Topology Optimization Design of Heterogeneous Materials and Structures

Cover......Page 1
Half-Title Page......Page 3
Title Page......Page 5
Copyright Page......Page 6
Contents......Page 7
 I.1. Background and motivations......Page 11
  I.2.1. Topology optimization methods......Page 14
  I.2.2. Material design and multiscale optimization......Page 17
  I.2.3. Fracture resistance design......Page 21
 I.3. Outline of the book......Page 23
PART 1: Multiscale Topology Optimization in the Context of Non-separated Scales......Page 25
 1. Size Effect Analysis in Topology Optimization for Periodic Structures Using the Classical Homogenization......Page 27
   1.1.1. Localization problem......Page 28
   1.1.2. Definition and computation of the effective material properties......Page 31
   1.1.3. Numerical implementation for the local problem with PER......Page 33
   1.2.1. Optimization model and sensitivity number......Page 34
   1.2.2. Finite element meshes and relocalization scheme......Page 36
   1.2.3. Optimization procedure......Page 38
  1.3. Numerical examples......Page 40
   1.3.1. Doubly clamped elastic domain......Page 41
   1.3.2. L-shaped structure......Page 43
   1.3.3. MBB beam......Page 48
  1.4. Concluding remarks......Page 49
 2. Multiscale Topology Optimization of Periodic Structures Taking into Account Strain Gradient......Page 53
   2.1.1. Definition of local and mesoscopic fields through the filter......Page 54
   2.1.2. Microscopic unit cell calculations......Page 57
   2.1.3. Mesoscopic structure calculations......Page 63
   2.2.1. Model definition and sensitivity numbers......Page 65
   2.2.2. Overall optimization procedure......Page 66
  2.3. Validation of the non-local homogenization approach......Page 67
  2.4. Numerical examples......Page 69
   2.4.1. Cantilever beam with a concentrated load......Page 70
   2.4.2. Four-point bending lattice structure......Page 76
  2.5. Concluding remarks......Page 79
 3. Topology Optimization of Meso-structures with Fixed Periodic Microstructures......Page 81
  3.1. Optimization model and procedure......Page 82
   3.2.1. A double-clamped beam......Page 85
   3.2.2. A cantilever beam......Page 88
  3.3. Concluding remarks......Page 90
PART 2: Topology Optimization for Maximizing the Fracture Resistance......Page 91
 4. Topology Optimization for Optimal Fracture Resistance of Quasi-brittle Composites......Page 93
   4.1.1. Phase field approximation of cracks......Page 95
   4.1.2. Thermodynamics of the phase field crack evolution......Page 96
   4.1.3. Weak forms of displacement and phase field problems......Page 99
   4.1.4. Finite element discretization......Page 100
   4.2.1. Model definitions......Page 102
   4.2.2. Sensitivity analysis......Page 104
   4.2.3. Extended BESO method......Page 109
  4.3. Numerical examples......Page 111
   4.3.1. Design of a 2D reinforced plate with one pre-existing crack notch......Page 112
   4.3.2. Design of a 2D reinforced plate with two pre-existing crack notches......Page 117
   4.3.3. Design of a 2D reinforced plate with multiple pre-existing cracks......Page 120
   4.3.4. Design of a 3D reinforced plate with a single pre-existing crack notch surface......Page 122
  4.4. Concluding remarks......Page 125
 5. Topology Optimization for Optimal Fracture Resistance Taking into Account Interfacial Damage......Page 127
   5.1.1. Regularized representation of a discontinuous field......Page 128
   5.1.2. Energy functional......Page 130
   5.1.3. Displacement and phase field problems......Page 132
   5.1.4. Finite element discretization and numerical implementation......Page 135
   5.2.1. Model definitions......Page 138
   5.2.2. Sensitivity analysis......Page 140
  5.3. Numerical examples......Page 143
   5.3.1. Design of a plate with one initial crack under traction......Page 144
   5.3.2. Design of a plate without initial cracks for traction loads......Page 147
   5.3.3. Design of a square plate without initial cracks in tensile loading......Page 149
   5.3.4. Design of a plate with a single initial crack under three-point bending......Page 152
   5.3.5. Design of a plate containing multiple inclusions......Page 154
  5.4. Concluding remarks......Page 157
 6. Topology Optimization for Maximizing the Fracture Resistance of Periodic Composites......Page 159
  6.1. Topology optimization model......Page 160
   6.2.1. Design of a periodic composite under three-point bending......Page 162
   6.2.2. Design of a periodic composite under non-symmetric three-point bending......Page 170
  6.3. Concluding remarks......Page 175
Conclusion......Page 177
References......Page 181
Index......Page 197
Other titles from iSTE in Numerical Methods in Engineering......Page 199
EULA......Page 203