Anisotropic Voronoi Diagrams and Guaranteed-Quality Anisotropic

Mesh Generation

François LabelleJonathan Richard Shewchuk

Computer Science Division University of California at Berkeley

Berkeley, California

Presented by Jessica Schoen

Outline

Anisotropic meshes

Anisotropic Voronoi diagrams

Algorithm for anisotropic mesh generation

Current research

I. Anisotropic Meshes

What Are Anisotropic Meshes?Meshes with long, skinny triangles (in the right places).

Why are they important?•Often provide better interpolation of multivariate functions with fewer triangles.

•Used in finite element methods to resolve boundary layers and shocks. Source: “Grid Generation by the Delaunay

Triangulation,” Nigel P. Weatherill, 1994.

Distance MeasuresMetric tensor Mp: distances & angles measured by p.

Deformation tensor Fp: maps physical to rectified space.

Mp = FpT

Fp.Physical Space

Fp Fq

FqFp-1

pq

p qFpFq

-1

Distance MeasuresMetric tensor Mp: distances & angles measured by p.

Deformation tensor Fp: maps physical to rectified space.

Mp = FpT

Fp.Physical Space

Fp Fq

FqFp-1

Every point wants to be in a “nice” triangle in rectified space.

pq

p qFpFq

-1

The Anisotropic Mesh Generation Problem

Given polygonal domain and metric tensor field M,

generate anisotropic mesh.

A Hard Problem (Especially in Theory)

• Quadtree-based methods can be adapted to horizontal and vertical stretching, but not to diagonal stretching.

Common approaches to guaranteed-quality mesh generation do not adapt well to anisotropy.

• Delaunay triangulations lose their global optimality properties when adapted to anisotropy. No “empty circumellipse” property.

Heuristic Algorithms forGenerating Anisotropic Meshes

Bossen-Heckbert [1996] George-Borouchaki [1998]

Li-Teng-Üngör [1999]Shimada-Yamada-Itoh [1997]

II. Anisotropic Voronoi Diagrams

Voronoi Diagram: DefinitionGiven a set V of sites in Ed, decompose Ed into cells. The cell Vor(v) is the set of points “closer” to v than to any other site in V.

Mathematically:

Vor(v) = {p in Ed: dv(p)≤ dw(p) for every w in V.}

distance from v to p as measured by v

Distance Function Examples

1. Standard Voronoi diagram

dv(p) = || p – v ||2

Distance Function Examples

2. Multiplicatively weighted Voronoi diagram

dv(p) = cv|| p – v ||2

Distance Function Examples

3. Anisotropic Voronoi diagram

dv(p) = [(p – v)TMv(p – v)]1/2

Anisotropic Voronoi Diagram

Duality

Two Sites Define a Wedge

Dual Triangulation Theorem

III. Anisotropic Mesh Generation

by Voronoi Refinement

Easy Case: M = constant

Easy Case: M = constant

Voronoi Refinement Algorithm

Voronoi Refinement Algorithm

Insert new sites on unwedged portions of arcs.

Islands

Voronoi Refinement Algorithm

Insert new sites on unwedged portions of arcs.

Orphan

Voronoi Refinement Algorithm

Encroachment

Special Rules for the Boundary

Special Rules for the Boundary

Main Result

Why Does It Work?

Why Does It Work?

Numerical Problem

Red Voronoi vertex is intersection of conic sections

Numerical Problem

Intersection is computed numerically

?

Numerical ProblemWhich side of the red line is the vertex on?

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Numerical ProblemWhich side of the red line is the vertex on?

Geometric predicates are not always truthfuland the program crashes.

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IV. My Current Research

Star of a Vertex: Definition

The star of a vertex v is the set of all simplices having v for a face.

Star Based Anisotropic Meshing

Each vertex computes its own star independently

Inconsistent StarsIf the arcs and vertices of the corresponding anisotropic Voronoi diagram are not all wedged,

the diagram may not dualize to a triangulation, and the independently constructed stars may not form a consistent triangulation.

Equivalence TheoremIf the arcs and vertices of the anisotropic Voronoi diagram are all wedged, then

the independently constructed star of v

contains the same sites as star(v) in the dual of the anisotropic Voronoi

diagram.

v v