Delaunay Triangulations for 3D Mesh Generation Shang-Hua Teng Department of Computer Science, UIUC Work with: Gary Miller, Dafna Talmor, Noel Walkington Siu-Wing Cheng, Tamal Dey, Herbert Edelsbrunner, Micheal Facello Xiang-Yang Li and Alper Üngör

Unstructured Meshes

Numerical Methods Point Set: Triangulation: ad hoc octreeDelaunay Domain, Boundary, and PDEs elementdifference volume Finite Ax=b direct method Mesh Generation geometric structures Linear System algorithm data structures Approximation Numerical Analysis Formulation Math+Engineering iterative method multigrid

Outline n Mesh Generation in 2D u Mesh Qualities u Meshing Methods u Meshes and Circle Packings n Mesh Generation in 3D u Slivers u Numerical Solution: Control Volume Method u Geometric Solution: Sliver Removal by Weighted Delaunay Triangulations

Badly Shaped Triangles

Aspect Ratio ( R/r )

Meshing Methods n Advancing Front n Quadtree and Octree Refinement n Delaunay Based u Delaunay Refinement u Sphere Packing u 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? n Maximizes the smallest angle in 2D. n Has efficient algorithms and data structures. n Delaunay refinement: u In 2D, it generates optimal size, natural looking meshes with 20.7 o (Jim Ruppert)

Delaunay Refinement (Jim Ruppert) 2D insertion1D 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 n Delaunay failed on aspect ratio n Quadtree becomes octree (Mitchell-Vavasis) n Meshes become much larger n 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 n 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 2 r

Constant Degree Lemma (3D) (Miller-Talmor-Teng-Walkington) n The vertex degree of the Delaunay triangulation with a constant radius-edge ratio is bounded by a constant.

Packing Algorithms

Well-Spaced Points

Packing in 3D n Pack 2D boundaries by quadtree approximation or Ruppert Refinement n Approximate the LFS by octree n 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 D Y L

Pumping Lemma (Cheng-Dey-Edelsbrunner-Facello-Teng) D Y z H r s p P 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. n 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 n One by one in an arbitrary ordering n fix the weight of each point n Implementation: flip and keep the best configuration.

Mesh Coarsening

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

Supports n DOE ASCI (Center for Simulation of Advanced Rocket) n NSF OPAAL (Center for Process Simulation and Design) n NSF CAREER n Alfred P. Sloan