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arXiv · 2406.08647

Optimized Dual-Volumes for Tetrahedral Meshes

Alec Jacobson
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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.

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Alec Jacobson. 2024-06-12. Optimized Dual-Volumes for Tetrahedral Meshes. https://arxiv.org/abs/2406.08647

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