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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]
  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.
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
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.
@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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Works referenced in this record:

On the Finite Element Solution of the Pure Neumann Problem
journal, January 2005


Covolume Solutions of Three-Dimensional Div-Curl Equations
journal, December 1997


An arbitrary order variationally consistent integration for Galerkin meshfree methods: A VARIATIONALLY CONSISTENT INTEGRATION FOR MESHFREE METHODS
journal, June 2013

  • Chen, Jiun-Shyan; Hillman, Michael; Rüter, Marcus
  • International Journal for Numerical Methods in Engineering, Vol. 95, Issue 5
  • DOI: 10.1002/nme.4512

Mimetic finite difference method
journal, January 2014

  • Lipnikov, Konstantin; Manzini, Gianmarco; Shashkov, Mikhail
  • Journal of Computational Physics, Vol. 257
  • DOI: 10.1016/j.jcp.2013.07.031

Direct Discretization of Planar Div-Curl Problems
journal, February 1992

  • Nicolaides, R. A.
  • SIAM Journal on Numerical Analysis, Vol. 29, Issue 1
  • DOI: 10.1137/0729003

Parallel multilevel preconditioners
journal, September 1990


Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations
journal, September 1982

  • Brooks, Alexander N.; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 32, Issue 1-3
  • DOI: 10.1016/0045-7825(82)90071-8

A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems
journal, November 1986

  • Hughes, Thomas J. R.; Mallet, Michel
  • Computer Methods in Applied Mechanics and Engineering, Vol. 58, Issue 3
  • DOI: 10.1016/0045-7825(86)90153-2

On generalized moving least squares and diffuse derivatives
journal, September 2011

  • Mirzaei, D.; Schaback, R.; Dehghan, M.
  • IMA Journal of Numerical Analysis, Vol. 32, Issue 3
  • DOI: 10.1093/imanum/drr030

A mixed finite volume method for elliptic problems
journal, January 2007

  • Mishev, Ilya D.; Chen, Qian-Yong
  • Numerical Methods for Partial Differential Equations, Vol. 23, Issue 5
  • DOI: 10.1002/num.20213

Natural discretizations for the divergence, gradient, and curl on logically rectangular grids
journal, February 1997


A stabilized mixed finite element method for Darcy flow
journal, August 2002

  • Masud, Arif; Hughes, Thomas J. R.
  • Computer Methods in Applied Mechanics and Engineering, Vol. 191, Issue 39-40
  • DOI: 10.1016/S0045-7825(02)00371-7

Element-free Galerkin methods
journal, January 1994

  • Belytschko, T.; Lu, Y. Y.; Gu, L.
  • International Journal for Numerical Methods in Engineering, Vol. 37, Issue 2
  • DOI: 10.1002/nme.1620370205

A Conservative Mesh-Free Scheme and Generalized Framework for Conservation Laws
journal, January 2012

  • Kwan-yu Chiu, Edmond; Wang, Qiqi; Hu, Rui
  • SIAM Journal on Scientific Computing, Vol. 34, Issue 6
  • DOI: 10.1137/110842740

On the Angle Condition in the Finite Element Method
journal, April 1976

  • Babuška, I.; Aziz, A. K.
  • SIAM Journal on Numerical Analysis, Vol. 13, Issue 2
  • DOI: 10.1137/0713021

An Analysis of Finite Volume, Finite Element, and Finite Difference Methods Using Some Concepts from Algebraic Topology
journal, May 1997


A High-Order Staggered Meshless Method for Elliptic Problems
journal, January 2017

  • Trask, Nathaniel; Perego, Mauro; Bochev, Pavel
  • SIAM Journal on Scientific Computing, Vol. 39, Issue 2
  • DOI: 10.1137/16M1055992

A new Control Volume Finite Element Method for the stable and accurate solution of the drift–diffusion equations on general unstructured grids
journal, February 2013

  • Bochev, Pavel; Peterson, Kara; Gao, Xujiao
  • Computer Methods in Applied Mechanics and Engineering, Vol. 254
  • DOI: 10.1016/j.cma.2012.10.009

A stabilized mixed discontinuous Galerkin method for Darcy flow
journal, May 2006

  • Hughes, Thomas J. R.; Masud, Arif; Wan, Jing
  • Computer Methods in Applied Mechanics and Engineering, Vol. 195, Issue 25-28
  • DOI: 10.1016/j.cma.2005.06.018