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Size Competitive Meshing without Large Angles

Size Competitive Meshing without Large Angles. Gary L. Miller Carnegie Mellon Computer Science Joint work with Todd Phillips and Don Sheehy. The Problem. Input: A Planar Straight Line Graph. The Problem. Input: A Planar Straight Line Graph. Output: A Conforming Triangulation.

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Size Competitive Meshing without Large Angles

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  1. Size Competitive Meshing without Large Angles Gary L. Miller Carnegie Mellon Computer Science Joint work with Todd Phillips and Don Sheehy Gary Miller Overlay Stitch Meshing

  2. The Problem Input: A Planar Straight Line Graph Gary Miller Overlay Stitch Meshing

  3. The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation Gary Miller Overlay Stitch Meshing

  4. The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation Gary Miller Overlay Stitch Meshing

  5. The Problem Input: A Planar Straight Line Graph Output: A Conforming Triangulation Quality Gary Miller Overlay Stitch Meshing

  6. What is a quality triangle? Gary Miller Overlay Stitch Meshing

  7. What is a quality triangle? No Large Angles No Small Angles Gary Miller Overlay Stitch Meshing

  8. What is a quality triangle? No Large Angles No Small Angles Implies triangles have bounded aspect ratio. Gary Miller Overlay Stitch Meshing

  9. What is a quality triangle? No Large Angles No Small Angles Implies triangles have bounded aspect ratio. Implies triangles have bounded largest angles. Gary Miller Overlay Stitch Meshing

  10. What is a quality triangle? No Large Angles No Small Angles Implies triangles have bounded aspect ratio. Implies triangles have bounded largest angles. Can be efficiently computed by Delaunay Refinement Gary Miller Overlay Stitch Meshing

  11. What is a quality triangle? No Large Angles No Small Angles Sufficient for many applications. Implies triangles have bounded aspect ratio. Implies triangles have bounded largest angles. Can be efficiently computed by Delaunay Refinement Gary Miller Overlay Stitch Meshing

  12. What is a quality triangle? No Large Angles No Small Angles Sufficient for many applications. Implies triangles have bounded aspect ratio. Can be asymptotically smaller then Delaunay Refinement triangulations. Implies triangles have bounded largest angles. Can be efficiently computed by Delaunay Refinement Gary Miller Overlay Stitch Meshing

  13. What is a quality triangle? No Large Angles No Small Angles Sufficient for many applications. Implies triangles have bounded aspect ratio. Can be asymptotically smaller then Delaunay Refinement triangulations. Implies triangles have bounded largest angles. More difficult to analyze. Can be efficiently computed by Delaunay Refinement Gary Miller Overlay Stitch Meshing

  14. In Defense of Quality Gary Miller Overlay Stitch Meshing

  15. In Defense of Quality Gary Miller Overlay Stitch Meshing

  16. In Defense of Quality Gary Miller Overlay Stitch Meshing

  17. In Defense of Quality What went wrong? Gary Miller Overlay Stitch Meshing

  18. In Defense of Quality What went wrong? Gary Miller Overlay Stitch Meshing

  19. Interpolation Problem No Large Angles Large Angles Large angles give large H1 errors that FEMs try to minimize. Gary Miller Overlay Stitch Meshing

  20. Paying for the spread Gary Miller Overlay Stitch Meshing

  21. Paying for the spread Spread = L/s L s Gary Miller Overlay Stitch Meshing

  22. Paying for the spread Optimal No-Large-Angle Triangulation Gary Miller Overlay Stitch Meshing

  23. Paying for the spread What if we don’t allow small angles? Gary Miller Overlay Stitch Meshing

  24. Paying for the spread What if we don’t allow small angles? Gary Miller Overlay Stitch Meshing

  25. Paying for the spread What if we don’t allow small angles? O(L/s) triangles! Gary Miller Overlay Stitch Meshing

  26. Paying for the spread What if we don’t allow small angles? O(L/s) triangles! Gary Miller Overlay Stitch Meshing

  27. Delaunay Refinement Gary Miller Overlay Stitch Meshing

  28. Delaunay Refinement Gary Miller Overlay Stitch Meshing

  29. On Point Sets, we only pay for the log of the spread. Delaunay Refinement • Theorem: Delaunay Refinement on point sets terminates and returns a triangulation with • all angles at least 30-e degrees • O(n log L/s) triangles Gary Miller Overlay Stitch Meshing

  30. Paterson’s Example Requires O(n2) points. O(n) points O(n) lines Gary Miller Overlay Stitch Meshing

  31. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Gary Miller Overlay Stitch Meshing

  32. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Gary Miller Overlay Stitch Meshing

  33. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Gary Miller Overlay Stitch Meshing

  34. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Gary Miller Overlay Stitch Meshing

  35. Paterson’s Example Requires (n2) points. O(n) points O(n) lines Gary Miller Overlay Stitch Meshing

  36. Past Results • O(n) triangles with 90o largest angles for polygons with holes. [Bern, Mitchell, Ruppert, 95] • (n2) lower bound for arbitrary PLSGs. [Paterson] • O(n2) triangles with 132o angles on PSLGs. [Tan, 96] Gary Miller Overlay Stitch Meshing

  37. Past Results Delaunay Refinement Methods No-Large-Angle Methods Cons Pros Cons Pros Smaller Meshes Not well-graded Complicated to Implement Good Theory Optimal Runtime Graded Mesh Simple to Implement Esthetically Nice Huge Meshes O(L/s) Require Hacks to handle small input angles. Size depends on smallest angle. Smaller Meshes Worst-Case Optimal Gary Miller Overlay Stitch Meshing

  38. Past Results OUR Delaunay Refinement Methods No-Large-Angle Methods Cons Pros Cons Pros Smaller Meshes Not well-graded Complicated to Implement Good Theory Optimal Runtime Graded Mesh Simple to Implement Esthetically Nice Huge Meshes O(L/s) Require Hacks to handle small input angles. Size depends on smallest angle. Smaller Meshes Worst-Case Optimal Gary Miller Overlay Stitch Meshing

  39. Past Results OUR Delaunay Refinement Methods No-Large-Angle Methods Cons Pros Cons Pros Only Worst-Case Bounds Not well-graded Complicated to Implement Good Theory Optimal Runtime Graded Mesh Simple to Implement Esthetically Nice Huge Meshes O(L/s) Require Hacks to handle small input angles. Size depends on smallest angle. Smaller Meshes Worst-Case Optimal Gary Miller Overlay Stitch Meshing

  40. Past Results OUR Delaunay Refinement Methods No-Large-Angle Methods Cons Pros Cons Pros Only Worst-Case Bounds Not well-graded Complicated to Implement Good Theory Optimal Runtime Graded Mesh Simple to Implement Esthetically Nice Huge Meshes O(L/s) Require Hacks to handle small input angles. Size depends on smallest angle. Smaller Meshes Worst-Case Optimal size Log L/s -competitive Graded on Average Gary Miller Overlay Stitch Meshing

  41. Past Results OUR Delaunay Refinement Methods No-Large-Angle Methods Cons Pros Cons Pros Only Worst-Case Bounds Not well-graded Complicated to Implement Good Theory Optimal Runtime Graded Mesh Simple to Implement Esthetically Nice Huge Meshes O(L/s) Require Hacks to handle small input angles. Size depends on smallest angle. Smaller Meshes Worst-Case Optimal size Log L/s -competitive Graded on Average Our Angle bounds are not as good, 170o versus ~140o Gary Miller Overlay Stitch Meshing

  42. Local Feature Size lfs(x) = distance to second nearest vertex. Note: lfs is defined on the whole plane. x lfs(x) Gary Miller Overlay Stitch Meshing

  43. The OSM Algorithm(Overlay Stitch Meshing) Gary Miller Overlay Stitch Meshing

  44. The OSM Algorithm(Overlay Stitch Meshing) Gary Miller Overlay Stitch Meshing

  45. The OSM Algorithm(Overlay Stitch Meshing) Gary Miller Overlay Stitch Meshing

  46. The OSM Algorithm(Overlay Stitch Meshing) Gary Miller Overlay Stitch Meshing

  47. The OSM Algorithm(Overlay Stitch Meshing) Gary Miller Overlay Stitch Meshing

  48. The OSM Algorithm(Overlay Stitch Meshing) Gary Miller Overlay Stitch Meshing

  49. The OSM Algorithm(Overlay Stitch Meshing) Gary Miller Overlay Stitch Meshing

  50. The OSM Algorithm(Overlay Stitch Meshing) Gary Miller Overlay Stitch Meshing

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