Mesh Adaptation for Computational Dynamics 2 is the second of two volumes and introduces topics including optimal control formulation, minimizing a goal function, and extending the steady algorithm to unsteady physics. Also covered are multi-rate strategies, steady inviscid flows in aeronautics and an extension to viscous flows.
This book will be useful to anybody interested in mesh adaptation pertaining to CFD, especially researchers, teachers and students.
Table of Contents
Acknowledgments ix
Introduction xi
Chapter 1 Nonlinear Corrector for CFD 1
1.1. Introduction 1
1.1.1. Linear correction 3
1.1.2. Nonlinear correction 4
1.2. Two correctors for the Poisson problem 5
1.2.1. Notations 5
1.2.2. A priori corrector for the PDE solution 6
1.2.3. Finer-grid DC corrector for the PDE solution 8
1.3. RANS equations 9
1.3.1. Vector form of the RANS system 9
1.3.2. Formal discretization 10
1.3.3. Notations for discretization 11
1.4. Nonlinear functional correction 13
1.4.1. Finite volume nonlinear corrector 13
1.4.2. Finite element corrector 15
1.5. Example: supersonic flow 17
1.6. Concluding remarks 18
1.7. Notes 20
Chapter 2 Multi-scale Adaptation for Unsteady Flows 21
2.1. Introduction 21
2.2. Mesh adaptation efficiency 23
2.2.1. Regular and singular unsteady model 23
2.2.2. Representativity of the spatial interpolation error 24
2.3. Transient fixed-point mesh adaptation scheme 25
2.3.1. Size of subintervals in a mesh convergence 28
2.3.2. Mesh adaptation for unsteady Euler/Navier-Stokes equations with thickened interface 29
2.3.3. Convergent transient fixed-point 33
2.4. 2D bi-fluid example 33
2.5. Example: impact of a 3D water column on a obstacle 35
2.6. Conclusion 39
2.7. Appendix: remarks about the adaptation of the time step 39
2.8. Notes 41
Chapter 3 Multi-rate Time Advancing 43
3.1. Introduction 43
3.2. Multi-rate time advancing by volume agglomeration 45
3.2.1. Finite volume Navier-Stokes 45
3.2.2. Inner and outer zones 46
3.2.3. MR time advancing 47
3.3. Elements of analysis 49
3.3.1. Stability 49
3.3.2. Accuracy 50
3.3.3. Efficiency 51
3.3.4. Toward many rates 52
3.3.5. Impact of our MR complexity on mesh adaption 52
3.3.6. Parallelism 53
3.4. Applications 55
3.4.1. Circular cylinder at very high Reynolds number 55
3.4.2. Mesh adaption for a contact discontinuity 58
3.5. Conclusion 59
3.6. Notes 60
Chapter 4 Goal-Oriented Adaptation for Inviscid Steady Flows 65
4.1. Introduction 65
4.1.1. What to do with this estimate? 67
4.1.2. Adjoint-L 1 approach 68
4.1.3. Outline 69
4.2. A more accurate nonlinear error analysis 69
4.2.1. Assumptions and definitions 69
4.2.2. A priori estimation 70
4.3. The case of the steady Euler equations 72
4.3.1. Variational analysis 72
4.3.2. Approximation error estimation 73
4.4. Error model minimization 74
4.5. Adaptative strategy 76
4.5.1. Adjoint solver 77
4.5.2. Optimal goal-oriented discrete metric 77
4.5.3. Controlled mesh regeneration 79
4.6. Numerical outputs 79
4.6.1. High-fidelity pressure prediction of an aircraft 79
4.7. Conclusion 82
4.8. Notes 82
Chapter 5. Goal-Oriented Adaptation for Viscous Steady Flows 85
5.1. Introduction 85
5.2. Case of an elliptic problem 86
5.2.1. A priori finite-element analysis (first estimate) 86
5.2.2. Goal-oriented adaptation according to lemma 5.1 89
5.2.3. Goal-oriented adaptation according to a second estimate 91
5.3. Error analysis for Navier-Stokes problem 92
5.3.1. Mesh adaptation problem statement 92
5.3.2. Linearized error system 93
5.3.3. First estimate for Navier-Stokes problem 94
5.3.4. Second estimate for Navier-Stokes problem 98
5.3.5. Optimal goal-oriented continuous mesh 101
5.4. From theory to practice 101
5.4.1. Computation of the optimal continuous mesh 103
5.5. An example of application to a turbulent flow 103
5.6. Conclusion 107
5.7. Notes 109
Chapter 6 Norm-Oriented Formulations 111
6.1. Introduction 111
6.2. A summary of previous analyses 114
6.2.1. Feature-based adaptation by interpolation error optimization 114
6.2.2. Implicit a priori error estimate and corrector 115
6.2.3. Goal-oriented analysis 116
6.3. Norm-oriented approach 118
6.4. Numerical elliptic examples 119
6.4.1. Numerical features 119
6.4.2. 2D boundary layer 122
6.4.3. Poisson problem with discontinuous coefficient 123
6.5. Application to flows 126
6.5.1. A comparison feature-oriented/norm 127
6.5.2. Application to a viscous flow 129
6.6. Conclusion 130
6.7. Notes 131
Chapter 7 Goal-Oriented Adaptation for Unsteady Flows 133
7.1. Introduction 133
7.2. Formal error analysis 134
7.3. Unsteady Euler models 135
7.3.1. Continuous state system and finite volume formulation 135
7.3.2. Continuous adjoint system and discretization 137
7.3.3. Impact of the adjoint: numerical example 141
7.4. Optimal unsteady adjoint-based metric 142
7.4.1. Error analysis for the unsteady Euler model 142
7.4.2. Continuous error model 144
7.4.3 Spatial minimization for a fixed t 146
7.4.4. Temporal minimization 146
7.4.5. Temporal minimization for time sub-intervals 150
7.5. Theoretical mesh convergence analysis 155
7.5.1. Smooth flow fields 155
7.6. From theory to practice 157
7.6.1. Choice of the GO metric 158
7.6.2. Global fixed-point mesh adaptation algorithm 158
7.6.3. Computing the GO metric 161
7.7. Numerical experiments 161
7.7.1. 2D Acoustic wave propagation 161
7.7.2. 3D blast wave propagation 163
7.8. Conclusion 165
7.9. Notes 166
Chapter 8 Third-Order Unsteady Adaptation 167
8.1. Introduction 167
8.2. Higher order interpolation and reconstruction 168
8.3. CENO approximation for the 2D Euler equations 170
8.3.1. Model 170
8.3.2. CENO formulation 171
8.3.3. Vertex-centered low dissipation CENO 2 174
8.4. Error analysis 175
8.5. Metric-based error estimate 178
8.6. Optimal metric 179
8.7. From theory to practical application 182
8.8. A numerical example: acoustic wave 183
8.9. Conclusion 186
8.10. Notes 186
References 189
Index 199
Summary of Volume 1 201
