Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC...

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Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC th: ller, Dafna Talmor, Noel Walkington g Cheng, Tamal Dey, Herbert Edelsbrunner, Micheal F ang Li and Alper Üngör
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Transcript of Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC...

Page 1: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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

Page 2: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Unstructured Meshes

Page 3: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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

Page 4: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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

Page 5: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Badly Shaped Triangles

Page 6: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Aspect Ratio (R/r)

Page 7: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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.

Page 8: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Balanced Quadtree Refinements

(Bern-Eppstein-Gilbert)

Page 9: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Quadtree Mesh

Page 10: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Delaunay Triangulations

Page 11: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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)

Page 12: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Delaunay Refinement

(Jim Ruppert)

2D insertion 1D insertion

Page 13: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Delaunay Mesh

Page 14: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Local Feature Spacing (f)

f: R

Page 15: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Well-Shaped Meshes and f

Page 16: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

f is 1-Lipschitz and Optimal

Page 17: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Sphere-Packing

Page 18: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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

Page 19: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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)

Page 20: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Part I: Meshes to Packings

Page 21: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Part II: Packings to Meshes

Page 22: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

3D Challenges

Delaunay failed on aspect ratio Quadtree becomes octree

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

Page 23: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Badly Shaped Tetrahedra

Page 24: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Slivers

Page 25: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Radius-Edge Ratio

(Miller-Talmor-Teng-Walkington)

R L

R/L

Page 26: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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.

Page 27: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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

Page 28: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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.

Page 29: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Packing Algorithms

Page 30: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Well-Spaced Points

Page 31: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Well-Spaced Points

Page 32: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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)

Page 33: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Delaunay Refinement in 3D

Page 34: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Slivers

Page 35: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Sliver: the geo-roach

Page 36: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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

Page 37: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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

Page 38: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Weighted Points and Distance

p z

Page 39: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Orthogonal Circles and Spheres

Page 40: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Weighted Bisectors

Page 41: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Weighted Delaunay

Page 42: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Weighted Delaunay and Convex Hull

Page 43: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Parametrizing Slivers

DY

L

Page 44: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Pumping Lemma

(Cheng-Dey-Edelsbrunner-Facello-Teng)

DY

z

H

r s

pP

q

Page 45: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

… 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

Page 46: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

The Stories of Balloons

Page 47: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Interval Lemma

0 N(p)/3

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

Page 48: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Sliver Removal by Flipping

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

configuration.

Page 49: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Mesh Coarsening

Page 50: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

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

Page 51: Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington.

Supports DOE ASCI (Center for Simulation of

Advanced Rocket) NSF OPAAL (Center for Process

Simulation and Design) NSF CAREER Alfred P. Sloan