Constructive quasi-uniform sequences over triangles

TL;DR

This paper introduces a Voronoi-guided greedy algorithm for constructing quasi-uniform point sequences over triangles, achieving optimal mesh ratios and demonstrating practical efficiency through numerical experiments.

Contribution

It presents a new constructive algorithm for generating quasi-uniform point sets on triangles with proven mesh ratio bounds and analyzes existing low-discrepancy sets for quasi-uniformity.

Findings

The algorithm guarantees a mesh ratio of at most 2 after finite iterations.

Existing low-discrepancy sets have uniformly bounded mesh ratios.

Numerical experiments confirm the method's efficiency and high quality of generated point sets.

Abstract

In this paper, we develop constructive algorithms for generating quasi-uniform point sets and sequences over arbitrary two-dimensional triangular domains. Our proposed method, called the \emph{Voronoi-guided greedy packing} algorithm, iteratively selects the point farthest from the current set among a finite candidate set determined by the Voronoi diagram of the triangle. Our main theoretical result shows that, after a finite number of iterations, the mesh ratio of the generated point set is at most~2, which is known to be optimal. We further analyze two existing triangular low-discrepancy point sets and prove that their mesh ratios are uniformly bounded, thereby establishing their quasi-uniformity. Finally, through a series of numerical experiments, we demonstrate that the proposed method provides an efficient and practical strategy for generating high-quality point sets on individual triangles.