Well-Centered Triangulation

TL;DR

This paper introduces an iterative optimization algorithm to transform meshes into well-centered configurations, enhancing dual mesh properties and improving angle quality in 2D and 3D triangulations.

Contribution

It presents the first optimization-based heuristic for well-centered meshes applicable in both two and three dimensions, with a new characterization and cost function.

Findings

Successfully transforms meshes into well-centered configurations

Preserves mesh connectivity and boundary vertices during optimization

Improves angle quality and maintains gradation in meshes

Abstract

Meshes composed of well-centered simplices have nice orthogonal dual meshes (the dual Voronoi diagram). This is useful for certain numerical algorithms that prefer such primal-dual mesh pairs. We prove that well-centered meshes also have optimality properties and relationships to Delaunay and minmax angle triangulations. We present an iterative algorithm that seeks to transform a given triangulation in two or three dimensions into a well-centered one by minimizing a cost function and moving the interior vertices while keeping the mesh connectivity and boundary vertices fixed. The cost function is a direct result of a new characterization of well-centeredness in arbitrary dimensions that we present. Ours is the first optimization-based heuristic for well-centeredness, and the first one that applies in both two and three dimensions. We show the results of applying our algorithm to small and large two-dimensional meshes, some with a complex boundary, and obtain a well-centered tetrahedralization of the cube. We also show numerical evidence that our algorithm preserves gradation and that it improves the maximum and minimum angles of acute triangulations created by the best known previous method.