Optimized Dual-Volumes for Tetrahedral Meshes
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Date
2024
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Publisher
The Eurographics Association and John Wiley & Sons Ltd.
Abstract
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.
Description
@article{10.1111:cgf.15133,
journal = {Computer Graphics Forum},
title = {{Optimized Dual-Volumes for Tetrahedral Meshes}},
author = {Jacobson, Alec},
year = {2024},
publisher = {The Eurographics Association and John Wiley & Sons Ltd.},
ISSN = {1467-8659},
DOI = {10.1111/cgf.15133}
}