Fast Adaptive Hybrid Mesh Generation Based on Quad-tree Decomposition

1. Fast Adaptive Hybrid Mesh Generation Based on Quad-tree Decomposition Mohamed Ebeida (msebeida@ucdavis.edu) Mechanical and Aeronautical Eng. Dept –UCDavis Bay Area Scientific Computing Day 2008 March 29, 2008

2. Motivation Unstructured Grids • Complex geometries  • Delaunay point insertion algorithms / advancing front • re-triangulation  mesh points can move • Agglomeration Multigrid solvers • Adaptation using quad-tree or oct-tree (FEM) • Grid quality Structured Grids • Relatively simple geometries • Algebraic – Elliptic – Hyperbolic methods • Line relaxation solvers  • Structured Multigrid solvers  • Adaptation using quad-tree or oct-tree decomp (FEM) • Grid quality

3. Motivation • Sophisticated Multiblock and Overlapping Structured Grid Techniques are required for Complex Geometries Overlapping grid system on space shuttle (Slotnick, Kandula and Buning 1994)

4. Motivation • Multigrid solvers • Multigrid techniques enable optimal O(N) solution complexity • Based on sequence of coarse and fine meshes • Originally developed for structured grids

5. Motivation • Agglomeration Multigrid solvers for unstructured meshes

6. Quad-tree decomposition • Fast • Adaptive • Grid Quality • Line solvers • Hanging nodes • Multigrid • Complex geometries

7. Our Goals • A fast technique • Quality • Complex geometries • Adaptive (geometries – solution variables) • Multigrid • Line relaxation solvers • No hanging nodes • Simple optimization steps (3D) • Parallelizable

8. Spatial Decomposition

9. Strategy Algorithm Algorithm 1 Adaptive grid based on the geometries Algorithm 2 Adaptive grid based on the Simulation

10. Algorithm 1 - Geometries • Start with a coarse Cartesian grid with aspect ratio = 1.0 • Dim: 30x30 Sp = 2.0 256 points

11. Algorithm 1 - Geometries • Perform successive refinements till you reach a level that resolves the curvature of the geometries of the domain

12. Algorithm 1 - Geometries • Level of refinements depend on the curvature of each shape

13. Algorithm 1 - Geometries • Define a buffer zone and delete any element with a node in that zone

14. Algorithm 1 - Geometries • Project nodes on the edge of the buffer zone orthogonally to the geometry

15. Algorithm 1 - Geometries • Move nodes on the edge of the buffer zone orthogonally to the geometry to adjust B.L. elements

16. Algorithm 1 - Geometries

17. Another way ! • Increase the width of the buffer zone and create boundary elements explicitly  better bounds!

19. Algorithm 1 - Geometries • Final mesh 22416 pts 22064 elem. Quad dom. 94.86% Min edge length 7.6 x 10 Max A.R. = 64 -6

20. Complex geometries

21. Testing Algorithm 1 output

22. Algorithm 2 – Simulation based • Use the output of Algorithm 1 as a base mesh for the spatial decomposition • Run the simulation for n time steps (unsteady) or n iterations (steady) • Perform Spatial decomposition on the base mesh based on a level set function. • Map the variables from the grid used in the last simulation

23. How about transition elements? • In order to ensure quality, transition element has to advance one step per spatial decomposition level x x

24. Results

26. Multigrid Levels • Spatial decomposition allows us to generate prolongation and restriction operators easily • How about the elements of each grid level?  We already have them

27. Multigrid Levels

28. Multigrid Results • For elliptic equations, the application of Multigrid is straight forward once we have the grid levels. • For convection diffusion equations, line solvers are crucial for good results

29. Checking our Goals • A fast technique • Quality • Complex geometries • Adaptive with a starting coarse grid • Multigrid • Line relaxation solvers • No hanging nodes • Simple optimization steps (3D) • Parallelizable

30. Thank you!