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101
Voronoi diagrams  a survey of a fundamental geometric data structure
 ACM COMPUTING SURVEYS
, 1991
"... This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. ..."
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Cited by 566 (5 self)
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This paper presents a survey of the Voronoi diagram, one of the most fundamental data structures in computational geometry. It demonstrates the importance and usefulness of the Voronoi diagram in a wide variety of fields inside and outside computer science and surveys the history of its development. The paper puts particular emphasis on the unified exposition of its mathematical and algorithmic properties. Finally, the paper provides the first comprehensive bibliography on Voronoi diagrams and related structures.
A Delaunay Refinement Algorithm for Quality 2Dimensional Mesh Generation
, 1995
"... We present a simple new algorithm for triangulating polygons and planar straightline graphs. It provides "shape" and "size" guarantees: All triangles have a bounded aspect ratio. The number of triangles is within a constant factor of optimal. Such "quality" triangulatio ..."
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Cited by 194 (0 self)
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We present a simple new algorithm for triangulating polygons and planar straightline graphs. It provides "shape" and "size" guarantees: All triangles have a bounded aspect ratio. The number of triangles is within a constant factor of optimal. Such "quality" triangulations are desirable as meshes for the nite element method, in which the running time generally increases with the number of triangles, and where the convergence and stability may be hurt by very skinny triangles. The technique we use  successive refinement of a Delaunay triangulation  extends a mesh generation technique of Chew by allowing triangles of varying sizes. Compared with previous quadtreebased algorithms for quality mesh generation, the Delaunay refinement approach is much simpler and generally produces meshes with fewer triangles. We also discuss an implementation of the algorithm and evaluate its performance on a variety of inputs.
Provably Good Mesh Generation
 J. Comput. Syst. Sci
, 1990
"... We study several versions of the problem of generating triangular meshes for finite element methods. We show how to triangulate a planar point set or polygonally bounded domain with triangles of bounded aspect ratio; how to triangulate a planar point set with triangles having no obtuse angles; how t ..."
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Cited by 186 (10 self)
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We study several versions of the problem of generating triangular meshes for finite element methods. We show how to triangulate a planar point set or polygonally bounded domain with triangles of bounded aspect ratio; how to triangulate a planar point set with triangles having no obtuse angles; how to triangulate a point set in arbitrary dimension with simplices of bounded aspect ratio; and how to produce a linearsize Delaunay triangulation of a multidimensional point set by adding a linear number of extra points. All our triangulations have size (number of triangles) within a constant factor of optimal, and run in optimal time O(n log n+k) with input of size n and output of size k. No previous work on mesh generation simultaneously guarantees wellshaped elements and small total size. 1. Introduction Geometric partitioning problems ask for the decomposition of a geometric input into simpler objects. These problems are fundamental in many areas, such as solid modeling, computeraided ...
Mesh Generation And Optimal Triangulation
, 1992
"... We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some cri ..."
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Cited by 179 (7 self)
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We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two and threedimensions. An optimal triangulation is a partition of the domain into triangles or tetrahedra, that is best according to some criterion that measures the size, shape, or number of triangles. We discuss algorithms both for the optimization of triangulations on a fixed set of vertices and for the placement of new vertices (Steiner points). We briefly survey the heuristic algorithms used in some practical mesh generators.
Tetrahedral Mesh Generation by Delaunay Refinement
 Proc. 14th Annu. ACM Sympos. Comput. Geom
, 1998
"... Given a complex of vertices, constraining segments, and planar straightline constraining facets in E 3 , with no input angle less than 90 ffi , an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradiustoshortest edge ratios are no greater than two ..."
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Cited by 114 (7 self)
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Given a complex of vertices, constraining segments, and planar straightline constraining facets in E 3 , with no input angle less than 90 ffi , an algorithm presented herein can generate a conforming mesh of Delaunay tetrahedra whose circumradiustoshortest edge ratios are no greater than two. The sizes of the tetrahedra can provably grade from small to large over a relatively short distance. An implementation demonstrates that the algorithm generates excellent meshes, generally surpassing the theoretical bounds, and is effective in eliminating tetrahedra with small or large dihedral angles, although they are not all covered by the theoretical guarantee. 1 Introduction Meshes of triangles or tetrahedra have many applications, including interpolation, rendering, and numerical methods such as the finite element method. Most such applications demand more than just a triangulation of the object or domain being rendered or simulated. To ensure accurate results, the triangles or tetr...
Parallel Construction of Quadtrees and Quality Triangulations
, 1999
"... We describe e#cient PRAM algorithms for constructing unbalanced quadtrees, balanced quadtrees, and quadtreebased finite element meshes. Our algorithms take time O(log n) for point set input and O(log n log k) time for planar straightline graphs, using O(n + k/ log n) processors, where n measure ..."
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Cited by 59 (6 self)
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We describe e#cient PRAM algorithms for constructing unbalanced quadtrees, balanced quadtrees, and quadtreebased finite element meshes. Our algorithms take time O(log n) for point set input and O(log n log k) time for planar straightline graphs, using O(n + k/ log n) processors, where n measures input size and k output size. 1. Introduction A crucial preprocessing step for the finite element method is mesh generation, and the most general and versatile type of twodimensional mesh is an unstructured triangular mesh. Such a mesh is simply a triangulation of the input domain (e.g., a polygon), along with some extra vertices, called Steiner points. Not all triangulations, however, serve equally well; numerical and discretization error depend on the quality of the triangulation, meaning the shapes and sizes of triangles. A typical quality guarantee gives a lower bound on the minimum angle in the triangulation. Baker et al. 1 first proved the existence of quality triangulations fo...
Aspects of Unstructured Grids and FiniteVolume Solvers for the Euler and NavierStokes Equations
 in: Lecture Notes Presented at the VKI Lecture Series
"... ..."
Numerical Schemes for the HamiltonJacobi and Level Set Equations on Triangulated Domains
, 1997
"... Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the HamiltonJacobi (HJ), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for certain forms of the HJ equations. Unf ..."
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Cited by 56 (7 self)
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Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the HamiltonJacobi (HJ), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for certain forms of the HJ equations. Unfortunately, the basic scheme lacks proper Lipschitz continuity of the numerical Hamiltonian. By employing a "virtual" edge ipping technique, Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and naturally exhibit proper Lipschitz continuity of the numerical Hamiltonian. Finally, a class of PetrovGalerkin approximations are considered. These schemes are stabilized via a leastsquares bilinear form. The PetrovGalerkin schemes do not possess a discrete...
Mesh Generation
 Handbook of Computational Geometry. Elsevier Science
, 2000
"... this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary. ..."
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Cited by 48 (6 self)
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this article, we emphasize practical issues; an earlier survey by Bern and Eppstein [24] emphasized theoretical results. Although there is inevitably some overlap between these two surveys, we intend them to be complementary.
Quality Mesh Generation in Higher Dimensions
, 1996
"... We consider the problem of triangulating a ddimensional region. Our mesh generation algorithm, called QMG, is a quadtreebased algorithm that can triangulate any polyhedral region including nonconvex regions with holes. Furthermore, our algorithm guarantees a bounded aspect ratio triangulation ..."
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Cited by 47 (7 self)
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We consider the problem of triangulating a ddimensional region. Our mesh generation algorithm, called QMG, is a quadtreebased algorithm that can triangulate any polyhedral region including nonconvex regions with holes. Furthermore, our algorithm guarantees a bounded aspect ratio triangulation provided that the input domain itself has no sharp angles. Finally, our algorithm is guaranteed never to overrefine the domain in the sense that the number of simplices produced by QMG is bounded above by a factor times the number produced by any competing algorithm, where the factor depends on the aspect ratio bound satisfied by the competing algorithm. The QMG algorithm has been implemented in C++ and is used as a mesh generator for the finite element method.