1 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

2 Benoît Hudson, CMU Joint work with Umut Acar, TTI-C Gary Miller and Todd Phillips, CMU Papers available at Mesh refinement: sequential, parallel, and dynamic

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

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

7 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

8 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

9 Input Points Segments Polygons

10 Input Points Segments Polygons ‘P’ courtesy of Shewchuk

11 Output ‘P’ courtesy of Shewchuk Conforming All features appear (subdivided)

12 Big angles are bad

13 No small angle ) no big angle

14 Need Steiner Points

15 Sizing Big features Big triangles No-small-angle ) Medium size between small, large features Small feature Small triangles Size-optimality Output O(m opt ) points

16 Formal problem Input: –Points 2 R d, Segments, Polygons, … –Quality bound: angle   Output: –Conforms: All features appear –Quality: No angle smaller than  –Size-optimal: O(m opt ) vertices

17 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

18 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

19 Delaunay Triangulation Maximizes minimum angle

20 Delaunay Triangulation Maximizes minimum angle May not be good enough

21 Ruppert (1992)

22 Ruppert: Identify skinny triangle

23 Ruppert: Find circumcenter

24 Ruppert: Snap to segment

25 Ruppert: Insert, retriangulate

26 Ruppert: Repeat until done

27 Evaluation criteria Feature handling Runtime 2d points 3d points segments polygons n2n2 n lg nn polylog(n) Miller04Ruppert92

28 Ruppert 3D: a bad example

29 Ruppert 3D: a bad example

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

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32 Evaluation criteria Feature handling Runtime 2d points 3d points segments polygons n2n2 n lg nn polylog(n) Mil04 Shewchuk98

33 Quadtree: Bern et al (1990)

34 Quadtree: Bern et al (1990)

35 Quadtree: Bern et al (1990)

36 Quadtree: Bern et al (1990)

37 Evaluation criteria Feature handling Runtime 2d points 3d points segments polygons n2n2 n lg nn polylog(n) Mil04 She98 BEG90 MV92

38 Compare and contrast 55 triangles, 30  86 triangles, 17 

39 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

40 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

41 Fast sequential meshing Hudson, Miller, Phillips 2006 Sparse Voronoi Refinement 15th International Meshing Roundtable

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

43 Sparse Delaunay Refinement

44 Add a bounding box

45 Triangulate just the box!

46 Apply splitting rules 1.If a triangle is skinny, split it. 2.If a triangle contains input, split it.

47 Apply splitting rules 1.If a triangle is skinny, split it. 2.If a triangle contains input, split it.

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

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

50 Apply splitting rules 1.If a triangle is skinny, split it. 2.If a triangle contains input, split it.

51 Apply splitting rules 1.If a triangle is skinny, split it. 2.If a triangle contains input, split it.

52 Apply splitting rules 1.If a triangle is skinny, split it. 2.If a triangle contains input, split it.

53 Apply splitting rules 1.If a triangle is skinny, split it. 2.If a triangle contains input, split it.

54 General flavour Like quadtree, refine top-down Like Ruppert, use input points, circumcenters Warp to input when possible.

55 Runtime proof (1) Quality always “good” –Never warp close to mesh vertex –No angle smaller than ® ’

56 Runtime proof (1) Quality always good (2) Quality ) bounded degree –360 degrees –degree 360/ ® ’

57 Runtime proof (1) Quality always good (2) Quality ) bounded degree (3) Bounded degree ) O(1) operations –insertions –range queries

58 Runtime proof (1) Quality always good (2) Quality ) bounded degree (3) Bounded degree ) O(1) operations (4) Quality ) divide & conquer –!!!

59 Quality ) divide & conquer p

60 Quality ) divide & conquer p e 0 (p) e 1 (p) e 0 (p) · · 

61 Quality ) divide & conquer p e 0 (p) e 2 (p) e 0 (p) · 2· 2

62 Quality ) divide & conquer p e 0 (p) e i (p) e 0 (p) · i· i

63 Quality ) divide & conquer p e 0 (p) e i (p) farthest(p) nearest(p) · k· k

64 Quality ) divide & conquer p farthest(p) nearest(q) ·  2k q

65 Packing Lemma p At most O(1) q i before farthest(p) falls by half. q2q2 q1q1 q3q3 q5q5 q4q4

66 s Packing Lemma At most O(1) q i before farthest(p) falls by half. farthest(p) falls by half at most log (L/s) times L

67 Runtime proof (1) Quality always good (2) Quality ) bounded degree (3) Bounded degree ) O(1) operations (4) Quality ) divide & conquer O(m + n log L/s)

68 Evaluation criteria Feature handling Runtime 2d points 3d points segments polygons n2n2 n lg nn polylog(n) Mil04 She98 BEG90 MV92 SVR (HMP06)

69 In practice: point clouds N points around ring N points on line Preparata (N=10 3, 10 4 ) Stanford Bunny (34834 points)

70 In practice: N=1,000 N points around circle N points along line Algorithm SVR Pyramid Max #tets 70K 1.014M Max degree 70 1,010 Output #tets 70K 87K Runtime 2s 21s 1 GHz Pentium III 1 GB RAM

71 (Calculate: (10 4 ) 2 = 10 8 tets in Delaunay, each tet is 8 pointers  ~ 3.2 GB) In practice: N=10,000 N points around circle N points along line SVR: outputs 722K tets in 22s Pyramid: thrashes, ^c after 4 hours

72 In practice: the bunny Algorithm SVR Pyramid Output #tets 430K 383K Runtime 12s 25s

73 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

74 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

75 Parallel SVR Hudson, Miller, Phillips 2007 Sparse Parallel Delaunay Refinement To appear, SPAA.

76 Why parallel? My laptop: two cores today –Paterson: expect 100 cores in 10 years National Labs: 41,000 cores last week Understanding the dependencies: –Helps with constant factors in sequential case –Helps with out-of-core, distributed, compression, dynamic case, …

77 Parallel Algorithms Comparison SVR [HMP07] Quadtree [BET93] Ruppert [STU02] any d 2d 3d O(lg (L/s) lg(m)) O(lg n) O(polylog(L/s)) n: number of input points, segments, … m: number of output points L/s: spread of the input O(n lg L/s + m) O(n lg n + m) O(m polylog(L/s))  (n 2 ) DepthWorkDimensionAlgorithm L: largest distance s: smallest distance L/s  spread Normally: L/s  poly(n) lg L/s  lg n

78 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

79 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

80 Dynamic meshing Acar, Hudson, 2007 (in submission) Dynamic mesh refinement using Quadtrees and Off-centers

81 O’Brien, Hodgins 1999

82 Stability

83 Stability

84 Stability

85 Stability

86 Stability

87 Stability bound: O(log L/s) Proof in paper: Newly split cells pack around new point

88 Self-adjusting computation: Acar et al, 2006 Update speed = O(# stability) Insert / delete are exactly symmetric Stability is O(log L/s) ) Quadtree update is O(log L/s) Implementation in SML

89 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

90 Outline 1.Precise problem description 2.Prior solutions 3.Sequential 4.Parallel 5.Dynamic 6.Many open problems

91 Visualize (1) Dynamic Simulation Model Partial Diff. Eqs. Mesh Solve

92 Visualize (1) Dynamic Simulation Model Partial Diff. Eqs. Mesh Solve

93 Visualize (1) Dynamic Simulation Model Partial Diff. Eqs. Mesh Solve

94 Visualize (1) Dynamic Simulation Model Partial Diff. Eqs. Mesh Solve New requirements: (1)Dynamic matrix assembly (2)Dynamic linear solver (3)Dynamic visualizer (4)Dynamic AMR

95 (2) Dynamic with Features Dynamic algorithm does not handle segments, polygons,... Dynamic SVR? –Likely coming soon

96 (3) Out of core refinement Engineers want billions of elements Doesn’t fit in memory Dynamic refinement allows partial meshing

97 (4) Distributed refinement Engineers want billions of elements Engineers have supercomputers Need to address: –Mesh partitioning –Load balancing

98 (5) Surface reconstruction Traditional approach: Compute Delaunay Skinny tet ) surface Refinement: Compute Delaunay Skinny tet ) refine

99 Summary Everyone needs a mesher Sequential: First optimal-time mesher –First sub-quadratic in 3d! –Implementation in progress Parallel: First optimal-work, shallow-depth Dynamic: First optimal-time mesher

100 Bibliography [Che89]: Chew “Guaranteed quality triangular meshes”, 1989 [BEG90]: Bern, Eppstein, Gilbert “Provably good mesh generation”, 1994 [MV92]: Mitchell, Vavasis “Quality mesh generation …”, 2000 [Rup92]: Ruppert “A Delaunay refinement algorithm for …”, 1995 [BET93]: Bern, Eppstein, Teng “Parallel construction …”, 1999 [She97]: Shewchuk “Delaunay refinement mesh generation”, 1997 [MPW02]: Miller, Pav, Walkington “Fully incremental …”, 2002 [STU02]: Spielman, Teng, Ungor “Parallel Delaunay …”, 2002 [Mil04]: Miller, “A time-efficient Delaunay Refinement …”, 2004 [HPU05]: Har-Peled, Ungor, “A time-optimal Delaunay …”, 2005 [HMP06]: Hudson, Miller, Phillips, “Sparse Voronoi Refinement”, 2006 [HMP07]: ~, “Sparse Parallel Delaunay Refinement”, 2007 [MPS07]: Miller, Phillips, Sheehy, “Size competitive …”, 2007 [HA07]: Acar, Hudson, “Dynamic quad-tree mesh refinement...”, submitted