Anisotropic simplicial meshing using local convex functions

TLDR

The authors propose a novel method for generating high‑quality anisotropic simplicial meshes. The method transforms the meshing problem into functional approximation by constructing convex functions whose Hessians match the prescribed Riemannian metric and iteratively adjusting vertex positions and connectivity to minimize the discrepancy between these functions and their piecewise‑linear interpolation. The approach generalizes optimal Delaunay triangulation, yielding a simple and efficient algorithm that outperforms state‑of‑the‑art methods in quality and speed across diverse domains and metrics.

Abstract

We present a novel method to generate high-quality simplicial meshes with specified anisotropy. Given a surface or volumetric domain equipped with a Riemannian metric that encodes the desired anisotropy, we transform the problem to one of functional approximation. We construct a convex function over each mesh simplex whose Hessian locally matches the Riemannian metric, and iteratively adapt vertex positions and mesh connectivity to minimize the difference between the target convex functions and their piecewise-linear interpolation over the mesh. Our method generalizes optimal Delaunay triangulation and leads to a simple and efficient algorithm. We demonstrate its quality and speed compared to state-of-the-art methods on a variety of domains and metrics.