Part I. Generalities and methods
1. Introduction ................................................. 3
Dominique Thevénin
References .................................................. 16
2. A Few Illustrative Examples of CFD-based Optimization ....... 17
Gábor Janiga
2.1. Introduction ........................................... 18
2.1.1. Purpose ......................................... 18
2.1.2. Heat Exchanger Optimization (Case A) ............ 19
2.1.3. Optimization Coupled with Chemical Reactions
(Case B) ........................................ 21
2.1.4. Determination of Turbulence Model Parameters
Based on Optimization (Case C) .................. 22
2.2. Evolutionary Algorithms for Multi-objective
Optimization ........................................... 23
2.2.1. Multi-objective Optimization .................... 23
2.2.2. The Concept of Pareto Dominance ................. 25
2.2.3. Evolutionary Algorithm for Multi-objective
Problems ........................................ 26
2.3. The Optimal Position of the Tubes in a Heat
Exchanger (Case A) ..................................... 28
2.3.1. Tube Bank Heat Exchanger ........................ 28
2.3.2. Problem Parameters .............................. 29
2.3.3. Opal (Optimization ALgorithms) Package .......... 29
2.3.4. Evaluation of the Objectives for Case A ......... 30
2.3.5. Parallelization ................................. 33
2.3.6. Computational Results ........................... 34
2.4. Multi-objective Optimization of a Laminar Burner
(Case B) ............................................... 38
2.4.1. Governing Equations ............................. 38
2.4.2. Numerical Solution .............................. 40
2.4.3. Optimization of the Laminar Burner .............. 42
2.5. Optimization of the Standard k-ω Turbulence
Model Parameters (Case C) .............................. 46
2.5.1. Governing Equations ............................. 48
2.5.2. Numerical Results ............................... 50
2.6. Conclusions ............................................ 52
References .................................................. 56
3. Mathematical Aspects of CFD-based Optimization .............. 61
Hans Georg Bock and Volker Schulz
3.1. Introduction ........................................... 62
3.2. Simultaneous Model-based Optimization .................. 63
3.2.1. Sequential Quadratic Programming (SQP) .......... 63
3.2.2. Modular SQP Methods ............................. 65
3.2.3. Multiple Set-point Optimization ................. 69
3.2.4. Multigrid Optimization .......................... 70
3.3. Unsteady Problems ...................................... 72
3.3.1. Time-domain Decomposition by Multiple
Shooting ........................................ 73
3.3.2. Parallel Multiple Shooting ...................... 75
3.3.3. Real-time Optimization and Nonlinear Model
Predictive Control .............................. 75
3.3.4. Sensitivity Driven Multiple Shooting ............ 76
References .................................................. 77
4. Adjoint Methods for Shape Optimization ...................... 79
Kyriakos C. Giannakoglou and Dimitrios I. Papadimitriou
4.1. Introduction ........................................... 80
4.2. Principles of the Adjoint Approach ..................... 83
4.2.1. The Discrete Adjoint Approach ................... 83
4.2.2. The Continuous Adjoint Approach ................. 84
4.2.3. Differences Between Discrete and Continuous
Adjoint ......................................... 85
4.3. Inverse Design Using the Euler Equations ............... 86
4.4. Inverse Design Using the Navier-Stokes Equations ....... 90
4.5. Viscous Losses Minimization in Internal Flows .......... 91
4.5.1. Minimization of Total Pressure Losses ........... 92
4.5.2. Minimization of Entropy Generation .............. 93
4.6. Computation of the Hessian Matrix ...................... 94
4.6.1. Discrete Direct-adjoint Approach for the
Hessian ......................................... 95
4.6.2. Continuous Direct-adjoint Approach for the
Hessian (Inverse Design) ........................ 96
4.7. Applications ........................................... 97
4.7.1. Gradient and Hessian-based Inverse Design
of a 2D Duct .................................... 97
4.7.2. Losses Minimization of a 2D Compressor
Cascade ......................................... 99
4.8. Conclusions ........................................... 104
References ................................................. 106
Part II. Specific Applications of CFD-based Optimization to
Engineering Problems
5. Efficient Deterministic Approaches for Aerodynamic Shape
Optimization ............................................... 111
Nicolas R. Gauger
5.1. Introduction .......................................... 112
5.2. Parameterization by Deformation ....................... 113
5.2.1. Surface Deformation ............................ 114
5.2.2. Grid Deformation ............................... 115
5.3. Sensitivity-based Aerodynamic Shape Optimization ...... 117
5.4. Sensitivity Computations .............................. 119
5.4.1. Finite Difference Method ....................... 119
5.4.2. Continuous Adjoint Formulation ................. 120
5.4.3. Algorithmic Differentiation (AD) ............... 122
5.5. Adjoint Flow Solvers .................................. 124
5.5.1. Continuous Adjoint Flow Solvers ................ 124
5.5.2. Discrete Adjoint Flow Solvers .................. 125
5.6. Automatic Differentiation Applied to an Entire
Design Chain .......................................... 126
5.6.1. Test Case Definition ........................... 127
5.6.2. Finite Differences ............................. 127
5.6.3. Automatic Differentiation ...................... 132
5.7. Adjoint Approch for Aero-Structure Coupling ........... 133
5.7.1. Adjoint Formulation for Aero-Structure
Coupling .............................................. 133
5.7.2. Implementation ................................. 140
5.7.3. Validation and Application ..................... 141
5.8. One-shot Methods ...................................... 142
References ................................................. 144
6. Numerical Optimization for Advanced Turbomachinery
Design ..................................................... 147
Réne A. Van den Braembussche
6.1. Introduction .......................................... 147
6.2. Optimization Methods .................................. 150
6.2.1. Search Mechanisms .............................. 150
6.2.2. Objective Function ............................. 156
6.2.3 Parameterization ........................... 159
6.3. Two-level Optimization ................................ 160
6.3.1. Artificial Neural Networks ..................... 162
6.3.2. Database ....................................... 164
6.4. Single Point Optimization of Turbine Blade ............ 166
6.4.1. 2D Blade Geometry Definition ................... 166
6.4.2. Penalty for Non-optimum Mach Number
Distribution PMach .............................. 168
6.4.3. Design of a Transonic Turbine Blade ............ 170
6.5. Multipoint Optimization of a Low Solidity Diffuser .... 172
6.6. Multidisciplinary Optimization ........................ 175
6.6.1. 3D Geometry Definition ......................... 176
6.6.2. Multidisciplinary Objective Function ........... 178
6.6.3. Design Conditions and Results .................. 180
6.7. Conclusions ........................................... 187
References ................................................. 188
7. CFD-based Optimization for Automotive Aerodynamics ......... 191
Laurent Dumas
7.1. Introducing Automotive Aerodynamics ................... 192
7.1.1. A Major Concern for Car Manufacturers .......... 192
7.1.2. Experiments on Bluff Bodies .................... 192
7.1.3. Wake Flow Behind a Bluff Body .................. 193
7.1.4. Drag Variation with the Slant Angle ............ 194
7.2. The Drag Reduction Problem ............................ 195
7.2.1. Drag Reduction in the Automotive Industry ...... 196
7.2.2. Numerical Modelization ......................... 197
7.3. Fast and Global Optimization Methods .................. 199
7.3.1. Evolutionary Algorithms ........................ 199
7.3.2. Adaptive Hybrid Methods (AHM) .................. 201
7.3.3. Genetic Algorithms with Approximated
Evaluations (AGA) .............................. 203
7.3.4. Validation on Analytic Test Functions .......... 205
7.4. Car Drag Reduction with Numerical Optimization ........ 207
7.4.1. Description of the Test Case ................... 207
7.4.2. Details of the Numerical Simulation ............ 207
7.4.3. Numerical Results .............................. 209
7.5. Another Possible Application of CFD-O:
Airplane Engines ...................................... 212
7.5.1. General Description of the Optimization Case ... 212
7.5.2. Details of the Computation ..................... 213
7.5.3. Obtained Results ............................... 213
7.6. Conclusion ............................................ 214
References ................................................. 214
8. Multi-objective Optimization for Problems Involving
Convective Heat Transfer ................................... 217
Marco Manzan, Enrico Nobile, Stefano Pieri and
Francesco Pinto
8.1. Introduction .......................................... 218
8.2. Literature Review ..................................... 219
8.3. Problem Statement ..................................... 222
8.3.1. Governing Equations ............................ 223
8.3.2. Fluid Dynamic Boundary Conditions .............. 224
8.3.3. Temperature Boundary Conditions ................ 225
8.4. Numerical Methods ..................................... 228
8.4.1. Fluid Dynamic Iterative Solution ............... 228
8.4.2. Thermal Field Iterative Solution ............... 229
8.5. Geometry Parametrization .............................. 231
8.5.1. Wavy Channels .................................. 231
8.5.2. CC Module ...................................... 234
8.6. Optimization Methods .................................. 235
8.6.1. Design of Experiment ........................... 238
8.7. Optimization Algorithms ............................... 239
8.7.1. Genetic Algorithm .............................. 242
8.7.2. Multi-objective Approaches ..................... 243
8.7.3. Multi-Criteria Decision Making (MCDM) .......... 246
8.7.4. Optimization Process ........................... 247
8.8. Results and Discussion ................................ 249
8.8.1. Linear Piecewise Optimization .................. 249
8.8.2. NURBS Optimization ............................. 250
8.8.3. Linear Piecewise versus NURBS .................. 255
8.8.4. Three-dimensional Analysis ..................... 256
8.8.5. CC Module ...................................... 258
8.9. Concluding Remarks .................................... 262
References ................................................. 263
9. CFD-based Optimization for a Complete Industrial
Process: Papermaking ....................................... 267
Jari Hämäläinen, Taija Hämäläinen, Elina Madetoja and
Henri Ruotsalainen
9.1. Introduction .......................................... 267
9.2. Optimal Shape Design of the Tapered Header ............ 269
9.3. Optimal Control of the Fiber Orientation in the
Slice Channel ......................................... 272
9.3.1. On Modeling Fiber Orientation .................. 272
9.3.2. HOCS Fiber - A Trouble Shooting Tool ........... 273
9.3.3. Depth-averaged Navier-Stokes Equations ......... 274
9.3.4. Validation of the Depth-averaged Navier-
Stokes Equations ............................... 276
9.4. Multi-objective Optimization of Papermaking ........... 278
9.4.1. Multi-objective Optimization ................... 279
9.4.2. Modeling and Optimizing the Complete
Papermaking Process ............................ 281
9.4.3. Numerical Examples ............................. 284
9.5. Towards Decision Support Systems ...................... 286
9.6. Conclusions ........................................... 287
References ................................................. 288
Index ......................................................... 291
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