2024/06/12 by Alec Jacobson, Jacobson, Alec · 1 citation
Computer Science · Engineering · #Advanced Materials and Mechanics #Advanced Theoretical and Applied Studies in Material Sciences and Geometry #Computational Geometry and Mesh Generation #FOS: Computer and information sciences #Graphics (cs.GR)
paper · pdf · doi:10.48550/arxiv.2406.08647
openalex publication_date 2024/06/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Constructing well-behaved Laplacian and mass matrices is essential for tetrahedral mesh processing. Unfortunately, the 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.