VoroCrust: Simultaneous Surface Reconstruction and Volume Meshing with Voronoi cells

Transcription

1 VoroCrust: Simultaneous Surface Reconstruction and Volume Meshing with Voronoi cells Scott A. Mitchell (speaker), joint work with Ahmed H. Mahmoud, Ahmad A. Rushdi, Scott A. Mitchell, Ahmad Abdelkader Abdelrazek, Chandrajit L. Bajaj, John D. Owens, Mohamed S. Ebeida Polytopal Element Methods in Mathematics and Engineering Atlanta, Georgia 28 Oct 2015, 10:55-11:20am (25 minutes)

2 Summary VoroCrust is meshing for polytopal cells talk has no finite element content, just geometry Output Produces 3D Voronoi cells convex planar facets no sharp dihedral angles good aspect ratio Surface (boundary) mesh is reconstructed (naturally, without clipping, snapping or cleanup) by the boundary between inside and outside 3D cells planar convex 2D surface facets convex 3D cells adjacent to boundary not cells clipped by the boundary Open quality goals no small edges, no sharp edge angles, no small area faces... did other workshop talks describe what was needed in a mesh?

3 VoroCrust ALGORITHM OVERVIEW Sandia National Laboratories

4 VoroCrust Primal-Dual-Primal Dance Input: domain with boundary (3D algorithm, illustrated in 2D) Theory for smooth manifolds Practical rules for sharp features

5 Primal-Dual-Primal Dance Create well-spaced sample points Required properties Weighted Voronoi balls cover bdy Spheres have uncovered interior and exterior point: north and south poles Sufficient for uncovered poles local feature size lfs spacing and no ball center inside another ball

6 Primal-Dual-Primal Dance Spheres Intersect at two points = unweighted seeds pairs red = exterior blue = interior Ideal: both intersection points lie outside all other balls Want seeds sufficient to reconstruct surface surface mesh nodes are Voronoi vertices surface mesh edges are Voronoi edges surface mesh triangles are Voronoi facets

7 Primal-Dual-Primal Dance Build unweighted Voronoi diagram of seeds. Boundary between interior and exterior cells = surface mesh, it reconstructs surface Properties surface mesh nodes are Voronoi vertices surface mesh edges are Voronoi edges surface mesh triangles are Voronoi facets

8 Mesh Quality Enhancement Freedom: Additional seeds outside balls. Achieve aspect ratio bound. Additional goals: lattice points for a hex-dominant mesh

9 Enhancement Freedom: Additional seeds outside balls. Achieve aspect ratio bound. Additional goals: lattice points for a hex-dominant mesh

10 Examples

11 Examples Wall of Hampton Inn Atlanta, GA

12 Armadillo

13 hex-dominant mesh is trivial extention interior seeds = lattice points (centers of hexes) Armadillo

14 Dragon

15 Bunny size graded mesh

16 Built-in Robustness No parameterization, complicated topology? Easy, overlap and coverage tests are purely local, sphere radius Not water tight? Easy, sample spheres fill the gaps

17 Fertility

18 Theory Why does this work?

19 Intersections and Duality s2 p2 c12 e12 p1 s1 c23 s3 e23 G e13 p3 c13 p2 G p1 G p3 Intersection point pairs 2D: 2 circles two points 3D: 3 spheres two points Intersection points on Voronoi edge dual to triangle Three overlapping spheres = wdel triangle, negative circumradius

20 Intersections and Duality Multiple seeds on a sphere surface, none inside a sphere Sphere center is equidistant to all these seeds and none are closer = definition of a Voronoi vertex G G Surface mesh is set of Voronoi facets that happen to be triangles

21 Naïve Mirroring/Ghosting Placing seeds exactly on spheres is important Bad normals result from mirrored pairs without cospherical seeds around surface points.

22 Challenge half-covered guide pairs Problem One intersection point is covered, other is uncovered It happens for sliver tetrahedra (3D) four points nearly cocircular and coplanar Solution It s not that bad Consequences Extra surface vertex New triangles, interpolating a sliver tetrahedra Surface mesh still provably close to input manifold normals?

23 vs. Filtering Some reconstruction techniques select 2 triangles of sliver p2 p2 upper pair p4 3 f1 2 f23 4 p3 p4 3 f1 2 f134 p1 p3 p1 lower pair p2 p4 f123 f134 light = triangle seen from below dark = triangle seen from above p1 p3

24 Sliver Tetrahedra G4 234 G3 124 p2 G 234 G 124 s2 s1 f13 s3 s4 G 2 g 1 g 2 f34 p4 f14 f24 3 f2 n f 12 p3 G1 123 c34 G2 134 p1 G3 124 G 1 p2 G4 234 f34 f13 G 134 G 123 p1 f14 n f23 p4f24 f12 G1 123 G2 134 p3

25 Sliver Tetrahedra p3 G3 124 surface mesh = four green triangles p2 f 14 G1 123 n f24 G f1g f2 p4 p1 G3 124 p2 G4 234 f34 f13 p1 f14 n f23 p4f24 f12 G1 123 G2 134 p3

26 images courtesy Nina Amenta, power crust author Power crust VoroCrust vs. famous power crust algorithm [1998] 1000 Google Scholar citations (6 papers) Densely sample boundary. (Same lfs density. vs. need sample weights = balls covering boundary.) Far unweighted Voronoi vertices lie near medial axis. (vs. close points from weighted Voronoi edges near surface.) Both have sliver issues, but they re different issues. VoroCrust allows additional seeds and a well-shaped mesh. Use as weighted seeds. (vs. unweighted seeds) Select weighted Delaunay triangles for mesh and surface reconstruction. (vs. Voronoi cells)

27 Challenge Problem: Theory was for smooth manifolds What about sharp CAD models? Solution: Careful placement at sharp features, akin to Delaunay based methods

28 Sharp Features ideal is to share overlaps ignoring sharp features results in bad sphere overlaps small spheres can avoid overlaps

29 Conclusion VoroCrust Robust polyhedral meshing with true Voronoi cells has arrived ideal polytopal mesh with convex cells, planar facets no clipping good aspect ratio provable surface convergence Status paper nearly complete reimplementing production software Open mesh quality short edges, sharp edge angles, small area faces mesh quality needed for polytopal elements? seeds interior to the volume, want surface ones too? (bad surface normals in pathological cases?)

30 backup slides

31 Primal-Dual-Primal dance red: 58, 218, 75, 329, 53, 432, 74, 509.5, 204, 718, 268, 69 blue 122, 218, 105, 329, 129, 430, 112, 509, 203, 564, 269, 58 tweaked: red: 58, 218, 75, 329, 53, 432, 74, 509.5, 204, 718, 268, 699, blue: 122, 218, 105, 329, 129, 430, 112, 509, 203, 564, 269, 588