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Delaunay refinement algorithms for triangular mesh generation. (English) Zbl 1016.68139
Summary: 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.
Although small angles inherent in the input geometry cannot be removed, one would like to triangulate a domain without creating any new small angles. Unfortunately, this problem is not always soluble. A compromise is necessary. A Delaunay refinement algorithm is presented that can create a mesh in which most angles are \(30^\circ\) or greater and no angle is smaller than \(\arcsin[(\sqrt{3}/2)\sin(\phi/2)]\sim (\sqrt{3}/4)\phi\), where \(\phi\leq 60^\circ\) is the smallest angle separating two segments of the input domain. New angles smaller than \(30^\circ\) appear only near input angles smaller than \(60^\circ\). In practice, the algorithm’s performance is better than these bounds suggest.
Another new result is that Ruppert’s analysis technique can be used to reanalyze one of Chew’s algorithms. Chew proved that his algorithm produces no angle smaller than \(30^\circ\) (barring small input angles), but without any guarantees on grading or number of triangles. He conjectures that his algorithm offers such guarantees. His conjecture is conditionally confirmed here: if the angle bound is relaxed to less than \(26.5^\circ\), Chew’s algorithm produces meshes (of domains without small input angles) that are nicely graded and size-optimal.

MSC:
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
Software:
Triangle
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