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. Voronoi/Delaunay

6. 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).

7. 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.

8. Delaunay Refinement for 2D point sets R/l > 1.0 30 degree l R

9. f(x) x Local Feature Size • Local feature size: radius of smallest ball that intersects two disjoint input elements. • Lipschitz property:

10. Delaunay Refinement with Boundary x L R >f(x) Circumcenter of skinny triangle encroaching edge. Conforming but still not Gabriel

11. No input angle is less than 90 degree Polyhedral Volumes and Surface[Shewchuk 98] Input PLC Final Mesh

12. Quality of Tetrahedra …… Thin Flat Sliver radius-edge-ratio:

13. 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.

14. Refinement Steps • Compute Delaunay of vertices Do the splits in the following order: • Split encroached subsegments • Split encroached subfacets • Let c be the circumcenter of a skinny tetrahedron • if c encroaches a subsegment or subfacet split it. • Else insert c.

15. > 2.0 Child-Parent and insertion radii

16. Polyhedral surface with any angle • Small angles allowed • Conforming : • Each input edge is the union of some mesh edges. • Each input facet is the union of some mesh triangles. • Quality guarantees.

17. History • No quality guarantee • Effective implementation [Shewchuk 00, Murphy et al. 00, Cohen-Steiner et al. 02]. • Quality guarantee • [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. • [Cheng, Dey, Ramos and Ray 04]

18. Main Result • Quality Meshing for Polyhedra with Small Angles [Cheng, Dey, Ramos, Ray 04] • 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.

19. Small angle problem

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

21. 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.

22. QualMesh 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/tetrahedra • Keep their circumcenters outside We do not want to compute f (center)

23. Refinement Cont.. • Split encroached subsegments and non-Delaunay subfacets. • Let c be the circumcenter of a skinny triangle/tetrahedra. • 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.

24. Positions of skinny triangle/tet

25. 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.

26. Results

27. Results

28. R/L Distribution

29. Dihedral Angle Distribution

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

31. History • Bern, Eppstein, Gilbert 94 - Quadtree meshing (Non-Delaunay) • Cheng, Dey, Edelsbrunner, Facello, Teng 2000 - Silver exudation (no boundary) • Li, Teng 2001 - Silver exudation with boundary (randomized extending Chew)

32. Weighted points and distances • Weighted point: • Weighted distance: • If

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

34. Silver Exudation • Delaunay refinement guarantees tetrahedra with bounded radius-edge-ratio • Vertices are pumped with weights Sliver Theorem [Cheng-Dey-Edelsbrunner-Facello-Teng]: 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,N(v)] for each vertex v in V such that () 0 and ()>0 for each tetrahedron  in the weighted Delaunay triangulation of V.

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

37. No Sliver • Weight property[]: each weight u  N(u) • Ratio property []: orthoradius-edge-ratio is at most . • Lemma : 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 : 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: Assume that Del V has ratio property []. The degree of every vertex in K(V) is bounded by some constant  depending on  and .

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

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

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

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

42. Time

43. Extending sliver exudations to polyhedra with small angles Cheng-Dey-Ray 2005 (Meshing Roundtable 2005) • 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.

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

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

46. f(x) • Medial axis • f(x) is the distance to medial axis Local Feature Size and ε-sample [ABE98] • Each x has a sample within f(x) distance

47. Previous Work • Chew 93: first Delaunay refinement for surfaces • Cheng-Dey-Edelsbrunner-Sullivan 01: Skin surface meshing, Ensure topological ball property by feature size • Boissonnat-Oudot 03: General implicit surfaces, Ensure TBP with local feature size • Cheng-Dey-Ramos-Ray 04: General implicit surface, no feature size computation.

48. Restricted Delaunay • Del Q|G :- Collection of Delaunay simplices whose corresponding dual Voronoi face intersects G.

49. Delaunay Refinement (Chew)

50. Topological Ball Property • A -dimensional Voronoi face intersects G in a -dimensional ball. • Theorem : [ES’97] The underlying space of the complex Del Q|G is homeomorphic to G if Vor Q has the topological ball property.