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Title: Robust, multidimensional mesh motion based on Monge-Kantorovich equidistribution

Abstract

Mesh-motion (r-refinement) grid adaptivity schemes are attractive due to their potential to minimize the numerical error for a prescribed number of degrees of freedom. However, a key roadblock to a widespread deployment of the technique has been the formulation of robust, reliable mesh motion governing principles, which (1) guarantee a solution in multiple dimensions (2D and 3D), (2) avoid grid tangling (or folding of the mesh, whereby edges of a grid cell cross somewhere in the domain), and (3) can be solved effectively and efficiently. In this study, we formulate such a mesh-motion governing principle, based on volume equidistribution via Monge-Kantorovich optimization (MK). In earlier publications [1, 2], the advantages of this approach in regards to these points have been demonstrated for the time-independent case. In this study, demonstrate that Monge-Kantorovich equidistribution can in fact be used effectively in a time stepping context, and delivers an elegant solution to the otherwise pervasive problem of grid tangling in mesh motion approaches, without resorting to ad-hoc time-dependent terms (as in moving-mesh PDEs, or MMPDEs [3, 4]). We explore two distinct r-refinement implementations of MK: direct, where the current mesh relates to an initial, unchanging mesh, and sequential, where the current mesh ismore » related to the previous one in time. We demonstrate that the direct approach is superior in regards to mesh distortion and robustness. The properties of the approach are illustrated with a paradigmatic hyperbolic PDE, the advection of a passive scalar. Imposed velocity flow fields or varying vorticity levels and flow shears are considered.« less

Authors:
 [1];  [1]
  1. Los Alamos National Laboratory
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
970960
Report Number(s):
LA-UR-09-06020; LA-UR-09-6020
Journal ID: ISSN 0021-9991; JCTPAH; TRN: US201003%%304
DOE Contract Number:  
AC52-06NA25396
Resource Type:
Journal Article
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Name: Journal of Computational Physics; Journal ID: ISSN 0021-9991
Country of Publication:
United States
Language:
English
Subject:
99; ADVECTION; DEGREES OF FREEDOM; DIMENSIONS; OPTIMIZATION; VELOCITY

Citation Formats

Delzanno, G L, and Finn, J M. Robust, multidimensional mesh motion based on Monge-Kantorovich equidistribution. United States: N. p., 2009. Web.
Delzanno, G L, & Finn, J M. Robust, multidimensional mesh motion based on Monge-Kantorovich equidistribution. United States.
Delzanno, G L, and Finn, J M. 2009. "Robust, multidimensional mesh motion based on Monge-Kantorovich equidistribution". United States. https://www.osti.gov/servlets/purl/970960.
@article{osti_970960,
title = {Robust, multidimensional mesh motion based on Monge-Kantorovich equidistribution},
author = {Delzanno, G L and Finn, J M},
abstractNote = {Mesh-motion (r-refinement) grid adaptivity schemes are attractive due to their potential to minimize the numerical error for a prescribed number of degrees of freedom. However, a key roadblock to a widespread deployment of the technique has been the formulation of robust, reliable mesh motion governing principles, which (1) guarantee a solution in multiple dimensions (2D and 3D), (2) avoid grid tangling (or folding of the mesh, whereby edges of a grid cell cross somewhere in the domain), and (3) can be solved effectively and efficiently. In this study, we formulate such a mesh-motion governing principle, based on volume equidistribution via Monge-Kantorovich optimization (MK). In earlier publications [1, 2], the advantages of this approach in regards to these points have been demonstrated for the time-independent case. In this study, demonstrate that Monge-Kantorovich equidistribution can in fact be used effectively in a time stepping context, and delivers an elegant solution to the otherwise pervasive problem of grid tangling in mesh motion approaches, without resorting to ad-hoc time-dependent terms (as in moving-mesh PDEs, or MMPDEs [3, 4]). We explore two distinct r-refinement implementations of MK: direct, where the current mesh relates to an initial, unchanging mesh, and sequential, where the current mesh is related to the previous one in time. We demonstrate that the direct approach is superior in regards to mesh distortion and robustness. The properties of the approach are illustrated with a paradigmatic hyperbolic PDE, the advection of a passive scalar. Imposed velocity flow fields or varying vorticity levels and flow shears are considered.},
doi = {},
url = {https://www.osti.gov/biblio/970960}, journal = {Journal of Computational Physics},
issn = {0021-9991},
number = ,
volume = ,
place = {United States},
year = {2009},
month = {1}
}