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Optimal Design of Material Microstructure For OPTIMAL DESIGN OF MATERIAL MICROSTRUCTURE FOR CONVECTIVE HEAT TRANSFER IN A SOLID-FLUID MIXTURE by Jeong Hun Seo A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering) in The University of Michigan 2009 Doctoral Committee: Professor Noboru Kikuchi, Chair Professor Panos Y. Papalambros Professor Scott J. Hollister Research Scientist Zheng-Dong Ma © Jeong Hun Seo 2009 DEDICATION To my family ii ACKNOWLEDGEMENTS I would like to express my appreciation to my advisor, Professor Noboru Kikuchi for his gentle guidance and great inspiration during my doctoral studies at the University of Michigan. I also would like to sincerely acknowledge the helpful suggestions and criticisms made by my doctoral committee members, Professor Panos Y. Papalambros, Professor Scott J. Hollister and Dr. Zheng-Dong Ma. I would like to express my gratitude to Professors Yoon Young Kim and Gil Ho Yoon who not only guided me to the field of topology optimization at Seoul National University, but has also been advising me even after my graduation. I am also grateful to Professor Krister Svanberg for providing his MMA code. I am also indebted to my colleagues and friends at the Computational Mechanics Laboratory at the University of Michigan, Jaewook Lee, Kyungjun Lee, Dr. Youngwon Hahn, Geunsoo Ryu, Dr. Ikjoong Lee, Dr. Jangho Shin, Dr. Tsuyoshi Higo, Dr. Minako Sekiguchi and many others with whom I shared memories during my studies in Ann Arbor. My special thanks are expressed to Dr. Peter Nagourney and Dr. Roann Altman for their tutoring me with writing in English. iii I acknowledge the financial support from Toyota Central Research and Development Laboratories, Inc. I would like to extend my thanks to Mr. Toru Matsushima at Toyota Motor Corporation. Last, I am very grateful to my parents for their continuous patience, support and encouragement during my doctoral studies at the University of Michigan. iv TABLE OF CONTENTS DEDICATION .............................................................................................................. ii ACKNOWLEDGEMENTS ........................................................................................ iii LIST OF FIGURES .................................................................................................. viii LIST OF APPENDICES .......................................................................................... xvi ABSTRACT .............................................................................................................. xvii CHAPTER 1. INTRODUCTION .................................................................................................... 1 1.1. Overview .............................................................................................................. 1 1.2. Designing material microstructures ..................................................................... 3 1.3. Homogenization of material microstructure ........................................................ 5 1.4. Topology optimization ....................................................................................... 10 1.5. Outline of dissertation ........................................................................................ 23 2. CONTROLLING MEMBER LENGTHS IN TOPOLOGY OPTIMIZATION ................................................................................................... 24 2.1. Introduction ........................................................................................................ 24 2.2. The filtering methods ......................................................................................... 26 2.3. Functionals measuring geometry ....................................................................... 52 v 2.4. Summary ............................................................................................................ 71 3. LEVEL-SET BASED TOPOLOGY OPTIMIZATION ...................................... 73 3.1. Representing the geometry of structures via level-set function ......................... 73 3.2. Level-set evolution by Hamilton-Jacobi equation ............................................. 75 3.3. Level-set evolution by optimization algorithm .................................................. 86 3.4. Numerical studies ............................................................................................... 90 3.5. Summary .......................................................................................................... 106 4. TOPOLOGY OPTIMIZATION OF FLUID MECHANICS PROBLEMS .... 107 4.1. Introduction ...................................................................................................... 107 4.2. Brinkman-type flow model of porous media ................................................... 112 4.3. Stabilized finite element methods .................................................................... 114 4.4. Interpolation of material properties.................................................................. 121 4.5. Topology optimization of fluid mechanics problem ........................................ 128 4.6. Summary .......................................................................................................... 142 5. TOPOLOGY OPTIMIZATION OF HEAT TRANSFER PROBLEMS ........ 144 5.1. Introduction ...................................................................................................... 144 5.2. Formulation of optimization problem .............................................................. 145 5.3. Numerical examples ......................................................................................... 147 5.4. Summary .......................................................................................................... 155 6. MULTI-FUNCTIONAL DESIGN OF MATERIAL MICROSTRUCTURE . 156 6.1. Introduction ...................................................................................................... 156 6.2. Effective properties of heterogeneous media ................................................... 158 6.3. Homogenization in a solid-fluid mixture ......................................................... 161 6.4. Calculation of effective properties ................................................................... 162 vi 6.5. Numerical result of effective properties........................................................... 168 6.6. Design optimization for desired effective properties ....................................... 171 6.7. Multi-objective formulation for desired multiple effective tensors ................. 174 6.8. Design of homogenized elasticity .................................................................... 181 6.9. Design of homogenized fluid permeability ...................................................... 195 6.10. Design of effective dispersivity ..................................................................... 207 6.11. Summary ........................................................................................................ 213 7. CONCLUSION ..................................................................................................... 214 7.1. Geometry-control in topology optimization .................................................... 214 7.2. Level-set based topology optimization ............................................................ 215 7.3. Topology optimization of fluid mechanics problems ...................................... 216 7.4. Topology optimization of heat transfer problem ............................................. 217 7.5. Multi-functional design of material microstructure ......................................... 217 APPENDICES .......................................................................................................... 219 BIBLIOGRAPHY .................................................................................................... 232 vii LIST OF FIGURES Figure 2.2.1. (a) Linear weighting function and (b) nonlinear weighting function (ε=0.01, p=4) ................................................................................................................................... 27 Figure 2.2.2. Design examples and boundary conditions. (a) Cantilever beam, (b) force inverter, and (c) periodic unit cell for homogenization. ................................................... 32 Figure 2.2.3. Design results of cantilever beam with linearly weighted filtering (filtering radius=2×element length). (a) Density filtering, (b) sensitivity filtering, and (c) filtering by convolution operator. ................................................................................................... 34 Figure 2.2.4. Design results of cantilever beam with nonlinearly weighted filtering (ε=0.01, p=4 and filtering radius=2×element length). (a) Density filtering, (b) sensitivity filtering, and (c) filtering by convolution operator. .......................................................... 35 Figure 2.2.5. Design results of force inverter with linearly weighted filtering (filtering radius=2×element length). (a) Density filtering, (b) sensitivity filtering, and (c) filtering by convolution operator. ................................................................................................... 38 Figure 2.2.6. Design results of force inverter with nonlinearly weighted filtering (ε=0.01, p=4 and filtering radius=2×element length). (a) Density filtering, (b) sensitivity filtering, and (c) filtering by convolution operator. ........................................................................
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