Tetrahedron Meshes. Linh Ha

Transcription

1 Tetrahedron Meshes Linh Ha

2 Overview Fundamental Tetrahedron meshing Meshing polyhedra Tetrahedral shape Delaunay refinements Removing silver Variational Tetrahedral Meshing

3 Meshing Polyhedra Is that easy? A Polyhedron is union of convex polyhedra P = i I H i

4 Faces are not necessarily simply connected

5 Facets and Segments Neighborhood of x N ε = ( x + ε.b ) P b : open unit ball Face figure of x Face of P: closure of maximal collection of point with identical face figures x + λ > 0 λ ( Nε ( x ) x )

6 What does it look like? 1- Faces Face figures a line Polyhedra segment 2- Faces Face figure a plane Polydedra facets

7 24 verties, 30 segments, 11s facets 6 segment and 3 faces are not non-connected

8 Tetrahedrization Tetrahedrization of P is a simplicial complex K whose underlying space is P: K =P Only bounded polyhedra have tetrahedrization Every vertex of P is vertex of K Is the number of vertices of K = P?

9 Is that easy? Prim easy : 3 tetrahedrons The Schonhardt polyhedron? How many?

10 C A B M c a b 8 tetrahedrons, 7verties, 12 edges, 8 triangles Ruppert and Seidel: the problem of deciding if a polyhedron can be tetrahedrized without inserting extra vertices is NP-hard and problem of deciding if a polyhedron can be tetrahedrized with only k additional vertices is NP-hard.

11 Fencing off - Simple methods Every bounded polyhedron has a tetrahedration

12 Fencing algorithms Step1: Erect the fence of each segment. Step2:Triangulate the bottom facet of every cylinder and erect fences from the new segments Step3:Decompose each wall into triangles and finally tetrahedrize each cylinder by constructing cones from an interior point to the boundary.

13 Limits Upper bound: 28m2 - number of tetrahedrons for m-segments Polyhedra Lower bound: (n+1)2 with n : number of cut m=14n+8

14 Tetraheral Shape What is good and bad tetrahedra? Bad tetrahedra

15 Bad tetrahedral classification Skinny: vertices are close to a line Flat : vertices are close to a plane

16 Quality Metrics 2D : minimum angle in the triangulation 3D : Radius-edge ratio R: outer radius L : shortest egde Aspect ratio r: inner radius Volume ratio V: volume

17 Ratio properties ρ A mesh of tetrahedra has ratio property if ρ 0 for all tetrahedra For all triangles ρ ρ 0 Characteristics: Claim A: if abc is a triangle in K then 1/ 2ρ 0 a b a c 2ρ 0 a b Proof: geometry Ratio properties a b 2R a b R/ρ 0 R / ρ 0 a c 2R 2R a b R/ρ 0

18 Ratio property ) Denote Claim B: if angle between ab and ap less than η 0 then 1 / 2 a b a p 2 a b ( η 0 = arctan 2 ρ 0 ρ 02 1 / 4 Proof: a v = R 2 R a b /4 2 = a b /4 R+ 2 R a b /4 bax = arctan 2( x v / a b ( arctan 2 ρ 0 = η0 ( ρ0 ) ρ o2 1 / 4 ) ) ρ 0 1/ 4. a b

19 Length Variation and Constant degree Length variation: if ab, ap are edges in K m0 = 2 / (1 cos(η 0 / 4) ) a b / v0 a p v0 a b Lengths bound v0 = 2 2 m0 1 ρ 0m0 1 of edges with common endpoint is Constant degree : Every vertex a in K belong to at most 3 ( 2 ) δ 0 = 2v0 + 1 However this constraint is too loose

20 Delaunay Triangulation Canonical, associated to any point set.

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22 Properties of Delaunay triangulation maximizing the minimum angle in the triangulation: Better lighting performance Improvements on the accuracy Even stronger: lexicographically maximized sequence of angles Every triangle of a Delaunay triangulation has an empty circumcircle

23 Delaunay Tetrahedration Every tetrahedra of a Delaunay Tetrahedration has an empty circumsphere. However max-min angle optimality in 2D dont s maintain Tetrahedra can have small dihedral angles.

24 Left: Delaunay tetrahedralization: have arbitrarily thin tetrahedron known as a sliver Right: non-delaunay tetrahedralization

25 Edge flip Incremental algorithms in 2D: edge flips In 3D: it is not so easy

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30 Problems Segment that not cover by the edges Faces are not covered by triangles of D Have to add new points and update Delaunary tetrahedration well-spaced points generate only round or sliver Delaunay tetrahedra

31 Delaunary refinement Construct Delaunary tetrahedration has ratio property Definition upon sub segment encroached upon sub facet V a b encroached V A B C

32 Refinement algorithms Rule1: If a subsegment is encroached upon, we split it by adding the mid- point as a new vertex to the Delaunay tetrahedrization Rule2 : If a subfacet is encroached upon, we split it by adding the circum-centre x as a new vertex to the Delaunay tetrahedrization Rule3: If a tetrahedron inside P has R / L > ρ 0 then split the tetrahedron by adding the circumcentre x as a new vertex to the Delaunay tetrahedrization Proprity : Rule1 > Rule2 > Rule3

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34 Local density Local feature size :f R3 R, f(x) f varies slowly with x Insertion radius : rx length of the shortest Delaunay edge with endpoint x immediately after adding x

35 Radii and Parrents Parent vertex: p(v) is the vertex that is responsible for the insertion of v

36 Radius claims Let x be a vertex of D and p is its parents, if it exits. Then rx f ( x ) or rx c.rp where c = 1 / 2 if x has type 1 or 2 and c = ρ 0

37 Graded meshes Ratio claim: Let x be a Delaunay vertex with parent and rx c.rp Then f ( x) / rx 1 + f ( p ) / ( c.rp ) Invariant: if x is a type I vertex in the Delaunay tetrahedrization, for i=1..3 then rx f ( x ) / Ci Conclusion: x y f ( x ) / (1 + C1 )

38 Silver exudation Silver exits frequently between well shaped tetrahedral inside Delaunay tetrahedration

39 Variational Tetrahedral Meshing Pierre Alliez David Cohen-Steiner Mariette Yvinec Mathieu Desbrun

40 Goal Tetrahedral mesh generation Focus on: quality: shape of elements control over sizing dictated by simulation constrained by boundary low number of elements

41 Quality Metrics 3D : Radius-edge Aspect ratio Volume ratio ratio R: outer radius L : shortest egde r: inner radius V: volume In the paper use Radius ratio = Aspect ratio fair to measure silver

42 Popular Meshing Approaches Advancing front Specific subdivision octree crystalline lattice Delaunay refinement sphere packing Optimization: spring energy aspect ratios dihedral angles solid angles volumes edge lengths containing sphere radii etc. [Freitag et al.; Amenta et al.]

43 Delaunay Triangulation Degree of freedom: vertex positions

44 Optimizing Vertex Placement Improve compactness of Voronoi cells by minimizing E= ρ ( x) x x j = 1..k x R j site centroid 2 j dx Necessary condition for optimality: Centroidal Voronoi Tessellation (CVT)

45 Optimizing Vertex Placement CVT [Du-Wang 03] best PL approximant compact Voronoi cells isotropic sampling Vi:local cell Best L1 approximation of paraboloid optimized dual

46 Alas, Harder in 3D... well-spaced points generate only round or sliver Delaunay tetrahedra [Eppstein 01]

47 Alas, Harder in 3D... well-spaced points generate only round or sliver Delaunay tetrahedra regular tetrahedron does not tile space (360 / = 5.1) dihedral angle of the regular tetrahedron = arcos(1/3) 70.53

48 Idea: Underlaid vs Overlaid CVT ODT [Chen 04] best PL approximant best PL interpolant compact Voronoi cells isotropic sampling compact simplices not dual isotropic meshing Best L1 approximation of paraboloid

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50 Rationale Behind ODT Approximation theory: linear interpolation: optimal shape of an element related to the Hessian of f [Shewchuk] Hessian( x 2) = Id regular tetrahedron best

51 Which PL interpolating mesh best approximates the paraboloid? for fixed vertex locations Delaunay triangulation is the optimal connectivity for fixed connectivity min of quadratic energy leads to the optimal vertex locations closed form as function of neighboring vertices

52 Optimizing Connectivity Delaunay triangulation is the optimal connectivity Optimal connectivities minimize EODT

53 Optimizing Vertex Position Simple derivation in xi lead to optimal position in its 1-ring Update vertex position

54 Optimal Update Rule 1 xi = τ j c j Ω i τ j Ω i * xi circumcenter

55 Optimization Alternate updates of connectivity (Delaunay triangulation) vertex locations

56 Optimization: Init good distribution of radius ratios

57 Optimization: Step 1 good distribution of radius ratios

58 Optimization: Step 2 bad good distribution of radius ratios

59 Optimization: Step 50 good distribution of radius ratios

60 Optimization: Step 50

61 Sizing Field Goal reminder: minimize number of elements better approximate the boundary while preserving good shape of elements Those are not independent! well-shaped elements iff K-Lipschitz sizing field [Ruppert, Miller et al.]

62 Automatic Sizing Field Properties: size lfs (local feature size) on boundary sizing field = maximal K-Lipschitz µ ( x) = inf [ K x y + lfs ( y )] y Ω parameter

63 Sizing Field: Example K=0.1 K=1 K=1 K=100

64 3D Example 1 xi = τ j c j Ω i τ j Ω i * local feature size sizing field

65 Algorithm

66 Result

67 Stanford Bunny

68 Hand local feature size

69 Hand: Radius Ratios

70 Torso (courtesy A.Olivier-Mangon & G.Drettakis)

71 Fandisk

72 Comparison with the Unit Mesh Approach

73 Conclusion Generate high quality isotropic tetrahedral meshes (improved aspect ratios) Simple alternated optimizations connectivity: Delaunay vertex positions: weighted circumcenters Good in practice Limit Approximate input boudary intead of comforming Theoretical guarantees to be developed

74 Thank you for your attention