Simulation technology, and computational fluid dynamics (CFD) in particular, is essential in the search for solutions to the modern challenges faced by humanity. Revolutions in CFD over the last decade include the use of unstructured meshes, permitting the modeling of any 3D geometry. New frontiers point to mesh adaptation, allowing not only seamless meshing (for the engineer) but also simulation certification for safer products and risk prediction.
Mesh Adaptation for Computational Dynamics 1 is the first of two volumes and introduces basic methods such as feature-based and multiscale adaptation for steady models. Also covered is the continuous Riemannian metrics formulation which models the optimally adapted mesh problem into a pure partial differential statement. A number of mesh adaptative methods are defined based on a particular feature of the simulation solution.
This book will be useful to anybody interested in mesh adaptation pertaining to CFD, especially researchers, teachers and students.
Author(s): Alain Dervieux, Frédéric Alauzet, Adrien Loseille, Bruno Koobus
Series: Numerical Methods in Engineering Series
Publisher: Wiley-ISTE
Year: 2022
Language: English
Pages: 245
City: London
Cover
Half-Title Page
Title Page
Copyright Page
Contents
Acknowledgments
Introduction
1. CFD Numerical Models
1.1. Compressible flow
1.1.1. Introduction
1.1.2. Spatial representation
1.1.3. Spatial second-order accuracy: MUSCL
1.1.4. Low dissipation advection schemes
1.1.5. Time advancing
1.1.6. Positivity of mixed element-volume formulations
1.2. Viscous compressible flows
1.2.1. Model for laminar flows
1.2.2. Boundary conditions spatial discretization
1.2.3. No-slip boundary condition
1.2.4. Slip boundary condition
1.2.5. Influence stencil
1.2.6. Spalart–Allmaras one equation turbulence model
1.2.7. SA one-equation model without trip and without ft2 term
1.2.8. “Standard” SA one-equation model (without trip)
1.2.9. “Full” SA one-equation model (with trip)
1.2.10. Mixed element-volume discretization of SA
1.2.11. Implicit time integration
1.3. A multi-fluid incompressible model
1.3.1. Introduction
1.3.2. Bi-fluid incompressible Navier–Stokes equations
1.3.3. Finite element approximation
1.3.4. Error estimate for the level set advection
1.3.5. Provisional conclusion on scheme accuracy
1.4. Appendix: circumcenter cells
1.4.1. Two-dimensional circumcenter cells
1.4.2. Three-dimensional circumcenter cells
1.5. Notes
2. Mesh Convergence and Barriers
2.1. Introduction
2.2. The early capturing property
2.2.1. Smoothness, non-smoothness, heterogeneity
2.2.2. Behavior of the uniform-mesh strategy
2.2.3. An example of 1D adaptation
2.3. Unstructured meshes in finite element method
2.3.1. Basics of finite element meshes
2.3.2. Anisotropy
2.4. Accuracy of an interpolation
2.5. Isotropic adaptative interpolation
2.5.1. The 2D case
2.5.2. A first 3D case
2.5.3. A limiting barrier for the isotropic 3D case
2.6. Anisotropic adaptative interpolation
2.6.1. Anisotropic adaptation of a Heaviside function
2.6.2. Heaviside function with curved discontinuity
2.7. Numerical illustration: anisotropic versus isotropic interpolation
2.8. CFD applications of anisotropic capture
2.8.1. Pressure with discontinuous gradient
2.8.2. Scramjet flow
2.9. Unsteady case
2.9.1. Barriers for second-order time-leveled case
2.9.2. Barriers for third-order time-leveled case
2.10. Conclusion
2.11. Notes
3. Mesh Representation
3.1. Introduction
3.2. An introductory example
3.3. Euclidean metric space
3.3.1. Geometric interpretation
3.3.2. Natural metric mapping
3.4. Riemannian metric space
3.5. Generation of adapted anisotropic meshes
3.5.1. Unit element
3.5.2. Geometric invariants
3.5.3. Global duality
3.5.4. Quantifying mesh anisotropy
3.6. Operations on metrics
3.6.1. Metric intersection
3.6.2. Metric interpolation
3.7. Computation of geometric quantities
3.7.1. Computation of lengths
3.7.2. Computation of volumes
3.8. Notes
3.8.1. A short history
4. Geometric Error Estimate
4.1. The 1D case
4.1.1. 1D metric
4.1.2. P1 Interpolation error bound
4.1.3. 1D optimal metric
4.1.4. Convergence order of the continuous metric model
4.2. Discrete-continuous duality for linear interpolation error
4.2.1. Interpolation error in L1 norm for quadratic functions
4.2.2. Linear interpolation on a continuous element
4.2.3. Continuous linear interpolate
4.3. Numerical validation of the continuous interpolation error
4.3.1. Continuous interpolation error calculation
4.3.2. Comparison with discrete interpolation error computation
4.3.3. Three-dimensional validation
4.3.4. Some conclusions
4.4. Optimal control of the interpolation error in Lp norm
4.4.1. Formal resolution
4.4.2. Uniqueness
4.4.3. Optimal orientations and main result
4.5. Multidimensional discontinuity capturing
4.6. Linear interpolate operator
4.7. A local L8 upper bound of the interpolation error
4.8. Metric construction for mesh adaptation
4.8.1. Handling degenerated cases
4.8.2. Isotropic mesh adaptation
4.9. Mesh adaptation for analytical functions
4.9.1. Algorithms
4.9.2. Examples of L8 adaptation
4.10. Conclusion
4.11. Notes
5. Multiscale Adaptation for Steady Simulations
5.1. Introduction
5.2. Definitions and notations (2D)
5.3. Solving the problematic of the unknown solution (2D/3D)
5.4. Numerical computation/recovery of the Hessian matrix
5.4.1. Numerical computation of nodal gradients (2D)
5.4.2. A double L2-projection method
5.4.3. A method based on the Green formula
5.4.4. A least-square approach
5.4.5. From our experience
5.4.6. Discrete-continuous interpolation
5.5. Solution interpolation
5.5.1. Localization algorithm
5.5.2. Classical polynomial interpolation
5.6. Mesh adaptation algorithm
5.7. Example of a CFD numerical simulation
5.8. Conclusion
5.9. Notes
5.9.1. A short review of mesh/PDE coupling
6. Multiscale Convergence and Certification in CFD
6.1. Introduction
6.2. A mesh convergence algorithm
6.2.1. Mesh adaptation with a fixed complexity
6.2.2. Transfers and numerical convergence
6.3. An academic test case
6.3.1. Uniform refinement study
6.3.2. Isotropic adaptation study
6.3.3. Anisotropic adaptation study
6.3.4. Error level
6.4. 3D multiscale anisotropic mesh adaptation
6.5. Conclusion
6.6. Notes
References
Index
Summary of Volume 2
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