BERN, M., AND EPPSTEIN, D. Mesh generation and optimal triangulation. Computing in Euclidean Geometry (1992), 23 --  Home/Search   Document Not in Database   Summary   Related Articles   Check

....computed in a conformal parameter space to produce the sampling. sis to Computational Geometry. Finite element mesh generation usually amounts to find a partition of a given domain that is optimal according to some criteria related to shape of elements, angles, sizes or complexity (see [3, 4]) In most cases only the frontier of the domain has to be given, the goal being to discretize this domain in accordance with an importance function. Our problem is slightly different since it is dealing with surface remeshing. The domain to discretize is now given by an original surface mesh that ....

....by an original surface mesh that has to be re discretized so that the result best matches some user specified properties. Meshes for numerical analysis Numerical analysts essentially focus on mesh quality, since it impacts the numerical accuracy of computations performed on the mesh elements [3, 4]. We distinguish between techniques that use a parameter space and techniques that act on an explicit mesh. The key idea of the first is to partition a parameter domain into sets of adjacent elements that have the same specified properties [14, 50, 23] The key idea of the second is to ....

BERN, M., AND EPPSTEIN, D. Mesh Generation and Optimal Triangulation. Computing in Euclidean Geometry, edited by D.-Z. Du and F. K. Hwang (1992), pp.23--90.

....extended to general PSLGs. Unlike Delaunay refinement algorithms, the Bern et al. algorithm is impractical because it generates far more triangles than are needed in practice; see Section 3 for a visual comparison. For a survey of provably good mesh generation algorithms, see Bern and Eppstein [3]. 2 The Key Idea Behind Delaunay Refinement In the finite element community, there are a wide variety of measures in use for the quality of a triangle, the most obvious being the smallest and largest angles of the triangle. Miller, Talmor, Teng, and Walkington [23] have pointed out that the most ....

Marshall Bern and David Eppstein. Mesh Generation and Optimal Triangulation. Computing in Euclidean Geometry (Ding-Zhu Du and Frank Hwang, editors), Lecture Notes Series on Computing, volume 1, pages 23--90. World Scientific, Singapore, 1992.

....to numerical optimization) and faster to compute. The weaker upper bound is tight to within a factor of three: for any triangle t, there is a function f such that = c t r circ . These bounds are interesting because the two dimensional Delaunay triangulation minimizes the max imum circumradius [9], just as it minimizes the maximum min containment radius. This property does not hold in three or more dimensions, unlike Rajan s min containment radius result. Hence, the twodimensional Delaunay triangulation is good (but not optimal) for controlling the worst case value of . The upper ....

Marshall Bern and David Eppstein. Mesh Generation and Optimal Triangulation. Computing in Euclidean Geometry (Ding-Zhu Du and Frank Hwang, editors), Lecture Notes Series on Computing, volume 1, pages 23--90. World Scientific, Singapore, 1992.

.... The quality of their shapes influences the quality of the finite element solution [18] This motivated the decade long research on generating meshes with guaranteed aspect ratio called quality meshes [1, 3, 4, 6, 7, 15, 16, 17] A considerable literature has built up on the subject, see the books [2, 9]. We review only a few of them in the context of the work in this paper. Bern, Eppstein and Gilbert [3] pioneered a quadtree based triangulation approach for producing quality meshes with close to optimal size in two dimensions. Mitchell and Vavasis [15] extended this technique to triangulate ....

....though the elements produced by this method have a biased alignment due to the axis parallel boxes used in quadtree octree subdivisions. Delaunay based triangulations do not have this problem and they are widely used in mesh generation for their uniqueness and many other nice properties, see [2, 9]. As a result researchers also concentrated on This research has been supported by the Research Grant Council, Hong Kong, China, project no. HKUST8088 99E and NSF under grant CCR9988216. Dept. of Computer Science, HKUST, Clear Water Bay, Hong Kong. e mail: scheng cs.ust.hk Dept. of CIS, ....

M. Bern and D. Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean Geometry, 2nd Ed., World Scientific, 1995, 47--123.

.... The quality of their shapes influences the quality of the finite element solution [18] This motivated the decade long research on generating meshes with guaranteed aspect ratio called quality meshes [1, 3, 4, 6, 7, 15, 16, 17] A considerable literature has built up on the subject, see the books [2, 9]. We review only a few of them in the context of the work in this paper. Bern, Eppstein and Gilbert [3] pioneered a quadtree based triangulation approach for producing quality meshes with close to optimal size in two dimensions. Mitchell and Vavasis [15] extended this technique to triangulate ....

....though the elements produced by this method have a biased alignment due to the axis parallel boxes used in quadtree octree subdivisions. Delaunay based triangulations do not have this problem and they are widely used in mesh generation for their uniqueness and many other nice properties, see [2, 9]. As a result researchers also concentrated on This research has been supported by the Research Grant Council, Hong Kong, China, project no. HKUST8088 99E and NSF under grant CCR9988216. y Dept. of Computer Science, HKUST, Clear Water Bay, Hong Kong. e mail: scheng cs.ust.hk z Dept. of CIS, ....

M. Bern and D. Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean Geometry, 2nd Ed., World Scientific, 1995, 47--123.

....than 7=9: However, any triangulation that achieves no small angles is doomed to use a nonpolynomial number of Steiner points, dependent on the input geometry. There are several previous algorithms that achieve similar results (by dissimilar techniques) for polygonal input. See Bern and Eppstein [5] for a summary. Edelsbrunner, Tan, and Waupotitsch [8] shows how to generate a constrained triangulation (one where no Steiner points are allowed) of a PSLG such that the maximum angle is minimized. The technique used is edge insertion, a global strategy that is a generalization of local edge ....

M. Bern and D. Eppstein, Mesh generation and optimal triangulation, Computing in Euclidean Geometry, D. Du, and F. Hwang, eds., World Scientific, 1992.

....an approximately minimum weight triangulation. No large and no small angles are achieved simultaneously in Baker, Gross and Rafferty [1988] and in Melissaratos and Souvaine [1992] However, the cardinality of a triangulation with no small angles is doomed to be dependent on the input geometry. Bern and Eppstein [1992] summarizes much of the Steiner triangulation literature. Mitchell[1993] presents the only other known algorithm for a covering triangulation with a provable bound on triangle shape. Given a PSLG, the algorithm generates a triangulation whose minimum angle is at least a constant factor times an ....

M. Bern and D. Eppstein [1992], Mesh generation and optimal triangulation, Computing in Euclidean Geometry, D. Du, and F. Hwang, eds., World Scientific.

....the minimum angle in Chew[1989] Bern, Eppstein and Gilbert [1990] and Rupert[1992] Mitchell and Vavasis [1992] solves this problem in three dimensions. Bern, Dobkin and Eppstein [1991] presents an algorithm that maximizes the minimum height, and also algorithms that minimize the maximum angle. Bern and Eppstein [1992] summarizes much of the computational geometry literature relevant to Steiner and constrained triangulations. A complete version of this paper appears in Mitchell [1993] 1.3 Application motivation Triangulation of polyhedral regions is a fundamental geometric problem for numerical analysis. ....

M. Bern, D. Eppstein [1992], Mesh Generation and Optimal Triangulation, Computing in Euclidean Geometry, D. Du, and F.

....convex polygons in three dimensional space which approximates the object s surface. However, in order to make a numerical analysis applicable, a suitable refinement of the coarse mesh is necessary. A large amount of research has been done in the area of mesh refinement into triangles, see [Ho88] [BE95] and [BP97] for surveys. In contrast, there is much less work on quadrilaterals, although meshes which consist solely of quadrilaterals are more appropriate in many applications, such as torsion problems and crash simulations [ZT89] Bra93] This is the background of our work, and therefore, in ....

M. Bern and D. Eppstein, Mesh generation and optimal triangulation, Computing in Euclidean Geometry, 2nd Edition (D.-Z. Du and F. Hwang, eds.), World Scientific, Singapore, 1995, pp. 47--123.

....arbitrary linear three dimensional region; some of the fundamental difficulties of doing so are described in Section 2.1.3. Nevertheless, the problem is reasonably well understood, and a thorough survey of the pertinent techniques, in both two and three dimensions, is offered by Bern and Eppstein [6]. A second goal of mesh generation is to offer as much control as possible over the sizes of elements in the mesh. Ideally, this control includes the ability to grade from small to large elements over a relatively short distance. The reason for this requirement is that element size has two ....

....segment is forced into the triangulation by deleting all the edges it crosses, inserting the new segment, and retriangulating the two resulting polygons (one on each side of the segment) as illustrated in Figure 2.16. For a proof that any polygon can be triangulated, see Bern and Eppstein [6]. Once a triangulation containing all the input segments is found, the flip algorithm may be applied, with the provision that segments cannot be flipped. The following results may be proven analogously to the proofs in Section 2.1.1. The only changes that need be made in the proofs is to ignore ....

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Marshall Bern and David Eppstein. Mesh Generation and Optimal Triangulation. Computing in Euclidean Geometry (Ding-Zhu Du and Frank Hwang, editors), Lecture Notes Series on Computing, volume 1, pages 23--90. World Scientific, Singapore, 1992.

....design, finite element methods, approximation theory, and computational geometry. Some prior surveys of related methods exist, notably a bibliography on approximation [45] a survey of spatial data structures for curves and surfaces [106] and surveys of triangulation methods with both theoretical [6] and scientific visualization [89] orientations. None of these surveys surface simplification in depth, however. The present paper attempts to survey all previous work on surface simplification and place the algorithms in a taxonomy. In this taxonomy, we intermix algorithms from various fields, ....

.... The philosophy of many of the feature approaches is that some knowledge about the nature of terrains is essential for good simplification [129, 108] In a feature approach, the chosen features become the vertex set, and the chosen break lines (if any) become edges in a constrained triangulation [6]. The most commonly used feature detectors are 2 2and33 linear or nonlinear filters, sometimes followed by a weeding process that discards features that are too close together, 9 such as a sequence of points along a ridge line. Such approaches were employed by Peucker Douglas and ChenGuevara ....

Marshall Bern and David Eppstein. Mesh generation and optimal triangulation. Technical report, Xerox PARC, March 1992. CSL-92-1. Also appeared in "Computing in Euclidean Geometry", F. K. Hwang and D.-Z. Du, eds., World Scientific, 1992.

....needs a tool as a prerequisite which converts a CAD model into a finite element mesh model suitable for a numerical analysis. Therefore, various algorithms for the generation of meshes have been developed, mostly decomposing surfaces into triangles and solid bodies into tetrahedra, for surveys see [BE95, BP97]. In many applications, however, quadrilateral and hexahedral meshes have numerical advantages. The potential savings gained from an all hexahedral meshing tool compared to an analysis based on tetrahedral meshing may be enormous (with estimations in the range of 75 time and cost reductions ....

M. Bern and D. Eppstein, Mesh generation and optimal triangulation, Computing in Euclidean Geometry, 2nd Edition (D.-Z. Du and F. Hwang, eds.), World Scientific, Singapore, 1995, pp. 47-- 123.

....and physical reasoning, and three dimensional visualization. Important goals in the construction of both surface triangular meshes and tetrahedral meshes (3D finite elements) is to reduce the number of triangles and tetrahedra as well as assume that these finite elements have good aspect ratio [5, 30]. In this paper we present implementations of two algorithms. One is to construct surface elements, and the other is to construct tetrahedral finite elements from planar contour data. Our algorithms are not limited to the biomedical field. They are applicable to other areas where the input can be ....

....is complicated by having the contours in planar slices (i.e. multiple sets of points on a plane and so not in general position for three dimensions) Extensive research has been conducted in the unstructured tetrahedral mesh generation. Chazelle et. al [10] Field [16] Lo [27] and Bern et al. [5] provide a good coverage of different approaches toward automatic mesh generation from a polyhedron. These methods include subdivision, octree, Delaunay based tetrahedral decompositions, and advancing fronts. The Delaunay based approaches [6, 8, 19, 39] and the advancing front approaches [9, 20, ....

[Article contains additional citation context not shown here]

M. Bern and D. Eppstein. Mesh Generation and Optimal Triangulation, Computing in Euclidean Geometry, edited by D.-Z. Du and F. K. Hwang, pages 23--90. World Scientific, 1992.

....is computed by adding the membership contributions (cf. Eq. 1) of each cluster over that configuration, giving an approximation of the SUF. However, in low dimensional problems, i.e. 2 d or 3d, an efficient discretization of this function is feasible by applying the Delaunay triangulation [2] on the cluster centers, building at the same time a connectivity graph [11] of the robot s free space. The 2 d Delaunay triangulation of a point set is the planar dual of the Voronoi diagram [1] of that set. The Voronoi diagram of a set of points in IR 2 is a partition of the plane into ....

M. Bern and D. Eppstein. Computing in Euclidean Geometry, pages 47--123. World Scientific, 2nd edition, 1995. Du, D.-Z. and Hwang, F. K. (eds.).

....the efficiency of its triangulation algorithms and data structures, so I discuss these first. I assume the reader is familiar with Delaunay triangulations, constrained Delaunay triangulations, and the incremental insertion algorithms for constructing them. Consult the survey by Bern and Eppstein [2] for an introduction. There are many Delaunay triangulation algorithms, some of which are surveyed and evaluated by Fortune [7] and Su and Drysdale [18] Their results indicate a rough parity in speed among the incremental insertion algorithm of Lawson [11] the divide and conquer algorithm of Lee ....

Marshall Bern and David Eppstein. Mesh Generation and Optimal Triangulation. Computing in Euclidean Geometry (Ding-Zhu Du and Frank Hwang, editors), Lecture Notes Series on Computing, volume 1, pages 23--90. World Scientific, Singapore, 1992.

....by having the contours in planar slices (i.e. multiple sets of points on a plane and hence not in general position for three dimensions) Extensive research has been conducted on unstructured tetrahedral mesh generation. Chazelle and Palios [9] Lo [19] Bajaj and Dey [2] Bern and Eppstein [3], and Field [11] provide a good coverage of different approaches toward automatic mesh generation from a polyhedron. These methods include octree decomposition, convex decomposition, Delaunay triangulation, and advancing front technique (AFT) Of these methods, the Delaunay based approaches [6, 5, ....

....provide a protection rule to reduce the chance of generating untetrahedralizable remaining parts. The characterization also leads to the classification of two common untetrahedralizable categories so they can be better postprocessed if they do occur. For example, a Schonhardt prism (Fig. 2(a) [3, 25] cannot be tetrahedralized without adding any Steiner point. It can be tetrahedralized by adding a Steiner point which is visible to all faces. Traditional AFT cannot characterize an untetrahedralizable shape. AFT keeps generating smaller tetrahedra, and the remaining part (Fig. 2(b) becomes too ....

M. Bern and D. Eppstein. Mesh Generation and Optimal Triangulation, Computing in Euclidean Geometry, edited by D.-Z. Du and F. K. Hwang, pages 23--90. World Scientific, 1992.

....assumptions, the cluster centers can be regarded as the centers of a Voronoi diagram [5] of the robot s free space. To exploit this clustering, a way for connecting the cluster centers in pairs must be found. For that, we triangulate the set of clusters centers by using the Delaunay triangulation [6], to produce a mesh on which subsequent tasks can easily be carried out. This mesh is a non directed graph on which path planning can be performed by the usual graph searching techniques [7] In our case, we apply a modified A algorithm on the graph, like in [8] We demonstrate some of the ....

....cell of an input point p contains all those loci whose nearest input point according to the Euclidean distance is p. In order for the clustering to be of practical use, connections must be established between pairs of cluster centers. To this direction, we apply the Delaunay triangulation (DT) [6] to the cluster centers. The DT of a point set is the planar dual of the Voronoi diagram of that set. Each edge of a DT triangle is the perpendicular bisector of a Voronoi cell edge, whereas the triangle vertices correspond to the Voronoi centers. The resulting mesh of triangles forms a ....

M. Bern and D. Eppstein, Computing in Euclidean Geometry, World Scientific, 2nd edn. (1995), Du, D.-Z. and Hwang, F. K. (eds.).

....such a method needs a tool which converts a CAD model into a finite element mesh model suitable for a numerical analysis. Therefore, various algorithms for the generation of meshes have been developed, mostly decomposing surfaces into triangles and solid bodies into tetrahedra, for surveys see [2, 3]. In many applications, however, quadrilateral and hexahedral meshes have numerical advantages. The potential savings gained from an all hexahedral meshing tool compared to an analysis based on tetrahedral meshing may be enormous (with estimations in the range of 75 time and cost reductions ....

M. Bern and D. Eppstein, Mesh generation and optimal triangulation, Computing in Euclidean Geometry, 2nd Edition (D.-Z. Du and F. Hwang, eds.), World Scientific, Singapore, 1995, pp. 47--123.

....A triangulation of a finite set of points S in 2 is a maximally connected, straight line planar graph with vertex set S. Each bounded face is a triangle, and the triangulation includes the boundary of the convex hull. Triangulations find use in areas such as finite element analysis [BeEp92, StFi73], computational geometry [PrSh85] and surface approximation [DLR90] Applications typically require triangulations with well shaped triangles, meaning for example that triangles with very small or large angles should be avoided. Taking a worst case approach, one can define the quality of a ....

....Though usually simple to verify, conditions (I) and (II) are somewhat restrictive. It would be interesting to find conditions weaker than (I) even though the price to pay may be implementations of the paradigm that take more than cubic time. Listings of optimality criteria can be found in [Barn77, BeEp92, Lind83, Schu87]. Furthermore, implementations for criteria satisfying (I) and (II) that run in time o(n 3 ) and o(n 2 log n) are sought. Acknowledgment The authors thank two anonymous referees for suggestions on improving the style of this paper. ....

M. W. Bern and D. Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean Geometry. World Scientific, Singapore, 1992.

....in ray shooting. Approximate MWST Eppstein [7] showed that a constant factor approximation of the MWST of n points or a convex polygon with n vertices can be computed in O(n log n) time. The problem of designing an approximation algorithm for general polygons was posed by Bern and Eppstein [5]. We present a solution in this paper. Average case ray shooting The MWST problem is recently related to the ray shooting problem by Aronov and Fortune [1] The ray shooting problem is to report the first obstacle hit by a query ray. In this paper, we assume that the query light source falls ....

M. Bern and D. Eppstein, Mesh generation and optimal triangulation, Computing in Euclidean Geometry, D.-Z. Du and F.Hwang (eds.), World Scientific, 1992, 23--90.

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M. Bern and D. Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean Geometry, 2nd edition, pp. 47--123. World Scientific, Lecture Notes Ser. Comput. 4, 1995.

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M. Bern and D. Eppstein. Mesh generation and optimal triangulation, Computing in Euclidean Geometry, D.-Z. Du and F.K. Hwang, eds., World Scientific, 1992.

....in part by NSF grant CCR 9258355 and by matching funds from Xerox Corp. 1 Introduction There has recently been a great deal of theoretical work on unstructured mesh generation for finite element methods, largely concentrating on triangulations and higher dimensional simplicial complexes [2]. However in the numerical community, where these meshes have been actually used, meshes of quadrilaterals or hexahedra (cuboids) are often preferred due to their numerical properties. For this reason many mesh generation researchers are working on systems for construction of hexahedral meshes. ....

M. Bern and D. Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean Geometry, 2nd ed., World Scientific (1995) 47--123.

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BERN, M., AND EPPSTEIN, D. Mesh generation and optimal triangulation. Computing in Euclidean Geometry (1992), 23 --

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Marshall Bern and David Eppstein. Mesh Generation and Optimal Triangulation. Computing in Euclidean Geometry (Ding-Zhu Du and Frank Hwang, editors), Lecture Notes Series on Computing, volume 1, pages 23--90. World Scientific, Singapore, 1992.

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Bern Marshall and David Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean geometry, World Scienti c, 23:46-123, 1992.

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Bern Marshall and David Eppstein. Mesh generation and optimal triangulation. Computing in Euclidean geometry, World Scienti c, 23:46-123, 1992.

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Bern, M., Eppstein, D. Mesh generation and optimal triangulation, Computing in Euclidean Geometry, World Scientific, 1992.

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M. Bern and D. Eppstein, \Mesh generation and optimal triangulation," in Computing In Euclidean Geometry, D.-Z. Du and F. Hwang, eds., pp. 23-90, World Scientic, 1992.

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