A Practical Delaunay Meshing Algorithm for a Large Class of Domains*

Summary

Recently a Delaunay refinement algorithm has been proposed that can mesh domains as general as piecewise smooth complexes [7]. This class includes polyhedra, smooth and piecewise smooth surfaces, volumes enclosed by them, and above all non-manifolds. In contrast to previous approaches, the algorithm does not impose any restriction on the input angles. Although this algorithm has a provable guarantee about topology, certain steps are too expensive to make it practical.

In this paper we introduce a novel modification of the algorithm to make it implementable in practice. In particular, we replace four tests of the original algorithm with only a single test that is easy to implement. The algorithm has the following guarantees. The output mesh restricted to each manifold element in the complex is a manifold with proper incidence relations. More importantly, with increasing level of refinement which can be controlled by an input parameter, the output mesh becomes homeomorphic to the input while preserving all input features. Implementation results on a disparate array of input domains are presented to corroborate our claims. %We present a Delaunay refinement algorithm for meshing a piecewise smooth complex in three dimensions with correct topology. The small angles between %the tangents of two meeting manifold patches pose difficulty. We protect these %regions with weighted points. The weights are chosen to mimic the local feature size and to satisfy a Lipschitz-like property. A Delaunay refinement using the weighted Voronoi diagram is shown to terminate with the recovery of the topology of the input. To this end, we present new concepts and results %including a new definition of local feature size and a proof for a %generalized topological ball property.