Homogenization and Structural Topology Optimization

Preface vii.- Acknowledgements xi.- Table of Contents xiii.- Notation xvii.- 1. Introduction.- 1.1 Overview.- 1.2 Mathematical description of optimization problem.- 1.3 Types of structural optimization.- 1.4 Aspects of topology optimization.- 1.5 Layout of the book.- References.- I: Homogenization.- 2. Homogenization Theory for Media with a Periodic Structure.- 2.1 Introduction.- 2.2 Periodicity and asymptotic expansion.- 2.3 One dimensional elasticity problem.- 2.4 General boundary value problem.- 2.5 Elasticity problem in cellular bodies.- References.- 3. Solution of Homogenization Equations for Topology Optimization.- 3.1 Introduction.- 3.2 Material models.- 3.2.1 Rectangular microscale voids.- 3.2.2 Ranked layered material cells.- 3.2.3 Artificial materials.- 3.3 Analytical Solution of the homogenization equation for rank laminate composites.- 3.3.1 Rank-1 materials.- 3.3.2 Rank-2 materials.- 3.3.3 Bi-material rank-2 composites.- 3.4 Numerical Solution of the homogenization equation for a cellular body with rectangular holes.- 3.4.1 Finite element formulation.- 3.4.2 Derivation of the boundary conditions from periodicity.- 3.4.3 Examples.- 3.4.4 Homogenization constitutive matrix for Square microcells with rectangular voids.- 3.4.5 Least squares smoothing.- References.- II: Topology Optimization.- 4. Structural Topology Optimization using Optimality Critieria Methods.- 4.1 Introduction.- 4.2 Kuhn-Tucker condition.- 4.3 Analytical optimality criteria.- 4.3.1 An illustrative example of variational analysis.- 4.3.2 An illustrative example of derivation of optimality criteria.- 4.4 Mathematical model for the topological structural optimization.- 4.5 Optimality criteria for the topological structural optimization.- 4.5.1 Optimality conditions.- 4.5.2 Updating scheme.- 4.5.3 A modified resizing scheme.- 4.6 Optimal Orientation.- 4.7 Algorithm.- 4.8 Examples.- References.- 5. Experiences in Topology Optimization of Plane Stress Problems.- 5.1 Introduction.- 5.2 Effect of material model.- 5.2.1 Material model with rectangular holes.- 5.2.2 Artificial material model.- 5.2.3 Rank-2 material model.- 5.3 Effect of resizing scheme.- 5.4 Effect of the orientation variable.- 5.5 Effect of finite element discretization.- 5.5.1 Continuation method.- 5.5.2 Unstructured mesh.- 5.7 Effect of material volume.- 5.8 Effect of resizing parameters.- 5.9 Examples.- 5.9.1 Bridge with support layout 1.- 5.9.2 Bridge with support layout 2.- 5.9.3 Bracket with a hole.- 5.9.4 Shear wall with openings.- References.- 6. Topological layout and Reinforcement Optimization of Plate Structures.- 6.1 Introduction.- 6.2 Selection of plate base cell model.- 6.3 A brief review of Mindlin-Reissner plate theory.- 6.4 Homogenization of the plate model microstructure.- 6.5 Optimization problem.- 6.6 The finite element method.- 6.7 Optimal rotation.- 6.8 Examples.- 6.8.1 Simple supported Square plate with a central point load.- 6.8.2 Simple supported Square plate subject to a uniform load.- 6.8.3 Square plate subject to four point loads.- 6.8.4 Square slab with a circular holes.- 6.8.5 Fiat slab of a multi-span floor.- References.- III: Other Methods and Integrated Structural Optimization.- 7. Alternative Approaches to Structural Topology Optimization.- 7.1 Introduction.- 7.2 Simulation of functional adaptation of bone mineralization.- 7.2.1 A remodelling scheme based on effective strain energy density.- 7.2.2 A scheme based on effective stresses.- 7.3 Evolutionary fully stressed design method.- References.- 8. Integrated Structural Optimization.- 8.1 Introduction.- 8.2 Overview of integrated structural optimization.- 8.3 Topology optimization module.- 8.3.1 Ground structure method.- 8.3.2 Bubble method.- 8.4 Image processing module.- 8.4.1 Elimination of mesh dependency and checkerboard Problems using noise cleaning techniques.- 8.5 Shape optimization module.- 8.5.1 Boundary Variation method.- 8.5.2 Adaptive growth method.- 8.6 Integrated adaptive topology and shape optimization.- 8.7 Final thoughts.- References.- Appendix A.- Appendix B.- Appendix C.- Appendix D HOMOG Manual.- Appendix E PLATO Manual.- Author Index.