9.3 Topological Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . 338
9.3.1 Methods Based on Density . . . . . . . . . . . . . . . . . . . . . . . . 338
9.3.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
9.4 Applying Topological Cell Optimization . . . . . . . . . . . . . . . . . . . 342
9.4.1 Flexibility Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . 342
9.4.2 Thermoelastic Structures . . . . . . . . . . . . . . . . . . . . . . . . . 343
9.4.3 Topological Optimization of Stress State . . . . . . . . . . . . . . 346
9.4.4 Topological Optimization in the Field
of Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
9.4.5 Interpolating Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
9.5 Reactions on Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
9.5.1 Reaction on Boundary and Their Sensitivity . . . . . . . . . . . 353
9.5.2 Formulating Problems of Topological Optimization . . . . . . 354
9.6 Topological Optimization of Thermoelastic Composites
with Maximized Stiffness and Heat Transfer . . . . . . . . . . . . . . . . 357
9.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
9.6.2 Determination of the Effective Tensor of Elastic
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
9.6.3 Determination of the Effective Heat Transfer
Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
9.6.4 Topological Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 365
9.6.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
9.7 Non-Linear Dynamics of Size-Dependent Euler-Bernoulli
Beams with Topologically Optimized Microstructure
and Subjected to Temperature Field . . . . . . . . . . . . . . . . . . . . . . . 373
9.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
9.7.2 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . 377
9.7.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 384
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
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9.3.1 Methods Based on Density . . . . . . . . . . . . . . . . . . . . . . . . 338
9.3.2 Regularization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
9.4 Applying Topological Cell Optimization . . . . . . . . . . . . . . . . . . . 342
9.4.1 Flexibility Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . 342
9.4.2 Thermoelastic Structures . . . . . . . . . . . . . . . . . . . . . . . . . 343
9.4.3 Topological Optimization of Stress State . . . . . . . . . . . . . . 346
9.4.4 Topological Optimization in the Field
of Thermoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348
9.4.5 Interpolating Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 351
9.5 Reactions on Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353
9.5.1 Reaction on Boundary and Their Sensitivity . . . . . . . . . . . 353
9.5.2 Formulating Problems of Topological Optimization . . . . . . 354
9.6 Topological Optimization of Thermoelastic Composites
with Maximized Stiffness and Heat Transfer . . . . . . . . . . . . . . . . 357
9.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
9.6.2 Determination of the Effective Tensor of Elastic
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
9.6.3 Determination of the Effective Heat Transfer
Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
9.6.4 Topological Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 365
9.6.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366
9.7 Non-Linear Dynamics of Size-Dependent Euler-Bernoulli
Beams with Topologically Optimized Microstructure
and Subjected to Temperature Field . . . . . . . . . . . . . . . . . . . . . . . 373
9.7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
9.7.2 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . 377
9.7.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . 384
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
Contents
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