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Mesh deformation with Inverse Distance Weighted method
So far, we have considered solving a linear-elasticity-like PDE to compute the mesh deformation. Another big family of methods for that step is interpolation methods, either direct or indirect.
These methods compute the displacement of the inner vertices as an algebraic interpolation function of the displacement of the boundary vertices, i.e. the volume displacement field is seen as some kind of average of the boundary displacement. The most widespread of these method is known as Radial Basis Functions [de Boer 2007]. But this is an implicit methods, which means that the coefficients of the interpolation function are found solving a linear system the size of the number of vertices. It can thus be time consuming. Explicit interpolation methods on the other hand directly build the interpolation function, but the interpolation functions must be designed with care to handle large displacement and avoid creating false elements. We chose to add such a direct interpolation method in our code, and to compare it with the elasticity analogy. The method that we chose, the Inverse Distance Weighted (IDW) method, is described at length in [Luke 2012]. In the following, I will recall this method, then I will explain how we plugged the code implementing it into our own code.
Boundary layers for deformable geometries
Dealing with boundary layer (BL) meshes for rigid bodies is rather easy: the whole mesh layers are rigidified and moved together with the body. However, it is a lot more difficult with deformable bodies, since the mesh structured layers must both follow the deformation of the body and keep their structure. It was a priori not certain that the movement provided by the elasticity or IDW step would be precise enough to preserve that structure. In the sequel we demonstrate that both methods can move one thick layer with a good accuracy, and that such layers can also be moved correctly with the elasticity method.
The Geometric Conservation Law
We need to make sure that the movement of the mesh is not responsible for any artificial alteration of the physical phenomena involved, or at least, to make our best from a numerical point of view for the mesh movement to introduce an error of the same order as the one introduced by the numerical scheme. If System (2.1) is written for a constant state, assuming Fext = 0, we get, for any arbitrary closed volume C = C(t): d (|C(t)|) dt − Z @C(t) (w · n) ds = 0. (2.8).
As the constant state is a solution of the Euler equations, if boundaries transmit the flux towards the outside as it comes, we find a purely geometrical relation inherent to the continuous problem. For any arbitrary closed volume C = C(t) of boundary @C(t), this relation is integrated into: |C(t + %t)| − |C(t)| = Z t+$t t Z @C(t) (w · n) dsdt, with t and t + %t 2 [0, T] , (2.9).
which is usually known as the Geometric Conservation Law (GCL). From a geometric point of view, this relation states that the algebraic variation of the volume of C between two instants equals the algebraic volume swept by its boundary.
The role of the GCL is still controversial, and many papers have tried to analyze its role in ALE simulations. Should Relation (2.8) be satisfied at the discrete level? What are the e↵ects of respecting this law at a discrete level on the consistency, stability and accuracy of the numerical scheme? How can it be enforced at a discrete level? A recent review on the subject can be found in [Etienne 2009]. It has been shown that the GCL is neither a necessary nor a sufficient condition to preserve time accuracy, however violating it can lead to numerical oscillation [Mavriplis 2006]. In [Farhat 2001] the authors show that compliance with the GCL guarantees an accuracy of at least the first order in some conditions. Therefore, most would agree that the GCL should be enforced at the discrete level for a large majority of cases.
Discrete GCL enforcement
A new approach to enforcing the Discrete GCL was proposed in [Mavriplis 2006, Yang 2005, Yang 2007], in which the authors proposed a framework to build ALE high order temporal schemes that reach approximately the design order of accuracy. The originality of this approach consists in precisely defining which ALE parameters are true degrees of freedom and which are not. In contrast to other approaches [Koobus 1999, Lesoinne 1996, Nkonga 2000], they consider that the times and configurations at which the fluxes are evaluated do not constitute a new degree of freedom to be set thanks to the ALE scheme. To maintain the design accuracy of the fixed-mesh temporal integration, the moment at which the geometric parameters, such as the cells’ interfaces’ normals or the upwind/downwind tetrahedra must be computed, is entirely determined by the intermediate configurations involved in the temporal scheme chosen. The only degree of freedom to be set by enforcing the GCL at the discrete level is « . Incidentally, it is implicitly stated that w is never involved alone but only hidden in the term « knk which represents the instantaneous algebraic volume swept.
MUSCL approach and RK schemes
As regards spatial accuracy, we have seen that the order of accuracy can be enhanced using the MUSCL technique coupled with a limiter, thereby guaranteeing that this ALE scheme is intrinsically TVD. However, one must determine how to compute upwind/downwind gradients which are necessary for the V4-schemes. Still following [Yang 2005], it is clear that preserving the expected order of accuracy in time imposes that the upwind/downwind elements are computed on the current Runge-Kutta configuration, i.e., on the mesh at ts.
Table of contents :
Conventions
Introduction
I 3D FSI moving mesh simulations
Introduction
1 Connectivity-change moving mesh strategy
1.1 Our moving mesh algorithm
1.1.1 A two step process
1.1.2 Mesh deformation step
1.1.3 Mesh optimization step
1.1.4 Optimization procedure
1.1.5 Handling of boundaries
1.1.6 Algorithm
1.1.7 Choice of the di↵erent parameters for a robust algorithm
1.2 Mesh deformation with Inverse Distance Weighted method
1.2.1 IDW method
1.2.2 Implementation
1.3 Examples
1.3.1 Rigid-body examples
1.3.2 Deformable-body examples
1.4 Boundary layers for deformable geometries
1.4.1 One boundary layer
1.4.2 Several boundary layers
1.5 Large volume variations
1.5.1 Engine piston
1.6 Conclusion
2 ALE solver
2.1 Euler equations in the ALE framework
2.2 Spatial discretization
2.2.1 Edge-based Finite-Volume solver
2.2.2 HLLC numerical flux
2.2.3 High-order scheme
2.2.4 Limiter
2.2.5 Boundary conditions
2.3 Time discretization
2.3.1 The Geometric Conservation Law
2.3.2 Discrete GCL enforcement
2.3.3 RK schemes
2.3.4 Application to the SSPRK(4,3) scheme
2.3.5 Practical computation of the volumes swept
2.3.6 MUSCL approach and RK schemes
2.3.7 Computation of the time step
2.3.8 Handling the swaps
2.4 FSI coupling
2.4.1 Movement of the geometries
2.4.2 Discretization
2.4.3 Explicit coupling
2.5 Implementation
2.5.1 Non-dimensionalization
2.5.2 Parallelization
2.6 Conclusion
3 Numerical Examples
3.1 Validation of the solver
3.1.1 Static vortex in a rotating mesh
3.1.2 Flat plate in free fall
3.1.3 Piston
3.2 Numerical examples
3.2.1 AGARD test cases
3.2.2 F117 nosing up
3.2.3 Two F117 aircraft crossing flight paths
3.2.4 Ejected cabin door
3.3 Parallel performance
3.4 Conclusion
Conclusion
II Extension of anisotropic metric-based mesh adaptation to moving mesh simulations
4 Basics of metric based mesh adaptation
4.1 State of the art
4.1.1 Meshing status
4.1.2 History of metric-based mesh adaptation
4.1.3 Other mesh adaptation approaches
4.2 Principle of metric-based adaptation
4.2.1 Euclidian and Riemannian metric spaces
4.2.2 Unit mesh
4.2.3 Operations on metrics
4.2.4 The non-linear adaptation loop
4.3 The continuous mesh framework
4.3.1 Duality discrete-continous: a new formalism
4.3.2 Continuous linear interpolation
4.3.3 Summary
4.4 Multiscale mesh adaptation
4.4.1 Optimal control of the interpolation error and optimal meshes
4.4.2 Control of the error in Lp norm
4.5 Conclusion
5 Unsteady mesh adaptation
5.1 Error estimate
5.1.1 Error model
5.1.2 Spatial minimization for a fixed t
5.1.3 Temporal minimization
5.1.4 Error analysis for time sub-intervals
5.1.5 Global fixed-point mesh adaptation algorithm
5.2 From theory to practice
5.2.1 Computation of the mean Hessian-metric
5.2.2 Choice of the optimal continuous mesh
5.2.3 Matrix-free P1-exact conservative solution transfer
5.2.4 The remeshing step
5.2.5 Software used
5.3 Choice of the mean Hessian-metric
5.4 Numerical examples
5.4.1 2D shock-bubble interaction
5.4.2 3D circular blast
5.4.3 3D shock-bubble interaction
5.4.4 3D blast on the London Tower Bridge
5.5 Parallelization of the mesh adaptation loop
5.5.1 Choices of implementation
5.5.2 Analysis of parallel timings
5.5.3 Conclusion
5.6 Conclusion
6 Extension of unsteady adaptation to moving meshes
6.1 ALE metric
6.2 Analytic examples
6.2.1 Procedure
6.2.2 Functions considered
6.2.3 Results
6.3 Update of the adaptation algorithm
6.3.1 Error analysis
6.3.2 Algorithm
6.3.3 Update of the metric for optimizations
6.3.4 Handling of the surface
6.4 3D numerical examples
6.4.1 Shock tube in expansion
6.4.2 Moving ball in a shock tube
6.4.3 Nosing-up f117
6.4.4 Two F117 aircraft flight paths crossing
6.5 Conclusion
Conclusion
Conclusion and perspectives
Acknowledgments
Appendices
A Intersection of metrics
A.1 In two dimensions
A.2 In three dimensions
B GPU acceleration of a 3D Finite Volume flow solver
B.1 Solver used
B.2 GPU acceleration
B.3 State of the art
B.4 Principles of implementation
B.5 Results and timings
B.6 Conclusion
Bibliography
