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Title: A conservative, consistent, and scalable meshfree mimetic method

Abstract

Mimetic methods discretize divergence by restricting the Gauss theorem to mesh cells. Because point clouds lack such geometric entities, construction of a compatible meshfree divergence remains a challenge. In this work, we define an abstract Meshfree Mimetic Divergence (MMD) operator on point clouds by contraction of field and virtual face moments. This MMD satisfies a discrete divergence theorem, provides a discrete local conservation principle, and is first-order accurate. We consider two MMD instantiations. The first one assumes a background mesh and uses generalized moving least squares (GMLS) to obtain the necessary field and face moments. This MMD instance is appropriate for settings where a mesh is available but its quality is insufficient for a robust and accurate mesh-based discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graph-Laplacian equations. This MMD instance does not require a background grid and is appropriate for applications where mesh generation creates a computational bottleneck. It allows one to trade an expensive mesh generation problem for a scalable algebraic one, without sacrificing compatibility with the divergence operator. We demonstrate the approach by using the MMD operator to obtain a virtual finite-volume discretization of conservation lawsmore » on point clouds. Finally, numerical results in the paper confirm the mimetic properties of the method and show that it behaves similarly to standard finite volume methods.« less

Authors:
 [1]; ORCiD logo [1];  [1]
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  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI Identifier:
1618096
Alternate Identifier(s):
OSTI ID: 1600080
Report Number(s):
SAND-2019-2722J
Journal ID: ISSN 0021-9991; 673358; TRN: US2106793
Grant/Contract Number:  
AC04-94AL85000; SC0000230927; NA0003525; SC0019453
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 409; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; meshfree; generalized moving least squares; compatible discretizations; mimetic methods

Citation Formats

Trask, Nathaniel, Bochev, Pavel, and Perego, Mauro. A conservative, consistent, and scalable meshfree mimetic method. United States: N. p., 2020. Web. doi:10.1016/j.jcp.2019.109187.
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Trask, Nathaniel, Bochev, Pavel, & Perego, Mauro. A conservative, consistent, and scalable meshfree mimetic method. United States. https://doi.org/10.1016/j.jcp.2019.109187
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Trask, Nathaniel, Bochev, Pavel, and Perego, Mauro. Fri . "A conservative, consistent, and scalable meshfree mimetic method". United States. https://doi.org/10.1016/j.jcp.2019.109187. https://www.osti.gov/servlets/purl/1618096.
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@article{osti_1618096,
title = {A conservative, consistent, and scalable meshfree mimetic method},
author = {Trask, Nathaniel and Bochev, Pavel and Perego, Mauro},
abstractNote = {Mimetic methods discretize divergence by restricting the Gauss theorem to mesh cells. Because point clouds lack such geometric entities, construction of a compatible meshfree divergence remains a challenge. In this work, we define an abstract Meshfree Mimetic Divergence (MMD) operator on point clouds by contraction of field and virtual face moments. This MMD satisfies a discrete divergence theorem, provides a discrete local conservation principle, and is first-order accurate. We consider two MMD instantiations. The first one assumes a background mesh and uses generalized moving least squares (GMLS) to obtain the necessary field and face moments. This MMD instance is appropriate for settings where a mesh is available but its quality is insufficient for a robust and accurate mesh-based discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graph-Laplacian equations. This MMD instance does not require a background grid and is appropriate for applications where mesh generation creates a computational bottleneck. It allows one to trade an expensive mesh generation problem for a scalable algebraic one, without sacrificing compatibility with the divergence operator. We demonstrate the approach by using the MMD operator to obtain a virtual finite-volume discretization of conservation laws on point clouds. Finally, numerical results in the paper confirm the mimetic properties of the method and show that it behaves similarly to standard finite volume methods.},
doi = {10.1016/j.jcp.2019.109187},
journal = {Journal of Computational Physics},
number = C,
volume = 409,
place = {United States},
year = {2020},
month = {1}
}
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https://doi.org/10.1016/j.jcp.2019.109187
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