Optimized Dual-Volumes for Tetrahedral Meshes
<|reference_start|>Optimized Dual-Volumes for Tetrahedral Meshes: 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.<|reference_end|>
@article{jacobson2024optimized,
    title={Optimized Dual-Volumes for Tetrahedral Meshes},
    author={Alec Jacobson},
    journal={arXiv preprint arXiv:2406.08647},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.08647},
    primaryClass={cs.GR}
}