Parallel Delaunay Refinement: Algorithms and Analyses

TL;DR

This paper presents provably efficient parallel algorithms for Delaunay mesh refinement that match the quality and size guarantees of sequential methods, significantly reducing computation time.

Contribution

It introduces the first polylogarithmic parallel time algorithms for Delaunay refinement that produce high-quality, well-shaped meshes with optimal size.

Findings

Parallel construction of independent point sets is efficient.

Number of iterations is O(log^2(L/s)) for general meshes.

For quasi-uniform meshes, iterations reduce to O(log(L/s)).

Abstract

In this paper, we analyze the complexity of natural parallelizations of Delaunay refinement methods for mesh generation. The parallelizations employ a simple strategy: at each iteration, they choose a set of ``independent'' points to insert into the domain, and then update the Delaunay triangulation. We show that such a set of independent points can be constructed efficiently in parallel and that the number of iterations needed is $O(\log^2(L/s))$, where $L$ is the diameter of the domain, and $s$ is the smallest edge in the output mesh. In addition, we show that the insertion of each independent set of points can be realized sequentially by Ruppert's method in two dimensions and Shewchuk's in three dimensions. Therefore, our parallel Delaunay refinement methods provide the same element quality and mesh size guarantees as the sequential algorithms in both two and three dimensions. For quasi-uniform meshes, such as those produced by Chew's method, we show that the number of iterations can be reduced to $O(\log(L/s))$. To the best of our knowledge, these are the first provably polylog$(L/s)$ parallel time Delaunay meshing algorithms that generate well-shaped meshes of size optimal to within a constant.