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 EngineeringAtlanta, Georgia

28 Oct 2015, 10:55-11:20am (25 minutes)

VoroCrust: Simultaneous Surface Reconstruction

and Volume Meshing with Voronoi cells

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?

ALGORITHM OVERVIEWVoroCrust

SandiaNationalLaboratories

VoroCrust Primal-Dual-Primal Dance

Input: domain with boundary(3D algorithm, illustrated in 2D)

Theory for smooth manifoldsPractical rules for sharp features

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

Primal-Dual-Primal Dance

Spheres Intersect at two points = unweighted seeds pairs

red = exteriorblue = 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

Primal-Dual-Primal Dance

Properties surface mesh nodes are Voronoi vertices surface mesh edges are Voronoi edges surface mesh triangles are Voronoi facets

Build unweighted Voronoi diagram of seeds. Boundary between interior and exterior cells= surface mesh, it reconstructs surface

Mesh Quality Enhancement

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

Enhancement

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

Examples

Examples

Wall of Hampton InnAtlanta, GA

Armadillo

Armadillo

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

Dragon

Bunny – size graded mesh

Built-in Robustness

Not water tight? Easy, sample spheres fill the gaps

No parameterization, complicated topology? Easy, overlap and coverage tests are purely local, sphere radius

Fertility

Theory

• Why does this work?

Intersections and Duality

• Intersection point pairs– 2D: 2 circles ∧ two points– 3D: 3 spheres ∧ two points

• Intersection points on Voronoi edgedual to triangle

• Three overlapping spheres =wDel triangle, negative circumradius

G↑

G↓p1

p3p2

s1

s3

s2

c12

c13

c23

e13

e23e12

G

p1p3

p2

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 facetsthat happen to be triangles

Naïve Mirroring/Ghosting

• Placing seeds exactly on spheres is important– Bad normals result from

mirrored pairswithout cospherical seeds around surface points.

• 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?

Challengehalf-covered guide pairs

vs. Filtering

• Some reconstruction techniques select2 triangles of sliver f123

p1

p3

p2 p4f234

f123

p1

p3

p2 p4

f134

light = triangle seen from belowdark = triangle seen from above

p1

p3

p2 p4

f134f123

upper pair

lower pair

Sliver Tetrahedra

G 2↑g 1↑

G 1↓

g 2↓

s1

s3s2

s4 c34

G 234↑

G 124↑

G 123↓

G 134↓

f14

G2 134↓

G4 234↑

f34

f12

G3 124↑

G1 123↓

f24f13 f23

p1

p3

p2 p4

n

f14

G2 134↓

G4 234↑

f34

f12

G3 124↑

G1 123↓

f24f13f23

p1

p3

p2

p4

n

Sliver Tetrahedra

f23

f14

G2 134↓

G4 234↑

f34

f12

G3 124↑

G1 123↓

f24f13f23

p1

p3

p2

p4

n

surface mesh = four green triangles

f13f24

f14

p4

p3

p2

p1

n G4 234↑

G3 124↑

G2 134↓

G1 123↓

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.)

Use as weighted seeds.(vs. unweighted seeds)Select weighted Delaunaytriangles for mesh and surface reconstruction.(vs. Voronoi cells)

Far unweighted Voronoi vertices lie near medial axis.(vs. close points from weightedVoronoi edges near surface.)

images courtesy Nina Amenta, power crust authorhttps://www.cs.utah.edu/~jeffp/8F-CG/namenta.ppt

Both have sliver issues, but they’re different issues.VoroCrust allows additional seeds and a well-shaped mesh.

Challenge

• Problem:– Theory was for smooth manifolds– What about sharp CAD models?

• Solution:– Careful placement at sharp features, akin to

Delaunay based methods

Sharp Features

ignoring sharp featuresresults in bad sphere overlaps

small spheres can avoid overlaps

ideal is to share overlaps

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?)

• backup slides

Primal-Dual-Primal dance

tweaked:red: 58, 218, 75, 329, 53, 432, 74, 509.5, 204, 718, 268, 699, 444, 668, 487, 689, 608, 681, 640, 526, 511, 494, 471, 492, 418, 446, 454, 400, 597, 358, 611, 266, 570, 183, 445, 190, 278, 136, 156, 146,blue: 122, 218, 105, 329, 129, 430, 112, 509, 203, 564, 269, 588, 444, 620, 486, 598, 609, 603, 505, 580, 494, 537, 463, 514, 390, 509, 388, 368, 446, 380, 574, 310, 587, 280, 559, 279, 437, 238, 263, 234, 149, 192 black: 163, 246, 254, 271, 175, 322, 263, 346, 344, 317, 436, 292, 192, 415, 161, 491, 220, 513, 452, 559

red: 58, 218, 75, 329, 53, 432, 74, 509.5, 204, 718, 268, 699, 444, 668, 487, 689, 608, 681, 640, 526, 511, 494, 471, 492, 418, 446, 454, 400, 597, 358, 611, 266, 570, 183, 445, 190, 278, 136, 156, 146,blue 122, 218, 105, 329, 129, 430, 112, 509, 203, 564, 269, 588, 444, 620, 486, 598, 609, 603, 505, 580, 494, 537, 463, 514, 390, 509, 388, 368, 446, 380, 574, 310, 587, 280, 559, 279, 437, 238, 263, 234, 149, 192