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

2. Motivation Structured Grids Unstructured 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
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

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
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 %
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!