Applied shape optimization for fluids

0199546908......Page 1
Contents......Page 10
1 Introduction......Page 16
2.1 Introduction......Page 21
2.2.1 Minimum weight of structures......Page 22
2.2.2 Wing drag optimization......Page 23
2.2.3 Synthetic jets and riblets......Page 26
2.2.4 Stealth wings......Page 27
2.2.5 Optimal breakwater......Page 30
2.2.6 Two academic test cases: nozzle optimization......Page 31
2.3.1 Topological optimization......Page 32
2.3.2 Suficient conditions for existence......Page 33
2.4.1 Gradient methods......Page 34
2.4.2 Newton methods......Page 35
2.4.3 Constraints......Page 36
2.5 Sensitivity analysis......Page 37
2.5.1 Sensitivity analysis for the nozzle problem......Page 40
2.5.2 Numerical tests with freefem++......Page 42
2.6 Discretization with triangular elements......Page 43
2.6.1 Sensitivity of the discrete problem......Page 45
2.7.1 Independence from the cost function......Page 48
2.7.3 Automatic differentiation......Page 49
2.8.1 Optimal shape design for Stokes flows......Page 50
2.8.2 Optimal shape design for Navier-Stokes flows......Page 51
References......Page 52
3.2.2 Conservation of momentum......Page 56
3.2.3 Conservation of energy and and the law of state......Page 57
3.3 Inviscid flows......Page 58
3.5 Potential flows......Page 59
3.6.2 Reynolds equations......Page 61
3.6.3 The k – ε model......Page 62
3.7 Equations for compressible flows in conservation form......Page 63
3.7.1 Boundary and initial conditions......Page 65
3.8.1 Generalized wall functions for u......Page 66
3.8.2 Wall function for the temperature......Page 68
3.9.1 Pressure correction......Page 69
3.9.2 Corrections on adiabatic walls for compressible flows......Page 70
3.9.3 Prescribing ρ[sub(w)]......Page 71
3.9.4 Correction for the Reichardt law......Page 72
3.10 Wall functions for isothermal walls......Page 73
References......Page 75
4.2.1 Flux schemes and upwinded schemes......Page 76
4.2.2 A FEM-FVM discretization......Page 77
4.2.3 Approximation of the convection fluxes......Page 78
4.2.5 Positivity......Page 79
4.2.6 Time integration......Page 80
4.2.8 Implementation of the boundary conditions......Page 81
4.2.10 Solid walls: implementation of wall laws......Page 82
4.3 Incompressible flows......Page 83
4.3.1 Solution by a projection scheme......Page 84
4.3.2 Spatial discretization......Page 85
4.3.4 Numerical approximations for the k – ε equations......Page 86
4.4.1 Delaunay mesh generator......Page 87
4.4.2 Metric definition......Page 88
4.4.3 Mesh adaptation for unsteady flows......Page 90
4.5 An example of adaptive unsteady flow calculation......Page 92
References......Page 93
5.1 Introduction......Page 96
5.2.2 Complex variables method......Page 98
5.2.4 Adjoint method......Page 99
5.2.5 Adjoint method and Lagrange multipliers......Page 100
5.2.6 Automatic differentiation......Page 101
5.2.7 A class library for the direct mode......Page 103
5.3 Nonlinear PDE and AD......Page 107
5.4 A simple inverse problem......Page 109
5.5 Sensitivity in the presence of shocks......Page 116
5.6 A shock problem solved by AD......Page 118
5.7 Adjoint variable and mesh adaptation......Page 119
5.9 Direct and reverse modes of AD......Page 121
5.10 More on FAD classes......Page 124
References......Page 128
6.2 Shape parameterization and deformation......Page 131
6.2.3 Based on a set of reference shapes......Page 132
6.2.4 CAD-free......Page 133
6.2.5 Level set......Page 137
6.3 Handling domain deformations......Page 142
6.3.1 Explicit deformation......Page 143
6.3.3 Transpiration boundary condition......Page 144
6.3.4 Geometrical constraints......Page 146
6.4 Mesh adaption......Page 148
6.5 Fluide-structure coupling......Page 151
References......Page 153
7.2.1 Examples of local search algorithms......Page 155
7.3 Global optimization......Page 157
7.3.1 Recursive minimization algorithm......Page 158
7.3.2 Coupling dynamical systems and distributed computing......Page 159
7.4 Multi-objective optimization......Page 160
7.4.1 Data mining for multi-objective optimization......Page 163
7.5 Link with genetic algorithms......Page 165
7.6 Reduced-order modeling and learning......Page 168
7.6.1 Data interpolation......Page 169
7.7 Optimal transport and shape optimization......Page 173
References......Page 176
8.1 Introduction......Page 179
8.2.1 Limitations when using AD......Page 180
8.2.2 Storage strategies......Page 181
8.2.3 Key points when using AD......Page 182
8.3.1 Equivalent boundary condition......Page 183
8.3.2 Examples with linear state equations......Page 184
8.3.3 Geometric pressure estimation......Page 186
8.3.5 Multi-level construction......Page 187
8.3.6 Reduced order models and incomplete sensitivities......Page 188
8.3.7 Redefinition of cost functions......Page 189
8.3.9 Incomplete sensitivities and the Hessian......Page 190
8.4 Time-dependent flows......Page 191
8.4.1 Model problem......Page 193
8.4.2 Data mining and adjoint calculation......Page 196
References......Page 198
9.2 Generalities......Page 199
9.3 Consistent approximations......Page 201
9.3.2 Algorithm: conceptual......Page 202
9.4 Application to a control problem......Page 203
9.4.2 Verification of the hypothesis......Page 204
9.5 Application to optimal shape design......Page 205
9.5.1 Problem statement......Page 206
9.5.3 Optimality conditions: the continuous case......Page 207
9.5.4 Optimality conditions: the discrete case......Page 208
9.5.5 Definition of ϑ[sub(h)]......Page 209
9.5.7 Algorithm: OSD with mesh refinement......Page 210
9.5.9 Numerical example......Page 211
9.5.10 A nozzle optimization......Page 212
9.5.11 Theorem......Page 214
9.5.13 Drag reduction for an airfoil with mesh adaptation......Page 215
9.6 Approximate gradients......Page 218
9.6.1 A control problem with domain decomposition......Page 219
9.6.2 Algorithm......Page 220
9.6.3 Numerical results......Page 222
9.8.4 Continuity of ϑ......Page 224
References......Page 225
10.1 Introduction......Page 227
10.3 Four-element airfoil optimization......Page 228
10.4 Sonic boom reduction......Page 230
10.5 Turbomachines......Page 232
10.5.1 Axial blades......Page 234
10.5.2 Radial blades......Page 237
References......Page 240
11.1 Introduction......Page 242
11.2 A model problem for passive noise reduction......Page 243
11.4 Control in multi-disciplinary context......Page 244
11.4.1 A model problem......Page 245
11.4.2 Coupling strategies......Page 251
11.4.3 Low-complexity structure models......Page 252
11.5 Stability, robustness, and unsteadiness......Page 256
11.6 Control of aeroelastic instabilities......Page 259
References......Page 260
12.1 Introduction......Page 261
12.3 Stacking......Page 262
12.5 Design of microfluidic channels......Page 264
12.6 Microfluidic mixing device for protein folding......Page 270
12.7 Flow equations for microfluids......Page 274
12.7.1 Coupling algorithm......Page 275
References......Page 276
13.1 Introduction......Page 278
13.2.1 An example in dimension 2......Page 279
13.3 Solution by penalty......Page 280
13.3.1 A semi-analytical example......Page 282
13.4.1 Application......Page 283
References......Page 285
14 Conclusions and prospectives......Page 287
F......Page 290
O......Page 291
W......Page 292