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Title: Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach

Abstract

We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.

Authors:
 [1];  [2]; ORCiD logo [2]; ORCiD logo [1]
  1. Univ. of California, Santa Barbara, CA (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States); Univ. of California, Santa Barbara, CA (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
OSTI Identifier:
1619207
Alternate Identifier(s):
OSTI ID: 1602294; OSTI ID: 1633912
Report Number(s):
SAND-2019-5752J
Journal ID: ISSN 0021-9991; 675748
Grant/Contract Number:  
AC04-94AL85000; SC0019246
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:
97 MATHEMATICS AND COMPUTING; Meshfree methods; High order numerical methods; Generalized Moving Least Squares; Manifolds; Exterior calculus; Curved fluid interfaces; Fluid mechanics

Citation Formats

Gross, B. J., Trask, N., Kuberry, P., and Atzberger, P. J.. Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach. United States: N. p., 2020. Web. https://doi.org/10.1016/j.jcp.2020.109340.
Gross, B. J., Trask, N., Kuberry, P., & Atzberger, P. J.. Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach. United States. https://doi.org/10.1016/j.jcp.2020.109340
Gross, B. J., Trask, N., Kuberry, P., and Atzberger, P. J.. Wed . "Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach". United States. https://doi.org/10.1016/j.jcp.2020.109340. https://www.osti.gov/servlets/purl/1619207.
@article{osti_1619207,
title = {Meshfree methods on manifolds for hydrodynamic flows on curved surfaces: A Generalized Moving Least-Squares (GMLS) approach},
author = {Gross, B. J. and Trask, N. and Kuberry, P. and Atzberger, P. J.},
abstractNote = {We utilize generalized moving least squares (GMLS) to develop meshfree techniques for discretizing hydrodynamic flow problems on manifolds. We use exterior calculus to formulate incompressible hydrodynamic equations in the Stokesian regime and handle the divergence-free constraints via a generalized vector potential. This provides less coordinate-centric descriptions and enables the development of efficient numerical methods and splitting schemes for the fourth-order governing equations in terms of a system of second-order elliptic operators. Using a Hodge decomposition, we develop methods for manifolds having spherical topology. We show the methods exhibit high-order convergence rates for solving hydrodynamic flows on curved surfaces. The methods also provide general high-order approximations for the metric, curvature, and other geometric quantities of the manifold and associated exterior calculus operators. The approaches also can be utilized to develop high-order solvers for other scalar-valued and vector-valued problems on manifolds.},
doi = {10.1016/j.jcp.2020.109340},
journal = {Journal of Computational Physics},
number = C,
volume = 409,
place = {United States},
year = {2020},
month = {2}
}

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Cited by: 2 works
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