CONFORMING CENTROIDAL VORONOI DELAUNAY TRIANGULATION FOR QUALITY MESH GENERATION

As the methodology of centroidal Voronoi tessellation (CVT) is receiving more and more attention in the mesh generation community, a clear characterization of the in∞uence of geometric constraints on the CVT-based meshing is becoming increasingly important. In this paper, we flrst give a precise deflnition of the geometrically conforming centroidal Voronoi Delau- nay triangulation (CfCVDT) and then propose an e-cient algorithm for its construction that involves projection and lifting processes in two dimensional space. Finally, we show the high-quality of CfCVDT meshes and the efiective- ness and robustness of our algorithm through extensive examples.

[1]  Qiang Du,et al.  Acceleration schemes for computing centroidal Voronoi tessellations , 2006, Numer. Linear Algebra Appl..

[2]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[3]  N. Weatherill,et al.  Efficient three‐dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints , 1994 .

[4]  M. Yvinec,et al.  Variational tetrahedral meshing , 2005, SIGGRAPH 2005.

[5]  Peter Hansbo,et al.  On advancing front mesh generation in three dimensions , 1995 .

[6]  Martin Isenburg,et al.  Centroidal Voronoi diagrams for isotropic surface remeshing , 2005, Graph. Model..

[7]  D. A. Field.  Laplacian smoothing and Delaunay triangulations , 1988 .

[8]  T. Baker.  Automatic mesh generation for complex three-dimensional regions using a constrained Delaunay triangulation , 1989, Engineering with Computers.

[9]  Qiang Du,et al.  Constrained Centroidal Voronoi Tessellations for Surfaces , 2002, SIAM J. Sci. Comput..

[10]  J. MacQueen.  Some methods for classification and analysis of multivariate observations , 1967 .

[11]  Desheng Wang,et al.  Tetrahedral mesh generation and optimization based on centroidal Voronoi tessellations , 2003 .

[12]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[13]  Qiang Du,et al.  Constrained boundary recovery for three dimensional Delaunay triangulations , 2004 .

[14]  S. P. Lloyd,et al.  Least squares quantization in PCM , 1982, IEEE Trans. Inf. Theory.

[15]  Per-Olof Persson,et al.  A Simple Mesh Generator in MATLAB , 2004, SIAM Rev..

[16]  Susanne C. Brenner,et al.  Multigrid methods for the computation of singular solutions and stress intensity factors III: interface singularities , 2003 .

[17]  D. A. Field.  Qualitative measures for initial meshes , 2000 .

[18]  Paresh Parikh,et al.  Generation of three-dimensional unstructured grids by the advancing-front method , 1988 .

[19]  Jonathan Richard Shewchuk,et al.  What is a Good Linear Element? Interpolation, Conditioning, and Quality Measures , 2002, IMR.

[20]  Weidong Zhao,et al.  Adaptive Finite Element Methods for Elliptic PDEs Based on Conforming Centroidal Voronoi-Delaunay Triangulations , 2006, SIAM J. Sci. Comput..

[21]  C. Taylor,et al.  Predictive medicine: computational techniques in therapeutic decision-making. , 1999, Computer aided surgery : official journal of the International Society for Computer Aided Surgery.

[22]  Q. Du,et al.  Recent progress in robust and quality Delaunay mesh generation , 2006 .

[23]  H. Borouchaki,et al.  Fast Delaunay triangulation in three dimensions , 1995 .

[24]  M. Gunzburger,et al.  Voronoi-based finite volume methods, optimal Voronoi meshes, and PDEs on the sphere ☆ , 2003 .

[25]  P. George,et al.  OPTIMAL DELAUNAY POINT INSERTION , 1996 .

[26]  Mark S. Shephard,et al.  Automatic three‐dimensional mesh generation by the finite octree technique , 1984 .

[27]  LongChen,et al.  OPTIMAL DELAUNAY TRIANGULATIONS , 2004 .

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

[29]  Qiang Du,et al.  Convergence of the Lloyd Algorithm for Computing Centroidal Voronoi Tessellations , 2006, SIAM J. Numer. Anal..

[30]  Qiang Du,et al.  Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations , 2002, Parallel Comput..

[31]  Qiang Du,et al.  Anisotropic Centroidal Voronoi Tessellations and Their Applications , 2005, SIAM J. Sci. Comput..

[32]  Qiang Du,et al.  Grid generation and optimization based on centroidal Voronoi tessellations , 2002, Appl. Math. Comput..

[33]  Qiang Du,et al.  Centroidal Voronoi Tessellations: Applications and Algorithms , 1999, SIAM Rev..