Surface and Volume Meshing with Delaunay Refinement

1. Surface and Volume Meshing with Delaunay Refinement Tamal K. Dey The Ohio State University

2. QualMesh based on Cheng-Dey-Ramos-Ray 04 (solved small angle problem effectively) Polyhedral Volumes and Surface Input PLC Final Mesh

3. Implicit surface F: R3 => R, Σ = F-1(0)

4. Polygonal surface

5. Polyhedral surface Given a polyhedron P find the Delaunay mesh • Extra vertices allowed. • Conforming : • Each input edge is the union of some mesh edges. • Each input facet is the union of some mesh triangles. • Quality guarantees.

6. Quality Measures • Radius-edge ratio: R- circumradius of a triangle. – shortest edge length.

7. Quality Tetrahedra …… Thin Flat Sliver Bounded radius-edge-ratio:

8. Related work • Delaunay refinement • Bounded radius-edge ratio [Chew 89, Ruppert 92, Shewchuk 98]. • Disallow acute input angles. • Allow small angles • Effective implementation [Shewchuk 00, Murphy et al. 00, Cohen-Steiner et al. 02]. • No quality guarantee. • Allow small angles. • [Cheng and Poon 03] • [Cheng, Dey, Ramos and Ray 04]

9. Related work • [Cheng and Poon 03] • Complex. • Protect input segments with orthogonal balls. • Need to mesh spherical surfaces. • Expensive. • Compute local feature/gap sizes at many points.

10. Main Result • A simpler Delaunay meshing algorithm • Local feature size needed only at the sharp vertices. • No spherical surfaces to mesh. • Quality Guarantees • Most tetrahedra have bounded radius-edge ratio. • Skinny tetrahedra will be provably close to the acute input angles. • Quality Meshing for Polyhedra with Small Angles [Cheng, Dey, Ramos, Ray]

11. Voronoi/Delaunay

12. Basics of Delaunay Refinement Chew 89, Ruppert 95 • Maintain a Delaunay triangulation of the current set of vertices. • If some property is not satisfied by the current triangulation, insert a new point which is locally farthest. • Burden is on showing that the algorithm terminates (shown by packing argument).

13. Delaunay refinement for quality • R/l = 1/(2sinθ)≥1/√3 • Choose a constant > 1 if R/l is greater than this constant, insert the circumcenter.

14. Delaunay refinement for input conformity • Diametric ball of a subsegment empty. • If encroached by a point p, insert the midpoint. • Subfacets: 2D Delaunay triangles of vertices on a facet. • If diametric ball of a subfacet encroached by a point p, insert the center.

15. Local Feature Size • Local feature size: radius of smallest ball that intersects two disjoint input elements. • Lipschitz property:

16. Small angle problem

17. Sharp vertex protection SOS-split [Cohen-Steiner et al. 02]

18. Subfacet Splitting • Trick to stop indefinite splitting of subfacets in the presence of small angles is to split only the non-Delaunay subfacets. • It can be shown that the circumradius of such a subfacet is large when it is split.

19. Algorithm • Protect sharp vertices • Construct a Delaunay mesh. • Loop: • Split encroached subsegments and non-Delaunay subfacets. • 2-expansion of diametrical ball of sharp segments. (Radius = O( f(center) ) ) • Refinement: • Eliminate skinny triangle • Keep their circumcenters outside We do not want to compute f (center)

20. Refinement • Split encroached subsegments and non-Delaunay subfacets. • Let c be the circumcenter of a skinny triangle. • If c lies inside the protecting ball of a sharp vertex or sharp subsegment then do nothing • Else if c encroaches a subsegment or subfacet split it. • Else insert c.

21. Positions of skinny triangle

22. Summary of results • A simpler algorithm and an implementation. • Local feature size needed at only the sharp vertices. • No spherical surfaces to mesh. • Quality guarantees • Most tetrahedra have bounded radius-edge ratio. • Any skinny tetrahedron is at a distance from some sharp vertex or some point on a sharp edge.

23. Results

24. Results

25. R/L Distribution

26. Dihedral Angle Distribution

27. Meshing Polyhedra with Sliver Exudations Quality Meshing with Weighted Delaunay Refinement by Cheng-Dey 02

28. History • Bern, Eppstein, Gilbert 94 - Quadtree meshing (Non-Delaunay) • Shewchunk 98 – Extended Delaunay refinement to 3D (Slivers remain) • Cheng, Dey, Edelsbrunner, Facello, Teng 2000 - Silver exudation (no boundary) • Li, Teng 2001 - Silver exudation with boundary (randomized extending Chew)

29. f(x) x Definitions • Input is a PLC with no acute input angle. • f(x): Local feature size • Weighted point: • Weighted distance: • If

30. Weighted Delaunay • Smallest orthospheres, orthocenters, orthoradius • Weighted Delaunay tetrahedra

31. Silver Exudation • Delaunay refinement guarantees tetrahedra with bounded radius-edge-ratio • Vertices are pumped with weights Sliver Theorem: Given a periodic point set V and a Delaunay triangulation of V with radius-edge ratio , there exists 0>0 and 0>0 and a weight assignment in [0,2N(v)2] for each vertex v in V such that () 0 and ()>0 for each tetrahedron  in the weighted Delaunay triangulation of V.

32. p p Encroachments again • Subsegment encroachments • Subfacet encroachments • Weight assignments • Vertex Gap Property:For each vertex u in V, the weight of u used for encroachment checking or pumping is at most 02f(u)2 and the Euclidean nearest neighbor distance of u in V is at least 20f(u).

33. Locations of Centers • Lemma 3.1: Suppose that the vertex gap property holds. If no weighted-subsegment or weighted-subfacet is encroached, no weighted vertex intersects a segment or a facet that does not contain p. • Lemma 3.2: Suppose that the vertex gap property holds. • A weighted-subsegment contains its orthocenter. • If no weighted-subsegment is encroached, a facet contains the orthocenter of any weighted-subfacet on it. • If no weighted-subsegment or weighted-subfacet is encroached, the input domain contains the orthocenter of any weighted Delaunay tetrahedron inside the input domain.

34. p Encroachments and Projection • Lemma 3.3: If the vertex gap property holds, then for any weighted-subsegment ab on an edge e of P, ab cannot be encroached by any vertex that lies on an edge adjacent to e or a facet adjacent but non-incident to e. • Lemma 3.4: Let abc be a weighted-subfacet on a facet F of P. If there is no encroached weighted-subsegment, then abc cannot be encroached by any vertex that lies on a facet adjacent to F or an edge adjacent but non-incident to F. • Lemma 3.5: If no weighted-subsegment is encroached and encroaches upon some weighted-subfacet on a facet F, then there exists a weighted-subfacet h on F which is encroached upon by and h contains the orthogonal projection of p on F.

35. QUALMESH algorithm • Compute the Delaunay triangulation of input vertices • Refine Rule 1: subsegment refinement Rule 2: subfacet refinement Rule 3: Tetrahedron refinement Rule 4: Weighted encroachment Check if weighted vertices encroach, if so refine. • Pump a vertex incident to silvers

36. Insertion radii • Parent-child: • Type 3: a vertex of split tetrahedron • Type 1, 2: Encroaching vertex • rxis the distance from the nearest vertex in the current V Lemma 4.3:Suppose that the vertex gap property holds. Let x be an input vertex or a vertex inserted or rejected. Let p be the parent of x, if it exists. • If x is an input vertex or p is an input vertex, then • Otherwise,

37. Inter-vertex distances Lemma 5.2: Let x be a vertex of P or a vertex inserted or rejected by QUALMESH. We have the following invariants for 0 > 4. • If x is a vertex of P or the parent of x is a vertex of P, then Otherwise, if x has type i, for 1 i  3, then rx  f(x)/Ci. • For any other vertex y that appears in V currently, • If x is inserted by QUALMESH, the vertex gap property holds afterwards.

38. Guarantees • Theorem 7.1 (Termination): QUALMESH terminates with a graded mesh. • Theorem 7.2 (Conformity): No weighted-subsegment or weighted-subfacet is encroached upon the completion of QUALMESH

39. No Sliver • Weight property[]: each weight u  N(u) • Ratio property []: orthoradius-edge-ratio is at most . • Lemma 6.2: Let V be a finite point set. Assume that Del V has ratio property [], has weight property [], and the orthocenter of each tetrahedron in Del lies inside Conv V. Then Del has ratio property [’] for some constant ’ depending on  and  • Lemma 7.1: Assume that Del V has ratio property []. The lengths of any two adjacent edges in K(V) is within a constant factor v depending on  and . • Lemma 7.2: Assume that Del V has ratio property []. The degree of every vertex in K(V) is bounded by some constant  depending on  and .

40. No Silver (continued) • A silver remains incident to p only within a subinterval of width O(0) during pumping. • Lemma 7.2 says that the intervals of slivers can be made arbitrary small by choosing 0. • if the weight interval contains subinterval [0, 2N()2] for some , exudation works.

41. No Sliver (continued) • Lemma 7.3: Let M be the mesh obtained at the end of step 2 of QUALMESH. For any vertex v in M, its nearest neighbor distance is at most . • Theorem 7.3: There is a constant 0 > 0 such that () > 0 for every tetrahedron  in the output mesh of QUALMESH.

42. Size Optimality • Output vertices • Output tetrahedra • Any mesh of D with bounded aspect ratio must have tetrahedra • Theorem 7.4: The output size of QUALMESH is within a constant factor of the size of any mesh of bounded aspect ratio for the same domain.

43. Example - Arm Slivers Input PLC Sliver Removal Final Mesh

44. Slivers Input PLC Sliver Removal Final Mesh Example - Cap

45. Example - Wrench Slivers Input PLC Final Mesh Sliver Removal

46. Example - Propellant Input PLC Slivers Sliver Removal Final Mesh

47. Time

48. Extending sliver exudations to polyhedra with small angles • Carry on all steps for meshing polyhedra with small angles • Add the sliver exudation step • All tetrahedra except the ones near small angles have bounded aspect ratio. • Cheng-Dey-Ray 2005 (Meshing Roundtable 2005)

49. Delaunay Meshing for Implicit Surfaces Cheng-Dey-Ramos-Ray 04

50. Implicit surfaces • Surface Σ is given by an implicit equation E(x,y,z)=0 • Surface is smooth, compact, without any boundary