1 Introduction ................................................. 1
2 Optimal shape design ......................................... 6
2.1 Introduction ............................................ 6
2.2 Examples ................................................ 7
2.2.1 Minimum weight of structures ..................... 7
2.2.2 Wing drag optimization ........................... 8
2.2.3 Synthetic jets and riblets ...................... 11
2.2.4 Stealth wings ................................... 12
2.2.5 Optimal breakwater .............................. 15
2.2.6 Two academic test cases: nozzle optimization .... 16
2.3 Existence of solutions ................................. 17
2.3.1 Topological optimization ........................ 17
2.3.2 Sufficient conditions for existence ............. 18
2.4 Solution by optimization methods ....................... 19
2.4.1 Gradient methods ................................ 19
2.4.2 Newton methods .................................. 20
2.4.3 Constraints ..................................... 21
2.4.4 A constrained optimization algorithm ............ 22
2.5 Sensitivity analysis ................................... 22
2.5.1 Sensitivity analysis for the nozzle problem ..... 25
2.5.2 Numerical tests with freefem++ .................. 27
2.6 Discretization with triangular elements ................ 28
2.6.1 Sensitivity of the discrete problem ............. 30
2.7 Implementation and numerical issues .................... 33
2.7.1 Independence from the cost function ............. 33
2.7.2 Addition of geometrical constraints ............. 34
2.7.3 Automatic differentiation ....................... 34
2.8 Optimal design for Navier-Stokes flows ................. 35
2.8.1 Optimal shape design for Stokes flows ........... 35
2.8.2 Optimal shape design for Navier-Stokes flows .... 36
References .................................................. 37
3 Partial differential equations for fluids ................... 41
3.1 Introduction ........................................... 41
3.2 The Navier-Stokes equations ............................ 41
3.2.1 Conservation of mass ............................ 41
3.2.2 Conservation of momentum ........................ 41
3.2.3 Conservation of energy and and the law of
state ........................................... 42
3.3 Inviscid flows ......................................... 43
3.4 Incompressible flows ................................... 44
3.5 Potential flows ........................................ 44
3.6 Turbulence modeling .................................... 46
3.6.1 The Reynolds number ............................. 46
3.6.2 Reynolds equations .............................. 46
3.6.3 The к - ε model ................................. 47
3.7 Equations for compressible flows in conservation
form ................................................... 48
3.7.1 Boundary and initial conditions ................. 50
3.8 Wall laws .............................................. 51
3.8.1 Generalized wall functions for u ................ 51
3.8.2 Wall function for the temperature ............... 53
3.8.3 к and ε ......................................... 54
3.9 Generalization of wall functions ....................... 54
3.9.1 Pressure correction ............................. 54
3.9.2 Corrections on adiabatic walls for
compressible flows .............................. 55
3.9.3 Prescribing ρω .................................. 56
3.9.4 Correction for the Reichardt law ................ 57
3.10 Wall functions for isothermal walls .................... 58
References .................................................. 60
4 Some numerical methods for fluids ........................... 61
4.1 Introduction ........................................... 61
4.2 Numerical methods for compressible flows ............... 61
4.2.1 Flux schemes and upwinded schemes ............... 61
4.2.2 A FEM-FVM discretization ........................ 62
4.2.3 Approximation of the convection fluxes .......... 63
4.2.4 Accuracy improvement ............................ 64
4.2.5 Positivity ...................................... 64
4.2.6 Time integration ................................ 65
4.2.7 Local time stepping procedure ................... 66
4.2.8 Implementation of the boundary conditions ....... 66
4.2.9 Solid walls: transpiration boundary condition ... 67
4.2.10 Solid walls: implementation of wall laws ........ 67
4.3 Incompressible flows ................................... 68
4.3.1 Solution by a projection scheme ................. 69
4.3.2 Spatial discretization .......................... 70
4.3.3 Local time stepping ............................. 71
4.3.4 Numerical approximations for the k - ε
equations ....................................... 71
4.4 Mesh adaptation ........................................ 72
4.4.1 Delaunay mesh generator ......................... 72
4.4.2 Metric definition ............................... 73
4.4.3 Mesh adaptation for unsteady flows .............. 75
4.5 An example of adaptive unsteady flow calculation ....... 77
References .................................................. 78
5 Sensitivity evaluation and automatic differentiation ........ 81
5.1 Introduction ........................................... 81
5.2 Computations of derivatives ............................ 83
5.2.1 Finite differences .............................. 83
5.2.2 Complex variables method ........................ 83
5.2.3 State equation linearization .................... 84
5.2.4 Adjoint method .................................. 84
5.2.5 Adjoint method and Lagrange multipliers ......... 85
5.2.6 Automatic differentiation ....................... 86
5.2.7 A class library for the direct mode ............. 88
5.3 Nonlinear PDE and AD ................................... 92
5.4 A simple inverse problem ............................... 94
5.5 Sensitivity in the presence of shocks ................. 101
5.6 A shock problem solved by AD .......................... 103
5.7 Adjoint variable and mesh adaptation .................. 104
5.8 Tapenade .............................................. 106
5.9 Direct and reverse modes of AD ........................ 106
5.10 More on FAD classes ................................... 109
References ................................................. 113
6 Parameterization and implementation issues ................. 116
6.1 Introduction .......................................... 116
6.2 Shape parameterization and deformation ................ 116
6.2.1 Deformation parameterization ................... 117
6.2.2 CAD-based ...................................... 117
6.2.3 Based on a set of reference shapes ............. 117
6.2.4 CAD-free ....................................... 118
6.2.5 Level set ...................................... 122
6.3 Handling domain deformations .......................... 127
6.3.1 Explicit deformation ........................... 128
6.3.2 Adding an elliptic system ...................... 129
6.3.3 Transpiration boundary condition ............... 129
6.3.4 Geometrical constraints ........................ 131
6.4 Mesh adaption ......................................... 133
6.5 Fluide-structure coupling ............................. 136
References ................................................. 138
7 Local and global optimization .............................. 140
7.1 Introduction .......................................... 140
7.2 Dynamical systems ..................................... 140
7.2.1 Examples of local search algorithms ............ 140
7.3 Global optimization ................................... 142
7.3.1 Recursive minimization algorithm ............... 143
7.3.2 Coupling dynamical systems and distributed
computing ...................................... 144
7.4 Multi-objective optimization .......................... 145
7.4.1 Data mining for multi-objective optimization ... 148
7.5 Link with genetic algorithms .......................... 150
7.6 Reduced-order modeling and learning ................... 153
7.6.1 Data interpolation ............................. 154
7.7 Optimal transport and shape optimization .............. 158
References ................................................. 161
8 Incomplete sensitivities ................................... 164
8.1 Introduction .......................................... 164
8.2 Efficiency with AD .................................... 165
8.2.1 Limitations when using AD ...................... 165
8.2.2 Storage strategies ............................. 166
8.2.3 Key points when using AD ....................... 167
8.3 Incomplete sensitivity ................................ 168
8.3.1 Equivalent boundary condition .................. 168
8.3.2 Examples with linear state equations ........... 169
8.3.3 Geometric pressure estimation .................. 171
8.3.4 Wall functions ................................. 172
8.3.5 Multi-level construction ....................... 172
8.3.6 Reduced order models and incomplete
sensitivities .................................. 173
8.3.7 Redefinition of cost functions ................. 174
8.3.8 Multi-criteria problems ........................ 175
8.3.9 Incomplete sensitivities and the Hessian ....... 175
8.4 Time-dependent flows .................................. 176
8.4.1 Model problem .................................. 178
8.4.2 Data mining and adjoint calculation ............ 181
References ................................................. 183
9 Consistent approximations and approximate gradients ........ 184
9.1 Introduction .......................................... 184
9.2 Generalities .......................................... 184
9.3 Consistent approximations ............................. 186
9.3.1 Consistent approximation ....................... 187
9.3.2 Algorithm: conceptual .......................... 187
9.4 Application to a control problem ...................... 188
9.4.1 Algorithm: control with mesh refinement ........ 189
9.4.2 Verification of the hypothesis ................. 189
9.4.3 Numerical example .............................. 190
9.5 Application to optimal shape design ................... 190
9.5.1 Problem statement .............................. 191
9.5.2 Discretization ................................. 192
9.5.3 Optimality conditions: the continuous case ..... 192
9.5.4 Optimality conditions: the discrete case ....... 193
9.5.5 Definition of θh ............................... 194
9.5.6 Implementation trick ........................... 195
9.5.7 Algorithm: OSD with mesh refinement ............ 195
9.5.8 Orientation .................................... 196
9.5.9 Numerical example .............................. 196
9.5.10 A nozzle optimization .......................... 197
9.5.11 Theorem ........................................ 199
9.5.12 Numerical results .............................. 200
9.5.13 Drag reduction for an airfoil with mesh
adaptation ..................................... 200
9.6 Approximate gradients ................................. 203
9.6.1 A control problem with domain decomposition .... 204
9.6.2 Algorithm ...................................... 205
9.6.3 Numerical results .............................. 207
9.7 Conclusion ............................................ 209
9.8 Hypotheses in Theorem 9.3.2.1 ......................... 209
9.8.1 Inclusion ...................................... 209
9.8.2 Continuity ..................................... 209
9.8.3 Consistency .................................... 209
9.8.4 Continuity of θ ................................ 209
9.8.5 Continuity of θh(αh) ........................... 210
9.8.6 Convergence .................................... 210
References ................................................. 210
10 Numerical results on shape optimization .................... 212
10.1 Introduction .......................................... 212
10.2 External flows around airfoils ........................ 213
10.3 Four-element airfoil optimization ..................... 213
10.4 Sonic boom reduction .................................. 215
10.5 Turbomachines ......................................... 217
10.5.1 Axial blades ................................... 219
10.5.2 Radial blades .................................. 222
10.6 Business jet: impact of state evaluations ............. 225
References ................................................. 225
11 Control of unsteady flows .................................. 227
11.1 Introduction .......................................... 227
11.2 A model problem for passive noise reduction ........... 228
11.3 Control of aerodynamic instabilities around rigid
bodies ................................................ 229
11.4 Control in multi-disciplinary context ................. 229
11.4.1 A model problem ................................ 230
11.4.2 Coupling strategies ............................ 236
11.4.3 Low-complexity structure models ................ 237
11.5 Stability, robustness, and unsteadiness ............... 241
11.6 Control of aeroelastic instabilities .................. 244
References ................................................. 245
12 From airplane design to microfluidics ...................... 246
12.1 Introduction .......................................... 246
12.2 Governing equations for microfluids ................... 247
12.3 Stacking .............................................. 247
12.4 Control of the extraction of infinitesimal
quantities ............................................ 249
12.5 Design of microfluidic channels ....................... 249
12.5.1 Reduced models for the flow .................... 255
12.6 Microfluidic mixing device for protein folding ........ 255
12.7 Flow equations for microfluids ........................ 259
12.7.1 Coupling algorithm ............................. 260
References ................................................. 261
13 Topological optimization for fluids ........................ 263
13.1 Introduction .......................................... 263
13.2 Dirichlet conditions on a shrinking hole .............. 264
13.2.1 An example in dimension 2 ...................... 264
13.3 Solution by penalty ................................... 265
13.3.1 A semi-analytical example ...................... 267
13.4 Topological derivatives for fluids .................... 268
13.4.1 Application .................................... 268
13.5 Perspective ........................................... 270
References ............................................ 270
14 Conclusions and prospectives ............................... 272
Index ......................................................... 275
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