topology optimization in engineering structure design · vi topology optimization in engineering...

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Chapter 1. Standard Material Layout Design . . . . . . . . . . . . 1

1.1. Basic formulations of topology optimization . . . . . . . . . . . . 1 1.2. Typical applications of standard topology optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3. Topology optimization of cellular materials and structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.1. Homogenization method and material microstructure designs . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2. Scale-effect of the material microstructure . . . . . . . . . . 12 1.3.3. Scale-related topology optimization . . . . . . . . . . . . . . 15 1.3.4. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

Chapter 2. Low-Density Areas in Topology Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1. Localized mode in low-density areas . . . . . . . . . . . . . . . . 27 2.2. Localized deformation . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.3. Polynomial interpolation model . . . . . . . . . . . . . . . . . . . 41 2.4. Breakdown issue in ESO . . . . . . . . . . . . . . . . . . . . . . . 51 2.5. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 3. Dynamic Problems . . . . . . . . . . . . . . . . . . . . . . 61

3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.2. Analysis methods for harmonic force excitations . . . . . . . . . 64

vi Topology Optimization in Engineering Structure Design

3.2.1. Mode displacement method . . . . . . . . . . . . . . . . . . . 65 3.2.2. Mode acceleration method . . . . . . . . . . . . . . . . . . . 66 3.2.3. Full method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.4. Comparative tests of harmonic analysis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3. Topology optimization under harmonic force excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3.1. Topology optimization formulation . . . . . . . . . . . . . . 74 3.3.2. Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 75 3.3.3. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 77

3.4. Analysis methods for stationary random force excitations . . . . . 87 3.4.1. Complete quadratic combination method . . . . . . . . . . . 87 3.4.2. Conventional pseudo-excitation method . . . . . . . . . . . 89 3.4.3. The combined method of PEM and MAM . . . . . . . . . . 90 3.4.4. Comparative tests of stationary random analysis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3.5. Topology optimization under stationary random force excitation . . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.5.1. Topology optimization formulation . . . . . . . . . . . . . . 95 3.5.2. Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 96 3.5.3. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 97

3.6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

Chapter 4. Thermo-Elastic Problems . . . . . . . . . . . . . . . . . 107

4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2. Thermo-elastic analysis . . . . . . . . . . . . . . . . . . . . . . . . 108 4.3. Thermo-elastic topology optimization with single material . . . . 111

4.3.1. Topology optimization formulation . . . . . . . . . . . . . . 111 4.3.2. Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 112 4.3.3. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 114

4.4. Thermo-elastic topology optimization with multiple materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4.4.1. Standard optimization formulation . . . . . . . . . . . . . . . 125 4.4.2. Sensitivity analysis . . . . . . . . . . . . . . . . . . . . . . . . 125 4.4.3. Mass constraint . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.4.4. Improved optimization formulation . . . . . . . . . . . . . . 131 4.4.5. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . 137

4.5. Distinction between mean compliance and elastic strain energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

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4.5.1. Formulations of mean compliance and elastic strain energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.5.2. Comparisons between mean compliance and elastic strain energy . . . . . . . . . . . . . . . . . . . . . . . . . 145 4.5.3. Effects of thermal and mechanical loads on the optimized configurations . . . . . . . . . . . . . . . . . . . . . 151

4.6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Chapter 5. Integrated Layout and Topology Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.1. Introduction to integrated optimization . . . . . . . . . . . . . . . 159 5.2. Finite-circle method . . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.2.1. Formulation of finite-circle method . . . . . . . . . . . . . . . 160 5.2.2. Improved adaptive constraint aggregation . . . . . . . . . . . 166

5.3. Density points and embedded meshing . . . . . . . . . . . . . . . 173 5.3.1. Definition of the density points . . . . . . . . . . . . . . . . . 173 5.3.2. Superelement and semi-analytical sensitivities . . . . . . . . 176 5.3.3. Decomposition optimization strategies . . . . . . . . . . . . . 180

5.4. MPC-based component-structure connections . . . . . . . . . . . 185 5.5. Integrated optimization based on implicit model . . . . . . . . . 194

5.5.1. Implicit representation of component geometry . . . . . . . 194 5.5.2. Sensitivity analysis and examples with implicit functions . 202 5.5.3. Integrated optimization based on XFEM . . . . . . . . . . . . 207

5.6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

Chapter 6. Optimization with Constraints on Multifastener Joint Loads . . . . . . . . . . . . . . . . . . . . . . . 217

6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.2. Joint load calculation and sensitivity analysis . . . . . . . . . . 219 6.3. Numerical examples and discussions . . . . . . . . . . . . . . . 222 6.3.1. Cantilever beam with experiments . . . . . . . . . . . . . . . 222 6.3.2. Two different wing boxes . . . . . . . . . . . . . . . . . . . . 230

6.4. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

Chapter 7. Potential Applications of Topology Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

7.1. Shape-preserving design . . . . . . . . . . . . . . . . . . . . . . . . 239 7.2. Smart structure design . . . . . . . . . . . . . . . . . . . . . . . . . 243

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7.3. Structural features design . . . . . . . . . . . . . . . . . . . . . . . 245 7.4. Topology optimization and additive manufacturing . . . . . . . 247

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273