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

Journal Article · · Journal of Computational Physics

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.

Research Organization:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
Grant/Contract Number:
AC04-94AL85000; SC0000230927; NA0003525; SC0019453
OSTI ID:
1618096
Alternate ID(s):
OSTI ID: 1600080
Report Number(s):
SAND-2019-2722J; 673358; TRN: US2106793
Journal Information:
Journal of Computational Physics, Vol. 409, Issue C; ISSN 0021-9991
Publisher:
ElsevierCopyright Statement
Country of Publication:
United States
Language:
English

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