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3D hodge decompositions of edge- and face-based vector fields

Zhao, Rundong and Desbrun, Mathieu and Wei, Guo-Wei and Tong, Yiying (2019) 3D hodge decompositions of edge- and face-based vector fields. ACM Transactions on Graphics, 38 (6). Art. No. 181. ISSN 0730-0301. https://resolver.caltech.edu/CaltechAUTHORS:20191210-100906607

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Abstract

We present a compendium of Hodge decompositions of vector fields on tetrahedral meshes embedded in the 3D Euclidean space. After describing the foundations of the Hodge decomposition in the continuous setting, we describe how to implement a five-component orthogonal decomposition that generically splits, for a variety of boundary conditions, any given discrete vector field expressed as discrete differential forms into two potential fields, as well as three additional harmonic components that arise from the topology or boundary of the domain. The resulting decomposition is proper and mimetic, in the sense that the theoretical dualities on the kernel spaces of vector Laplacians valid in the continuous case (including correspondences to cohomology and homology groups) are exactly preserved in the discrete realm. Such a decomposition only involves simple linear algebra with symmetric matrices, and can thus serve as a basic computational tool for vector field analysis in graphics, electromagnetics, fluid dynamics and elasticity.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1145/3355089.3356546DOIArticle
ORCID:
AuthorORCID
Desbrun, Mathieu0000-0003-3424-6079
Additional Information:© 2019 Association for Computing Machinery. Partial funding for this work was provided by NSF grants DMS-1721024 and IIS-1900473, and Pixar Animation Studios. Rundong wishes to thank Jin Huang at Zhejiang University for hosting him during the final editing phase of this paper.
Funders:
Funding AgencyGrant Number
NSFDMS-1721024
NSFIIS-1900473
Pixar Animation StudiosUNSPECIFIED
Issue or Number:6
Record Number:CaltechAUTHORS:20191210-100906607
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20191210-100906607
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:100260
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:10 Dec 2019 21:04
Last Modified:10 Dec 2019 21:04

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