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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 firstorder 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 meshbased discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graphLaplacian 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 finitevolume discretization of conservation lawsmore »
 Authors:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), 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):
 SAND20192722J
Journal ID: ISSN 00219991; 673358
 Grant/Contract Number:
 AC0494AL85000; 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 00219991
 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.
Trask, Nathaniel, Bochev, Pavel, & Perego, Mauro. A conservative, consistent, and scalable meshfree mimetic method. United States. doi:10.1016/j.jcp.2019.109187.
Trask, Nathaniel, Bochev, Pavel, and Perego, Mauro. Fri .
"A conservative, consistent, and scalable meshfree mimetic method". United States. doi:10.1016/j.jcp.2019.109187.
@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 firstorder 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 meshbased discretization. The second MMD operator retains the GMLS field moments but defines virtual face moments using computationally efficient weighted graphLaplacian 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 finitevolume 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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