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Title: On differentiable local bounds preserving stabilization for Euler equations

Abstract

This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. Furthermore, the importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. Additionally, in the case of transient problem, we also observe a reduction in the computational cost.

Authors:
 [1];  [2];  [3];  [4]
+ Show Author Affiliations
  1. Monash Univ., Clayton, VIC (Australia); Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Castelldefels (Spain)
  2. Centre Internacional de Mètodes Numèrics en Enginyeria (CIMNE), Castelldefels (Spain); Univ. Politecnica de Catalunya, Barcelona (Spain)
  3. Clemson Univ., SC (United States)
  4. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States). Center for Computing Research; Univ. of New Mexico, 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); La Caixa Foundation
OSTI Identifier:
1639088
Alternate Identifier(s):
OSTI ID: 1637576
Report Number(s):
SAND-2020-6831J
Journal ID: ISSN 0045-7825; 687134
Grant/Contract Number:  
AC04-94AL85000; NA0003525; LCF/BQ/DE15/10360010
Resource Type:
Accepted Manuscript
Journal Name:
Computer Methods in Applied Mechanics and Engineering
Additional Journal Information:
Journal Volume: 370; Journal ID: ISSN 0045-7825
Publisher:
Elsevier
Country of Publication:
United States
Language:
English
Subject:
42 ENGINEERING; 97 MATHEMATICS AND COMPUTING; Shock capturing; Euler equations; hyperbolic systems; positivity preservation

Citation Formats

Badia, Santiago, Bonilla, Jesús, Mabuza, Sibusiso, and Shadid, John N. On differentiable local bounds preserving stabilization for Euler equations. United States: N. p., 2020. Web. doi:10.1016/j.cma.2020.113267.
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Badia, Santiago, Bonilla, Jesús, Mabuza, Sibusiso, & Shadid, John N. On differentiable local bounds preserving stabilization for Euler equations. United States. https://doi.org/10.1016/j.cma.2020.113267
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Badia, Santiago, Bonilla, Jesús, Mabuza, Sibusiso, and Shadid, John N. Thu . "On differentiable local bounds preserving stabilization for Euler equations". United States. https://doi.org/10.1016/j.cma.2020.113267. https://www.osti.gov/servlets/purl/1639088.
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@article{osti_1639088,
title = {On differentiable local bounds preserving stabilization for Euler equations},
author = {Badia, Santiago and Bonilla, Jesús and Mabuza, Sibusiso and Shadid, John N.},
abstractNote = {This work presents the design of nonlinear stabilization techniques for the finite element discretization of Euler equations in both steady and transient form. Implicit time integration is used in the case of the transient form. A differentiable local bounds preserving method has been developed, which combines a Rusanov artificial diffusion operator and a differentiable shock detector. Nonlinear stabilization schemes are usually stiff and highly nonlinear. This issue is mitigated by the differentiability properties of the proposed method. Moreover, in order to further improve the nonlinear convergence, we also propose a continuation method for a subset of the stabilization parameters. The resulting method has been successfully applied to steady and transient problems with complex shock patterns. Numerical experiments show that it is able to provide sharp and well resolved shocks. Furthermore, the importance of the differentiability is assessed by comparing the new scheme with its non-differentiable counterpart. Numerical experiments suggest that, for up to moderate nonlinear tolerances, the method exhibits improved robustness and nonlinear convergence behavior for steady problems. Additionally, in the case of transient problem, we also observe a reduction in the computational cost.},
doi = {10.1016/j.cma.2020.113267},
journal = {Computer Methods in Applied Mechanics and Engineering},
number = ,
volume = 370,
place = {United States},
year = {Thu Oct 01 00:00:00 EDT 2020},
month = {Thu Oct 01 00:00:00 EDT 2020}
}
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https://doi.org/10.1016/j.cma.2020.113267
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