFT plot of a clamped composite beam submerged in water.

0.1 0.09 0.08 Absolute value

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

10

20

30 40 50 Frequency (Hz)

FIGURE 10.45 The FFT plot of a clamped composite beam submerged in water with centrally attached mass and spring.

very small compared to unity. In other words, ωn should be much greater than ω. With ωn chosen to be 100 Hz, the vibrational frequency ω was set to 10 Hz, which was lower than the first natural frequencies for all the cases studied previously for the clamped composite beam. All the plots that follow were normalized for the unit force, that is, Fo = 1N. With the selected values, the composite beam was analyzed. The bending strain at the center of the beam was computed for the composite beam with its time history as plotted in Figure 10.46. Likewise, the bending strain at the boundary is also plotted in Figure 10.47. The bending strains were calculated on the beam side in contact with water. The deformation of a clamped beam has opposite curvatures at the center and the boundary so that the bending strains have opposite signs, as seen in Figures 10.46 and 10.47. The magnitudes of the bending strains at both elements were similar to those without water. The FSI effect

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4

×10–6 w/o FSI w/ FSI

3 2

Strain

1 0

–1 –2 –3 –4

0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.46 Strain at the center element for a clamped composite beam subjected to a 10-Hz vibrating force.

5

×10–6

4 3

Strain

2 1 0

–1 –2 –3

w/o FSI w/ FSI

–4 –5 0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.47 Strain at the boundary element for a clamped composite beam subjected to a 10-Hz vibrating force.

significantly increased the bending strains in the composite beam. The bending strain was more affected by FSI at the boundary than at the center. The maximum peak strain was increased by approximately 50% at the center and 85% at the boundary because of FSI. Also, the variation in peaks and valleys of the strain history was more profound with FSI. The result clearly suggests that the FSI effect on the composite beam in contact with water will significantly reduce the fatigue life of the structure. To further understand the FSI effect, the force induced in the spring is plotted in Figure 10.48. The figure shows that the force in the spring was not affected by the FSI. The velocity

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4

Spring force (N)

3 2 1 0

–1 –2

w/o FSI w/ FSI

–3 –4 0

0.05

0.1

0.2 0.25 Time (s)

0.15

0.3

0.35

0.4

FIGURE 10.48 Spring force for a clamped composite beam subjected to a 10-Hz vibrating force.

at the center of the composite beam is plotted in Figure 10.49. The FSI influenced the velocity significantly. Interestingly, the plot of the central deflection of the beam was similar to the strain plot at the center element as shown in Figure 10.46. Therefore, the deflection plot is omitted here. Both steel and aluminum beams were also tested under the same condition; their responses are plotted in Figures 10.50 through 10.53. The steel beam had almost no FSI effect on both the bending strain and velocity as seen in Figures 10.50 and 10.51, respectively. Likewise, the aluminum beam also had a negligible FSI effect on the bending strain, as shown in Figure 10.52. The time history of the central velocity was close for the with and without FSI conditions in terms of the phase and magnitude. However, FSI induced

Velocity (m/s)

5

×10–3 w/o FSI w/ FSI

0

–5

0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.49 Velocity at the center for a clamped composite beam subjected to a 10-Hz vibrating force.

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×10–7

Strain

5

w/o FSI w/ FSI

0

–5 0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.50 Strain at the center for a clamped steel beam subjected to a 10-Hz vibrating force.

×10–4

Velocity (m/s)

1

0

w/o FSI w/ FSI

–1 0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.51 Velocity at the center for a clamped steel beam subjected to a 10-Hz vibrating force.

additional high-frequency responses in the velocity of the aluminum beam, like the composite beam. Such a high-frequency response was also observed during the free vibrational experiment of a composite cantilever beam [9]. Comparing the three different materials, the composite beam showed the largest FSI effect because of its lowest density and stiffness. In the following study, some parameters in the system were varied to determine which parameter yielded the most critical FSI effect. As the first parameter, the equipment vibrating frequency was varied from 10 to 5 Hz. Comparison of Figures 10.46 and 10.54 suggests that the lower vibrating frequency resulted in a little less FSI effect on the bending strain at the beam center because 10 Hz is closer to the natural frequency of the composite beam in water.

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×10–6 1

w/o FSI w/ FSI

Strain

0.5 0

–0.5 –1 0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.52 Strain at the center for a clamped aluminum beam subjected to a 10-Hz vibrating force. ×10–4 w/o FSI w/ FSI

Velocity (m/s)

4 2 0

–2 –4 0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.53 Velocity at the center for a clamped aluminum beam subjected to a 10-Hz vibrating force.

To further investigate the effect of the exciting vibrational frequency on the dynamic response of the composite beam, the vibrational frequency was varied gradually, and the ratio of the maximum dynamic deflection to the static deflection at the center of the clamped composite beam was computed as plotted in Figure 10.55. The figure shows two resonance frequencies for the beam submerged in water and one resonance frequency for the dry beam up to 35 Hz. Those resonance frequencies agreed with the natural frequencies discussed previously. Comparing the deflection ratios between the conditions with and without FSI, as shown in Figure 10.55, clearly suggests that the FSI effect significantly increased the deflection of the composite beam (i.e., the greater bending strains). The following study changed the boundary condition of the beam: The beam was simply supported instead of clamped. The vibrating frequency was 5 Hz so it was much lower than its natural frequency in water for the simply supported beam. Bending strains at

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3

×10–6

2

Strain

1 0

–1 w/o FSI w/ FSI

–2 –3

0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.54 Strain at the center for a clamped composite beam subjected to a 5-Hz vibrating force.

Deflection ratio

15

10

w/o FSI w/ FSI

5

0

0

5

10

15 20 Frequency (Hz)

25

30

35

FIGURE 10.55 Plot of the deflection ratio as a function of the exciting vibrational frequency.

the center and the boundary are plotted in Figures 10.56 and 10.57, respectively. The more flexible boundary, such as the simply supported one, yielded a much larger difference between the strains with and without the FSI effect. Such a difference was more profound at the simply supported boundary, as seen in Figure 10.57. The FSI effect resulted in a bending strain nearly three times greater at the boundary along with very-high-frequency components in the strain-time history. An experimental study confirmed the numerical results qualitatively. For example, threepoint bending tests were conducted with cyclic loading while beams were in air or in water, respectively, under the same loading conditions. The numbers of cycles to failure were compared between the dry and wet tests. Table 10.5 shows that there was a significant reduction in the number of cycles to failure in water due to FSI. The failure cycle was about 50% greater in air than in water.

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×10–6

Strain

5

0

w/o FSI w/ FSI

–5

0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.56 Strain at the center for a simply supported composite beam subjected to a 5-Hz vibrating force. ×10–6 w/o FSI w/ FSI

1

Strain

0.5 0

–0.5 –1 0

0.05

0.1

0.15

0.2 0.25 Time (s)

0.3

0.35

0.4

FIGURE 10.57 Strain at the boundary for a simply supported composite beam subjected to a 5-Hz vibrating force.

TABLE 10.5 Ratio of Number of Cycles to Failure in Air versus in Water under Cyclic Loading Average of ratios Standard deviation

10 Hz

5 Hz

1.43 0.22

1.52 0.16

Finally, a clamped composite plate was also studied. The material was the same E-glass composite as used for the beam, and its dimension was 1 × 1 m and it was 0.02 m thick. The vibrating equipment was located at the center of the square plate with the same single mass and spring ratio of K/M = 1000 as before. The first natural frequency of the dry composite plate without the single mass and spring was computed from the following formula:

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Eh2 ρL4 (1 − ν2 )

ω n = 1.655

(10.6)

where E and ν are the elastic modulus and Poisson ratio, ρ is the volume density, and L and h are the length and thickness of the square plate, respectively. The natural frequency of the dry plate without the single mass and spring was 110 Hz (from Equation 10.6). When the spring and mass were included, the natural frequency was computed from the eigenvalue analysis of the finite element model, and it became 16 Hz. The natural frequency of the plate in contact with water was also computed using the numerical modal analysis, which resulted in a frequency of 36 Hz. As a result, the exiting frequency of the equipment was selected as either 10 or 20 Hz. The transverse deflection was computed at multiple locations on the plate and compared between the dry and wet vibrations. The selected locations on the plate are shown in Figure 10.58. The ratio of the maximum transverse deflection of the wet plate to the dry plate is provided in Table 10.6 for two different excitation frequencies. The results showed that the largest difference between the wet and dry plate deflections occurred at

Loc. D

Loc. E

Loc. F

Loc. B

Loc. C Loc. A

FIGURE 10.58 Locations on the clamped plate where displacements were compared between dry and wet vibrations.

TABLE 10.6 Ratio of Maximum Displacement between the Wet and Dry Clamped Composite Plate (Please see Figure 10.58 for the locations) Excitation Frequency Location A B C D E F

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10 Hz

20 Hz

1.20 1.14 1.16 1.11 1.14 1.16

1.41 1.31 1.34 1.24 1.29 1.33

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3

×10−6 w/o FSI w/ FSI

Displacement (m)

2 1 0

–1

–2 –3 0

0.05

0.1

0.15 Time (s)

0.2

0.25

0.3

FIGURE 10.59 Displacement at the center of a clamped composite plate subjected to a 20-Hz vibrating force.

location A in Figure 10.58, while the least difference occurred at location D. These data suggest that the FSI effect was larger at a location closer to the clamped boundary and smaller at a location nearer the plate center. This statement can also be confirmed by comparing locations B and C, as well as the locations D, E, and F, respectively. At location A close to the boundary corner, the deflections of the wet plate were 20% and 41% greater than that of the dry plate for the two different excitation frequencies, respectively. The vibrating equipment with 20 Hz resulted in more than a 40% increase in the deflection at location A. The transverse deflection at the center of the plate is plotted in Figure 10.59 to illustrate the vibrational motion of the plate.

10.6 Hydrodynamic Loading In this section, transient fluid flow around a flexible composite plate is studied [11]. Because the fluid flows, the Navier-Stokes equation was used to model the fluid domain. The details of the computer models and their results are discussed. 10.6.1 Model Description A box-shape structure 1 m3 was considered. The front side was a flexible E-glass composite plate that interacted with fluid flow, while the remaining five sides were assumed to be rigid. The box-shape structure was varied further in terms of the shape and size for a series of parametric studies. The structural element size remained relatively constant for different cases. The fluid domain was of a sufficient size to minimize interference from the sides of the fluid domain. The fluid domain extended 1 m beyond the cube from its top, sides, and the rear face and 2 m from the front face. The fluid domain had dimensions of 3 × 3 × 4 m. The

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3

Velocity (m/s)

2.5 2 1.5 1 0.5 0

0

0.5

1 Time (s)

1.5

2

FIGURE 10.60 Inlet velocity profile of the base model.

structure was surrounded by water only on its outside. There was no fluid inside the box. The water was assumed to have a density of 1000 kg/m3 and a kinematic viscosity of 1.0 × 10 –6 m2/s. The Reynolds number for the base model of the box structure was 2.0 × 106. The interfaces between the structure and fluid were defined as follows: The front interface had the FSI. The other five sides of the interface had no-slip boundary conditions because the structural parts were held fixed in space. The inlet side of the fluid domain had a prescribed velocity as a function of time. The outlet of the fluid domain had a prescribed constant pressure. Full-slip boundary conditions were also applied to all other exterior surfaces of the fluid domain to mitigate boundary-layer effects from the fluid domain onto the cubic structure. The model described here was called the base model. The base model considered transient fluid flow at the inlet under a constant acceleration for the first 0.5 s until a terminal velocity of 2 m/s was reached, at which point the fluid velocity remained constant for the duration of the simulation, as shown in Figure 10.60. A time step size of 0.01 s was used for plotting the results even though the actual time step size used for computation was much smaller. 10.6.2 Numerical Results with Constant Acceleration As the first example, the flexible front plate of the base model was replaced by a rigid plate so that no FSI existed at the front plate. Then, the two models with and without FSI were compared to examine the effect of FSI on fluid flow. The numerical results showed that the center node of the flexible base model had a maximum 45-mm inward displacement at 0.09 s and a maximum 15-mm outward displacement at 0.58 s followed by a steady-state 5-mm inward displacement at 2.0 s. These local maximum/minimum and steady-state displacement should have the largest effect on fluid flow around the cube. At a time of 0.09 s, the inward deflection of the deformable plate minimized overall resistance on the upstream flow field and allowed upstream fluid to progress at a faster average velocity than the rigid model. The maximum outward deflection of the flexible plate occurred at 0.58 s and gave the cube a slight streamlined effect that resulted in an average velocity around the cube of 2.32 m/s. The average fluid velocity of the rigid body around the cube at 0.58 s was 2.22 m/s, slower than the flexible model. The velocity contours

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between the two models at steady state (i.e., at 2.0 s) were virtually identical, with only minor velocity differences of approximately 0.002 m/s. Once the effect of FSI was investigated using the base model, the flexible base model was further analyzed to investigate the effect of transient flow on the flexible front plate. From this point in the discussion, the term base model refers to the box with the flexible front plate. To determine the maximum stress/strain in the structural domain caused by transient flow, it was necessary to first verify the location of peak stress or strain. The numerical result showed that the maximum stress/strain occurred near the clamped boundaries. One of the four maximum strain locations was selected, and from this point is called the boundary point. Figure 10.61 compares the elastic equivalent strains at the boundary point and the center as a function of time. The elastic equivalent strain is like the von Mises stress in terms of strain components. This plot shows that the boundary point had a much highest strain value than at the center. The ratio of the maximum transient value to the steady-state value of the elastic equivalent strain was a little more than 8 at both the center and the boundary nodes. This ratio remained the same for the von Mises stress as well as the maximum displacement, as expected. Nodal acceleration and displacement at the center are plotted in Figure 10.62. The plate displacement corresponded closely to the acceleration, with an expected minimal lag between the two variables. There was an average 0.04-s delay of the local maximum and minimum displacement values to the acceleration values. Peak values for structural stress and strain occurred during times when displacement was at either a local maximum or minimum. Last, at 0.5 s, the fluid acceleration suddenly changed to zero, as shown in Figure 10.60, and there was an immediate response in structural nodal acceleration as well as in displacement. Fluid pressure at the center of the plate was also evaluated from the base model and is plotted in Figure 10.63. In addition, the average fluid pressure over the entire flexible plate is also shown in the same plot. Multiplication of the average pressure by the plate surface area resulted in the total fluid force on the plate. The average pressure plot closely matched the pressure plot at the center node. The ratio of the maximum to the steady-state values was also computed for the two kinds of fluid pressure (i.e., pressure at the plate center and the average pressure). The ratio for the fluid pressure at the plate center was close to that obtained for the stress and strain values (i.e., a little over 8). However, the ratio for the average pressure was 14% greater than that for the central pressure.

Elastic equivalent strain

0.004 Boundary

Center

0.003 0.002 0.001 0

0

0.5

1 Time (s)

1.5

2

FIGURE 10.61 Plot of the time history of elastic equivalent strain at two locations of the front plate of the base model.

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Acceleration (m/s)

Displacement

0.06

200

0.04

100

0.02 0

0

–100

–0.02

–200

–0.04

–300

0

0.2

0.4

0.6

0.8

1 1.2 Time (s)

1.4

1.6

1.8

2

Displacement (m)

Acceleration

300

–0.06

FIGURE 10.62 Plot of the time history of acceleration and displacement at the center of the base model. 30,000

Center press.

Avg. press. interface

0.5

1 Time (s)

Pressure (Pa)

20,000 10,000 0

–10,000 –20,000

0

1.5

2

FIGURE 10.63 Plot of fluid pressure (press.) at the center and average (Avg.) fluid pressure over the entire plate of the base model.

One more thing to note here is that the negative pressure in Figure 10.63 suggests potential local cavitation. However, such cavitation did not affect the initial maximum transient response because it occurred before the cavitation. Likewise, the final steady-state response long after cavitation would not affected by the cavitation. As a result, potential cavitation was neglected in this study. 10.6.3 Numerical Results with Nonlinear Acceleration In the next study, the inlet fluid velocity was varied as shown in Figure 10.64. The first case modeled a monotonically increasing acceleration, and the second case was a monotonically decreasing acceleration. Together, they showed the effects of nonlinear acceleration during the period of velocity variation. The monotonically increasing acceleration closely resembled the profile of an accelerating ship, where acceleration increases with time. On the other hand, the monotonically decreasing model was a near mirror image.

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Base

3

Monotonic decreasing

Monotonic increasing

Velocity (m/s)

2.5 2 1.5 1 0.5 0

0

0.5

1.5

1 Time (s)

2

FIGURE 10.64 Inlet velocity profiles of monotonically varying accelerations.

As seen in Figure 10.64, the monotonically decreasing model had an almost-linear acceleration for the first 0.25 s, and the plate responded with oscillations similar to the base model, with a linear acceleration as shown in Figure 10.65. However, the monotonically decreasing model had a larger slope than the base model, which resulted in oscillations with a higher amplitude. During the final 0.25 s of the velocity transient, acceleration slowly reduced to zero in the monotonically decreasing model. This caused the oscillations to be reduced prior to achieving the steady-state velocity so that the amplitude of stress became lower for the monotonically decreasing model than for the base model. On the other hand, the monotonically increasing acceleration yielded an almost-linear increase in the center displacement. The monotonically decreasing model had the largest ratio of peak to steady-state stresses, which was nearly 14; the monotonically increasing model had a ratio of slightly larger than 10, and the linear model had a ratio of 8. This was expected based on the initial linearity of the fluid acceleration, with a greater slope than the base model. The monotonically decreasing and linear models reached peak displacements in the negative direction at 0.08 and 0.09 s, respectively. The monotonically increasing model reached a maximum inward Base

Displacement (mm)

50

Monotonic decreasing

Monotonic increasing

0

–50

–100

0

0.5

1 Time (s)

FIGURE 10.65 Plot of displacement at the center for monotonically varying accelerations.

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2

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displacement at 0.5 s, and the ratio was higher than for the base model because its slope was higher during the second half of the transient. When a monotonically increasing acceleration is followed by a monotonically decreasing acceleration, such a combined velocity profile portrays a transient velocity curve with an inflection point and reduced acceleration slopes at both ends. This combination brings the advantages of each curve and avoids the initial transient oscillation while encouraging rapid oscillation decay prior to reaching steady-state velocity. This type of transient velocity curve is also most representative of a ship at sea and proves that the attributes of such a transient acceleration are beneficial for the structure. 10.6.4 Numerical Results with Free Surface The boundary condition applied to the top surface of the fluid domain was changed in this study. Previously, it was the free-slip boundary, but it was changed to the free boundary with zero pressure. Then, the structure was placed inside the fluid with the varying depth from the free surface. The n-meter model means the distance from the free surface of the fluid domain to the top surface of the structural box is n meters. Although the time for peak pressure on the 1-m model occurred at 0.07, which is slightly earlier than that for the base model without the free fluid surface, the 1-m model resulted in a far different response due to the significant effect of the free surface. Fluid along the upper half moved freely and resulted in much lower pressures above the cube. This also reduced the hydraulic forces applied to the fluid-structure interface. The base model represented an infinite water depth, while the 1-m model represented the shallow-water depth because the hydrostatic loading was neglected in the study. The 2-m and 3-m models showed increased fluid pressure in the region above the plate, indicating that the rise in depth increased fluid pressure and stress on the front plate. Figure 10.66 compares the central displacement of the plate with different water depths. The stress and strain responded similar to the displacement for different water depths. Comparing the maximum to the steady-state values was also conducted for the base, 1-m, 2-m, and 3-m models. The base model resulted in the highest ratio and was symmetric about the center. The 1-m model had the smallest ratio at the upper boundary point, and the ratio increased along the plate toward its bottom boundary. The ratios for the 2-m and 3-m models Base

Displacement (mm)

30

1-meter

3-meter

2-meter

0

–30

–60

0

0.5

1 Time (s)

1.5

FIGURE 10.66 Comparison of the central displacement with different water depths from the free surface.

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Infinite depth

Max. to steady state ratio

10 8 6 4 2 0

1

2

3

4

8 9 5 6 7 Distance from surface (m)

10

11

12

FIGURE 10.67 Plot of the ratios of the maximum to the steady-state stress/strain against the water depth from the free surface.

also tracked accordingly with depth. Additional models with different depths were also analyzed to draw a curve as shown in Figure 10.67, which suggests that any water depth greater than 9 m can be treated as the infinite depth in a practical sense when the water depth effect is considered on the ratio of the maximum to steady-state values. The 1-m model had a ratio of 2 approximately, and the ratio increased as a function of the water depth until it reached a plateau at around 9 m. This result was similar with the data provided in Reference 12, which investigated energy scavenging from tidal forces and reported that normalized velocity deficits in the water column decreased to near zero as depth exceeded 7 m. 10.6.5 Numerical Results with Different Material Properties This case considered different composite materials subjected to the same boundary conditions and acceleration profile as the base model. The base model had a density of 2000 kg/m3 and a Young’s modulus of 20 GPa; the first and second models changed Young’s modulus to 50 GPa and 100 GPa, respectively. The third model changed the density to 3000 kg/m3 and the Young’s modulus to 50 GPa. Peak stresses for the base model occurred at 0.08 s; all others occurred at 0.07 s. The three models (basic model, model at 2000 kg/m3/50 GPa, model at 2000 kg/m3/100 GPa) had the same mass density but different elastic moduli. Figure 10.68 compares the fluid pressure at the center of the front plate. It shows almost the same magnitudes but small changes in the frequency of the response. Even though the pressure had a small variation, the plate displacement at the center was influenced by the plate material properties, as shown in Figure  10.69. As expected, the lower modulus resulted in higher displacement if the structure had the same density. However, the resultant stress was higher with the higher modulus, as shown in Figure 10.70. On the other hand, the two models (with 2000 kg/m3/50 GPa and with 3000 kg/m3/ 50 GPa) could be compared to determine the effect of the mass density. The results suggest that the lower density resulted in greater strain and stress. The lower-density model showed approximately 15% higher stress than the higher-density model. This indicates that a lighter composite structure can have greater stress and strain under the same hydrodynamic loading, which should be considered in designs of marine composite structures.

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Base

Pressure (Pa)

30,000

2000/50

2000/100

3000/50

20,000 10,000 0

–10,000

0

0.5

1.5

1 Time (s)

2

FIGURE 10.68 Comparison of the pressure at the center of the plate with different material properties. Base

Displacement (mm)

30

2000/50

3000/50

2000/100

0

–30

–60

0

0.5

1 Time (s)

1.5

2

FIGURE 10.69 Comparison of displacements at the center of the plate with different material properties. Base (2000/20)

von Mises stress (MPa)

150

2000/50

3000/50

2000/100

120 90 60 30 0

0

0.5

1 Time (s)

1.5

FIGURE 10.70 Comparison of von Mises stresses at the center of the plate with different material properties.

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When the ratios of the maximum to steady-state stresses were compared for all models, the base model had a slightly higher ratio than the other models. However, the difference was less than a couple percent, so that it was negligible in the practical sense because material properties influenced both the transient peak stress/strain and the steady-state stress/ strain at almost the same proportion.

10.7 Hydrodynamic Ram 10.7.1 What Is Hydrodynamic Ram? Hydrodynamic ram (HRAM) refers to the damage process resulting from high pressures generated when a projectile with a high velocity penetrates a compartment or vessel containing a fluid [13]. The large internal fluid pressure that acts on the walls of the fluid-filled tank can result in severe structural damage, especially at the entrance and exit walls. The study of HRAM effects on fuel tanks used on military aircraft is one good example. In most nonexploding projectile impacts with penetration and traveling through a fluidfilled tank, the HRAM phenomenon can be described in four distinct phases [14]: • Shock phase: initial impact of a projectile into an entry wall of a fuel tank • Drag phase: movement of a projectile through fluid • Cavitation phase: development of a cavity behind the projectile as it moves through the fluid and the subsequent cavity oscillation and collapse • Exit phase: projectile penetrates the exit wall and leaves the tank only when there is sufficient energy remaining Each phase contributes to structural damage of the tank walls via a different mechanism, and the extent of damage depends on numerous factors, such as projectile shape and velocity, fluid level in the impacted tank, obliquity of impact, and fuel tank material. The amount of structural damage can be significant, with large-scale peeling and tearing of the entry and exit walls. 10.7.2 Numerical Models The HRAM model consisted of three parts: tank structure, projectile, and fluid inside the structure. The simulation of HRAM required a very fine Euler mesh and small sampling times to capture the propagation of shock waves in the fluid. For a simplified computational model, a generic cubic tank 200 × 200 × 200 mm, impacted by a 10-mm diameter spherical projectile was studied. Some parametric studies were also conducted of the effect of the selected parameter on HRAM. The projectile impacting at the center of the entry wall of the tank was a rigid sphere with a mass of 4 g and 10-mm diameter. One reason for selecting a spherical projectile was to prevent tumbling of the projectile during the drag phase, which would result in significant pressure fluctuations in the fluid, causing an erratic response to the structure. As shown in Figure 10.71, two models were constructed for this study [15]. The first model was for the investigation of the shock phase of the HRAM with the projectile outside the tank, impacting the entry wall at a prescribed velocity. For this model, model 1, the

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Model 1

Model 2

y

x FIGURE 10.71 Schematic of models 1 and 2.

displacement of the tank walls due to projectile impact and the subsequent ram pressure of the propagating hemispherical shock wave in the fluid from the impact point would be of interest. For the second model, model 2, the initial starting position of the projectile was at the inner surface of the entry wall of the impact point to simulate the projectile movement just after penetrating the entry wall. The initial velocity of the projectile was less due to retardation of the projectile by the entry wall. Model 2 would be used to study the fluid pressures and the tank wall response during the drag phase. By considering two models separately, the complex penetration process in the entry wall could be neglected. In addition, we could study how each phase affected the dynamic responses of the system individually even though the actual HRAM included both of them, one after the other sequentially. The structure was constrained at the bottom face of the box shape and stationary in the beginning. The fluid was also stationary, while an initial velocity was assigned to the projectile. For model 1, the initial velocity of the projectile was 300 m/s; the velocity was 250 m/s for model 2. An important aspect of the FSI problem is the coupling of the interfaces between the structure and fluid mesh. In both models, FSI was defined at the interface of the structural tank and the internal fluid. In addition, the contact/impact condition was applied to the sphere and the outside of the structure for model 1. On the other hand, the projectile was coupled to the fluid for model 2. 10.7.3 Numerical Results The baseline model 1 simulation was set up for a 100%, 80%, or 60% water-filled tank impacted without penetration at the center of the entry wall by a spherical rigid projectile with an initial velocity of 300 m/s [15]. Even though this was a hypothetical situation because the projectile would likely penetrate the entry wall in an actual experiment, this simulation provided some insight to the tank wall behavior during the initial shock phase of the HRAM event. The event was simulated for 1 ms with a sampling rate of 20 μs for data collection to plot the event time history. For comparison, the following discussion compares model 1 to an empty tank impacted under the same conditions. The entry wall x displacement is plotted at the center of the plate in Figure 10.72. The x direction corresponds to the major component of the entry wall because the direction of projectile velocity impacting the entry wall was in the positive x direction. It can be observed that the peak displacement of the entry wall for the 100% filled baseline model 1 was higher at around 9 mm as compared to 7 mm for the empty tank. An interesting

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1.00E–02 XDIS - empty

Displacement (m)

5.00E–03

XDIS - 100%

0.00E+00 0.00E+00

2.00E–04

4.00E–04

6.00E–04

8.00E–04

1.00E–03

–5.00E–03

–1.00E–02

Time (s)

FIGURE 10.72 Entry wall x displacement (XDIS) at the center for model 1.

phenomenon observed for model 1 is the entry wall displacing in the negative x direction at around 0.06 ms after impact, which indicates the entry wall bulging outward. The x component velocity indicates a much larger peak value of around 210 m/s in the negative x direction right after projectile impact. This corresponded to the time when the entry wall started to bulge. The effective entry wall stress (i.e., the von Mises stress) reached a higher peak value for model 1 but over a shorter duration of time than for the empty tank. The exit wall response to HRAM was of main interest in this study as it is an area where main structural components and load-bearing members are likely to be located. Graphs for exit wall response were plotted from data collected from the center node of the exit wall panel. The x displacement plot in Figure 10.73 shows a peak displacement of around 2 mm experienced by the exit wall at the end of the simulation, a value much higher than 5.00E−03

Displacement (m)

4.00E−03 3.00E−03

XDIS - empty XDIS - 100%

2.00E−03 1.00E−03 0.00E+00 0.00E+00 –1.00E−03

2.00E−04

–2.00E−03

4.00E−04

Time (s)

FIGURE 10.73 Exit wall x displacement (XDIS) at the center for model 1.

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6.00E−04

8.00E−04

1.00E−03

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that experienced by the empty tank. The exit wall for model 1 started deforming earlier, at approximately 0.13 ms. This was approximately the time when the initial shock wave due to projectile impact at the entry wall impinged onto the exit wall, causing it to displace. The presence of fluid in the tank actually resulted in a much smaller velocity and effective stress at the exit wall. Peak stress at the center of the exit wall registered a much lower value of approximately 100 MPa, as compared to 500 MPa for the empty tank. Besides the propagation of a shock wave through the aluminum tank structure, a hemispherical shock wave was observed to propagate in the fluid toward the exit wall. This ram pressure generated by the impact of the projectile in the shock phase was computed at the center of the fluid. The peak pressure of 1.6 MPa occurred for the 100% filled tank in the beginning. On the other hand, the 60% filled tank showed the peak pressure of 1.4 MPa around 0.7 ms instead of the beginning. Simulation for baseline model 2 was set up for a 100%, 80%, or 60% water-filled tank 2 mm thick with an initial projectile velocity of 250 m/s. Model 2 was developed to assist in understanding the structural response of the tank walls during the drag-and-cavitation phase of HRAM. All displacement, velocity, and effective stress values plotted were obtained from the center node or element of the tank walls. Because the collapse of the cavity would most likely occur at a much later time, the cavitation collapse pressure and its subsequent effect on the tank walls were omitted from this study. Instead, the effects on tank walls due to drag-phase pressure and the formation of a cavity in the fluid would be the main interest. The exit wall response graphs are plotted up to 2 ms. However, the projectile impacted the exit wall approximately at 1.5 ms. Thus, the results are only of interest before impacting the exit wall. The exit wall started to move and deform at approximately 0.13 ms into the simulation due to the initial shock wave impinging onto the exit wall. At approximately 1 ms into the simulation, the rate of displacement of the exit wall registered an increase, as can be observed from the steeper gradient of the displacement time history plot of the exit wall as illustrated in Figure 10.74. This was due to the projectile approaching the exit wall and the high-pressure region in front of the projectile during the drag phase exerting a greater pressure and prestressing the exit wall before projectile impact. Likewise, the 1.00E−02

Displacement (m)

8.00E−03

Fluid level - 100% Fluid level - 80% Fluid level - 60%

6.00E−03 4.00E−03 2.00E−03 0.00E+00 0.00E+00

5.00E−04

–2.00E−03

1.00E−03

Time (s)

FIGURE 10.74 Exit wall resultant displacement at the center for model 2.

© 2016 by Taylor & Francis Group, LLC

1.50E−03

2.00E−03

2.50E−03

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Multiscale and Multiphysics Modeling

6.00E+06 Fluid level - 100% Fluid level - 80% Fluid level - 60%

Pressure (Pa)

5.00E+06 4.00E+06 3.00E+06 2.00E+06 1.00E+06 0.00E+00 0.00E+00

5.00E−04

–1.00E+06

1.00E−03

1.50E−03

2.00E−03

2.50E−03

Time (s)

FIGURE 10.75 Drag-phase fluid pressure output from fluid location 2 for model 2.

effective stress had a peak value after 1 ms when the projectile approached the exit wall. The exit wall reached a peak velocity of 7 m/s and a peak stress of around 94 MPa prior to projectile impact. Figure 10.75 shows the drag-phase fluid pressure recorded in the middle of the fluid. A peak pressure of around 5 MPa was obtained as the projectile approached the fluid location at around 0.5 ms. The drag-phase pressure rise was gradual and occurred over a longer period of time as compared to the initial shock-phase pressure. As the projectile moved past the fluid center, the pressure record went to zero, indicating the formation of a cavity behind the projectile path. The cavitation phase of HRAM, which includes the oscillation and the subsequent collapse of the cavity, was not a part of this study because it would occur at a later time after the simulation ended. An interesting parameter that the numerical simulation provided for the drag-phase analysis was the cavity evolution when the projectile traversed the fluid toward the exit wall. The maximum cavity diameter measured from the fringe plot at 2 ms was approximately 60 mm. The bulging of the entry and exit walls can also be observed.

© 2016 by Taylor & Francis Group, LLC

11 Multiscale Analysis of Electromechanical System

11.1 Introduction One of the common engineering systems is an electromechanical system based on both electrical and mechanical principles. An electric car is one example, and a rail gun launcher is another example. This chapter introduces a multiphysics-based modeling technique for a rail gun launcher using the finite element method. The multiphysics modeling was conducted for a rail gun launcher to predict the exit velocity of the launch object, temperature distribution, and thermal contact stress distribution. For this modeling, electromagnetic field analysis, heat transfer analysis, thermal stress analysis, and dynamic analysis were conducted for a system consisting of two parallel rails and a moving armature. Especially, a contact theory was used to estimate the electric as well as thermal conductivities at the interface.

11.2 Principle of Operation of a Rail Gun Launcher The principle of operation of the simplest type of rail gun launchers is discussed. The rail gun launcher is a type of projectile weapon [1]. The basic structure of the rail gun launcher is shown in Figure 11.1. The simple rail gun launcher consists of two parallel conducting rails and an armature between the rails, which accelerates a projectile between the rails to a high speed in a short time using electromagnetic force. A large electric current flows to one of the two parallel conducting rails, travels through the conducting armature between the rails, and then goes back to the electric current source through the second rail. A projectile to be fired lies on the outer side of the armature and fits loosely between the rails. The electric current in the rails produces a magnetic field between the parallel rails. The magnetic fields are directed normal to the plane containing the two rails. The resultant magnetic field exerts a force on the armature due to the electric current that flows through it. This is called the Lorentz force [2]. The electromagnetic force points outward along the rails and pushes the projectile, accelerates it, and launches it at a very high speed. There are different types of rail guns to enhance the launch power. One is made using a solid armature, and another one is constructed of a plasma armature. In addition, there are series augmented rails or parallel augmented rails.

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Parallel rails (conductors)

Power supply unit

Armature (conductor)

Projectile

FIGURE 11.1 Basic structure of a rail gun launcher.

11.3 Previous Study of Rail Gun Launchers Rail gun launchers have been studied previously [3–7]. Because the electromagnetic field is the major player of rail gun launchers, most studies examined electromagnetic fields. Various formulations have been developed to solve the electromagnetic problems of moving conductors [8–15]. Most of them applied the finite element method; some others utilized the boundary element method [14] and the coupled finite and boundary element methods [11]. Furthermore, a parallel algorithm was also investigated [15]. Some researchers investigated coupled problems, mostly electromagnetic and thermal analyses together [3,10] because electric currents generate heat. Heat is an important aspect in the rail gun launcher. The contact interface condition between two conductors, such as an armature and a rail, was studied [16] because those conditions affect the electromagnetic as well as thermal fields significantly. The following sections present mathematical models for multiphysics analyses, a contact theory for electric and thermal conductivities at the contact interfaces, a description of analysis models, and example problems.

11.4 Mathematical Models for Multiphysics Analysis The overall schematic of the multiscale analysis of a rail gun launcher is given in Figure 11.2 and described next [16]. The whole analysis is a time-dependent transient problem. With the initial location of the armature between two rails, electromagnetic wave analysis is first conducted. From the electromagnetic analysis, the Lorentz force, which is the driving force of the armature and the projectile, is calculated. Then, Joule’s law is used to determine the heat generation, which is utilized for the transient heat transfer analysis. Once the temperature distribution is computed, thermal stress analysis is undertaken and the contact load is computed between the armature and the rails. Then, the acceleration of the armature is determined by applying Newton’s second law to the armature and

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Mathematical program of rail gun launcher

Is the armature inside the rail?

No

Stop time increment

Time increment loop

Yes Increment of time step Electromagnetic field analysis to compute Lorentz force Heat transfer analysis with electric heat generation Thermal stress analysis to compute contact and frictional forces Rigid dynamic analysis to determine acceleration, velocity, and new position FIGURE 11.2 Schematic of multiscale analysis of a rail gun launcher.

the projectile with Lorentz force and friction force between the armature and rails. The acceleration is integrated twice over time to calculate the new position of the armature. If the armature is still located inside the parallel rails, the whole analysis repeats itself. Otherwise, the program is terminated, and the exit velocity is determined. Electric and thermal conductivities at the interface between the armature and rails contribute significantly to the electromagnetic and temperature fields. As a result, it is necessary to estimate those properties accurately. To determine those interface properties, a contact theory is also considered. Each analysis is described in more detail subsequently.

11.5 Electromagnetic Theory An electromagnetic system with moving conductors obeys five basic integral laws [10]: 1. Ampere’s circuital law 2. Faraday’s induction law

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3. Law of source free magnetic flux 4. Gauss’s law 5. Law of charge conservation Only four of them are independent. They are described in the following: 



  d J c ⋅ n da + dt ∂Ω( t )

∫ H ⋅ d l = ∫∫

Γ (t )

  d E⋅d l = − dt Γ (t )

∫

 ∫∫

∂Ω( t )

 ∫∫

∂Ω( t )

∂Ω( t )

∂Ω( t )

  B ⋅ n da

  B ⋅ n da = 0

  G ⋅ n da =

  d J ⋅ n da + dt ∂Ω( t )

 ∫∫

∫∫

∫∫

  G ⋅ n da

∫∫∫

Ω( t )

∫∫∫

Ω( t )

(11.1)

(11.2)

(11.3)

ρe dv

ρe dv = 0

(11.4)

(11.5)

   where intensity, B denotes magnetic flux, E denotes electric intensity, H is the magnetic   G is electric flux, J c indicates conduction current density, ρe is the volume charge density, t is time, Γ(t) is the moving curve, ∂Ω(t) indicates the surface, and Ω(t) denotes the volume. Time-dependent configurations in integral Equations 11.1 through 11.5 imply that the conductor undergoes motion. Governing differential equations of an electromagnetic system with moving conductors can be deduced from the integral equations, and their forms depend on the chosen description of field variables. The field variables in the Lagrangian description are expressed as a function of time and reference positions of the particles. Therefore, the integrations can be performed over a conductor reference configuration that is fixed in space. Thus, convective terms involving velocity components drop out of the equations. Moreover, physical dimensions of electromagnetic systems in the applications are much shorter than the wavelength of electromagnetic waves, so the displacement current can be neglected. By virtue of the Gauss divergence theorem and Stokes theorem, quasi-static Maxwell’s equations in the Lagrangian form can be obtained from the integral equations and have the following forms:

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  ∇ × H = JΓ

(11.6)

  ∂B ∇×E= − ∂t

(11.7)

Multiscale Analysis of Electromechanical System

347

 ∇×B= 0

(11.8)

 ∇ × JΓ = 0

(11.9)

Elimination of the convective term greatly simplifies the numerical analysis as far as storage requirements and numerical stability are concerned. A further consequence of adopting the Lagrangian description is the position information on conductor boundaries available at all times during the motion. This is especially important in this analysis where accurate data are needed at all times on the locations of rails and projectile boundaries. The materials of interest are assumed to be isotropic and nonferromagnetic but with temperature-dependent electrical conductivity. The associated constitutive relations are expressed as follows:   B = µ0 H

(11.10)

  J = σ(T )E

(11.11)

where T is temperature, μ0 (= 4π × 10 –7) is the permeability of free space, and σ(T) is the temperature-dependent electrical conductivity. By expressing magnetic flux as the curl of magnetic vector potential and electrical intensity as the negative sum of the time derivative of magnetic vector potential and the gradient of electric scalar potential, a set of magnetic diffusion equations can be deduced from quasi-static Maxwell’s equations with constitutive relations as follows:   B=∇×A

(11.12)

  ∂A E=− − ∇Φ ∂t

(11.13)

   ∂A 1 σ +∇× ∇ × A + σ∇Φ = J s ∂t µ0

(11.14)

   ∂A ∇ ⋅  −σ − σ∇Φ = 0   ∂t

(11.15)

  where A is the magnetic vector potential, Φ is the electrical scalar potential, and J s is the impressed current density. For nonconductive regions, the diffusion Equations 11.14 and 11.15 can be reduced to one equation due to vanishing  electric conductivity and impressed current density. The Coulomb gauge condition ∇A = 0 is imposed to uniquely determine the magnetic vector  potential A. The electromagnetic and temperature fields are coupled because the electrical conductivity is temperature dependent and the Joule heating is generated due to the electrical

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Multiscale and Multiphysics Modeling

resistivity. To obtain accurate magnetic fields, especially in high-current devices, it is necessary to include the thermal effect. From Fourier’s law and energy balance, the thermal diffusion equation from the Lagrangian viewpoint is expressed as ∇ ⋅ ( k∇T ) + R = Cp

∂T ∂t

  J ⋅J R= σ(T )

(11.16)

(11.17)

where R is the heat source, k is the temperature-dependent thermal conductivity, and Cp is the temperature-dependent specific heat. Moreover, changes in the magnetic field are assumed to only weakly depend on changes in the instantaneous body configuration as a first approximation. Furthermore, the body is assumed to be rigid so that the effect of deformations of the body is neglected. The magnetic field is only affected by the rigid body motion of the conductor. The position and velocity of the conductor are updated throughout the entire analysis. The equations of motion are described in the following:  Mγ =

∫∫∫

Ω( t )

  J × B dv − Ff

(11.18)

 where M is the conductor mass, γ is the acceleration, and Ff is the frictional force, discussed further in the chapter. Three sets of equations—magnetic diffusion Equations 11.6 through 11.9, thermal diffusion Equations 11.16 and 11.17, and the equation of motion, Equation 11.18—derived previously with constitutive equations form the theoretical basis of the mathematical formulation modeling.

11.6 Thermal Stress Analysis From the electrical heat generation, thermal analysis is conducted using Equations 11.16 and 11.17. The heat transfer in the launcher is also affected by the contact interface condition. Therefore, the same contact theory used for electrical conductivity at the interface is also utilized for thermal conductivity at the interface. Using the temperature distribution in the launcher, thermal stress analysis is undertaken with the following set of equations. The stress equilibrium for two dimensions is expressed as ∂τ xx ∂τ xy + =0 ∂x ∂y

(11.19)

and ∂τ xy ∂x

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+

∂τ yy ∂y

=0

(11.20)

Multiscale Analysis of Electromechanical System

349

where τij is the stress tensor. Stresses and strains are related for an isotropic material using the generalized Hooke’s law as follows: ε xx =

1 (τ xx − ντ yy ) + αT Y

(11.21)

ε yy =

1 (τ yy − ντ xx ) + αT Y

(11.22)

1 τ xy 2G

(11.23)

ε xy =

where εij is the strain tensor, Y is the elastic modulus, G is the shear modulus, ν is the Poisson ratio, and α is the coefficient of thermal expansion. Finally, the strain-displacement relationship is written as ε xx =

∂u ∂x

(11.24)

ε yy =

∂v ∂y

(11.25)

∂u ∂v + ∂y ∂x

(11.26)

2ε xy =

where u and v are the displacements in the x and y directions, respectively. Equations 11.19 through 11.26 are solved using the finite element method with a known temperature field [17]. From the stress analysis, the contact force between the armature and the rails is computed. Then, using Coulomb’s frictional law, the frictional force at the interface is calculated. Eventually, the net force along the parallel rails is determined by subtracting the frictional force from the Lorentz force. Using the net force, rigid dynamic analysis is performed to find the acceleration of the armature by applying Newton’s second law by modifying Equation 11.18. In other words, the frictional force is subtracted on the right side of the equation. The initial time integration of the acceleration yields velocity, and the next time integration of the velocity results in the new position of the armature. The Euler integration scheme is utilized for time integration. As long as the armature is still in contact with the rails, the whole analysis starting from electromagnetic analysis to dynamic analysis is repeated.

11.7 Contact Theory As the armature and the rails are in contact with each other, electricity and heat pass through the contact surfaces. Therefore, both electric and thermal conductivities of the

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Multiscale and Multiphysics Modeling

contact interface play an important role in the electromagnetic and temperature fields, which also influences the exit velocity of the projectile. To estimate conductivities (or sensitivities) at the contact interface, a statistics-based contact theory [18] is considered. A nominally flat surface has a surface roughness (e.g., peaks and valleys) at the microscale. When two nominally flat surfaces meet each other, those peaks and valleys encounter one another as actual contacts. Therefore, the nominal contact interface consists of two parts: the actual contact area and the void area. The void area usually contains air, which has much less conductivity compared to metallic materials. Therefore, the overall conductivities of the interface depend on the amount of actual contact area. A nominally flat surface is considered to have a large enough nominal area so that individual contacts of asperity are dispersed and the forces acting on neighboring contact spots do not affect one another. The actual area of contact between two nominally flat surfaces is determined by the elastic or plastic deformation of their highest asperities [18]. This leads to the result that the actual area of contact is directly proportional to the contact load. Furthermore, the contact deformation depends on the topography of the surface. To further simplify the problem, a nominally flat surface is assumed to have a large number of asperities, which are spherical (or circular in a two-dimensional [2-D] case), at least near their peaks. In addition, it is assumed that all the asperity peaks have the same radius β, and that their heights vary randomly. The probability that a particular asperity has a height between z and z + dz above the given reference plane can be expressed as ϕ(z)dz. If two surfaces become close together until their reference planes are separated by a distance d, any set of asperities whose sum of their height was originally greater than d will contact one another. Thus, the probability of having contact at any given asperity, of height z, is expressed as ∞

prob( z > d) =

∫ φ(z) dz

(11.27)

d

If there are N asperities total in the contact surface, the expected number of contacts becomes ∞

∫

n = N φ( z) dz

(11.28)

d

From the Hertz contact solution, the actual contact area of two bodies with the same radius β is expressed as A=

π βw 2

(11.29)

The corresponding contact load is P=

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1

3

8 Ecβ 2 w 2 3

(11.30)

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Multiscale Analysis of Electromechanical System

where w is the compliance (the distance at which points outside the deforming zone move together during the deformation), which is equal to z – d. In addition, 1 1 − ν12 1 − ν22 = + Ec E1 E2

(11.31)

in which E1 and E2 are the elastic moduli of two bodies in contact, and ν1 and ν2 are their Poisson ratios, respectively. Then, the mean contact area is ∞

∫ πβ(z − d)φ(z) dz

(11.32)

d

and the expected total area of contact can be written as ∞

π Atot = Nβ ( z − d)φ( z) dz 2

∫

(11.33)

d

The expected total load is 1 ∞

3

8 Ptot = NEcβ 2 ( z − d) 2 φ( z) dz 3

∫

(11.34)

d

A single asperity contact has electrical conductivity of 2a/ρc, where ρc is the average of resistivity of two contact bodies. Furthermore, it is assumed that microcontacts are sufficiently separated so that the current flow through them can be also independent. Then, the electrical contact over the total area can be expressed as −1 c

G = Nρ β

1 ∞ 2

∫

1

( z − d) 2 φ( z) dz

(11.35)

d

The thermal conductivities can be computed from a similar expression. For convenience, normalized variables are introduced. For example, the surface density of asperities η is introduced such that N = ηS, in which S is the nominal contact area. The normalized separation distance h = d/s is also used, where s is the standard deviation of the asperity height distribution. Then, Equations 11.33 through 11.35 can be rewritten as Atot =

Ptot =

© 2016 by Taylor & Francis Group, LLC

π ηSβsF1 ( h) 2

(11.36)

1 3

8 ηSEcβ 2 s 2 F3 ( h) 3 2

(11.37)

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Multiscale and Multiphysics Modeling

1 1

G = 2 ηSρ−c 1β 2 s 2 F1 ( h)

(11.38)

2

where ∞

Fµ =

∫ (s − d) φ*(s)ds µ

(11.39)

h

Here, ϕ*(s) is the normalized height distribution, that is, the height distribution scaled to make its standard deviation unity [18]. For the present study, the Gaussian distribution of asperities is assumed.

11.8 Analysis Model A 2-D finite element analysis model of a rail gun launcher is shown in Figure 11.3. The figure shows two rails and an armature between them, and the armature’s shape was simplified as a rectangular shape. The rails were 0.5 m long and 9.5 mm wide; the armature had a height of 19 mm and a width of 10 mm. The armature was assumed located initially at 0.05 m from the left of the rails with zero initial velocity. An electric potential 6.5 kV was applied to the rails at the initial room temperature of 300 K. As materials, aluminum, copper, and steel were selected for the present study. Their material properties are available in Reference 19, so that they are omitted here. 0.05 0.04 0.03 0.02

Rails

Armature

0.01 0 −0.01 −0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

FIGURE 11.3 Finite element mesh of the armature and rails (a crude mesh is shown for visual clarity even though more refined meshes were used for actual computations).

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Multiscale Analysis of Electromechanical System

TABLE 11.1 Statistical Values for Contact Surface Roughness Surface Parameters

Surface Condition A

Surface Condition B

Surface Condition C

1.37 μm 13 μm

0.065 μm 0.24 mm

0.01 μm 0.5 mm

Standard deviation Mean radius of peaks

As far as the contact surface between the armature and rails is concerned, three different surface conditions were considered [18]. Their statistical values for the Gaussian distribution are provided in Table 11.1. Furthermore, the rails are assumed to be constrained at both outer edges from any movement and are insulated.

11.9 Example Problems The first model of the rail gun launcher was constructed using an aluminum armature and copper rails. The contact surface condition A in Table 11.1 was considered. Because the electric and thermal conductivities at the contact surface played an important role, those properties were examined first. Considering different normalized separation distances, we calculated electric conductivity, thermal conductivity, and the contact force at the armature/rail interface, which are listed in Table 11.2. Once we knew the normalized separation distance or the contact force, we could estimate both electric and thermal conductivities at the interface. After the interface properties were determined, the multiphysics-based computations  as outlined in Figure 11.2 were executed to determine the armature exit velocity, temperature distribution, and thermal stresses in the rail gun launcher. Table 11.3 shows the results of the rail gun constructed of an aluminum armature and copper rails. The results show that as the distance between the reference planes became smaller (i.e., larger contact load), the electric and thermal conductivities become larger, which eventually resulted in a higher exit velocity with a larger Lorentz force and higher temperature. Figures 11.4 and 11.5 plot the distance between the reference planes of the contact surfaces as a function of the contact load and the electric conductivity, respectively. The figures show that as the contact load increased, the relative distance between the reference planes decreased, as expected. Then, such a decrease in the relative distance increased electric conductivity. Both plots show nonlinear relationships. TABLE 11.2 Computed Values of Electric Conductivity, Thermal Conductivity, and Contact Load for Different Normalized Separation Distances Normalized Separation Distance

0.985

1.000

1.015

1.029

1.044

Electric conductivity (106 mho/m) Thermal conductivity (W/m-degrees kelvin) Contact load (N)

1.139 1.879 824

1.111 1.832 799

1.083 1.787 775

1.056 1.742 752

1.029 1.698 730

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TABLE 11.3 Computer Simulation Results for Copper Rails and Aluminum Armature for Different Distances between Reference Planes of Contact Surfaces Distance between Reference Planes (μm)

1.35

1.37

1.39

1.41

1.43

Exit velocity (m/s) Total launch time (ms) Lorentz force (kN) Average temperature of contact elements (degrees K)

4446 0.203 275 784

4332 0.212 262 769

4237 0.217 250 754

4132 0.222 238 745

4040 0.228 226 729

Distance between reference planes d (m)

2.4

×10−6 Contact load P

2.2 2 1.8 1.6 1.4 1.2 1 0.8 200

400

600

800 Load P (N)

1200

1000

1400

FIGURE 11.4 Plot of distance between the reference planes of two contact surfaces versus contact load.

Distance between reference planes d (m)

2.4

×10–6 Conductance G

2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.4

0.6

0.8 1 1.2 1.4 Conductance G [(Ohm–m)−1]

1.6

1.8 ×106

FIGURE 11.5 Plot of electric conductivity versus distance between the reference planes of two contact surfaces.

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Multiscale Analysis of Electromechanical System

Figure 11.6 compares the projectile velocities of three different armature materials (i.e., aluminum, mild steel, and polished steel) inside copper rails with the contact surface condition A in Table 11.1. Comparing the results, the aluminum armature yielded the highest projectile velocity. Even if the analysis program already included frictional effect, the friction coefficient was assumed to be zero for the present study. The next case switched the rail and armature materials. In other words, copper was used for the armature and aluminum was selected for rails. Then, this case was compared to the previous case before exchanging the materials, as shown in Figure 11.7. The aluminum armature/copper rail case resulted in a higher velocity than for the opposite combination. From comparing the values of Lorentz forces that were exerted on the armature, no significant change was observed between the two cases. However, the mass of the aluminum armature was smaller than that of copper because the volume of the armature was kept constant in the analysis. The lighter mass of aluminum resulted in a higher velocity according to Newton’s second law. However, heat generation and heat transfer were quite different between the two cases, which gave very different average temperatures at the contact interfaces, as shown in Figure 11.8. The average temperature at the interfaces was lower in the case of copper armature/aluminum rail than that in the case of the opposite material combination. On the other hand, the former case had a higher rate of temperature rise at the interface as the armature moved along the rail compared to the latter case. To further investigate this trend of temperature, the rail was extended to 1.0 m or 1.5 m long for the next study. The results are discussed next. First, the velocity was higher for the longer rail, as tabulated in Table 11.4. Figure 11.9 compares the velocity of the two cases with the rail length of 1.5 m. The increase of the velocity followed a parabolic orbit. Hence, there must be an upper limit at the length of the rail (also called a barrel) beyond which any further increase does not significantly affect the value of the exit velocities and makes the design impractical. Rail-copper

5000

Aluminum Mild steel Polished steel

4500 4000

Velocity (m/s)

3500 3000 2500 2000 1500 1000 500 0 0.05

0.1

0.15

0.2

0.25 0.3 0.35 0.4 Length of barrel (m)

0.45

0.5

0.55

FIGURE 11.6 Velocity as a function of the location in the copper rails for three different types of armature: aluminum, dotted curve; mild steel, dashed curve; and polished steel, rigid curve.

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Velocities

6000

Rail-Cu/proj-Al Rail-Al/proj-Cu

5000

Velocity (m/s)

4500 3000 2000 1000 0 0.05

0.15

0.1

0.2

0.45

0.3 0.25 0.35 0.4 Length of the rail (m)

0.5

0.55

FIGURE 11.7 Velocity as a function of the distance along the rail for two different cases of materials.

Average temperatures

800 700 Temperature (K)

600 500 400

Rail-Cu/proj-Al Rail-Al/proj-Cu

300 200 100 0 0.05

0.1

0.15

0.2

0.25 0.35 0.3 0.4 Length of the barrel (m)

0.45

0.5

0.55

FIGURE 11.8 Comparison of average temperatures at the interface between the rails and the armature for two different material combinations.

TABLE 11.4 Exit Velocities for Three Different Rail Lengths Rail Length

Copper Rail/Aluminum Armature

Aluminum Rail/Copper Armature

4100 m/s 6200 m/s 7900 m/s

2500 m/s 3500 m/s 4500 m/s

0.5 m 1.0 m 1.5 m

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Velocities

8000

Rail-Cu/proj-Al Rail-Al/proj-Cu

7000

Velocity (m/s)

6000 5000 4000 3000 2000 1000 0

0

0.5 1 Length of the barrel (m)

1.5

FIGURE 11.9 Velocity as a function of the displacement in the 1.5-m rail.

Calculation of the kinetic energy from the exit velocity is given in Table 11.5. The kinetic energy of the aluminum armature was lower than that of the copper one because of its lighter mass. The total mass of the launch object was the mass of the armature and the mass of the projectile. That means that the projectile mass can be determined based on either the desired exit velocity or the kinetic energy. Some combination of the two criteria may be also used for designing the weight of the projectile. Furthermore, of greater interest is what happened at the distribution of the temperature at the interfaces. The two temperature curves intersect as the rail length became about 0.8 m beyond which the aluminum rail with a copper armature had a higher temperature than the copper rail with an aluminum armature. The maximum values of average temperatures for the two cases are shown in Table 11.6. Two questions now arise. Why was the initial average temperature of the aluminum armature case higher than that of the copper armature case? In addition, why was the rate of increase of the former lower than that of the latter? Taking into consideration the specific design selected previously, the electric resistance of the case of the aluminum armature/copper rail was lower than that of the opposite case. This resulted in greater heat generation for the former case than the latter case. Therefore, the temperature at the interface was higher for the aluminum armature/copper rail case in the beginning. TABLE 11.5 Kinetic Energy for Three Different Rail Lengths Rail Length

Copper Rail/Aluminum Armature

Aluminum Rail/Copper Armature

82 kJ 188 kJ 304 kJ

100 kJ 198 kJ 327 kJ

0.5 m 1.0 m 1.5 m

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TABLE 11.6 Maximum Average Temperatures (degrees kelvin) at the Interfaces Rail Length

Copper Rail/Aluminum Armature

Aluminum Rail/Copper Armature

760 860 1010

660 920 1430

0.5 m 1m 1.5 m

As the armature moved down the rails with time, the heat generated from the electric current was conducted along the rails and dissipated. Copper has higher thermal conductivity than aluminum. This caused faster heat removal from the rail gun launcher for the copper rails. For this reason, the rate of increase of the temperature profile at the interfaces was lower for the aluminum armature/copper rail case.

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12 Multiphysics Analysis of Biomechanics

12.1 Introduction One of the examples for fluid-structure interaction (FSI) is blood flow in the heart and blood vessels of the human body. Blood is a viscous fluid, and the heart and blood vessels are generally viscoelastic solids. Pulsating motions in the heart and blood circulate blood in the body. This chapter discusses the interaction between blood and a blood vessel. In particular, the focus of the study is the aneurysm. An aneurysm is a focal dilation of a blood vessel, which may rupture, leading to death of the patient. There are two major aneurysms in the human body. One is a cerebral artery aneurysm (CAA), and the other is the abdominal aortic aneurysm (AAA). The AAA occurs in the infrarenal aorta and has a diameter greater than 3 cm; it can grow to 9 cm in length [1,2]. Numerical modeling and simulation was conducted to investigate what effects influence an aneurysm.

12.2 Review of Previous Work Most of the previous studies on aneurysms considered either already existing realistic aneurysms or idealized aneurysms with focuses on understanding and predicting ruptures of blood vessels. To predict ruptures, stresses in blood vessels must be determined. As a result, blood vessel stresses were calculated as a function of the vessel diameter [3,4], wall thickness [5], asymmetry [6], tortuosity [7], material property [8], calcification [9], and blood flow [10]. Blood vessel strength was measured by ex vivo studies [11,12]. Researchers considered different constitutive models for investigating inception and growth of aneurysms. Vena et al. presented an anisotropic model for early stages of the aneurysm to study growth and remodeling [13]. Schmid and his colleagues investigated the effects of differences in elastic properties, fiber orientations, and metabolic activities on aneurysm formation and rupture in a layer-specific structural artery model [14]. A few computational studies considered the early stages of cerebral aneurysms. Inception and growth of aneurysms in an idealized carotid artery were modeled in Reference 15. They examined the initiation of the aneurysm by decreasing the modulus of elasticity at a local region, where smooth muscle cell relaxation was assumed. A similar assumption was used in Reference 16 so that high wall shear stresses could cause wall weakening. The modulus of elasticity was decreased at the regions of high wall shear stresses in curved and straight idealized intracranial arterial geometries. Another study [17] used a 359 © 2016 by Taylor & Francis Group, LLC

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hyperelastic (HE) material model with a large strain formulation at the bifurcation where most saccular cerebral aneurysms exist; other regions of the vessel were modeled as linear elastic (LE) material. They also assumed that smooth muscle cell relaxation could cause aneurysm formation. The thickness of the aorta consists of three layers: intima, media, and adventitia [18]. However, most studies assumed a single layer with a uniform material property for simplification even though in vivo studies showed that each layer had a different contribution to the material property of the blood vessel [19–21]. This chapter presents the results of a numerical study to determine the effects of material properties of a three-dimensional (3-D), idealized, three-layer abdominal aorta on aneurysm initiation and fully developed aneurysm [22]. To this end, the numerical model considered three individual layers in the blood vessel. Both LE and HE material properties were considered for healthy and locally degenerated sections of the blood vessel. The FSI between the blood and vessel was also considered in the study.

12.3 Description of Numerical Models Idealized 3-D models were generated for a three-layer abdominal aorta for aneurysm initiation as well as for an already-developed aneurysm [22]. The first model was for aneurysm initiation; it had three concentric cylinders that were secured to one another. The actual length of the blood vessel under study was 12 cm. However, the numerical model was 24 cm long so that the effect of the end boundary conditions could be neglected for the flow characterization. The in vivo wall thickness of the infrarenal aorta is between 0.14 and 0.15 cm [21]. The ratio for the intima, media, and adventitia layers is 20:47:33. As a result, this study considered a constant wall thickness of 0.15 cm with thicknesses of 0.03, 0.075, and 0.045 cm, respectively, for the three layers. The lumen diameter was assumed to be 2 cm. The vessel wall was modeled using two different materials. One material model was nearly incompressible, isotropic, and LE. The elastic modulus was 1.2 MPa, and the Poisson ratio was 0.49 [23]. The other material model used a HE wall and the coefficients in Reference 24. It was a model of the Mooney-Rivlin type. The strain energy density function is given as W = C10 ( I i − 3) + C20 ( I1 − 3)2 +

( J − 1)2 d

(12.1)

where W is the strain energy density of the material, C10 and C20 are the material constants, I1 is the first deviatoric strain invariant, J is the ratio of the deformed elastic volume over the undeformed volume of materials, and the parameter d is the material incompressibility parameter. Some of the materials were related as follows: K=

2 µ = d 2(1 − 2 ν)

(12.2)

in which μ is the initial shear modulus, ν is the Poisson ratio, and K is the bulk modulus. The material properties for each layer used in the study are listed in Table 12.1.

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TABLE 12.1 Material Properties for Each Layer of a Healthy Blood Vessel

Intima Media Adventitia

Elastic Model

Hyperelastic Model

Thickness Ratio

E (MPa)

ν

C10 (MPa)

C20 (MPa)

1 3 2

0.522 1.566 1.044

0.490 0.490 0.490

0.034 0.101 0.067

0.363 1.090 0.727

d 1.190 0.397 0.595

The aneurysmal blood vessel section was assumed to have reduced material properties compared to a healthy blood vessel. The same reduction ratio was considered for both LE and HE materials. The wall density was 1120 kg/m3. The blood was assumed to have the characteristics of a Newtonian, laminar, and incompressible flow. The density of blood was 1050 kg/m3, and its dynamic viscosity was 0.0035 Pa s [25]. Young’s modulus ratio for intima/media/adventitia was assumed to be a ratio of 1:3:2 as shown in Table 12.1. The degenerated material properties were applied on the media layer by decreasing its modulus of elasticity or the coefficients of the strain energy function by 1/20 [26]. Four different sizes of degenerated regions were considered. The model called degeneration in region A had local degeneration consisting of two medial arcs (100 elements) located near the center of the longitudinal axis. The arcs were half circles (i.e., 180°). The other three models had degenerated circular rings. The second model, degeneration in region B, had one circular medial ring (400 elements); the third model, degeneration in region C, had two circular medial rings (800 elements); and the final model, called degeneration in region D, had three consecutive medial rings (1200 elements). The four cases are called DR case A, DR case B, DR case C, and DR case D, respectively, from this point. As boundary conditions for blood flow, the inlet had time-dependent, fully developed laminar flow velocity; the outlet had time-dependent pressure [27]. Peak systolic pressure occurred at t = 0.53 s (15,594.5 kg/ms2), and peak systolic flow was obtained at t = 0.45 s (0.437886 m/s). Figure 12.1 shows the pressure time history of two cycles. Because the initial condition of the blood flow was not known, it was assumed to be zero. To provide the proper initial condition, the numerical computation was conducted for two cycles such 20

Pressure (kPa)

15 10 1st cycle

5 0

0

0.5

2nd cycle

1

1.5 Time (s)

FIGURE 12.1 Pressures applied to the blood.

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that the solution after the first cycle could be naturally the initial condition for the second cycle. Then, the solution during the second cycle was used for comparison. In the plot of the result, the starting point of the second cycle was set to zero. For the solid domain, both the inlet and outlet of the domain were fixed for all degrees of freedom. To minimize the effect of the constrained blood vessel on the solution, the blood vessel was modeled much longer than the actual length of the vessel under examination. At the interface between fluid and solid, a no-slip boundary condition was applied.

12.4 Comparison of Models with and without FSI First, the result of the FSI model was compared to that of the model without FSI, which did not consider FSI between the blood vessel and the blood. Then, the results of FSI were further discussed [22]. Five different locations through the blood vessel thickness were selected. Location 1 was at the interface of the intima and blood, location 2 was at the interface of the media and intima, location 3 was at the middle of the media, location 4 was at the interface of the adventitia and media, and location 5 was at the upper side of the adventitia. Figure 12.2 compares the applied pressure on the lumen sac for the analysis without FSI (called noFSI) to the calculated pressure at the blood-intima interface for the FSI analysis. The calculated pressure waveform for the FSI analysis (dashed line) was at the middle of the blood vessel. The two pressure waveforms were similar to each other except for a phase shift between them. Figure 12.3 compares the von Mises stresses at the interfaces of the layers: the intimamedia interface and media-adventitia interface. The analysis results with FSI and without FSI were significantly different from each other. The noFSI model overestimated the peak von Mises stress by 52% at the intima-media interface and by 22% at the media-adventitia interface. In addition, the FSI result showed a gradual continuous stress variation through the thickness of the blood vessel, as shown in Figure 12.4; the noFSI result showed a sharp change at the interface of each layer through the vessel thickness, as shown in Figure 12.5. 20

Pressure (kPa)

15 10 Lumen Sac-noFSI

5

Interface FSI-2nd cycle 0

0

0.5

Time (s)

FIGURE 12.2 Comparison of pressures between noFSI and FSI cases.

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200

Intima/media noFSI Intima/media FSI Media/adventitia noFSI

von Mises stress (kPa)

150

Media/adventitia FSI

100

50

0

0

0.2

0.4

0.6 Time (s)

0.8

1

1.2

FIGURE 12.3 Comparison of von Mises stresses between noFSI and FSI results. 400 1st mesh

von Mises stress (kPa)

350

2nd mesh

300

3rd mesh

250 200 150 100 50 0

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.4 von Mises stress distribution through the thickness of the blood vessel with FSI.

The figures also show the mesh sensitivity studies. All the meshes resulted in the same solutions, so that the mesh size effect could be neglected. In single-layer models, the von Mises stresses between FSI and noFSI models were small. However, in the three-layer model, the two results were quite different in the von Mises stresses at the interface of the layers [22]. Because blood flow does exist in human arteries, FSI analysis is more realistic than noFSI analysis. Therefore, the blood flow should be taken into consideration in studying mechanical properties of the arteries.

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von Mises stress (kPa)

200

Mesh 1 Mesh 2

150

Mesh 3

100 50 0

0

0.0004

0.0008 Radial distance (m)

0.0012

0.0016

FIGURE 12.5 von Mises stress distribution through the thickness of the blood vessel without FSI.

12.5 Results of the Aneurysm Initiation Studies Four different models of aneurysm initiation were compared to the healthy abdominal blood vessel without degeneration in terms of stresses and strains at the initiation site of the three-layer abdominal aorta. The aneurysm initiation models may have had a combination of LE and HE materials in the healthy and degenerated sections of the blood vessel. The combinations were the LE-vessel/LE-degeneration model, which used the LE material for the blood vessel except for the degenerated area and the reduced LE material used for the degenerated area; the LE-vessel/HE-degeneration model, which used the LE material for the vessel except for the degenerated area and the reduced HE (i.e., Mooney-Rivlin) material for the degenerated area; the HE-vessel/LE-degeneration model, which utilized the HE material for the vessel except for the degenerated area and the reduced LE material for the degenerated area; and the HE-vessel/HE-degeneration model, which had the HE material used for the vessel except for the degenerated area and the reduced HE (i.e., Mooney-Rivlin) material used for the degenerated area. 12.5.1 Blood Vessel Wall Modeling Hoop strains are compared among different material models. Figure 12.6 shows the results of the blood vessel with the LE material for the healthy part and the reduced LE material for the degenerated part acting as initiation of aneurysm. As expected, the blood vessels with local degeneration had greater hoop strains compared to the healthy blood vessel. A larger degeneration area resulted in a larger hoop strain. The strain variation along the vessel thickness was linear, but the slope of the line increased as the degeneration zone became larger. A similar observation was made for the LE healthy vessel with the reduced HE degenerated zones, as seen in Figure 12.7. When the healthy vessel was modeled as the HE material while the degenerated zone was modeled as the elastic material, the effect of the degeneration size on the increase in the hoop strain was smaller compared to the previous cases. This can be shown when Figure 12.8 is compared to Figures 12.6 and 12.7. The increase in the hoop strain in the LE vessel was about 40% from one circular degenerated ring (DR case B) to two rings (DR case C) and about 20% from two circular degenerated rings (DR case C) to three degenerated rings

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Intact vessel LE DR case B LE DR case C LE DR case D

0.4 0.35

Hoop strain

0.3 0.25 0.2 0.15 0.1 0.05 0

0

0.0004

0.0008 Radial distance (m)

0.0012

0.0016

FIGURE 12.6 Comparison of hoop strains for LE-vessel/LE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings. 0.4

Intact vessel MR DR case B

0.35

MR DR case C

Hoop strain

0.3

MR DR case D

0.25 0.2 0.15 0.1 0.05 0

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.7 Comparison of hoop strains for LE-vessel/HE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

(DR case D). The HE-vessel/LE-degeneration model showed about a 15% increase in the hoop strain from DR case B to DR case C and less than a 5% increase between DR case C and DR case D. The hoop strains varied similarly at least qualitatively through the vessel thickness of different healthy and degenerated models. As a result, it was difficult to detect any possible initiation of aneurysm in terms of the hoop strain behavior. The von Mises stresses were compared among the healthy and degenerated models. Figure 12.9 plots the von Mises stresses for the blood vessels whose healthy parts were modeled using the LE material and the degenerated sections were modeled using the reduced LE material. The figure shows an interesting result. Aneurysm initiation modeled as a reduced LE modulus decreased the von Mises stresses in the media layer and increased them in the intima and adventitia layers. In other words, the media layer had a

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0.4

Intact vessel

0.35

LE DR case B LE DR case C

Hoop strain

0.3

LE DR case D

0.25 0.2 0.15 0.1 0.05 0

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.8 Comparison of hoop strains for HE-vessel/LE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings. 400 Intact vessel

von Mises stress (kPa)

350

LE DR case B LE DR case C LE DR case D

300 250 200 150 100 50 0

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.9 Comparison of von Mises stresses for LE-vessel/LE-degeneration models with different degeneration areas. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

higher von Mises stress than the intima and adventitia layers in the healthy aorta, but the initiation of an aneurysm, modeled as a loss in the medial layer, revised that behavior. The medial stress decreased, and the intima and adventitia carried more loads. The collagenrich adventitia had the highest stress values. This was consistent with the previous finding in Reference 28, which stated that fibroblasts and smooth muscle cells increased the synthesis of collagen due to the mechanical loading, which played a role in the growth and remodeling of the AAA. All different material models showed similar von Mises stress distributions resulting from aneurysm initiation. One minor observation is that the von Mises stress at the medial

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region increased as the area of the degeneration region increased for the models for the LE-vessel/HE-degeneration model, as seen in Figure 12.10. Hoop and longitudinal stresses were similar to von Mises stresses with a little change in magnitude. The longitudinal strain increased along the blood vessel thickness when an aneurysm initiated in both LE-vessel and HE-vessel models. However, the responses of the two material properties were different from each other. Figure 12.11 compares the longitudinal

von Mises stress (kPa)

400 350

Intact vessel

300

MR DR case B MR DR case C

250

MR DR case D

200 150 100 50 0

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.10 Comparison of von Mises stresses for LE-vessel/HE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings. 0.1

Intact vessel LE DR case A

0.08

Longitudinal strain

LE DR case B LE DR case C

0.06

LE DR case D 0.04 0.02 0 –0.02

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.11 Comparison of longitudinal strain for LE-vessel/LE-degeneration models with different degeneration sizes. DR case A: one medial arc; DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

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0.06 Intact vessel

Longitudinal strain

0.05

LE DR case B LE DR case C

0.04

LE DR case D

0.03 0.02 0.01 0

–0.01

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.12 Comparison of longitudinal strain for HE-vessel/LE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

strains for the healthy vessel made of the LE material and the degenerated parts made of the reduced LE material. Likewise, Figure 12.12 plots the longitudinal strains for the healthy vessel made of the HE material and the degenerated parts made of the reduced LE material. Figure 12.13 is for the blood vessel models made of HE-vessel/LE-degeneration material.

0.1

Intact vessel MR DR case A

Longitudinal strain

0.08

MR DR case B MR DR case C

0.06

MR DR case D

0.04 0.02 0 –0.02

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.13 Comparison of longitudinal strain for LE-vessel/HE-degeneration models with different degeneration sizes. DR case A: one medial arc; DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

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Regardless of what materials were used for the healthy and degenerated regions of the blood vessels, DR case C and DR case D showed similar longitudinal strain distributions, but they were different from DR case A and DR case B, both of which also had similar distributions. The medial longitudinal strain in the DR case B was much smaller than those of DR case C and DR case D for the HE-vessel/LE-degeneration model, whereas the magnitude of the medial strain in DR case B was close to the magnitudes in DR case C and DR case D in the LE-vessel/LE-degeneration model. On the other hand, for HE-degeneration models, the material property of the blood vessel also influenced the longitudinal strain distribution through the vessel thickness but not as much as in the case of LE-degeneration models. Furthermore, the distributions of longitudinal strains of DR case A and DR case B were not the same, unlike those of the hoop strain and von Mises stress. Radial stresses in the two models of blood vessels like LE-vessel and HE-vessel were also different from each other. As shown in Figure 12.14, the LE-vessel/LE-degeneration model showed sharp changes in the radial stress at the interfaces of the layers for DR case B. In DR case C, the change in radial stress at the interfaces was larger at the interface between media and adventitia than at the interface between intima and media. For the largest area of degeneration, like the DR case D model, the radial stress had smaller changes at the layer interfaces than those of less-degenerated areas. In the HE-vessel/ LE-degeneration model, as seen in Figure 12.15, the DR case B showed oscillations of the radial stress value through the vessel thickness. At the HE-vessel/HE-degeneration model with DR case B and DR case C, the radial stresses did not change at the interfaces of the layers, unlike the LE-vessel/HE-degeneration model. Figures 12.16 and 12.17 support the statement. The numerical results also indicated that the radial stresses were the largest in the adventitia for all models considered, and the adventitial radial stresses in the

40 Intact vessel

30

LE DR case B

Radial stress (kPa)

20

LE DR case C LE DR case D

10 0 –10 –20 –30 –40

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.14 Comparison of radial stresses for LE-vessel/LE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

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5

Radial stress (kPa)

0

–5

Intact vessel

–10

LE DR case B LE DR case C

–15

LE DR case D –20

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.15 Comparison of radial stresses for HE-vessel/LE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings. 40

Intact vessel

Radial stress (kPa)

30

MR DR case A MR DR case B

20

MR DR case C

10

MR DR case D

0

–10 –20 –30 –40

0

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

0.0016

Radial distance (m) FIGURE 12.16 Comparison of radial stresses for LE-vessel/HE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

HE-vessel/LE-degeneration and HE-vessel/HE-degeneration models were significantly larger than those in the intima and media layers. Figures 12.18 through 12.21 show all the plots for the radial strains. The radial strain distribution resulting from aneurysm initiation was different for the HE-vessel and LE-vessel models. The radial strains were almost constant in the LE-vessel/LE-degeneration

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5

Radial stress (kPa)

0 –5

–10 Intact vessel MR DR case B

–15 –20

MR DR case C 0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.17 Comparison of radial stresses for HE-vessel/HE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings.

0 –0.05 –0.1 Radial strain

–0.15 –0.2

–0.25 –0.3

Intact vessel LE DR case A LE DR case B LE DR case C LE DR case D

–0.35 –0.4 –0.45 –0.5

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.18 Comparison of radial strains for LE-vessel/LE-degeneration models with different degeneration sizes. DR case A: one medial arc; DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

model, while the magnitude depended on the degeneration size. On the other hand, the HE-vessel/LE-degeneration model showed the reduced radial strain in the medial layer. The LE-vessel/HE-degeneration and HE-vessel/HE-degeneration models also showed different distributions of the radial strains. The LE-vessel/HE-degeneration model had the increased radial strain in the media layer for DR case C and DR case D, while the HE-vessel/HE-degeneration model had reduced radial strains for DR case B and DR case C.

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0 –0.05

Radial strain

–0.1 –0.15 –0.2 –0.25

Intact vessel LE DR case B LE DR case C LE DR case D

–0.3 –0.35 –0.4

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.19 Comparison of radial strains for HE-vessel/LE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

0 –0.05 –0.1 Radial strain

–0.15 –0.2 –0.25 Intact vessel

–0.3

MR DR case A

–0.35

MR DR case B

–0.4

MR DR case C

–0.45 –0.5

MR DR case D 0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.20 Comparison of radial strains for LE-vessel/HE-degeneration models with different degeneration sizes. DR case A: one medial arc; DR case B: one medial ring; DR case C: two medial rings; DR case D: three medial rings.

The human abdominal aorta has nonlinear material characteristics, and it is believed that the Mooney-Rivlin-type material is more realistic than the LE-type material. When the initiation of an aneurysm was modeled as a reduction in the material properties, four different material models resulted in significantly different distributions of stresses and strains through the blood vessel. Therefore, it was considered proper that HE material modeling of the blood vessel should be selected in modeling aneurysm initiation.

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0 –0.05 –0.1 Radial strain

–0.15 –0.2 Intact vessel

–0.25

MR DR case B

–0.3

MR DR case C

–0.35 –0.4

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.21 Comparison of radial strains for HE-vessel/HE-degeneration models with different degeneration sizes. DR case B: one medial ring; DR case C: two medial rings.

12.5.2 Blood Viscosity Effect It has been proposed that blood viscosity increases with age and diabetes. A study in Reference 29 stated that the viscosity of blood increased with age in approximately 7%. The effect of blood viscosity on the initiation of aneurysm was studied using the LE-vessel/ LE-degeneration model with the DR case C model. The dynamic viscosity of blood was assumed to be 0.0038 Pa s. The numerical data suggested that the viscosity increase by 7% did not yield any noticeable change in the stress, strain, and deformation distribution through the blood vessel. A previous study showed the effect of the viscosity by changing the kinematic viscosity from 0.0027 Pa s, and 0.0097 Pa s made a small change in the blood vessel diameter from 52.407 to 52.408 mm and increased the peak wall shear stress of the aneurysmal wall [30]. However, the change in the blood viscosity was much greater in the previous study, while it was small in the present study. Therefore, an increase in blood viscosity by 7% is not believed to influence aneurysm formation at the initial stages in terms of the stress and strain perspectives.

12.6 Already-Developed Abdominal Aortic Aneurysm This section models the already-developed AAA. This numerical model was almost the same as the previous initiation model except that a locally bulged section was included at the center of the blood vessel. The maximum diameter of the bulged section was assumed to be 6 cm. The material modeling of the healthy abdominal aorta without any degeneration or aneurysm was compared to the already-developed aneurysm model in terms of stress-strain characteristics. The von Mises stresses were compared among different intact

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Multiscale and Multiphysics Modeling

von Mises stress (kPa)

and aneurysmal vessels. The von Mises stress was the highest in the media layer and the lowest in the intima layer for all the models, as shown in Figure 12.22. These results were consistent with the results from a previous study [20]. Although the magnitudes of stress were not the same due to differences in types of vessels, the von Mises stress distribution through layers was consistent with the present study. The range of stress from intima to adventitia was from 40 to 100 kPa without aneurysm, while the present study showed the stress ranged from 50 to 160 kPa. The material property of the vessel did not change the qualitative distribution of the von Mises stress in the intact or aneurysmal vessels as well as the magnitude of increase in von Mises stresses from intact to aneurysmal vessels. However, Figure 12.23 shows that an increase in hoop strain from intact to aneurysmal vessels was larger for the HE model than the LE model. Likewise, the HE vessels yielded larger radial displacements than for the LE vessels. The strain value was smaller for the aneurysmal vessel than the healthy vessel because the aneurysmal vessel was stiffer than the healthy aorta.

400

LE-intact V.

350

LE-aneurysmal V.

300

MR-intact V.

250

MR-aneurysmal V.

200 150 100 50 0

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.22 Comparison of von Mises stresses through the thickness of the intact and aneurysmal vessels using different material models. LE-intact V.

0.2

LE-aneurysmal V. MR-intact V.

Hoop strain

0.15

MR-aneurysmal V.

0.1 0.05 0

0

0.0004

0.0008

0.0012

0.0016

Radial distance (m) FIGURE 12.23 Comparison of hoop strains through the thickness of the intact and aneurysmal vessel.

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References

Chapter 1 1. Y. W. Kwon and H.-C. Bang. The finite element method using MATLAB. 2nd ed. Boca Raton, FL: CRC Press, 2000. 2. J. E. Akin. Finite element analysis for undergraduates. London: Academic Press, 1986. 3. O. C. Zienkiewicz and R. L. Taylor. The finite element method. 4th ed. London: McGraw-Hill, 1991. 4. D. A. Rapport. The art of molecular dynamics simulation. 2nd ed. Cambridge: Cambridge University Press, 2004. 5. J. M. Haile. Molecular dynamics simulation: Elementary methods. New York: Wiley, 1997. 6. S. Wolfram. Cellular automata and complexity. Reading, MA: Addison-Wesley, 1994. 7. D. A. Wolf-Gladrow. Lattice-gas cellular automata and lattice Boltzmann models: Introduction. Berlin: Springer-Verlag, 2000. 8. S. Succi. The lattice Boltzmann equation: For fluid dynamics and beyond. Oxford, UK: Clarendon Press, 2001. 9. Z. Guo and C. Shu. Lattice Boltzmann method and its applications in engineering. Singapore: World Scientific, 2013.

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Chapter 5 1. D. A. Rapport. The art of molecular dynamics simulation. 2nd ed. Cambridge: Cambridge University Press, 2004. 2. J. M. Haile. Molecular dynamics simulation: Elementary methods. New York: Wiley, 1997. 3. Y. W. Kwon and S. H. Jung. Atomic model and coupling with continuum model for static equilibrium problems. Computers and Structures, vol. 82, no. 23–26, 2004, pp. 1993–2000. 4. J. E. Lennard-Jones. The determination of molecular fields. I. From the variation of the viscosity of a gas with temperature. Proceedings of the Royal Society (London), vol. 106A, 1924, pp. 441–462. 5. J. E. Lennard-Jones. The determination of molecular fields. II. From the equation of state of a gas. Proceedings of the Royal Society (London), vol. 106A, 1924, pp. 463–477. 6. C. Hsieh and R. Thomson. Lattice theory of fracture and crack creep. Journal of Applied Physics, vol. 44, 1973, pp. 2051–2063. 7. G. C. Abell. Empirical chemical pseudopotential theory of molecular and metallic bonding. Physical Review B, vol. 31, 1984, pp. 6184–6196. 8. J. Tersoff. New empirical approach for the structure and energy of covalent systems. Physical Review B, vol. 37, 1987, pp. 6991–7000. 9. J. Tersoff and R. S. Ruoff. Structural properties of a carbon-nanotube crystal. Physical Review Letters, vol. 73, August 1994, pp. 676–679. 10. D. W. Brenner. Empirical potential for hydrocarbons for use in simulating the chemical vapor deposition of diamond films. Physical Review B, vol. 42, 1990, pp. 9458–9471. 11. W. G. Wilder, L. C. Venema, A. G. Rinzler, R. E. Smalley, and C. Dekker. Electronic structure of atomically resolved carbon nanotubes. Nature, vol. 391, 1998, pp. 59–62. 12. T. W. Odom, J.-L. Huang, P. Kim, and C. M. Lieber. Atomic structure and electronic properties of single walled carbon nanotubes. Nature, vol. 391, 1998, pp. 62–64. 13. S. T. Thornton and A. Rex. Modern physics for scientists and engineers. Orlando, FL: Sanders College, 2000. 14. S. O. Kasap. Principles of electronic materials and devices. New York: McGraw-Hill, 2002. 15. J. J. Oh. Determination of Young’s modulus of carbon nanotubes using MD simulation. MS thesis, Naval Postgraduate School, Monterey, CA, December 2003. 16. Y. W. Kwon, C. Manthena, J. J. Oh, and D. Srivastava. Vibrational characteristics of carbon nanotubes as nanomechanical resonators. Journal of Nanoscience and Nanotechnology, vol. 5, no. 5, May 2005, pp. 703–712. 17. M. S. Dresselhaus, G. Dresselhaus, and P. C. Eklund. Science of fullerenes and carbon nanotubes. New York: Academic Press, 1996. 18. C. Schönenberger, A. Bachtold, C. Strunk, J.-P. Salvetat, and L. Forró. Interference and interaction in multiwall carbon nanotubes. Applied Physics A, vol. 69, 1999, pp. 283–295. 19. A. Krishnan, E. Dujardin, T. W. Ebbesen, P. N. Yianilos, and M. M. J. Treacy. Young’s modulus of single-walled nanotubes. Physical Review B, vol. 58, 1998, pp. 14013–14019.

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Chapter 6 1. Y. W. Kwon. Discrete atomic and smeared continuum modeling for static analysis. Engineering Computations, vol. 20, no. 8, 2003, pp. 964–978. 2. Y. W. Kwon and S. H. Jung. Atomic model and coupling with continuum model for static equilibrium problems. Computers and Structures, vol. 82, no. 23–26, 2004, pp. 1993–2000.

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Chapter 7 1. Y. W. Kwon, D. H. Allen, and R. Talreja, eds. Multiscale modeling and simulation of composite materials and structures. New York: Springer, 2008. 2. Y. W. Kwon. Calculation of effective moduli of fibrous composites with micro-mechanical damage. Composite Structures, vol. 25, 1993, pp. 187–192. 3. Y. W. Kwon and J. M. Berner. Micromechanics model for damage and failure analyses of laminated fibrous composites. Engineering Fracture Mechanics, vol. 52, no. 2, 1995, pp. 231–242. 4. Y. W. Kwon and J. M. Berner. Matrix damage analysis of fibrous composites: Effects of thermal residual stresses and layer sequences. Computers and Structures, vol. 64, no. 1–4, 1997, pp. 375–382. 5. Y. W. Kwon and A. Altekin. Multi-level, micro-macro approach for analysis of woven fabric composites. Journal of Composite Materials, vol. 36, no. 8, pp. 1005–1022. 6. Y. W. Kwon and K. Roach. Unit-cell model of 2/2-twill woven fabric composites for multi-scale analysis. Computer Modeling in Engineering and Sciences, vol. 5, no. 1, 2004, pp. 63–72. 7. Y. W. Kwon and W. M. Cho. Multi-scale thermal stress analysis of woven fabric composite. Journal of Thermal Stresses, vol. 27, 2004, pp. 59–73. 8. Y. W. Kwon and L. E. Craugh. Progressive failure modeling in notched cross-ply fibrous composites. Applied Composite Materials, vol. 8, no. 1, January 2001, pp. 63–74. 9. Y. W. Kwon and C. Kim. Micromechanical model for thermal analysis of particulate and fibrous composites. Journal of Thermal Stresses, vol. 21, 1998, pp. 21–39. 10. Y. W. Kwon. Analysis of laminated and sandwich composite structures using solid-like shell elements. Applied Composite Materials, vol. 20, no. 4, 2013, pp. 355–373. 11. Y. W. Kwon and M. S. Park. Versatile micromechanics model for multiscale analysis of composite structures. Applied Composite Materials, vol. 20, no. 4, 2013, pp. 673–692. 12. M. S. Park and Y. W. Kwon. Elastoplastic micromechanics model for multiscale analysis of metal matrix composite structures. Computers and Structures, vol. 123, 2013, pp. 28–38.

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Chapter 10 1. J. N. Reddy and A. Miravite. Practical analysis of composite laminates. Boca Raton, FL: CRC Press, 1995. 2. Y. W. Kwon. Study of fluid effects on dynamics of composite structures. Journal of Pressure Vessel Technology, vol. 133, no. 3, 2011, 031301. 3. Y. W. Kwon, A. C. Owens, A. S. Kwon, and J. M. Didoszak. Experimental study of impact on composite plates with fluid-structure interaction. International Journal of Multiphysics, vol. 4, no. 3, 2010, pp. 259–271. 4. Y. W. Kwon, M. A. Violette, R. D. McCrillis, and J. M. Didoszak. Transient dynamic response and failure of sandwich composite structures under impact loading with fluid structure interaction. Applied Composite Materials, vol. 19, no. 6, 2012, pp. 921–940. 5. Y. W. Kwon and R. P. Conner. Low velocity impact on polymer composite plate in contact with water. International Journal of Multiphysics, vol. 6, no. 3, 2012, pp. 179–197. 6. L. E. Craugh and Y. W. Kwon. Coupled finite element and cellular automata methods for analysis of composite structures with fluid-structure interaction. Composite Structures, vol. 102, August 2013, pp. 124–137. 7. D. J. Inman. Engineering vibration. Englewood Cliffs, NJ: Prentice Hall, 1994. 8. F. S. Tse, I. E. Morse, and R. T. Hinkle. Mechanical vibrations, theory and applications. 2nd ed. Boston: Allyn and Bacon, 1978. 9. Y. W. Kwon, E. M. Priest, and J. H. Gordis. Investigation of vibrational characteristics of composite beams with fluid-structure interaction. Composite Structures, vol. 105, November 2013, pp. 269–278. 10. Y. W. Kwon. Dynamic responses of composite structures in contact with water while subjected to harmonic loads. Applied Composite Materials, vol. 21, 2014, pp. 227–245. 11. Y. W. Kwon and S. C. Knutton. Computational study of effect of transient fluid force on composite structures submerged in water. International Journal of Multiphysics, vol. 8, no. 4, 2014, pp. 367–395. 12. X. Sun. Numerical and experimental investigation on tidal and current energy extraction. PhD dissertation, University of Edinburgh, 2008. 13. K. D. Kimsey. Numerical simulation of hydrodynamic ram. ARBRL-TR-02217. Adelphi, MD: US Army Ballistic Research Laboratory, February 1980. 14. D. Varas, J. López-Puente, and R. Zaera. Experimental analysis of fluid-filled aluminum tubes subjected to high-velocity impact. International Journal of Impact Engineering, vol. 36, 2009, pp. 81–91. 15. K. Yang, Y. W. Kwon, C. Adams, and D. Liu. Modeling and simulation of hydrodynamic ram for aircraft survivability. Aircraft Survivability. (in print).

Chapter 11 1. H. D. Fair. Electromagnetic propulsion: A new initiative. IEEE Transactions on Magnetics, vol. Mag-18, no. 1, January 1982, pp. 4–6. 2. D. Halliday, R. Resnick, and J. Walker. Fundamentals of physics. 9th ed. New York: Wiley, 2010.

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Chapter 12 1. J. D. Humphrey and C. A. Taylor. Intracranial and abdominal aortic aneurysms: Similarities, differences, and need for a new class of computational models. Annual Review of Biomedical Engineering, vol. 10, 2008, pp. 221–246.

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Index A AAA, see Abdominal aortic aneurysm Abdominal aortic aneurysm (AAA), 359 already-developed, 373–374 growth and remodeling of, 366 Abell-Tersoff-Brenner (A-T-B) potential, 150–154 Acoustic wave equation, 47–50 ambient fluid density, 47 bulk modulus, 48 condensation, 47 fluid-structure interaction, 48 Newton’s 2nd law, 47 triangular element, area of, 49 Aneurysm abdominal aortic, 359, 373 cerebral artery, 359 initiation studies, results of, 364–373 investigation of, 1 AoS, see Array of structures Array of structures (AoS), 98 Artificial recoloring process, 68 Asymptotic homogenization theory, 256 A-T-B potential, see Abell-Tersoff-Brenner potential Atomic model, stiffness matrix of, 195 Axial bar and beam, 13–24 axial member, 13–17 backward difference technique, 23 beam, 17–21 boundary condition, 16 Crank-Nicholson solution technique, 22 deflection and slope, 18 diagonal mass matrices, 23 elemental stiffness matrix, 20 explicit solver, 23 finite difference method, 22 free-body diagram, 13 Galerkin finite element formulation, 14 governing equation, 18 Hooke’s law, 14 load intensity function, 21 moment equilibrium, 17 Newmark solution method, 24 nodal variable, 15, 19 shape functions, 18, 19 solution accuracy, 24

solution techniques, 21–24 unknown reaction force, 21 weak formulation, 18 B Bamboo shape nanotube (BSNT), 156 Bamboo-structured single-wall carbon nanotube (BSCNT), 160 models, Young’s modulus ratio concept for, 165 natural frequencies and mode shapes for, 172–173 Benchmark values (LBM scaling), 82, 88 Biomaterials, multiscale analysis of, 267–295 further adjustments in models, 292–295 adjustment results, 293–294 bone loss results, 295 hydroxyapatite crystal, dimensions of, 293 minerals, 292 modified hierarchy, 292–293 optimal adjustment results, 294–295 Reuss model, 292, 294 Voigt model, 292, 294 macroscale model, 281–292 cancellous bone, 288–289 cancellous bone model, 290–291 cancellous bone results, 291 coordinate transformation technique, 285 cortical bone, 284–285 cortical bone model, 285–286 cortical bone model results, 286–288 finite element method, 288 Haversian systems, 284, 285 lamellar bone, 281–282, 284 lamellar model, 282–283 lamellar model results, 283–284 misnomer, 285 osteons, 284 preferential orientation, 286 primary bone, 284 result of trabecular bone, 289–290 secondary bone, 284 smooth orientation, 286 soap bubble formation, modeling of, 290 stiffness matrix, 286 tetrakaidecahedral unit cell, 290

393 © 2016 by Taylor & Francis Group, LLC

394

trabeculae, 288 trabecular bone model, 289 microscale model, 272–281 amino acids, 272 bone fiber, 277–278 collagen network, 272 extrafibrillar mineralization, 278 fiber results, 280–281 fibril bundles, 277 fibrillogenesis, 272 fibril results, 276–277 gap, 273 hydroxyapatite crystals, growth of, 273 linear fibril subunit, 275 micromechanical fiber model, 278–279 micromechanics fibril model, 274–276 overlap, 273 tensile data, 281 three-dimensional fibril structure, 273–274 twisting crystalline structure, 274 twisting fibril subunit, 275 two-dimensional fibril structure, 272–273 nanoscale model, 267–271 bone mineral, 267 helical spring, 269–270 helical twist, 271 hydroxyapatite, 267–269 nanoindentation testing on, 269 results, 270–271 tropocollagen, 269 Biomechanics, multiphysics analysis of, 359–374 abdominal aortic aneurysm, 359 already-developed abdominal aortic aneurysm, 373–374 hoop strains, 374 strain value, 374 von Mises stress, 374 aneurysm initiation studies, results of, 364–373 AAA, growth and remodeling of, 366 blood vessel wall modeling, 364–373 blood viscosity effect, 373 Mooney-Rivlin-type material, 372 radial stresses, 369 von Mises stresses, 365 cerebral artery aneurysm, 359 comparison of models with and without FSI, 362–364 blood-intima interface, calculated pressure at, 362 single-layer models, 363 von Mises stresses, 363 hyperelastic material, 360

© 2016 by Taylor & Francis Group, LLC

Index

linear elastic material, 360 numerical models, description of, 360–362 aneurysm initiation, 360 cases, 361 degeneration model, 361 Mooney-Rivlin type model, 360 peak systolic pressure, 361 Poisson ratio, 360 vessel wall, modeling of, 360 review of previous work, 359–360 cerebral aneurysms, 359 modulus of elasticity (aneurysm), 359 numerical study, 360 Blood circulation system, 1 Boltzmann constant, 159 Bone fiber, 277–278 Boron/epoxy plain weave composites, 241 Brinkman equation, 55 BSCNT, see Bamboo-structured single-wall carbon nanotube, 160 BSNT, see Bamboo shape nanotube Burgers equation, 55 C CA, see Cellular automata CAA, see Cerebral artery aneurysm Cancellous bone, 288–289 Carbon nanotubes (CNTs), 145, 151 bond energy for, 151 elastic modulus, 156–168 bamboo shape nanotube, 156 basic structures of carbon nanotubes, 157–158 Boltzmann constant, 159 chiral nanotubes, 158 comparative results of equilibrium and nonequilibrium simulations, 165–168 elastic moduli, difference in, 165 elastic modulus of CNTs under equilibrium, 162–165 equilibrium and vibration motion of CNTs, 159–162 freestanding thermal vibration method, 159 Maxwell-Boltzman velocity distribution, 163 multiwall carbon nanotubes, 157 simulation time step, 158 single-wall carbon nanotube, 157 thermal vibration method, 161 time evolution method, 163 Young’s modulus ratio concept, 165

Index

equilibrated structure, 151 external tensile load, 153 multiwall, 157 physical properties of, 157 single-wall, 157 vibrational mode shapes, 168–173 comparison of radial breathing modes of armchair SWCNTs, 168 discrete Fourier transform approaches, 168 hollow beam, bending of, 171 natural frequencies and mode shapes for BSCNTs, 172–173 natural frequencies and mode shapes for MWCNTs, 171–172 natural frequencies and mode shapes for SWCNTs, 168–170 radial breathing modes, 172 Tersoff-Brenner potential, 171 Young’s modulus of, 152, 159 Cauchy-Born hypothesis, 256 Cellular automata (CA), 101–144, 203 Boolean quantities, 103 boundary cells, 102 boundary conditions, 116–117 D’Alembert solution, 116 waste of computational time, 116 wave propagation, 117 configuration, 101 convergence, 118–119 point-by-point error, 118 semi-infinite fluid-filled space, 118 wave propagation, 118 coupling technique, 203–205 discretization and model fidelity, 117–118 critical discretization versus initial perturbation width, 118 error norms, 117 Gaussian wave, 117 initial perturbations of varying widths, 118 nonreflecting boundary conditions, 117 “game of life,” 103 lattice Boltzmann method, 103 modeling moving objects using, 103–108 evolution rule, 105 Hamiltonian formalism, 107 internal particles, rule for time evolution of, 104 lattice coordinates, 105 left-end particles, 105–106 particle position, 104 right-end particles, 106–107

© 2016 by Taylor & Francis Group, LLC

395

spring constant, 105–107 string momentum, 107 string velocity, 107 velocity, mass, momentum, and energy, 107–108 Moore neighborhood, 101 multigrid technique, 133–144 example problems, 137–144 global-local technique, 134–137 local domain grids, 135 memory allocation issues, 134 multigrid method, 137 nodal points, 143 revisions, 137 uniform grid, 139, 140 wave propagation, onset time for, 137 physical examples using, 108–115 constrained particles, 111 D’Alembert solution, 115 finite element method solutions, 114 infinite string, 115 longitudinal vibration of long uniform rod, 109–110 Newton’s second law, 112 one-dimensional wave equation, 114–115 physical properties of string, 113 string with force applied on the middle, 112–114 string plucked at midpoint, 111–112 time-scaling factor, 110 time steps, 109 transverse vibration of string, 111–114 prevention of instability, 103 rule, 101 standard lattice Boltzmann method and, 55 strengths and weaknesses, 103 synchronous dynamics, 102 three-dimensional wave problem, 122–126 analytical solution, 122 Cartesian radius, 123 Simpson’s rule, 123 true point source, 123 two-dimensional wave problem, 119–122 cellular automata in two dimensions, 119–120 corner particle, rule for, 120 example of membrane vibration, 120–122 underwater acoustics, application to, 126–133 air-to-waterline boundary, 129 confined water channel, wave propagation in, 133 propagation losses, 127, 128 reflected wave, 132

396

root mean square pressure, 127 transmission loss development, 127–129 various boundary conditions, 129 wave propagation, 126 across flat-bottom ocean floor, 129–131 over curved hill, 131–132 over sloping bottom, 132–133 von Neumann neighborhood, 101 Central processing units (CPUs), 96, 98 Cerebral artery aneurysm (CAA), 359 CG technique, see Continuous Galerkin technique Chiral nanotubes, 158 Classical LBM (CLBM), 65 CLBM, see Classical LBM CNTs, see Carbon nanotubes Coefficients of thermal expansions (CTEs), 229 Color fluid model, 68–69 Composite structures, multiphysics analysis of, 297–342 fatigue loading with FSI, 318–330 beam system, 320 example problems, 320–330 problem description, 318–320 spring-to-mass ratio, 320 fluid-structure interaction modeling, 297–298 finite element method, 298 fluid flow, 297 lattice Boltzmann method, 297, 298 Navier-Stokes equation, 297 nodal variables, 298 wave equation, 297 hydrodynamic loading, 330–338 base model, 331 boundary point, 332 fluid pressure, 332 model description, 330–331 numerical results with constant acceleration, 331–333 numerical results with different material properties, 336–338 numerical results with free surface, 335–336 numerical results with nonlinear acceleration, 333–335 hydrodynamic ram, 338–342 description, 338 drag-phase fluid pressure, 342 entry wall x displacement, 339 hemispherical shock wave, 341 numerical models, 338–339

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Index

numerical results, 339–342 von Mises stress, 340 low-velocity impact with FSI, 298–306 comparison to experimental data, 304–306 continuous Galerkin technique, 299 discontinuous Galerkin technique, 299 single-layer plate model, 299–300 three-layer plate model, 302–304 two-layer plate model, 300–301 vibration with FSI, 306–318 example problems, 308–318 fast Fourier transform, 307 finite element eigenvalue analysis, 307 numerical modal analysis, 307 velocimeters, 307 verification study, 307–308 Composite structures, multiscale analysis of, 207–254 binding material, 207 common fibers, 207 coordinate transformation, 222–224 global and local coordinate axes, 223 stiffness transformation, 222, 223 stress and strain transformations, 222 transformation matrices, 223 elastoplastic analysis of composite materials, 243–246 Hollomon equation, 243 Kronecker delta, 243 Ludwik equation, 243 Newton’s iteration method, 244 return mapping algorithm, 243 state matrix, 245 subcell stress increment, 243 elastoplastic analysis of composite materials, examples of, 246–254 fibrous composite plate with preexisting crack, 251–254 fibrous metal matrix composite, 250–251 laminated composite plate, 254 microflaws, 251 particulate composite made of SiC and Cu alloy, 248 particulate metal matrix composite made of SiC and aluminum, 246–248 residual stress analysis, 253 strengthening mechanisms, 247 thermal residual stresses for whisker composite, 249–250 von Mises yield criterion, 251 exchange of critical information, 207

Index

fabric model, 214–222 bidirectional passage, 222 functions, 214 plain weave composite, 215–219 shear components, 218 stress/strain components, 215 subcells, strain compatibility conditions among, 216 twill composite, 219–222 unit cell, 215, 220 weaving information, 214 fabric model, examples of, 239–242 boron/epoxy plain weave composites, 241 coefficients of thermal expansions, 240 finite element method, 239 glass/epoxy plain weave composites, 241 thermal stresses, comparison of, 242 fiber model, 209–214 constitutive equations, 212 elastoplastic deformation, 211 incremental stress and strain tensors, 210 material stiffness matrix, 211 nonelastic problems, 210 shear stresses, 210–211 subcells, 209 unit cell level, thermal strain at, 214 fiber model, examples of, 225–239 coefficients of thermal expansions, 229 cubic inclusion, variational asymptotic method for, 226 delamination zone, 240 fibrous composite made of boron fibers and aluminum matrix, 226–227 fibrous composite made of graphite fiber and epoxy matrix, 227 fibrous composite plate with a hole, 234–235 fibrous composite with statistical consideration, 237–239 fibrous composite for strength, 239 filament wound cylinder, 233 Gaussian statistical distribution, 237 glass fiber/epoxy matrix fibrous composite under tensile load, 231–232 hierarchical composite, 229–230 particulate composite made of Al2O3 particle/aluminum matrix, 225–226 particulate composite made of SiC particle/aluminum matrix, 226 particulate composite plate with a hole, 236 Poisson ratio, 237

© 2016 by Taylor & Francis Group, LLC

397

porous material made of Al2O3, 228 rule of mixtures, 231 short-fiber composite made of SiC whiskers and aluminum matrix, 227–228 spherical inclusion, theory for, 226 thermal stress of glass fiber/epoxy matrix fibrous composite, 229 ultrafine grain aluminum, 229 lamination model, 224–225 bending stiffness properties, 224 constitutive equation, 224 numerical integration points, 224 postloop, 225 preloop, 225 matrix materials, 207 multiscale modeling of composite structures, 207–209 fabric model, 208 fiber model, 208 fiber splitting, 208 interlayer delamination, 208 lamination model, 208 matrix cracking, 208 stiffness loop, 208 strength loop, 208 transverse matrix cracking, 208 reinforcing material, 207 Constrained particles, 111 Continuous Galerkin (CG) technique, 299 Copper, behavior under conditions of cyclic loading, 183 Cortical bone, 284–285 Coulomb interaction among charged particles, 69 Coulomb’s frictional law, 349 Coupling techniques, 187–205 example problems of coupled finite element and atomic models, 188–194 atomic arrangement, kinds of, 191 atomic array embedded in finite element mesh with a crack, 191–192 atomic behavior at crack tip, 193–194 hexagonal array of atoms with dislocation, 189–190 iteration procedure, 189 nodal displacements, 192 shear loading, 190 example problems for smeared atomic model, 196–200 axial mode vibration shape, 199 bending mode vibration shape, 199 continuum model, mass matrix of, 197

398

interatomic potential, 198 natural frequencies, 197 shear mode vibration shape, 199 stiffness matrix, 196 vibration of atoms in one-dimensional arrangement, 196–198 vibration of atoms in two-dimensional arrangement, 198–200 finite element and atomic models, 187–188 continuum domain, 187 exchange of information, 187 matrix equation, 187 nodal displacement, 188 overlapped interface domain, 187 finite element and cellular automata models, 203–205 coupling acoustic domains, 203–204 coupling the acoustic and structural domains, 204–205 nodal variables, 204 time step size, 204 wave propagation, 203 finite element and lattice Boltzmann models, 200–203 coupling acoustic domains, 200–201 coupling fluid and structural domains, 201–203 element shape functions, 202 fluid stress tensor, 202 fluid-structure interaction, 201 Kronecker delta, 202 LBM grid points, 200 two-way coupling problem, 201 velocity compatibility condition, 202 homogenization of atomic model into continuum model, 194–196 atomic model, stiffness matrix of, 195 discrete atoms, 195 finite element stiffness matrix, 195 matrix equation, 195 nodal displacements, 195 shape functions, 195 lattice Boltzmann method, 200 wave propagation problem, 200 C programming language, 96 CPUs, see Central processing units CTEs, see Coefficients of thermal expansions D DFT, see Discrete Fourier transform DG technique, see Discontinuous Galerkin technique

© 2016 by Taylor & Francis Group, LLC

Index

Diagonal mass matrices, 23 Dirac delta functions, 63 Discontinuous Galerkin (DG) technique, 299 Discrete Fourier transform (DFT), 168 D2Q9 lattice, 56, 60 E Effective carbon bond length, 151 EFLBM, see Element-free-based lattice Boltzmann method EFM, see Extrafibrillar mineralization Elasticity (solid element), theory of, 28–30 constitutive equations, 29 force equilibrium, 29 isotropic material, 29 kinematic equations, 30 normal stresses, 28 shear stresses, 28 Electromechanical system, multiscale analysis of, 343–358 analysis model, 352–353 armature, 352 surface conditions, 355 contact theory, 349–352 asperities, 350, 351 expected total load, 351 height distribution, 352 Hertz contact solution, 350 interface parts, 350 electromagnetic theory, 345–348 Coulomb gauge condition, 347 equations, 348 field variables in Lagrangian description, 346 Fourier’s law, 348 integral laws, 345–346 magnetic field, changes in, 348 Stokes theorem, 346 example problems, 353–358 armature materials, 355 barrel, 355 electric resistance, 357 kinetic energy, 357 Lorentz forces, 355 rail gun, 353 Lorentz force, 343 mathematical models for multiphysics analysis, 344–345 electric and thermal conductivities, 245 Joule’s law, 344 Lorentz force, 344 Newton’s second law, 344

Index

rail gun launchers previous study of, 344 principle of operation of, 343 thermal stress analysis, 348–349 contact theory, 348 Coulomb’s frictional law, 349 Euler integration scheme, 349 Hooke’s law, 349 Lorentz force, 349 Element-free-based lattice Boltzmann method (EFLBM), 64–65 coefficient vector, 64 interpolation function vector, 64, 65 spline function, 65 weighting function, 64 Embedded atom method, 183 potential, 154 Euler integration scheme, 349 Explicit solver, 23 Extrafibrillar mineralization (EFM), 278, 280 F Fabric model, 208, 214–222, 239–242 bidirectional passage, 222 examples of, 239–242 functions, 214 plain weave composite, 215–219 shear components, 218 stress/strain components, 215 subcells, strain compatibility conditions among, 216 twill composite, 219–222 unit cell, 215, 220 weaving information, 214 Fast Fourier transform (FFT), 307 Fatigue of metals, 183–186 FELBM, see Finite element–based lattice Boltzmann method FEM, see Finite element method FFT, see Fast Fourier transform Fiber model, 208, 209–214 constitutive equations, 212 elastoplastic deformation, 211 examples of, 225–239 incremental stress and strain tensors, 210 material stiffness matrix, 211 nonelastic problems, 210 shear stresses, 210–211 subcells, 209 unit cell level, thermal strain at, 214 Fiber splitting, 208

© 2016 by Taylor & Francis Group, LLC

399

Fibril bundles, 277 Fibrillogenesis, 272, 274 Fibrous metal matrix composites (FMMCs), 250–251 Finite element–based lattice Boltzmann method (FELBM), 61–63 Crank-Nicholson time integration technique, 63 Dirac delta functions, 63 finite element mesh, 62 Galerkin method, 63 nodal variables, 62 single-relaxation time operator, 62 unknown variable, 62 weighted residual equation, 62 weighting function, 63 Finite element method (FEM), 2, 5–53 acoustic wave equation, 47–50 ambient fluid density, 47 bulk modulus, 48 condensation, 47 fluid-structure interaction, 48 Newton’s 2nd law, 47 triangular element, area of, 49 atomic models and, 187–188 axial bar and beam, 13–24 axial member, 13–17 backward difference technique, 23 beam, 17–21 boundary condition, 16 Crank-Nicholson solution technique, 22 deflection and slope, 18 diagonal mass matrices, 23 elemental stiffness matrix, 20 explicit solver, 23 finite difference method, 22 free-body diagram, 13 Galerkin finite element formulation, 14 governing equation, 18 Hooke’s law, 14 load intensity function, 21 moment equilibrium, 17 Newmark solution method, 24 nodal variable, 15, 19 shape functions, 18, 19 solution accuracy, 24 solution techniques, 21–24 unknown reaction force, 21 weak formulation, 18 cellular automata models and, 203–205 comparison of CA and, 114 cortical bone model, 288 Crank-Nicholson solution technique, 22

400

fabric model, 239 fluid-structure interaction modeling, 298 frame, 26–28 diagonal mass matrix, 28 element stiffness matrix, 26 global coordinate system, element matrix in, 27 transformation matrix, 26 Galerkin finite element formulation, 9–13 axial member, 14 linear shape functions, 10 nodal coordinates, 12 nodal variable, 9 shape functions, 9 strong formulation, 11 test function, 11 weak formulation, 11 interaction of structure with acoustic domain, 50–53 boundary condition, 52 final matrix expression, 50 fluid-solid interface, 51 fluid speed of sound, 51 fluid-structure interaction interface, 53 harmonic solutions for the rod, assumption of, 51 incompressible fluid, 51 nodal variables, 53 one-dimensional case, 50–51 submatrix, 53 two-dimensional case, 52–53 vibration theory, 51 isoparametric formulation, 36–41 chain rule, 39 element stiffness matrix, 40 Gauss-Legendre quadrature rule, 40 Jacobian matrix, 39 natural coordinate domain, 36 nodal variable, 38 physical coordinate domain, 36 shape functions, 36, 37 metallic materials, multiscale analysis of, 257 method of weighted residual, 5–9 boundary conditions, 5 cubic function, 7 Galerkin method, 7 polynomial function, 6 residual, 6 test function, 7 trial function, 6, 8 weighted residual, 7

© 2016 by Taylor & Francis Group, LLC

Index

plate and shell structures, 41–47 advantage, 42 bending energy, 46 constitutive equation, 41, 43 elastic modulus, 44 element stiffness matrix, 47 in-plane displacements, 42 natural coordinate system, 43 Poisson’s ratio, 44 shell element, 42 transverse displacement, 42 transverse normal strain, 45 transverse shear strains, 43 rail gun launcher, 343, 344 solid element, 28–36 constitutive equations, 29 diagonal mass matrix, 34 discretized problem domain, 31 elemental stiffness matrix, 34 equations of motion, 30 finite element formulation, 30–36 force equilibrium, 29 isotropic material, 29 kinematic equations, 30, 33 nodal displacements, 31 nodal variable, 31 normal stresses, 28 shape functions for triangular element, 31 shear stresses, 28 test functions, 30 theory of elasticity, 28–30 traction force vector, 35 triangular element, 32 structural domain, 204 transverse vibration of string, 114 truss, 24–26 coordinate systems, 24, 25 displacements in coordinates, 24 element, 25 global coordinate system, 24 local coordinate system, 24 stiffness matrix, 25 strain energy, 25 structure, 25 Fluid-structure interaction (FSI), 201 fatigue loading with, 318–330 beam system, 320 example problems, 320–330 problem description, 318–320 spring-to-mass ratio, 320 hybrid lattice Boltzmann formulation, 68 interface, 53

401

Index

low-velocity impact with, 298–306 comparison to experimental data, 304–306 continuous Galerkin technique, 299 discontinuous Galerkin technique, 299 single-layer plate model, 299–300 three-layer plate model, 302–304 two-layer plate model, 300–301 modeling, composite structures, 297–298 vibration with, 306–318 example problems, 308–318 fast Fourier transform, 307 finite element eigenvalue analysis, 307 numerical modal analysis, 307 velocimeters, 307 verification study, 307–308 FMMCs, see Fibrous metal matrix composites Fourier’s law, 348 Frame, 26–28 diagonal mass matrix, 28 element stiffness matrix, 26 global coordinate system, element matrix in, 27 transformation matrix, 26 Free-energy model, 69 FSI, see Fluid-structure interaction G Galerkin finite element formulation, 9–13 linear shape functions, 10 nodal coordinates, 12 nodal variable, 9 shape functions, 9 strong formulation, 11 test function, 11 weak formulation, 11 “Game of life,” 103 Gauss-Legendre quadrature rule, 40 Gear predictor-corrector method, 147 Generalized lattice Boltzmann equation, 60 Glass/epoxy plain weave composites, 241 GPUs, see Graphics processing units Graphics processing units (GPUs), 96–100 array of structures, 98 basic implementation on graphics processing units, 97–99 central processing units, 96 coalesced memory access, 99 collision operator, 96 computational requirements, 96–97 C programming language, 96 data layout, 98–99

© 2016 by Taylor & Francis Group, LLC

graphics processing units, 96 LBM routine, 97 Message Passing Interface, 96 NVIDIA CUDA GPU computing platform, 96 performance benchmark, 99–100 structure of arrays, 98 Visualization Toolkit file formats, 97 H HA, see Hydroxyapatite Hamiltonian dynamics, 147 Hamiltonian formalism, 107 Haversian systems, 284 HE material, see Hyperelastic material Hertz contact solution, 350 HLBM, see Hybrid LBM Hollomon equation, 243 Hooke’s law, 14, 349 HRAM, see Hydrodynamic ram Hybrid LBM (HLBM), 65–68 Hydrodynamic loading, 330–338 Hydrodynamic ram (HRAM), 338–342 description, 338 drag-phase fluid pressure, 342 entry wall x displacement, 339 hemispherical shock wave, 341 numerical models, 338–339 numerical results, 339–342 von Mises stress, 340 Hydroxyapatite (HA), 267–269, 294 chemical formula of, 268 crystal dimensions of, 293 lattice, 268 description of, 267 nanoindentation testing on, 269 Hyperelastic (HE) material, 360 I Interatomic potential energy function, 149–154 Abell-Tersoff-Brenner potential, 150–154 bond energy, 151 effective carbon bond length, 151 embedded atom potential, 154 equilibrated carbon nanotube structure, 151 kinetic molecular theory, 152 Lennard-Jones potential, 149 Morse potential, 150 short-range interaction potential, 153

402

Interlayer delamination, 208 Interparticle potential model, 69–70 Isoparametric formulation, 36–41 chain rule, 39 element stiffness matrix, 40 Gauss-Legendre quadrature rule, 40 Jacobian matrix, 39 natural coordinate domain, 36 nodal variable, 38 physical coordinate domain, 36 shape functions, 36, 37 J Jacobian matrix, 39 Joule’s law, 344 K Kinetic molecular theory, 152, 159 Korteweg-de Vries equation, 55 Kronecker delta, 202, 243 L Lamellar bone, 281–282 Laminated plate theory, 224 Lamination model, 208, 224–225 bending stiffness properties, 224 constitutive equation, 224 numerical integration points, 224 postloop, 225 preloop, 225 Lattice Boltzmann equation, generalized, 60 Lattice Boltzmann method (LBM), 2, 55–100, 200 boundary condition, 71–79 bounce-back boundary conditions, on-grid version of, 73 cumbersome data, 78 D2Q9 lattice, 74 dry nodes, 73 expressions requiring intensive bookkeeping, 78 fixed rigid boundary, 72–73 nodal variable, 71 periodic boundary, 72 pressure boundary condition, 73–74 required bookkeeping, 78 velocity boundary condition, 74–79 Brinkman equation, 55 Burgers equation, 55 cellular automata, 103

© 2016 by Taylor & Francis Group, LLC

Index

coupling between FEM and, 200 element-free-based lattice Boltzmann method, 64–65 coefficient vector, 64 interpolation function vector, 64, 65 spline function, 65 weighting function, 64 example problems, 85–96 backward-facing step, 87–88 benchmark stipulation, 89 channel flow over cylinder, 92–96 halfway bounce-back boundary condition, 86 lattice units, fluid viscosity in, 87 lid-driven cavity, 88–92 on-grid bounce-back technique, 89 Poiseuille flow, 85–87 Reynolds number, 94 Strouhal number, 94 trailing vortex, 92, 94 Von Karman street, 94 vorticity plot for cylinder, 95 finite element–based lattice Boltzmann method, 61–63 Crank-Nicholson time integration technique, 63 Dirac delta functions, 63 finite element mesh, 62 Galerkin method, 63 nodal variables, 62 single-relaxation time operator, 62 unknown variable, 62 weighted residual equation, 62 weighting function, 63 fluid-structure interaction modeling, 297 graphics processing units, implementation on, 96–100 array of structures, 98 basic implementation on graphics processing units, 97–99 central processing units, 96 coalesced memory access, 99 collision operator, 96 computational requirements, 96–97 C programming language, 96 data layout, 98–99 LBM routine, 97 Message Passing Interface, 96 NVIDIA CUDA GPU computing platform, 96 performance benchmark, 99–100 structure of arrays, 98

403

Index

hybrid lattice Boltzmann formulation, 65–68 classical LBM, 65 dispersion errors, 66 dissipation error, 66 first-order time integration schemes, 66 fluid-structure interaction, 68 hybrid LBM, 65 subdomain, 66 time and space integration methods, 66 Korteweg-de Vries equation, 55 macroscale conservation laws, 55 multicomponent flow, 68–71 artificial recoloring process, 68 color fluid model, 68–69 computational fluid dynamics methods, 69 Coulomb interaction among charged particles, 69 free-energy model, 69 immiscible multicomponent lattice Boltzmann procedures, 70–71 interparticle potential model, 69–70 LBM theory, models using, 68 mean field theory model, 69 nonequilibrium thermodynamics, Cahn-Hilliard’s approach for, 69 Rayleigh-Taylor instability, simulation of, 69 recoloring step, 68 surface tension effect, 69 Vlasov equation, 69 multiple relaxation lattice Boltzmann formulation, 60–61 collision operator, 60 D2Q9 lattice, 60 fixed Prandtl number, 60 generalized lattice Boltzmann equation, 60 Navier-Stokes equations, 55 scaling, 81–85 benchmark values, 82 inlet boundary condition, 82 lattice units, 81 LBM time step flowchart, 84 pressure boundary condition scaling, 83–85 Reynolds number, 81 velocity boundary condition scaling, 83 viscosity scaling, 83 Schrodinger equation, 55 standard lattice Boltzmann method, 55–60 cellular automaton, 55 collision process, 60 computational procedure, 60

© 2016 by Taylor & Francis Group, LLC

discrete velocity vector, 56 D2Q9 lattice structure, 56, 57 lattice definition, 56 lattice site update, rule for, 56 redistribution process, 60 streaming process, 60 turbulent flow, 79–80 eddy viscosity, 79 Prandtl mixing length approach, 80 relaxation constant, 79 von Karman constant, 80 Visualization Toolkit file formats, 97 wave equation, 80–81 refraction index, 81 wave attenuation factor, 81 wave propagation speed, 81 LBM, see Lattice Boltzmann method LE material, see Linear elastic material Lennard-Jones potential, 149 Linear elastic (LE) material, 360 Linear shape functions, 10, 11 LJ potential method, 171, 175 Lorentz force, 343, 349, 355 Ludwik equation, 243 M Maxwell-Boltzman velocity distribution, 163 MD, see Molecular dynamics Mean field theory model, 69 Message Passing Interface (MPI), 96 Metallic materials, multiscale analysis of, 255–265 example problems, 261–265 deformation state, 261 discrete atomic region, 264 elastic modulus, 261 grain boundary stiffness, 262 interface spring constants, 260, 262 no-defect case, 263 Poisson ratio, 261 smeared continuum region, 265 macroscale analysis, 257–258 displacements, 258 finite element method, 257 information obtained, 257 mesoscale analysis, 258–260 finite element mesh, 259 grain boundaries, 259 unit cell, 258 Voronoi diagram, 258 microscale analysis, 260

404

nanoscale analysis, 260–261 boundary conditions, 261 finite element mesh, 261 inner subdomain, 260 polycrystalline metals, 256–257 coupling, 257 intergrain boundary characteristics, 256 single-grain domain, 256 previous study, 255–256 asymptotic homogenization theory, 256 atomistic modeling, 256 Cauchy-Born hypothesis, 256 enhanced mesh superposition method, 256 grain deformation, 256 linking of models, 256 Metals, fatigue of, 183–186 Method of weighted residual (MWR), 5–9 boundary conditions, 5 choices, 63 cubic function, 7 Galerkin method, 7 polynomial function, 6 residual, 6 test function, 7 trial function, 6, 8 weighted residual, 7 Molecular dynamics (MD), 145–186 carbon nanotubes (elastic modulus), 156–168 bamboo shape nanotube, 156 basic structures of carbon nanotubes, 157–158 Boltzmann constant, 159 chiral nanotubes, 158 comparative results of equilibrium and nonequilibrium simulations, 165–168 elastic moduli, difference in, 165 elastic modulus of CNTs under equilibrium, 162–165 equilibrium and vibration motion of CNTs, 159–162 freestanding thermal vibration method, 159 Maxwell-Boltzman velocity distribution, 163 multiwall carbon nanotubes, 157 simulation time step, 158 single-wall carbon nanotube, 157 thermal vibration method, 161 time evolution method, 163 Young’s modulus ratio concept, 165

© 2016 by Taylor & Francis Group, LLC

Index

carbon nanotubes (vibrational mode shapes), 168–173 comparison of radial breathing modes of armchair SWCNTs, 168 discrete Fourier transform approaches, 168 hollow beam, bending of, 171 natural frequencies and mode shapes for BSCNTs, 172–173 natural frequencies and mode shapes for MWCNTs, 171–172 natural frequencies and mode shapes for SWCNTs, 168–170 radial breathing modes, 172 Tersoff-Brenner potential, 171 classical formulation, 145–146 atoms, interactive forces among, 145 Hamiltonian, 145 Newton’s second law, 145 fatigue of metals, 183–186 box boundary, atoms moving outside of, 183 copper, behavior under conditions of cyclic loading, 183 embedded atom method, 183 examples, 184–186 impurity models, 185 modeling, 183 output data, 183 simulation units, 183 interatomic potential energy function, 149–154 Abell-Tersoff-Brenner potential, 150–154 bond energy, 151 effective carbon bond length, 151 embedded atom potential, 154 equilibrated carbon nanotube structure, 151 kinetic molecular theory, 152 Lennard-Jones potential, 149 Morse potential, 150 short-range interaction potential, 153 molecular mechanics formulation, 154–156 displacement vectors, 154 external loads, 154 matrix expression, 156 nanoscale fluid flow, 179–183 boundary conditions, 179 fluid flow simulation, 179 geometry used, 179 particle-fluid interaction, 180–183 velocity profiles, 180, 182 polymers, 173–178 allocation of polymers, 174–175 autocorrelation, 176

405

Index

composite material, simulation representing, 178 cross-linking of polymers, 173–174 examples, 176–178 LJ potential method, 175 potential energy for polymers, 175–176 quasi-atom concept, 175 randomly dispersed particles, 173 time integration technique, 146–149 correction, 148–149 evaluation, 148 finite difference method, 147 Gear predictor-corrector method, 147 Hamiltonian dynamics, 147 prediction step, 148 Mooney-Rivlin type model, 360 Moore neighborhood, 101 Morse potential, 150 MPI, see Message Passing Interface Multiphysics and multiscale modeling, introduction to, 1–3 book organization, 2–3 computational methods, 2 cellular automata, 2 finite element method, 2 lattice Boltzmann method, 2 overview, 1–2 aneurysms, investigation of, 1 blood circulation system, 1 living systems in nature, 1 man-made systems, 1 Multiwall carbon nanotubes (MWCNTs), 157, 171–172 MWCNTs, see Multiwall carbon nanotubes MWR, see Method of weighted residual N Nanoropes (SWCNT), 157 Nanoscale fluid flow, 179–183 boundary conditions, 179 fluid flow simulation, 179 geometry used, 179 particle-fluid interaction, 180–183 velocity profiles, 180, 182 Natural coordinate domain, 36 Navier-Stokes equations, 55, 297 Newmark method (FEM), 24 Nodal variable, 9 axial member, 15 beam, 19 FELBM, 62

© 2016 by Taylor & Francis Group, LLC

fictitious particle densities as, 298 finite element formulation, 31 fluid-structure interaction modeling, 298 Galerkin finite element formulation, 9 isoparametric formulation, 38 LBM boundary condition, 71 structural domain (FEM), 204 Nonequilibrium thermodynamics, Cahn-Hilliard’s approach for, 69 NVIDIA CUDA GPU computing platform, 96 O On-grid bounce-back technique, 89 Osteons, 284 P Particulate metal matrix composite (PMMC), 246 Physical coordinate domain, 36 Plate and shell structures, 41–47 advantage, 42 bending energy, 46 constitutive equation, 41, 43 elastic modulus, 44 element stiffness matrix, 47 in-plane displacements, 42 natural coordinate system, 43 Poisson’s ratio, 44 shell element, 42 transverse displacement, 42 transverse normal strain, 45 transverse shear strains, 43 PMMC, see Particulate metal matrix composite Polycrystalline metals, 256–257 Polymers, 173–178 allocation of, 174–175 autocorrelation, 176 composite material, simulation representing, 178 cross-linking of, 173–174 examples, 176–178 LJ potential method, 175 potential energy for, 175–176 quasi-atom concept, 175 randomly dispersed particles, 173 Q Quasi-atom concept, 175

406

R Rail gun launcher, 343 Rayleigh-Taylor instability, simulation of, 69 Return mapping algorithm (RMA), 243 Reuss model, 294 RMA, see Return mapping algorithm RMS pressure, see Root mean square pressure ROM, see Rule of mixtures Root mean square (RMS) pressure, 127 Rule of mixtures (ROM), 231 S Schrodinger equation, 55 Shape functions atomic model, 195 beam element, 18 boundary integral, 35 continuum model, 195 coupling fluid and structural domains, 202 derivatives, 15, 39 displacements, 33 FELBM, 62 finite element nodal displacements, 188 Galerkin method, 9, 63 isoparametric formulation, 36, 37 linear, 10, 11 natural coordinate system, 43 triangular element, 31 Simpson’s rule, 123 Single-wall carbon nanotube (SWCNT), 157, 168–170 SoA, see Structure of arrays Solid element, 28–36 finite element formulation, 30–36 diagonal mass matrix, 34 discretized problem domain, 31 elemental stiffness matrix, 34 equations of motion, 30 kinematic equation, 33 nodal displacements, 31 nodal variable, 31 shape functions for triangular element, 31 test functions, 30 traction force vector, 35 triangular element, 32 theory of elasticity (solid element), 28–30 constitutive equations, 29 force equilibrium, 29 isotropic material, 29 kinematic equations, 30

© 2016 by Taylor & Francis Group, LLC

Index

normal stresses, 28 shear stresses, 28 Stokes theorem, 346 String infinite, 115 momentum of, 107 physical properties of, 113 transverse vibration of, 111–114 velocity, 107 vibration, modeling of, 103 Strouhal number, 94 Structure of arrays (SoA), 98 SWCNT, see Single-wall carbon nanotube T TEM, see Transmission electron microscope Test function, 7 Theory of elasticity (solid element), 28–30 constitutive equations, 29 force equilibrium, 29 isotropic material, 29 kinematic equations, 30 normal stresses, 28 shear stresses, 28 Time-scaling factor, 110 Transmission electron microscope (TEM), 159 Transverse matrix cracking, 208 Trial function, 6 Tropocollagen, 267, 269, 271 Truss, 24–26 coordinate systems, 24, 25 displacements in coordinates, 24 element, 25 global coordinate system, 24 local coordinate system, 24 stiffness matrix, 25 strain energy, 25 structure, 25 U UFG aluminum, see Ultrafine grain aluminum Ultrafine grain (UFG) aluminum, 229 Underwater acoustics, 126–133 air-to-waterline boundary, 129 confined water channel, wave propagation in, 133 propagation losses, 127, 128 reflected wave, 132 root mean square pressure, 127 transmission loss development, 127–129 various boundary conditions, 129

407

Index

wave propagation across flat-bottom ocean floor, 129–131 wave propagation over curved hill, 131–132 wave propagation over sloping bottom, 132–133 V Velocimeters, 307 Visualization Toolkit (VTK) file formats, 97 Vlasov equation, 69 Voigt model, 294 Von Karman street, 94 von Mises stress, 374 von Mises yield criterion, 251 von Neumann neighbors, 101 Voronoi diagram, 258 VTK file formats, see Visualization Toolkit file formats

© 2016 by Taylor & Francis Group, LLC

W Wave propagation across flat-bottom ocean floor, 129–131 CA rule for, 118 coupling acoustic domains, 203 one-dimensional, 117 over curved hill, 131–132 over sloping bottom, 132–133 problem, 200 speed, 81 underwater acoustics, 126 Weighted residual, 7 Y Young’s modulus (CNTs), 152, 159