Fast Sizing Calculations for Meshing

Provably correct algorithms for meshing difficult domains in three dimensions have been recently developed in the literature. These algorithms handle the problem of sharp angles (< π/2) between segments and between facets by constructing protective collars around these regions. The collars are approximately sized according to the local feature size of the input. With the eventual goal of developing time-efficient algorithms for the same mesh generation problems, we give a method for estimating the feature size of a 3D piecewise-linear-complex of size n on domain Ω in time O(n log∆+m), where∆ is the spread of the input. The linear termm ∈ O( ∫ Ω 1/ lfs) is bounded above by the output size of a quality generated mesh. Our algorithm is based on early termination of the Sparse-VoronoiRefinement (SVR) meshing algorithm, which is not guaranteed to terminate in the presence of sharp angles. 1 Local Sizing Functions There are several algorithms [1, 3, ?, ?] for computing quality meshes of three-dimensional PLCs that all use some form of “collars” as protective regions around sharp angles. The sizing for these collars is determined by some variant of the following two functions. The first is the local feature size (lfs(x)), the smallest x-centered ball that intersects two disjoint features of C, originally due to Ruppert [4]. The second is the gap-size (gs(x)), the smallest x-centered ball that intersects two features of C, one of which does not contain x . The lfs is bounded away from zero everywhere and is 1-Lipschitz. The gs may be very discontinuous, but it is 1-Lipschitz along the interior of any feature. The definitions imply that gs ≤ lfs everywhere. Meshing algorithms wish to know these values at corners and along creases. Some algorithms take these functions as given, naively requiring brute-force computations taking Ω(n). Later algorithms calculate approximations to sizing procedurally. This works well in practice, however the methods used do not have good runtime guarantees. Our contribution is a work-efficient procedure for approximating sizing functions. The goal of this research is to develop a provably efficient algorithm for generating 3D meshes of arbitrary PLCs. The ability to estimate sizing functions quickly is an important first step. We will return a pointwise sample of the domain with the exact values of lfs and gs at every point. The Lipschitz conditions will provides enough smoothness to give appropriate guarantees on the quality of our sample. As desired by most of these algorithms, our sample includes as a subset all the corners of of the PLC and a good sample along all the input segments.