Grid generation and optimization based on centroidal Voronoi tessellations

Centroidal Voronoi tessellations (CVTs) are Voronoi tessellations of a region such that the generating points of the tessellations are also the centroids of the corresponding Voronoi regions. Such tessellations are of use in very diverse applications, including data compression, clustering analysis, cell biology, territorial behavior of animals, and optimal allocation of resources. In this paper, we explore the use of CVTs in grid generation in connection with finite element approximations of partial differential equations. We being by describing these tessellations and methods for their determination. We then discuss their application to mesh generation and finish with some examples of their use for the solution of partial differential equations.

[1]  P. Eiseman,et al.  Adaptive grid generation , 1987 .

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

[3]  Rolf Klein,et al.  Concrete and Abstract Voronoi Diagrams , 1990, Lecture Notes in Computer Science.

[4]  Kelly R. Laflin,et al.  Dynamic Grid Adaption and Grid Quality , 1998 .

[5]  V. T. Rajan.  Optimality of the Delaunay triangulation in ℝd , 1994, Discret. Comput. Geom..

[6]  Houman Borouchaki,et al.  Delaunay Tetrahedralization using an Advancing-Front Approach , 1996 .

[7]  Theo Pilkington,et al.  Adaptive grid generation , 1992, 1992 14th Annual International Conference of the IEEE Engineering in Medicine and Biology Society.

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

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

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

[11]  D. Du,et al.  Computing in Euclidean Geometry , 1995 .

[12]  Bharat K. Soni,et al.  Handbook of Grid Generation , 1998 .

[13]  David L. Neuhoff,et al.  Quantization , 2022, IEEE Trans. Inf. Theory.

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

[15]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[16]  Jim Ruppert,et al.  A Delaunay Refinement Algorithm for Quality 2-Dimensional Mesh Generation , 1995, J. Algorithms.

[17]  Mark T. Jones,et al.  A Parallel Algorithm for Mesh Smoothing , 1999, SIAM J. Sci. Comput..

[18]  P. George,et al.  3D Delaunay mesh generation coupled with an advancing-front approach , 1998 .

[19]  W. Rheinboldt,et al.  Error Estimates for Adaptive Finite Element Computations , 1978 .

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

[21]  David Eppstein,et al.  MESH GENERATION AND OPTIMAL TRIANGULATION , 1992 .

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

[23]  Leonidas J. Guibas,et al.  Randomized Incremental Construction of Delaunay and Voronoi Diagrams , 1990, ICALP.

[24]  V. T. Rajan,et al.  Optimality of the Delaunay triangulation in Rd , 1991, SCG '91.

[25]  Allen Gersho,et al.  Asymptotically optimal block quantization , 1979, IEEE Trans. Inf. Theory.

[26]  David P. Dobkin,et al.  The quickhull algorithm for convex hulls , 1996, TOMS.

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