Quality Mesh Generation in Three Dimensions

Abstract

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.