An introduction to the topological derivative method

Preface......Page 7
Contents......Page 8
1 Introduction......Page 10
 1.1 The Topological Derivative Concept......Page 11
 1.2 Evaluation of the Topological Derivative......Page 17
  1.2.1 Adjoint Sensitivity Method......Page 18
  1.2.2 Auxiliary Result......Page 19
  1.2.3 A Simple Example......Page 20
 1.3 Organization of the Book......Page 23
 1.4 Exercises......Page 25
 2.1 Problem Formulation......Page 26
 2.2 Variation of the Energy Shape Functional......Page 28
 2.3 Topological Derivative Evaluation......Page 29
  2.3.1 Neumann Boundary Condition on the Hole......Page 30
  2.3.2 Dirichlet Boundary Condition on the Hole......Page 34
 2.4 Summary of the Results......Page 42
 2.5 Exercises......Page 43
 3.1 Problem Formulation......Page 44
 3.2 Existence of the Topological Derivative......Page 46
 3.3 Variation of the Compliance Shape Functional......Page 48
  3.4.1 Perturbation on the Right-Hand Side......Page 49
  3.4.2 Perturbation on the Lower Order Term......Page 51
  3.4.3 Perturbation on the Higher Order Term......Page 52
 3.5 Summary of the Results......Page 59
 3.6 Exercises......Page 60
 4.1 Coupled System......Page 62
  4.1.1 Non-perturbed Problem......Page 63
  4.1.2 Perturbed Problem......Page 64
 4.2 Domain Decomposition Technique......Page 65
  4.2.1 Compactness of the Asymptotic Expansion......Page 69
  4.2.2 Asymptotic Expansion of the Solution......Page 73
  4.2.3 Asymptotic Expansion of the Shape Functional......Page 74
 4.3 Exercises......Page 75
5 Topology Design Optimization......Page 76
 5.1 Model Problem in Elasticity......Page 77
  5.1.1 Existence of the Topological Derivative......Page 81
  5.1.2 Variation of the Shape Functional......Page 82
  5.1.3 Asymptotic Analysis of the Solution......Page 84
  5.1.4 Topological Derivative Evaluation......Page 89
 5.2 Topology Design Algorithm......Page 91
 5.3 Numerical Results......Page 93
  5.3.1 Structural Compliance Topology Optimization......Page 94
  5.3.2 Topology Design of Compliant Mechanisms......Page 97
 5.4 Final Remarks......Page 98
 5.5 Exercises......Page 101
 A.1 Inner, Vector, and Tensor Products......Page 103
 A.2 Gradient, Divergence, and Curl......Page 105
 A.3 Integral Theorems......Page 106
 A.4 Some Useful Decompositions......Page 107
 A.5 Polar and Spherical Coordinate Systems......Page 110
References......Page 114
Index......Page 119