Ultra-weak symmetry of stress for augmented mixed finite element formulations in continuum mechanics

Javier A. Almonacid, Gabriel N. Gatica, Ricardo Ruiz-Baier

Research output: Contribution to journalArticleResearchpeer-review

Abstract

In this paper we propose a novel way to prescribe weakly the symmetry of stress tensors in weak formulations amenable to the construction of mixed finite element schemes. The approach is first motivated in the context of solid mechanics (using, for illustrative purposes, the linear problem of linear elasticity), and then we apply this technique to reduce the computational cost of augmented fully-mixed methods for thermal convection problems in fluid mechanics, in the case where several additional variables are defined. We show that the new approach allows to maintain the same structure of the mathematical analysis as in the original formulations. Therefore we only need to focus on ellipticity of certain bilinear forms, as this property provides feasible ranges for the stabilization parameters that complete the description of augmented methods. In addition, we present some numerical examples to show that these methods perform better than their counterparts that include vorticity, and emphasize that the reduction in degrees of freedom (and therefore, in computational cost) does not affect the quality of numerical solutions.

Original languageEnglish
Article number2
Number of pages25
JournalCalcolo
Volume57
Issue number1
DOIs
Publication statusPublished - 1 Mar 2020

Keywords

  • A priori error estimates
  • Boussinesq equations
  • Linear elasticity
  • Mixed finite element methods
  • Ultra-weakly imposed symmetry

Cite this

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title = "Ultra-weak symmetry of stress for augmented mixed finite element formulations in continuum mechanics",
abstract = "In this paper we propose a novel way to prescribe weakly the symmetry of stress tensors in weak formulations amenable to the construction of mixed finite element schemes. The approach is first motivated in the context of solid mechanics (using, for illustrative purposes, the linear problem of linear elasticity), and then we apply this technique to reduce the computational cost of augmented fully-mixed methods for thermal convection problems in fluid mechanics, in the case where several additional variables are defined. We show that the new approach allows to maintain the same structure of the mathematical analysis as in the original formulations. Therefore we only need to focus on ellipticity of certain bilinear forms, as this property provides feasible ranges for the stabilization parameters that complete the description of augmented methods. In addition, we present some numerical examples to show that these methods perform better than their counterparts that include vorticity, and emphasize that the reduction in degrees of freedom (and therefore, in computational cost) does not affect the quality of numerical solutions.",
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Ultra-weak symmetry of stress for augmented mixed finite element formulations in continuum mechanics. / Almonacid, Javier A.; Gatica, Gabriel N.; Ruiz-Baier, Ricardo.

In: Calcolo, Vol. 57, No. 1, 2, 01.03.2020.

Research output: Contribution to journalArticleResearchpeer-review

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AU - Gatica, Gabriel N.

AU - Ruiz-Baier, Ricardo

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