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Delaunay Meshing for Piecewise Smooth Complexes

Delaunay Meshing for Piecewise Smooth Complexes. Tamal K. Dey The Ohio State U. Joint work: Siu-Wing Cheng, Joshua Levine, Edgar A. Ramos. Sharp Edges. Non-manifold. Piecewise Smooth Complexes. Piecewise Smooth Complexes. D is a piecewise smooth complex (PSC) if

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Delaunay Meshing for Piecewise Smooth Complexes

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  1. Delaunay Meshing for Piecewise Smooth Complexes Tamal K. Dey The Ohio State U. Joint work: Siu-Wing Cheng, Joshua Levine, Edgar A. Ramos

  2. Sharp Edges Non-manifold Piecewise Smooth Complexes

  3. Piecewise Smooth Complexes • D is a piecewise smooth complex (PSC) if • Each k-dimensional element is a manifold and compact subset of a smooth (C2) k-manifold, 0≤k≤2. • The k-th stratum, Dk : set of k-dim elements of D. • D0 – vertices, D1 – 1-faces, D2 – 2-faces. • D≤k = D0 …  Dk. • D satisfies usual reqs for being a complex. • Interiors of elements are disjoint and for σ D, bd σ D. • For any σ, D, either σ  =  or σ  D .

  4. Delaunay refinement : History • Chew89, Ruppert92, Shewchuk98 (Linear domains with no small angle) • Cohen-Steiner-Verdiere-Yvinec02, Cheng-Dey-Ramos-Ray04 (polyhedral domains with small angle) • Chew93 (surface without guarantees) • Cheng-Dey-Edelsbrunner-Sullivan01 (skin surfaces) • Boissonnat-Oudot03 and Cheng-Dey-Ramos-Ray04 (smooth surface) • Boissonnat-Oudot06 (Lipschitz surfaces) • Oudot-Rineau-Yvinec06 (Volumes)

  5. Basics of Delaunay Refinement • Chew 89, Ruppert 92, Shewchuk 98 • 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).

  6. Challenges for PSC • Topology • Polyhedral case (input conformity,topology trivial). • Curved elements (topology is an issue). • Topological Ball Property (TBP) was used for smooth manifolds [BO03,CDRR04]. • We need extended TBP for nonmanifolds. • Nonsmoothness • Lipschitz surfaces [BO06], Remeshing [DLR05]. • Small angles • Delaunay refinement is hard [CP03, CDRR05, PW04].

  7. Topological Ball Property • For a weighted point set S, let Vor S and Del S denote the weighted Voronoi and Delaunay diagrams. • S has the TBP for σDi if σ intersects any k-face in Vor S either in emptyset or in a closed topological (i+k-3)-ball.

  8. CW-Complexes • A CW-complex R is a collection of closed (topological) balls whose interiors are pairwise disjoint and whose boundaries are the union of other closed balls in R. • Our algorithm builds a CW-complex, Vor S||D|, to satisfy an extended TBP[ES97].

  9. Extended TBP • S  |D| has the extended TBP (eTBP) for D if there is a CW-complex R with |R| = |D| s.t. • (C1) The restricted Voronoi face F  |D| is the underlying space of a CW-complex R’  R. • (C2) The closed balls in R’ are incident to a unique closed ball bF  R. • (C3) If bF is a j-ball then bF  bd F is a (j-1)-sphere. • (C4) Each k-ball in R’, except bF, intersects bd F in a (k-1)-ball.

  10. Extended TBP • For a 1- or 2-face σ, let Del S|σ denote the Delaunay subcomplex restricted to σ. • Del S||Di| = σDiDel S|σ. • Del S||D| = σDDel S|σ. • Theorem. If S has the eTBP for D then the underlying space of Del S||D| is homeomorphic to |D| [ES97].

  11. Feature Size • For analysis, we require a feature size which is 1-Lipschitz and non-zero. • For any x  |D|, let f(x) = min{m(x), g(x)}. • For any σ  D, f() is 1-Lipschitz over int σ. • For δ  (0,1] and x  |D|, • if x  D0, lfsδ(x) = δf(x). • if x  int |Di|, for i ≥ 1, lfsδ(x) = max{δf(x), maxybd|Di| {lfsδ(y)-||x-y||}}.

  12. Protecting D1 • Any 2 adjacent balls on a 1-face must overlap significantly without containing each others centers • No 3 balls have a common intersection • For a point p σ D1, if we enlarge any protecting ball Bp by a factor c ≤ 8, forming B’: • B’ intersects σ in a single curve, and intersects all  D2 adjacent to σ in a topological disk. • For any q in B’  σ, the tangent variation between p and q is bounded. • For any q in B’   ( D2 adjacent to σ), the normal variation between p and q is bounded.

  13. Admissible Point Sets • Protecting balls are turned into weighted points • We call a point set S admissible if • S contains all weighted points placed on D1. • Other points in S are unweighted and they lie outside of the protecting balls (the weighted points). • We maintain an admissible point set at each step of the algorithm.

  14. D1 conformation • Lemma. Let S is an admissible point set. For a 1-face σ, if p and q are adjacent weighted vertices spanning segment σpq on σ then Vpq is the only Voronoi facet which intersects σpq and it does so exactly once.

  15. Meshing PSCs • Meshing algorithm uses four tests to detect eTBP violations. • Upon violation, we insert points outside of protected balls of weighted vertices.

  16. Test 1: Multi-Intersection(q,σ) • For a point qS on a 2-face σ, find a triangle t  Del S|σ incident to q s.t. Vt intersects σ multiple times. • If no t exists, return null, otherwise return the furthest (weighted) intersection point from q.

  17. Test 2: Normal-Deviation(q,σ,Θ) • For a point q S on a 2-face σ, check nσ(p), nσ(q) < Θ for all points p  Vq|σ. • 2ω ≤ Θ ≤ /6. • If so return null. • Otherwise return a point p where nσ(p), nσ(q) = Θ .

  18. Test 3: Infringement(q,σ) • We say q is infringed w.r.t. σ if • σ is a 2-face containing q s.t. pq  Del S|σ for some p  σ. • σ is a 2-face and there is a 1-face in bd σ containing q and a non-adjacent vertex p s.t. pq  Del S|σ. • For q S  σ, return null if q is not infringed, otherwise let pq be the infringing edge. • If the boundary edges of Vpq intersect int σ, return any intersection point. • Else, Vpq  σ is a collection of closed curves, return a critical point of Vpq  σ in a direction parallel to Vpq.

  19. Test 4: No-Disk(q,σ) • If the star of q in Del S|σ is a topological disk, return null. • Otherwise, find the triangle t  Del S|σ incident to q which has the furthest (weighted) intersection point in Vt|σ from q and return the intersection point.

  20. Meshing Algorithm • Protect elements in D≤1 with weighted points. Insert a point in each element of D2 outside of protected regions. Let S be this point set. • For any σ D2 and point q S  σ: • If Infringed(q,σ), Multi-Intersection(q,σ), Normal-Deviation(q,σ,Θ), or No-Disk(q,σ) (checked in that order) return a point x, insert x into S. • Repeat 2. until no points are inserted. • Return Del S|D.

  21. Admissibility is Invariant Lemma. The algorithm never attempts to insert a point in any protecting ball • Since no 3 weighted points intersect, • all surface points (intersections of dual Voronoi edges and D) lie outside of every protecting ball

  22. Initialization • The algorithm must initialize with a few points from each patch in D2 • Otherwise, components can be missed.

  23. Termination • Each point x inserted is Ω(lfsδ(x)) away from all other points. • Standard packing argument follows.

  24. Topology Preservation • To satisfy C1-C4 of eTBP, we show each Voronoi k-face F = Vp1 … Vp(4-k) has: • (P1) If F  σ ≠ , for σ Dj, the intersection is a (k+j-3)-ball • (P2) There is a unique lowest dimensional σF s.t. p1, …, p(4-k)σF. • (P3) F intersects σF and only incident elements of σF. • Theorem. If S satisfies P1-P3 then S satisfies C1-C4 of eTBP.

  25. Feature Preservation • h:|D|  |Del S|D| can be constructed which respects each Di [ES97]. • Thus hi:|Di|  |Del S|Di| also a homeomorphism with vertex restrictions, ensuring that the nonsmooth features are preserved.

  26. Delaunay Refinement made practical for PSCs S.-W. Cheng, Tamal K. Dey, Joshua Levine

  27. Definitions • For a patch σ Di, • When sampled with S • Del S|σ is the Delaunay subcomplex restricted to σ • Skli S|σ is the i-dimensional subcomplex of Del S|σ, • Skli S|σ = closure { t | t  Del S|σ is an i-simplex} • Skli S|Di = σ Di Skli S|σ

  28. Disk Condition • For a point p on a 2-face σ, • UmbD(p) is the set of triangles in Skl2 S|D2 incident to p. • Umbσ(p) is the set of triangles in Skl2 S|σ incident to p. • Disk_Condition(p) requires: • UmbD(p) = σ, pσ Umbσ(p) • For each σ containing p, Umbσ(p) is a 2-disk where p is in the interior iff p  int σ

  29. Meshing Algorithm DelPSC(D, r) • Protect elements of D≤1. • Mesh2Complex – Repeatedly insert surface points for triangles in Skl2 S|σ for some σ if either • Disk_Condition(p) violated for p σ, or • A triangle has orthoradius > r. • Mesh3Complex – Repeatedly insert orthocenters of tetrahedra in Skl3 S|σ for some σ if • A tetrahedra has orthoradius > r and its orthocenter does not encroach any surface triangle in Skl2 S|D2. • Return i Skli S|Di.

  30. Termination Properties • Curve Preservation • For each σ  D1, Skl1 S|σ σ. Two vertices are joined by an edge in Skl1 S|σ iff they were adjacent in σ. • Manifold • For 0 ≤ i ≤ 2, and σ  Di, Skli S|σ is a manifold with vertices only in σ. Further, bd Skli S|σ = Skli-1 S|bd σ. • For i=3, the above holds when Skli S|σ is nonempty after Mesh2Complex. • Strata Preservation • There exists some r > 0 so that the output of DelPSC(D, r) is homeomorphic to D. • This homeomorphism respects stratification.

  31. Voronoi Cells Intersect “Discly” • Given a vertex p on a 2-face σ, if • Triangles incident to p in Skl2 S|σ are small enough. • Then, • Vp|σ is a topological disk, • Any edge of Vp|σ intersects σ at most once, and • Any facet of Vp|σ which intersects σ does so in an open curve.

  32. TBP holds globally • if • All triangles incident in Skl2 S|σ are smaller than a bound for all 2-faces, • Then • TBP holds globally • This leads to the proof of ETBP and more…topic of a new unpublished paper.

  33. Adjusting MaxRad Example

  34. Adjusting MaxRad Example

  35. Examples

  36. Examples

  37. Examples

  38. Examples

  39. Examples

  40. Examples

  41. Examples

  42. Examples

  43. Examples

  44. Examples

  45. Sharp Example

  46. Conclusions • Delaunay meshing for PSC with guarantees. • Feature preservation is an extra `feature’. • Making computations easier, faster? • Analyzing size complexity?

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