We survey the computational geometry relevant to finite element mesh generation. We especially focus on optimal triangulations of geometric domains in two- and three-dimensions. 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.

We show how to triangulate a three dimensional polyhedral region with holes. Our triangulation is optimal in the following two senses. First, our triangulation achieves the best possible aspect ratio up to a constant. Second, for any other triangulation of the same region into m triangles with bounded aspect ratio, our triangulation has size n = O(m). Such a triangulation is desired as an initial mesh for a finite element mesh refinement algorithm. Previous three dimensional triangulation schemes either worked only on a restricted class of input, or did not guarantee well-shaped tetrahedra, or were not able to bound the output size. We build on some of the ideas presented in previous work by Bern, Eppstein, and Gilbert, who have shown how to triangulate a two dimensional polyhedral region with holes, with similar quality and optimality bounds. 1 Introduction Triangulation of polyhedral regions is a fundamental geometric problem for numerical analysis. In particular, if on...

We give an algorithm for triangulating n-vertex polygonal regions (with holes) so that no angle in the nal triangulation measures more than pi/2. The number of triangles in the triangulation is only O(n), improving a previous bound of O(n²), and the worst-case running time is O(n log² n). The basic technique used in the algorithm, recursive subdivision by disks, is new and may have wider application in mesh generation. We also report on an implementation of our algorithm.

We consider the problem of triangulating a d-dimensional region. Our mesh generation algorithm, called QMG, is a quadtree-based 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.

A plane geometric graph C in ! 2 conforms to another such graph G if each edge of G is the union of some edges of C. It is proved that for every G with n vertices and m edges, there is a completion of a Delaunay triangulation of O(m 2 n) points that conforms to G. The algorithm that constructs the points is also described. Keywords. Discrete and computational geometry, plane geometric graphs, Delaunay triangulations, point placement. Appear in: Discrete & Computational Geometry, 10 (2), 197--213 (1993) 1 Research of the first author is supported by the National Science Foundation under grant CCR-8921421 and under the Alan T. Waterman award, grant CCR-9118874. Any opinions, finding and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation. Work of the second author was conducted while he was on study leave at the University of Illinois. 2 Department of Computer Scienc...

We describe methods for triangulating polygonal regions of the plane so that no triangle has a large angle. Our main result is that a polygon with n sides can be triangulated with O(n 2 ) nonobtuse triangles. We also show that any triangulation (without Steiner points) of a simple polygon has a refinement with O(n 4 ) nonobtuse triangles. Finally we show that a triangulation whose dual is a path has a refinement with only O(n 2 ) nonobtuse triangles. Keywords: Computational geometry, mesh generation, triangulation, angle condition. 1. Introduction One of the classical motivations for problems in computational geometry has been automatic mesh generation for finite element methods. In particular, mesh generation has motivated a number of triangulation algorithms, such as finding a triangulation that minimizes the maximum angle. 1 A triangulation algorithm takes a geometric input, typically a point set or polygonal region, and produces an output that is a triangulation of ...

We show that any PSLG with v vertices can be triangulated with no angle larger than 7 =8 by adding O(v²log v) Steiner points in O(v²log² v) time. We first triangulate the PSLG with an arbitrary constrained triangulation and then refine that triangulation by adding additional vertices and edges. We follow a lazy strategy of starting from an obtuse angle and exploring the triangulation in search of a sequence of Steiner points that will satisfy a local angle condition. Explorations may either terminate successfully (for example at a triangle vertex), or merge. Some PSLGs require Ω(v) Steiner points in any triangulation achieving any largest angle bound less than π. Hence the number of Steiner points added by our algorithm is within a log v factor of worst case optimal. For most inputs the number of Steiner points and running time would be considerably smaller than in the worst case.

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We consider the following problem: given a planar straight-line graph, nd a covering triangulation whose maximum angle is as small as possible. A covering triangulation is a triangulation whose vertex set contains the input vertex set and whose edge set contains the input edge set. The covering triangulation problem di ers from the usual Steiner triangulation problem in that we may not add a vertex on any input edge. Covering triangulations provide a convenient method for triangulating multiple regions sharing a common boundary, as each region can be triangulated independently. We give an explicit lower bound opt on the maximum angle in any covering triangulation of a particular input graph in terms of its local geometry. Our algorithm produces a covering triangulation whose maximum angle is provably close to opt: Bounding by a constant times opt is trivial: We prove something signi cantly stronger. Specifically, we show that; min(; opt 2 6 i.e., our is not much closer to than is opt. Toour knowledge, this result represents the rst nontrivial bound on a covering triangulation's maximum angle. Our algorithm adds O(n) Steiner points and runs in time O(n log² n), where n is the number of vertices of the input. We have implemented an O(n²)time version of our algorithm.