Topological Derivatives in Shape Optimization

Cover......Page 1
Front matter......Page 2
Introduction......Page 20
The Topological Derivative Concept......Page 22
Relationship between Shape and Topological Derivatives......Page 29
The Topological-Shape Sensitivity Method......Page 31
An Example of Topological Derivative Evaluation......Page 33
Monograph Organization......Page 39
Exercises......Page 42
Domain Derivation in Continuum Mechanics......Page 44
Material and Spatial Descriptions......Page 45
Gradient of Vector Fields......Page 47
Spatial Description of Velocity Fields......Page 48
Material Derivatives of Spatial Fields......Page 49
Derivative of the Gradient of a Scalar Field......Page 50
Derivative of the Gradient of a Vector Field......Page 51
Domain Integral......Page 52
Boundary Integral......Page 54
Summary of the Derived Formulae......Page 57
The Eshelby Energy-Momentum Tensor......Page 59
Exercises......Page 62
Material and Shape Derivatives for Boundary 
Value Problems......Page 65
Preliminaries......Page 66
Sobolev-Slobodetskii Spaces......Page 67
Elliptic Regularity......Page 68
Elliptic Problems in Nonsmooth Domains......Page 69
Shape Derivatives......Page 70
Weak Material Derivatives for the Dirichlet Problem......Page 73
Strong Material Derivatives for the Dirichlet Problem......Page 78
Material Derivatives for the Neumann Problem......Page 80
Shape Derivatives for the Dirichlet Problem......Page 82
Shape Derivatives for the Neumann Problem......Page 84
Problem Formulation......Page 86
Material Derivatives for Elasticity......Page 89
Shape Derivatives for Elasticity......Page 91
Shape Derivatives for Interfaces......Page 94
Material and Shape Derivatives for Kirchhoff Plates......Page 95
Problem Formulation......Page 96
Material Derivatives for the Kirchhoff Plate......Page 97
Shape Derivatives for the Kirchhoff Plate......Page 98
The Adjugate Matrix Concept......Page 100
Material Derivatives for the Stationary, Homogeneous Navier-Stokes Problem......Page 105
Exercises......Page 107
Singular Perturbations of Energy Functionals......Page 108
Problem Formulation......Page 109
Shape Sensitivity Analysis......Page 111
Asymptotic Analysis of the Solution......Page 114
Topological Derivative Evaluation......Page 117
Examples with Explicit Form of Topological Derivatives......Page 123
Additional Comments and Summary of the Results......Page 127
Problem Formulation......Page 129
Shape Sensitivity Analysis......Page 131
Asymptotic Analysis of the Solution......Page 134
Topological Derivative Evaluation......Page 137
Fourth Order Elliptic Equation: The Kirchhoff Problem......Page 139
Problem Formulation......Page 140
Shape Sensitivity Analysis......Page 142
Asymptotic Analysis of the Solution......Page 146
Topological Derivative Evaluation......Page 149
Exercises......Page 153
Configurational Perturbations of Energy 
Functionals......Page 154
Problem Formulation......Page 155
Shape Sensitivity Analysis......Page 157
Asymptotic Analysis of the Solution......Page 160
Topological Derivative Evaluation......Page 162
Numerical Example......Page 164
Problem Formulation......Page 165
Shape Sensitivity Analysis......Page 168
Asymptotic Analysis of the Solution......Page 170
Topological Derivative Evaluation......Page 174
Numerical Example......Page 176
Fourth Order Elliptic Equation: The Kirchhoff Problem......Page 177
Problem Formulation......Page 178
Shape Sensitivity Analysis......Page 181
Asymptotic Analysis of the Solution......Page 184
Topological Derivative Evaluation......Page 189
Numerical Example......Page 190
Estimates for the Remainders......Page 192
Exercises......Page 197
Problem Formulation......Page 198
Shape Sensitivity Analysis......Page 201
Asymptotic Analysis of the Solution......Page 206
Asymptotic Expansion of the Direct State......Page 207
Asymptotic Expansion of the Adjoint State......Page 208
Topological Derivative Evaluation......Page 209
Exercises......Page 211
Topological Derivative for Steady-State 
Orthotropic Heat Diffusion Problems......Page 212
Problem Formulation......Page 213
Shape Sensitivity Analysis......Page 215
Asymptotic Analysis of the Solution......Page 216
Topological Derivative Evaluation......Page 218
Problem Formulation......Page 220
Shape Sensitivity Analysis......Page 223
Asymptotic Analysis of the Solution......Page 224
Topological Derivative Evaluation......Page 228
Numerical Example......Page 230
Multiscale Modeling in Solid Mechanics......Page 234
The Homogenized Elasticity Tensor......Page 236
Sensitivity of the Macroscopic Elasticity Tensor to Topological Microstructural Changes......Page 238
Compound Asymptotic Expansions for Spectral 
Problems......Page 241
Preliminaries and Examples......Page 242
Dirichlet Laplacian in Domains with Small Cavities......Page 244
First Order Asymptotic Expansion......Page 245
Second Order Asymptotic Expansion......Page 248
Complete Asymptotic Expansion......Page 252
Neumann Laplacian in Domains with Small Caverns......Page 255
First Boundary Layer Corrector......Page 258
Second Boundary Layer Corrector......Page 262
Correction Term of Regular Type......Page 266
Multiple Eigenvalues......Page 271
Configurational Perturbations of Spectral Problems in Elasticity......Page 272
Anisotropic and Inhomogeneous Elastic Body......Page 273
Vibrations of Elastic Bodies......Page 274
Formal Construction of Asymptotic Expansions......Page 277
Polarization Matrices......Page 289
Topological Asymptotic Analysis for Semilinear 
Elliptic Boundary Value Problems......Page 292
Topological Derivatives in R2......Page 294
Formal Asymptotic Analysis......Page 295
Formal Asymptotics of Shape Functional......Page 299
Asymptotic Approximation of Solutions......Page 302
Asymptotics of Shape Functional......Page 307
Exercises......Page 310
Topological Derivatives for Unilateral Problems......Page 313
Preliminaries......Page 314
Domain Decomposition and the Steklov-Poincaré Operator......Page 316
Domain Decomposition Technique......Page 318
Steklov-Poincaré Pseudodifferential Boundary Operators......Page 319
Problem Formulation......Page 322
Hadamard Differentiability of Minimizer......Page 324
Topological Derivative Evaluation......Page 325
Cracks on Boundaries of Rigid Inclusions......Page 328
Problem Formulation......Page 329
Approximation of a Rigid Inclusion with Contrast Parameter......Page 331
Hadamard Differentiability of Solutions to Variational Inequalities......Page 334
Topological Derivative Evaluation......Page 335
Exercises......Page 338
Back matter......Page 339