1 / 100

Mesh refinement: sequential, parallel, and dynamic

Mesh refinement: sequential, parallel, and dynamic. Benoît Hudson, CMU Joint work with Umut Acar, TTI-C Gary Miller and Todd Phillips, CMU. Papers available at http://www.cs.cmu.edu/~bhudson. Mesh refinement: sequential, parallel, and dynamic. Benoît Hudson, CMU

yul
Download Presentation

Mesh refinement: sequential, parallel, and dynamic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Mesh refinement:sequential, parallel,and dynamic Benoît Hudson, CMU Joint work with Umut Acar, TTI-CGary Miller and Todd Phillips, CMU Papers available at http://www.cs.cmu.edu/~bhudson 1

  2. Mesh refinement:sequential, parallel,and dynamic Benoît Hudson, CMU Joint work with Umut Acar, TTI-CGary Miller and Todd Phillips, CMU Papers available at http://www.cs.cmu.edu/~bhudson 2

  3. 3

  4. 4

  5. Mesh Visualize Solve Finite Element Simulation Overview Partial Diff. Eqs. Model 5

  6. Our results • First optimal-time sequential mesher • Fast in implementation • First provably fast parallel mesher • First optimal-time dynamic mesher 6

  7. Outline • Precise problem description • Prior solutions • Sequential • Parallel • Dynamic • Many open problems 7

  8. Outline • Precise problem description • Prior solutions • Sequential • Parallel • Dynamic • Many open problems 8

  9. Input Points Segments Polygons 9

  10. Input Points Segments Polygons ‘P’ courtesy of Shewchuk 10

  11. Output ConformingAll features appear (subdivided) ‘P’ courtesy of Shewchuk 11

  12. Big angles are bad 12

  13. No small angle ) no big angle 13

  14. Need Steiner Points 14

  15. Small feature Small triangles Big features Big triangles No-small-angle )Medium size betweensmall, large features Sizing Size-optimality Output O(mopt) points 15

  16. Formal problem • Input: • Points 2Rd, Segments, Polygons, … • Quality bound: angle ³a • Output: • Conforms: All features appear • Quality: No angle smaller than a • Size-optimal: O(mopt) vertices 16

  17. Outline • Precise problem description • Prior solutions • Sequential • Parallel • Dynamic • Many open problems 17

  18. Outline • Precise problem description • Prior solutions • Sequential • Parallel • Dynamic • Many open problems 18

  19. Delaunay Triangulation Maximizes minimumangle 19

  20. Delaunay Triangulation Maximizes minimumangle May not be good enough 20

  21. Ruppert (1992) 21

  22. Ruppert: Identify skinny triangle 22

  23. Ruppert: Find circumcenter 23

  24. Ruppert: Snap to segment 24

  25. Ruppert: Insert, retriangulate 25

  26. Ruppert: Repeat until done 26

  27. Evaluation criteria polygons segments Ruppert92 Miller04 Feature handling 3d points 2d points n2 n polylog(n) n lg n Runtime 27

  28. Ruppert 3D: a bad example 28

  29. Ruppert 3D: a bad example 29

  30. Ruppert 3D: W(n2) n/2 points along line n/2 points around circle Delaunay has n2/4 tets 30

  31. 31

  32. Evaluation criteria polygons Shewchuk98 segments Mil04 Feature handling 3d points 2d points n2 n polylog(n) n lg n Runtime 32

  33. Quadtree: Bern et al (1990) 33

  34. Quadtree: Bern et al (1990) 34

  35. Quadtree: Bern et al (1990) 35

  36. Quadtree: Bern et al (1990) 36

  37. Evaluation criteria She98 polygons MV92 segments Mil04 BEG90 Feature handling 3d points 2d points n2 n polylog(n) n lg n Runtime 37

  38. Compare and contrast 86 triangles, 17° 55 triangles, 30° 38

  39. Outline • Precise problem description • Prior solutions • Sequential • Parallel • Dynamic • Many open problems 39

  40. Outline • Precise problem description • Prior solutions • Sequential • Parallel • Dynamic • Many open problems 40

  41. Fast sequential meshing Hudson, Miller, Phillips 2006Sparse Voronoi Refinement15th International Meshing Roundtable 41

  42. The intuition Quadtree’s fast runtime: top-down. Intermediate meshes are good quality. SVR: always good quality, find features quickly Ruppert’s small size: bottom-up,good feature recovery. 42

  43. Sparse Delaunay Refinement 43

  44. Add a bounding box 44

  45. Triangulate just the box! 45

  46. Apply splitting rules If a triangle is skinny,split it. If a triangle containsinput, split it. 46

  47. Apply splitting rules If a triangle is skinny,split it. If a triangle containsinput, split it. 47

  48. Split If a triangle is skinny,split it. If a triangle containsinput, split it. Split(t) 1. Draw circle 2. Shrink by k 3. Choose a point 4. Insert it, retriangulate 48

  49. Apply splitting rules If a triangle is skinny,split it. If a triangle containsinput, split it. Split(t) 1. Draw circle 2. Shrink by k 3. Choose a point 4. Insert it, retriangulate 49

  50. Apply splitting rules If a triangle is skinny,split it. If a triangle containsinput, split it. 50

More Related