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Delaunay Refinement Algorithms for Triangular Mesh Generation
 Computational Geometry: Theory and Applications
, 2001
"... Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of tria ..."
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Delaunay refinement is a technique for generating unstructured meshes of triangles for use in interpolation, the finite element method, and the finite volume method. In theory and practice, meshes produced by Delaunay refinement satisfy guaranteed bounds on angles, edge lengths, the number of triangles, and the grading of triangles from small to large sizes. This article presents an intuitive framework for analyzing Delaunay refinement algorithms that unifies the pioneering mesh generation algorithms of L. Paul Chew and Jim Ruppert, improves the algorithms in several minor ways, and most importantly, helps to solve the difficult problem of meshing nonmanifold domains with small angles.
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 51 (8 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.
Provably Good Surface Sampling and Approximation
, 2003
"... We present an algorithm for meshing surfaces that is a simple adaptation of a greedy "farthest point" technique proposed by Chew. Given a surface S, it progressively adds points on S and updates the 3dimensional Delaunay triangulation of the points. The method is very simple and works in ..."
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Cited by 36 (0 self)
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We present an algorithm for meshing surfaces that is a simple adaptation of a greedy "farthest point" technique proposed by Chew. Given a surface S, it progressively adds points on S and updates the 3dimensional Delaunay triangulation of the points. The method is very simple and works in 3dspace without requiring to parameterize the surface. Taking advantage of recent results on the restricted Delaunay triangulation, we prove that the algorithm can generate good samples on S as well as triangulated surfaces that approximate S. More precisely, we show that the restricted Delaunay triangulation Del # S of the points has the same topology type as S, that the Hausdorff distance between Del # S and S can be made arbitrarily small, and that we can bound the aspect ratio of the facets of Del # S . The algorithm has been implemented and we report on experimental results that provide evidence that it is very effective in practice. We present results on implicit surfaces, on CSG models and on polyhedra. Although most of our theoretical results are given for smooth closed surfaces, the method is quite robust in handling smooth surfaces with boundaries, and even nonsmooth surfaces.
Mesh Generation for Domains with Small Angles
 Proc. 16th Annu. Sympos. Comput. Geom
, 2000
"... Nonmanifold geometric domains having small angles present special problems for triangular and tetrahedral mesh generators. Although small angles inherent in the input geometry cannot be removed, one would like to find a way to triangulate a domain without creating any new small angles. Unfortunately ..."
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Nonmanifold geometric domains having small angles present special problems for triangular and tetrahedral mesh generators. Although small angles inherent in the input geometry cannot be removed, one would like to find a way to triangulate a domain without creating any new small angles. Unfortunately, this problem is not always soluble. I discuss how mesh generation algorithms based on Delaunay refinement can be modified to ensure that they always produce a mesh, and to ensure that poor quality triangles or tetrahedra appear only near small input angles. 1 Introduction The Delaunay refinement algorithms for triangular mesh generation introduced by Jim Ruppert [4] and Paul Chew [1] are almost entirely satisfying in theory and in practice. However, one unresolved problem has limited their applicability: they do not always mesh domains with small angles wellor at allespecially if these domains are nonmanifold. This problem is not just true of Delaunay refinement algorithms; it ste...
Robust Three Dimensional Delaunay Refinement
 IN THIRTEENTH INTERNATIONAL MESHING ROUNDTABLE
, 2004
"... The Delaunay Refinement Algorithm for quality meshing is extended to three dimensions. The algorithm accepts input with arbitrarily small angles, and outputs a Conforming Delaunay Tetrahedralization where most tetrahedra have radiustoshortestedge ratio smaller than some user chosen µ > 2. Thos ..."
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Cited by 22 (4 self)
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The Delaunay Refinement Algorithm for quality meshing is extended to three dimensions. The algorithm accepts input with arbitrarily small angles, and outputs a Conforming Delaunay Tetrahedralization where most tetrahedra have radiustoshortestedge ratio smaller than some user chosen µ > 2. Those tets with poor quality are in well defined locations: their circumcenters are describably near input segments. Moreover, the output mesh is well graded to the input: short mesh edges only appear around close features of the input. The algorithm has the added advantage of not requiring a priori knowledge of the "local feature size," and only requires searching locally in the mesh.
Lecture Notes on Delaunay Mesh Generation
, 1999
"... purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the ..."
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Cited by 21 (0 self)
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purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the
Generalized delaunay mesh refinement: From scalar to parallel
 IN PROCEEDINGS OF THE 15TH INTERNATIONAL MESHING ROUNDTABLE
, 2006
"... The contribution of the current paper is threefold. First, we generalize the existing sequential point placement strategies for guaranteed quality Delaunay refinement: instead of a specific position for a new point, we derive a selection disk inside the circumdisk of a poor quality triangle. We pr ..."
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The contribution of the current paper is threefold. First, we generalize the existing sequential point placement strategies for guaranteed quality Delaunay refinement: instead of a specific position for a new point, we derive a selection disk inside the circumdisk of a poor quality triangle. We prove that any point placement algorithm that inserts a point inside the selection disk of a poor quality triangle will terminate and produce a sizeoptimal mesh. Second, we extend our theoretical foundation for the parallel Delaunay refinement. Our new parallel algorithm can be used in conjunction with any sequential point placement strategy that chooses a point within the selection disk. Third, we implemented our algorithm in C++ for shared memory architectures and present the experimental results. Our data show that even on workstations with a few cores, which are now in common use, our implementation is significantly faster the best sequential counterpart.
ThreeDimensional SemiGeneralized Point Placement Method for Delaunay Mesh Refinement
"... A number of approaches have been suggested for the selection of the positions of Steiner points in Delaunay mesh refinement. In particular, one can define an entire region (called picking region or selection disk) inside the circumscribed sphere of a poor quality element such that any point can be ..."
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A number of approaches have been suggested for the selection of the positions of Steiner points in Delaunay mesh refinement. In particular, one can define an entire region (called picking region or selection disk) inside the circumscribed sphere of a poor quality element such that any point can be chosen for insertion from this region. The two main results which accompany most of the point selection schemes, including those based on regions, are the proof of termination of the algorithm and the proof of good gradation of the elements in the final mesh. In this paper we show that in order to satisfy only the termination requirement, one can use larger selection disks and benefit from the additional flexibility in choosing the Steiner points. However, if one needs to keep the theoretical guarantees on good grading then the size of the selection disk needs to be smaller. We introduce two types of selection disks to satisfy each of these two goals and prove the corresponding results on termination and good grading first in two dimensions and then in three dimensions using the radiusedge ratio as a measure of element quality. We call the point placement method semigeneralized because the selection disks are defined only for mesh entities of the highest dimension (triangles in two dimensions and tetrahedra in three dimensions); we plan to extend these ideas to lowerdimensional entities in the future work. We implemented the use of both two and threedimensional selection disks into the available Delaunay refinement libraries and present one example (out of many choices) of a point placement method; to the best of our knowledge, this is the first implementation of Delaunay refinement with point insertion at any point of the selection disks (picking regions).