Moving Mesh Adaptation Techniques

1. Moving Mesh Adaptation Techniques Todd Phillips Gary Miller Mark Olah Lamps of Aladdin

2. Introduction • The Meshing Problem • Discretize a Spatial Domain • Minimize Size (Number of Triangles) • Maximize Quality (Shape of Triangles) • Mesh Adaptation • Introduce More Elements • Base Decisions on the Function being Interpolated • How do we make such a decision? Lamps of Aladdin

3. The Basic Setup • A Function Exists Lamps of Aladdin

4. The Basic Setup • A Mesh Also Exists Lamps of Aladdin

5. The Basic Setup • We Approximate the Function with the Mesh Lamps of Aladdin

6. The Basic Setup • We can adapt the Mesh to Better Approximate the Function Lamps of Aladdin

7. Background: Local Feature Size • What is Local Feature Size? • Roughly, the Size of Triangles We Should Be Using • The Smallest Distance to Two Geometric Domain Features • Local Feature Size Should be k-Lipschitz • Doesn’t change too fast • For all p,q lfs(p) <= lfs(q) + k*dist(p,q) • This is like having derivative bounded by k. Lamps of Aladdin

8. Optimal Meshing and Adaptive Meshing • Optimal Meshing • Consider the LFS of the Input Domain • Output a Mesh • Triangle Size is within a constant of LFS • Triangle Aspect Ratio is bounded • Note the Impossabilty of this if LFS is not Lipschitz • Adaptive Meshing • LFS Accounts for Other ‘Features’ Provided by Some Oracle • Introduce Geometric Features to Accommodate • Perturb the Original Mesh to obtain a new Optimal Mesh Lamps of Aladdin

9. Example of Modifying Local Feature Size • A Sample Geometric Domain Lamps of Aladdin

10. Example of Modifying Local Feature Size • LFS of the Sample Domain Lamps of Aladdin

11. Example of Modifying Local Feature Size • Modify the Geometric Domain to Capture a New Feature Lamps of Aladdin

12. Example of Modifying Local Feature Size • LFS of the Modified Domain Lamps of Aladdin

13. How can we Define an Oracle? • Function-Angle Based Refinement • Some Function is Approximated by the Mesh • This Embeds the Mesh as a surface in one-higher dimension • Observe the angle between faces of this surface • Introduce Features Where This Angle is Small • Insert Circumcenters, Split Edges, etc. Lamps of Aladdin

14. An Example in One Dimension: Function Lamps of Aladdin

15. An Example in One Dimension: Function and Uniform Mesh Lamps of Aladdin

16. An Example in One Dimension: Function, Mesh, and Error Lamps of Aladdin

17. An Example in One Dimension: Function and Adaptive Mesh Lamps of Aladdin

18. An Example in One Dimension: Adaptive Mesh LFS Lamps of Aladdin

19. Will this work? • Original Feature Size must be Small Enough • Actual Function must be Differentiable Lamps of Aladdin

20. Practicalities • Real Mesh Adaptation Requires Coarsening As Well • Sometimes Functional Features Move or Disappear • Thrashing is a dirty word • Real Solutions aren’t always differentiable • Give up after a Minimum Feature Size Lamps of Aladdin

21. Why is this so great for Moving Meshes? • Unlike the One Dimensional Movie • Fixed Mesh ( Grid points didn’t move left/right ) • Constantly ‘Chasing’ the Feature • Mesh moves with the Velocity Function • Hence, Mesh moves with the Velocity features • After features are captured, very little Mesh Adaptation Lamps of Aladdin

22. How does this compare to other methods? • Sensitivity Analysis and a posteriori Error Estimates • Double the Mesh Size Everywhere, compare solutions • Require multiple meshes of different sizing be maintained • Require multiple solves at every iteration • Don’t take advantage of Moving Mesh • (Persistance of Features) Lamps of Aladdin

23. Some 2-D Examples Lamps of Aladdin

24. Some 2-D Examples Lamps of Aladdin

25. Conclusion • Summary • Use an oracle based on Function-Angles • Introduce new geometric features • Use existing geometrically driven meshing algorithms • Future • Tighter Coupling of Oracle with Meshing Algorithms • Adaptive Time Refinement Lamps of Aladdin