Delaunay Triangulations for 3D Mesh Generation

Shang-Hua Teng

Department of Computer Science, UIUC

Work with:Gary Miller, Dafna Talmor, Noel WalkingtonSiu-Wing Cheng, Tamal Dey, Herbert Edelsbrunner, Micheal FacelloXiang-Yang Li and Alper Üngör

Unstructured Meshes

Numerical Methods

Point Set:Triangulation:

ad hoc

octree Delaunay

Domain, Boundary, and PDEs

element difference

volume

Finite

Ax=b

direct method

Mesh Generationgeometric structures

Linear Systemalgorithm

data structures

ApproximationNumerical Analysis

FormulationMath+Engineering

iterative method

multigrid

Outline Mesh Generation in 2D

Mesh Qualities Meshing Methods Meshes and Circle Packings

Mesh Generation in 3D Slivers Numerical Solution: Control Volume Method Geometric Solution: Sliver Removal by Weighted

Delaunay Triangulations

Badly Shaped Triangles

Aspect Ratio (R/r)

Meshing Methods

Advancing Front Quadtree and Octree Refinement Delaunay Based

Delaunay Refinement Sphere Packing Weighted Delaunay Triangulation

The goal of a meshing algorithm is to generate a well-shaped mesh that is as small as possible.

Balanced Quadtree Refinements

(Bern-Eppstein-Gilbert)

Quadtree Mesh

Delaunay Triangulations

Why Delaunay? Maximizes the smallest angle in 2D. Has efficient algorithms and data structures. Delaunay refinement:

In 2D, it generates optimal size, natural looking meshes with 20.7o (Jim Ruppert)

Delaunay Refinement

(Jim Ruppert)

2D insertion 1D insertion

Delaunay Mesh

Local Feature Spacing (f)

f: R

Well-Shaped Meshes and f

f is 1-Lipschitz and Optimal

Sphere-Packing

p

-Packing a Function f

No large empty gap: the radius of the largest empty sphere passing q is at most f(q).

f(p)/2

q

The Delaunay triangulation of a -packing is a well-shaped mesh of optimal size.

Every well-shaped mesh defines a -packing.

The Packing Lemma (2D)

(Miller-Talmor-Teng-Walkington)

Part I: Meshes to Packings

Part II: Packings to Meshes

3D Challenges

Delaunay failed on aspect ratio Quadtree becomes octree

(Mitchell-Vavasis) Meshes become much larger Research is more interesting?

Badly Shaped Tetrahedra

Slivers

Radius-Edge Ratio

(Miller-Talmor-Teng-Walkington)

R L

R/L

The Packing Lemma (3D)

(Miller-Talmor-Teng-Walkington)

The Delaunay Triangulation of a -packing is a well-shaped mesh (using radius-edge ratio) of optimal size.

Every well-shaped (aspect-ratio or radius-edge ratio) mesh defines a -packing.

Uniform Ball Packing In any dimension, if P is a maximal packing

of unit balls, then the Delaunay triangulation of P has radius-edge at most 1.

||e|| is at least 2

r is at most 2r

Constant Degree Lemma (3D)

(Miller-Talmor-Teng-Walkington)

The vertex degree of the Delaunay triangulation with a constant radius-edge ratio is bounded by a constant.

Packing Algorithms

Well-Spaced Points

Well-Spaced Points

Packing in 3D

Pack 2D boundaries by quadtree approximation or Ruppert Refinement

Approximate the LFS by octree Locally sample the region to create a

well-spaced point set

• 3D Delaunay refinement also generates meshes with a good edge-radius ratio (Shewchuck)

Delaunay Refinement in 3D

Slivers

Sliver: the geo-roach

Coping with Slivers: Control-Volume-Method(Miller-Talmor-Teng-Walkington)

Sliver Removal by Weighted Delaunay (Cheng-Dey-Edelsbrunner-Facello-Teng)

Weighted Points and Distance

p z

Orthogonal Circles and Spheres

Weighted Bisectors

Weighted Delaunay

Weighted Delaunay and Convex Hull

Parametrizing Slivers

DY

L

Pumping Lemma

(Cheng-Dey-Edelsbrunner-Facello-Teng)

DY

z

H

r s

pP

q

… under Assumptions

Property []: the radius-edge ratio the Delaunay triangulation is .

Property []: for any two points p and q, their weights P, Q < ||p-q|| / 3.

Boundary: The Delaunay mesh is periodic

The Stories of Balloons

Interval Lemma

0 N(p)/3

Constant Degree: The union of all weighted Delaunay triangulations with Property [] and Property [] has a constant vertex degree

Sliver Removal by Flipping

One by one in an arbitrary ordering fix the weight of each point Implementation: flip and keep the best

configuration.

Mesh Coarsening

Related and Future Research Meshing with a moving boundary Sphere-packing and advancing front Sphere-packing and Hex meshes Meshing for time-and-space Boundary handling in three dimensions Mesh smoothing and improvement Mesh coarsening in three dimensions Software, Software, Software!!! What are the constants in theory and practice

Supports DOE ASCI (Center for Simulation of

Advanced Rocket) NSF OPAAL (Center for Process

Simulation and Design) NSF CAREER Alfred P. Sloan