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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.08647 |
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| _version_ | 1866929384577826816 |
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| author | Jacobson, Alec |
| author_facet | Jacobson, Alec |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08647 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimized Dual-Volumes for Tetrahedral Meshes Jacobson, Alec Graphics 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. |
| title | Optimized Dual-Volumes for Tetrahedral Meshes |
| topic | Graphics |
| url | https://arxiv.org/abs/2406.08647 |