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Title: Optimization-based mesh correction with volume and convexity constraints

Abstract

In this study, we consider the problem of finding a mesh such that 1) it is the closest, with respect to a suitable metric, to a given source mesh having the same connectivity, and 2) the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh. This volume correction problem arises in important simulation contexts, such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi-Lagrangian transport schemes. In this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective, while the prescribed cell volumes, mesh validity and/or cell convexity specify the constraints. We solve this problem numerically using a sequential quadratic programming (SQP) method whose performance scales with the mesh size. To achieve scalable performance we develop a specialized multigrid-based preconditioner for optimality systems that arise in the application of the SQP method to the volume correction problem. Numerical examples illustrate the importance of volume correction, and showcase the accuracy, robustness and scalability of our approach.

Authors:
 [1];  [1];  [1];  [1];  [2]
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  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Publication Date:
Research Org.:
Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1240603
Alternate Identifier(s):
OSTI ID: 1254321; OSTI ID: 1348256
Report Number(s):
LA-UR-15-22070; SAND-2015-1521J
Journal ID: ISSN 0021-9991; PII: S0021999116001224
Grant/Contract Number:  
14-017511; AC52-06NA25396; AC04-94AL85000
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Computational Physics
Additional Journal Information:
Journal Volume: 313; Journal Issue: C; Journal ID: ISSN 0021-9991
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; lagrangian motion; incremental remap; semi-lagrangian transport; departure volume correction; volume fraction; passive tracer transport; 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS

Citation Formats

D'Elia, Marta, Ridzal, Denis, Peterson, Kara J., Bochev, Pavel, and Shashkov, Mikhail. Optimization-based mesh correction with volume and convexity constraints. United States: N. p., 2016. Web. doi:10.1016/j.jcp.2016.02.050.
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D'Elia, Marta, Ridzal, Denis, Peterson, Kara J., Bochev, Pavel, & Shashkov, Mikhail. Optimization-based mesh correction with volume and convexity constraints. United States. https://doi.org/10.1016/j.jcp.2016.02.050
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D'Elia, Marta, Ridzal, Denis, Peterson, Kara J., Bochev, Pavel, and Shashkov, Mikhail. Wed . "Optimization-based mesh correction with volume and convexity constraints". United States. https://doi.org/10.1016/j.jcp.2016.02.050. https://www.osti.gov/servlets/purl/1240603.
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@article{osti_1240603,
title = {Optimization-based mesh correction with volume and convexity constraints},
author = {D'Elia, Marta and Ridzal, Denis and Peterson, Kara J. and Bochev, Pavel and Shashkov, Mikhail},
abstractNote = {In this study, we consider the problem of finding a mesh such that 1) it is the closest, with respect to a suitable metric, to a given source mesh having the same connectivity, and 2) the volumes of its cells match a set of prescribed positive values that are not necessarily equal to the cell volumes in the source mesh. This volume correction problem arises in important simulation contexts, such as satisfying a discrete geometric conservation law and solving transport equations by incremental remapping or similar semi-Lagrangian transport schemes. In this paper we formulate volume correction as a constrained optimization problem in which the distance to the source mesh defines an optimization objective, while the prescribed cell volumes, mesh validity and/or cell convexity specify the constraints. We solve this problem numerically using a sequential quadratic programming (SQP) method whose performance scales with the mesh size. To achieve scalable performance we develop a specialized multigrid-based preconditioner for optimality systems that arise in the application of the SQP method to the volume correction problem. Numerical examples illustrate the importance of volume correction, and showcase the accuracy, robustness and scalability of our approach.},
doi = {10.1016/j.jcp.2016.02.050},
journal = {Journal of Computational Physics},
number = C,
volume = 313,
place = {United States},
year = {Wed Feb 24 00:00:00 EST 2016},
month = {Wed Feb 24 00:00:00 EST 2016}
}
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