Parallel Delaunay Refinement: Algorithms and Analyses

We present a parallel Delaunay refinement algorithm for generating well-shaped meshes in both two and three dimensions. Like its sequential counterparts, the parallel algorithm iteratively improves the quality of a mesh by inserting new points, the Steiner points, into the input domain while maintaining the Delaunay triangulation. The Steiner points are carefully chosen from a set of candidates that includes the circumcenters of poorly-shaped triangular elements. We introduce a notion of independence among possible Steiner points that can be inserted simultaneously during Delaunay refinements and show that such a set of independent points can be constructed efficiently and that the number of parallel iterations is O(log2Δ), where Δ is the spread of the input — the ratio of the longest to the shortest pairwise distances among input features. In addition, we show that the parallel insertion of these set of points can be realized by sequential Delaunay refinement algorithms such as by Ruppert's algorithm in ...

[1]  Mark T. Jones,et al.  Parallel Algorithms for Adaptive Mesh Refinement , 1997, SIAM J. Sci. Comput..

[2]  R. Seidel.  Backwards Analysis of Randomized Geometric Algorithms , 1993 .

[3]  J. Shewchuk,et al.  Delaunay refinement mesh generation , 1997 .

[4]  Rainald Löhner,et al.  Parallel Advancing Front Grid Generation , 1999, IMR.

[5]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

[6]  Herbert Edelsbrunner,et al.  Sliver exudation , 1999, SCG '99.

[7]  F. P. Preparata,et al.  Convex hulls of finite sets of points in two and three dimensions , 1977, CACM.

[8]  Jim Ruppert,et al.  A new and simple algorithm for quality 2-dimensional mesh generation , 1993, SODA '93.

[9]  Guy E. Blelloch,et al.  Design and Implementation of a Practical Parallel Delaunay Algorithm , 1999, Algorithmica.

[10]  Tiow Seng Tan,et al.  An upper bound for conforming Delaunay triangulations , 1992, SCG '92.

[11]  Michael Luby,et al.  A simple parallel algorithm for the maximal independent set problem , 1985, STOC '85.

[12]  Can C. Özturan,et al.  Parallel Automatic Adaptive Analysis , 1997, Parallel Comput..

[13]  L. Paul Chew,et al.  Parallel Constrained Delaunay Meshing , 2007 .

[14]  David Eppstein,et al.  Provably Good Mesh Generation , 1994, J. Comput. Syst. Sci..

[15]  John H. Reif,et al.  Parallel Computational Geometry: An Approach using Randomization , 2000, Handbook of Computational Geometry.

[16]  Jonathan Richard Shewchuk,et al.  Delaunay refinement algorithms for triangular mesh generation , 2002, Comput. Geom..

[17]  Guy E. Blelloch,et al.  Compact representations of simplicial meshes in two and three dimensions , 2005, Int. J. Comput. Geom. Appl..

[18]  Mariette Yvinec,et al.  Conforming Delaunay triangulations in 3D , 2002, SCG '02.

[19]  Xiang-Yang Li,et al.  Smoothing and cleaning up slivers , 2000, STOC '00.

[20]  Steven Fortune,et al.  Voronoi Diagrams and Delaunay Triangulations , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[21]  Jaime Peraire,et al.  Parallel unstructured mesh generation , 1996 .

[22]  Scott A. Mitchell,et al.  Quality mesh generation in three dimensions , 1992, SCG '92.

[23]  L. Paul Chew,et al.  Constrained Delaunay triangulations , 1987, SCG '87.

[24]  Nikos Chrisochoides,et al.  A new approach to parallel mesh generation and partitioning problems , 2002 .

[25]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[26]  L. Paul Chew,et al.  Guaranteed-Quality Triangular Meshes , 1989 .

[27]  Alper Üngör,et al.  Parallel delaunay refinement and space-time meshing , 2002 .

[28]  David Eppstein,et al.  Parallel Construction of Quadtrees and Quality Triangulations , 1993, WADS.

[29]  Nikos Chrisochoides,et al.  Simultaneous Mesh Generation and Partitioning for Delaunay Meshes , 2000, IMR.

[30]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[31]  Leonidas J. Guibas,et al.  Randomized incremental construction of Delaunay and Voronoi diagrams , 1990, Algorithmica.

[32]  D. Eppstein,et al.  MESH GENERATION AND OPTIMAL TRIANGULATION , 1992 .

[33]  Herbert Edelsbrunner,et al.  Sink-insertion for mesh improvement , 2001, SCG '01.

[34]  Nancy M. Amato,et al.  Parallel algorithms for higher-dimensional convex hulls , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.

[35]  Duc Thai Nguyen.  Parallel-Vector Equation Solvers for Finite Element Engineering Applications , 2002, Springer US.

[36]  Jonathan Richard Shewchuk,et al.  Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator , 1996, WACG.

[37]  David M. Mount,et al.  A point-placement strategy for conforming Delaunay tetrahedralization , 2000, SODA '00.

[38]  Michael Luby.  A Simple Parallel Algorithm for the Maximal Independent Set Problem , 1986, SIAM J. Comput..

[39]  Gary L. Miller,et al.  Control Volume Meshes Using Sphere Packing , 1998, IRREGULAR.

[40]  M. Rivara.  Algorithms for refining triangular grids suitable for adaptive and multigrid techniques , 1984 .

[41]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.

[42]  David Eppstein,et al.  Parallel Construction of Quadtrees and Quality Triangulations , 1993, Int. J. Comput. Geom. Appl..

[43]  Jonathan Richard Shewchuk,et al.  Tetrahedral mesh generation by Delaunay refinement , 1998, SCG '98.

[44]  Nikos Chrisochoides,et al.  Guaranteed: quality parallel delaunay refinement for restricted polyhedral domains , 2002, SCG '02.

[45]  Richard M. Karp,et al.  Parallel Algorithms for Shared-Memory Machines , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[46]  Shang-Hua Teng,et al.  Unstructured Mesh Generation: Theory, Practice, and Perspectives , 2000, Int. J. Comput. Geom. Appl..

[47]  Asad I. Khan,et al.  Parallel Finite Element Computations , 1999 .

[48]  L. Paul Chew,et al.  Guaranteed-quality Delaunay meshing in 3D (short version) , 1997, SCG '97.

[49]  Gary L. Miller,et al.  A Delaunay based numerical method for three dimensions: generation, formulation, and partition , 1995, STOC '95.

[50]  Herbert Edelsbrunner,et al.  Geometry and Topology for Mesh Generation , 2001, Cambridge monographs on applied and computational mathematics.