Optimized Dual-Volumes for Tetrahedral Meshes

TL;DR

This paper analyzes various finite-volume methods for tetrahedral meshes, identifying key properties and proposing a new convex optimization-based dual-volume construction that ensures desirable mathematical properties.

Contribution

It unifies existing methods under a common framework and introduces a novel dual-volume construction that guarantees all essential properties for mesh processing.

Findings

Existing methods differ mainly in simplex center choices.

The proposed method explicitly satisfies all key properties.

The new approach improves the robustness and accuracy of mesh processing.

Abstract

Constructing well-behaved Laplacian and mass matrices is essential for tetrahedral mesh processing. Unfortunately, the \emph{de facto} standard linear finite elements exhibit bias on tetrahedralized regular grids, motivating the development of finite-volume methods. In this paper, we place existing methods into a common construction, showing how their differences amount to the choice of simplex centers. These choices lead to satisfaction or breakdown of important properties: continuity with respect to vertex positions, positive semi-definiteness of the implied Dirichlet energy, positivity of the mass matrix, and unbiased-ness on regular grids. Based on this analysis, we propose a new method for constructing dual-volumes which explicitly satisfy all of these properties via convex optimization.