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