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Computational Methods in Applied Mathematics

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Equilibrated Stress Tensor Reconstruction and A Posteriori Error Estimation for Nonlinear Elasticity

Michele Botti
  • Corresponding author
  • University of Montpellier, Institut Montpéllierain Alexander Grothendieck, 34095 Montpellier, France
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/ Rita Riedlbeck
Published Online: 2018-06-20 | DOI: https://doi.org/10.1515/cmam-2018-0012


We consider hyperelastic problems and their numerical solution using a conforming finite element discretization and iterative linearization algorithms. For these problems, we present equilibrated, weakly symmetric, H(div)-conforming stress tensor reconstructions, obtained from local problems on patches around vertices using the Arnold–Falk–Winther finite element spaces. We distinguish two stress reconstructions: one for the discrete stress and one representing the linearization error. The reconstructions are independent of the mechanical behavior law. Based on these stress tensor reconstructions, we derive an a posteriori error estimate distinguishing the discretization, linearization, and quadrature error estimates, and propose an adaptive algorithm balancing these different error sources. We prove the efficiency of the estimate, and confirm it on a numerical test with an analytical solution. We then apply the adaptive algorithm to a more application-oriented test, considering the Hencky–Mises and an isotropic damage model.

Keywords: A Posteriori Error Estimate; Equilibrated Stress Reconstruction; Arnold–Falk–Winther Finite Element; Nonlinear Elasticity; Hyperelasticity

MSC 2010: 65N15; 65N30; 74B20; 74S05


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About the article

Received: 2018-02-12

Revised: 2018-05-31

Accepted: 2018-06-03

Published Online: 2018-06-20

Funding Source: Agence Nationale de la Recherche

Award identifier / Grant number: 2014-2-006

Funding Source: Bureau de Recherches Géologiques et Minières

Award identifier / Grant number: RP18DRP019

The work of Michele Botti was partially supported by Labex NUMEV (ANR-10-LABX-20) ref. 2014-2-006 and by the Bureau de Recherches Géologiques et Minières (Project RP18DRP019).

Citation Information: Computational Methods in Applied Mathematics, ISSN (Online) 1609-9389, ISSN (Print) 1609-4840, DOI: https://doi.org/10.1515/cmam-2018-0012.

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