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

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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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Author notes

  1. Javier A. Almonacid

    Present address: Department of Mathematics, Simon Fraser University, 8888 University Drive, Burnaby, BC, V5A 1S6, Canada

  2. Ricardo Ruiz-Baier

    Present address: School of Mathematical Sciences, Monash University, 9 Rainforest walk, Clayton, VIC, 3800, Australia

Authors and Affiliations

  1. Centro de Investigación en Ingeniería Matemática (CI²MA), Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Javier A. Almonacid

  2. CI²MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Gabriel N. Gatica

  3. Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG, UK

    Ricardo Ruiz-Baier

Authors
  1. Javier A. Almonacid
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  2. Gabriel N. Gatica
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  3. Ricardo Ruiz-Baier
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Correspondence to Gabriel N. Gatica.

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This research was partially supported by CONICYT-Chile through the project AFB170001 of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal, and Fondecyt project 1161325; by Centro de Investigación en Ingeniería Matemática (\(\hbox {CI}^2\)MA), Universidad de Concepción; and by the Oxford Centre for Doctoral Training in Industrially Focused Mathematical Modelling.

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Almonacid, J.A., Gatica, G.N. & Ruiz-Baier, R. Ultra-weak symmetry of stress for augmented mixed finite element formulations in continuum mechanics. Calcolo 57, 2 (2020). https://doi.org/10.1007/s10092-019-0351-2

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  • DOI: https://doi.org/10.1007/s10092-019-0351-2

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