Mesh Adaptation for Computational Fluid Dynamics, Volume 1

Acknowledgments ix

Introduction xi

Chapter 1 CFD Numerical Models 1

1.1. Compressible flow 1

1.1.1. Introduction 1

1.1.2. Spatial representation 4

1.1.3. Spatial second-order accuracy: MUSCL 13

1.1.4. Low dissipation advection schemes 16

1.1.5. Time advancing 17

1.1.6. Positivity of mixed element-volume formulations 20

1.2. Viscous compressible flows 27

1.2.1. Model for laminar flows 27

1.2.2. Boundary conditions spatial discretization 31

1.2.3. No-slip boundary condition 31

1.2.4. Slip boundary condition 31

1.2.5. Influence stencil 32

1.2.6. Spalart-Allmaras one equation turbulence model 33

1.2.7. SA one-equation model without trip and without f_t2 term 33

1.2.8. "Standard" SA one-equation model (without trip) 35

1.2.9. "Full" SA one-equation model (with trip) 35

1.2.10. Mixed element-volume discretization of SA 35

1.2.11. Implicit time integration 39

1.3. A multi-fluid incompressible model 40

1.3.1. Introduction 40

1.3.2. Bi-fluid incompressible Navier-Stokes equations 40

1.3.3. Finite element approximation 42

1.3.4. Error estimate for the level set advection 44

1.3.5. Provisional conclusion on scheme accuracy 46

1.4. Appendix: circumcenter cells 47

1.4.1. Two-dimensional circumcenter cells 47

1.4.2. Three-dimensional circumcenter cells 48

1.5. Notes 49

Chapter 2 Mesh Convergence and Barriers 51

2.1. Introduction 51

2.2. The early capturing property 53

2.2.1. Smoothness, non-smoothness, heterogeneity 53

2.2.2. Behavior of the uniform-mesh strategy 54

2.2.3. An example of 1D adaptation 56

2.3. Unstructured meshes in finite element method 58

2.3.1. Basics of finite element meshes 58

2.3.2. Anisotropy 59

2.4. Accuracy of an interpolation 60

2.5. Isotropic adaptative interpolation 61

2.5.1. The 2D case 61

2.5.2. A first 3D case 62

2.5.3. A limiting barrier for the isotropic 3D case 64

2.6. Anisotropic adaptative interpolation 64

2.6.1. Anisotropic adaptation of a Heaviside function 64

2.6.2. Heaviside function with curved discontinuity 66

2.7. Numerical illustration: anisotropic versus isotropic interpolation 67

2.8. CFD applications of anisotropic capture 68

2.8.1. Pressure with discontinuous gradient 68

2.8.2. Scramjet flow 68

2.9. Unsteady case 71

2.9.1. Barriers for second-order time-leveled case 72

2.9.2. Barriers for third-order time-leveled case 74

2.10. Conclusion 75

2.11. Notes 76

Chapter 3 Mesh Representation 77

3.1. Introduction 77

3.2. An introductory example 78

3.3. Euclidean metric space 81

3.3.1. Geometric interpretation 84

3.3.2. Natural metric mapping 85

3.4. Riemannian metric space 85

3.5. Generation of adapted anisotropic meshes 90

3.5.1. Unit element 90

3.5.2. Geometric invariants 92

3.5.3. Global duality 95

3.5.4. Quantifying mesh anisotropy 103

3.6. Operations on metrics 104

3.6.1. Metric intersection 104

3.6.2. Metric interpolation 106

3.7. Computation of geometric quantities 108

3.7.1. Computation of lengths 108

3.7.2. Computation of volumes 110

3.8. Notes 110

3.8.1. A short history 110

Chapter 4 Geometric Error Estimate 113

4.1. The 1D case 114

4.1.1. 1D metric 114

4.1.2. P 1 Interpolation error bound 115

4.1.3. 1D optimal metric 116

4.1.4. Convergence order of the continuous metric model 118

4.2. Discrete-continuous duality for linear interpolation error 120

4.2.1. Interpolation error in L ¹ norm for quadratic functions 121

4.2.2. Linear interpolation on a continuous element 124

4.2.3. Continuous linear interpolate 126

4.3. Numerical validation of the continuous interpolation error 133

4.3.1. Continuous interpolation error calculation 134

4.3.2. Comparison with discrete interpolation error computation 138

4.3.3. Three-dimensional validation 142

4.3.4. Some conclusions 146

4.4. Optimal control of the interpolation error in L p norm 147

4.4.1. Formal resolution 147

4.4.2. Uniqueness 150

4.4.3. Optimal orientations and main result 151

4.5. Multidimensional discontinuity capturing 154

4.6. Linear interpolate operator 155

4.7. A local L 8 upper bound of the interpolation error 156

4.8. Metric construction for mesh adaptation 159

4.8.1. Handling degenerated cases 161

4.8.2. Isotropic mesh adaptation 162

4.9. Mesh adaptation for analytical functions 162

4.9.1. Algorithms 162

4.9.2. Examples of L 8 adaptation 163

4.10. Conclusion 170

4.11. Notes 171

Chapter 5 Multiscale Adaptation for Steady Simulations 173

5.1. Introduction 173

5.2. Definitions and notations (2D) 174

5.3. Solving the problematic of the unknown solution (2D/3D) 176

5.4. Numerical computation/recovery of the Hessian matrix 179

5.4.1. Numerical computation of nodal gradients (2D) 179

5.4.2. A double L ² -projection method 180

5.4.3. A method based on the Green formula 181

5.4.4. A least-square approach 181

5.4.5. From our experience 183

5.4.6. Discrete-continuous interpolation 183

5.5. Solution interpolation 183

5.5.1. Localization algorithm 183

5.5.2. Classical polynomial interpolation 187

5.6. Mesh adaptation algorithm 189

5.7. Example of a CFD numerical simulation 190

5.8. Conclusion 191

5.9. Notes 191

5.9.1. A short review of mesh/PDE coupling 191

Chapter 6 Multiscale Convergence and Certification in CFD 195

6.1. Introduction 195

6.2. A mesh convergence algorithm 197

6.2.1. Mesh adaptation with a fixed complexity 198

6.2.2. Transfers and numerical convergence 199

6.3. An academic test case 201

6.3.1. Uniform refinement study 201

6.3.2. Isotropic adaptation study 203

6.3.3. Anisotropic adaptation study 203

6.3.4. Error level 204

6.4. 3D multiscale anisotropic mesh adaptation 205

6.5. Conclusion 206

6.6. Notes 208

References 211

Index 225

Summary of Volume 2 227