Show Summary Details
More options …

# Computational Methods in Applied Mathematics

Editor-in-Chief: Carstensen, Carsten

Managing Editor: Matus, Piotr

4 Issues per year

IMPACT FACTOR 2017: 0.658

CiteScore 2017: 1.05

SCImago Journal Rank (SJR) 2017: 1.291
Source Normalized Impact per Paper (SNIP) 2017: 0.893

Mathematical Citation Quotient (MCQ) 2016: 0.75

Online
ISSN
1609-9389
See all formats and pricing
More options …

# 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
• Email
• Other articles by this author:
/ Rita Riedlbeck
Published Online: 2018-06-20 | DOI: https://doi.org/10.1515/cmam-2018-0012

## Abstract

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, $H\left(\mathrm{div}\right)$-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.

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

## References

• [1]

M. Ainsworth and J. T. Oden, A Posteriori Error Estimation in Finite Element Analysis, Pure Appl. Math. (New York), Wiley-Interscience, New York, (2000). Google Scholar

• [2]

M. Ainsworth and R. Rankin, Guaranteed computable error bounds for conforming and nonconforming finite element analysis in planar elasticity, Internat. J. Numer. Methods Engrg. 82 (2010), 1114–1157.

• [3]

T. Arbogast and Z. Chen, On the implementation of mixed methods as nonconforming methods for second-order elliptic problems, Math. Comp. 64 (1995), no. 211, 943–972. Google Scholar

• [4]

D. N. Arnold, R. S. Falk and R. Winther, Mixed finite element methods for linear elasticity with weakly imposed symmetry, Math. Comp. 76 (2007), 1699–1723.

• [5]

D. N. Arnold and R. Winther, Mixed finite elements for elasticity, Numer. Math. 92 (2002), 401–419.

• [6]

A. M. Barrientos, N. G. Gatica and P. E. Stephan, A mixed finite element method for nonlinear elasticity: Two-fold saddle point approach and a-posteriori error estimate, Numer. Math. 91 (2002), no. 2, 197–222.

• [7]

M. Botti, D. A. Di Pietro and P. Sochala, A hybrid high-order method for nonlinear elasticity, SIAM J. Numer. Anal. 55 (2017), no. 6, 2687–2717.

• [8]

D. Braess, V. Pillwein and J. Schöberl, Equilibrated residual error estimates are p-robust, Comput. Methods Appl. Mech. Engrg. 198 (2009), 1189–1197.

• [9]

D. Braess and J. Schöberl, Equilibrated residual error estimator for edge elements, Math. Comp. 77 (2008), no. 262, 651–672. Google Scholar

• [10]

F. Brezzi, J. Douglas and L. D. Marini, Recent results on mixed finite element methods for second order elliptic problems, Vistas in Applied Mathematics. Numerical Analysis, Atmospheric Sciences, Immunology, Optimization Software Inc., New York (1986), 25–43. Google Scholar

• [11]

M. Čermák, F. Hecht, Z. Tang and M. Vohralík, Adaptive inexact iterative algorithms based on polynomial-degree-robust a posteriori estimates for the stokes problem, Numer. Math. 138 (2017), no. 4, 1027–1065.

• [12]

M. Cervera, M. Chiumenti and R. Codina, Mixed stabilized finite element methods in nonlinear solid mechanics: Part II: Strain localization, Comput. Methods Appl. Mech. Engrg. 199 (2010), no. 37–40, 2571–2589.

• [13]

L. Chamoin, P. Ladevèze and F. Pled, On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples, Internat. J. Numer. Methods Engrg. 88 (2011), no. 5, 409–441.

• [14]

L. Chamoin, P. Ladevèze and F. Pled, An enhanced method with local energy minimization for the robust a posteriori construction of equilibrated stress field in finite element analysis, Comput. Mech. 49 (2012), 357–378.

• [15]

M. Destrade and R. W. Ogden, On the third- and fourth-order constants of incompressible isotropic elasticity, J. Acoust. Soc. Amer. 128 (2010), no. 6, 3334–3343.

• [16]

P. Destuynder and B. Métivet, Explicit error bounds in a conforming finite element method, Math. Comp. 68 (1999), no. 228, 1379–1396.

• [17]

P. Dörsek and J. Melenk, Symmetry-free, p-robust equilibrated error indication for the hp-version of the FEM in nearly incompressible linear elasticity, Comput. Methods Appl. Math. 13 (2013), 291–304.

• [18]

L. El Alaoui, A. Ern and M. Vohralík, Guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems, Comput. Methods Appl. Mech. Engrg. 200 (2011), 2782–2795.

• [19]

A. Ern and M. Vohralík, A posteriori error estimation based on potential and flux reconstruction for the heat equation, SIAM J. Numer. Anal. 48 (2010), no. 1, 198–223.

• [20]

A. Ern and M. Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, SIAM J. Sci. Comput. 35 (2013), no. 4, A1761–A1791.

• [21]

A. Ern and M. Vohralík, Polynomial-degree-robust a posteriori estimates in a unified setting for conforming, nonconforming, discontinuous Galerkin, and mixed discretizations, SIAM J. Numer. Anal. 53 (2015), no. 2, 1058–1081.

• [22]

A. Ern and M. Vohralík, Broken stable ${H}^{1}$ and $H\left(\mathrm{div}\right)$ polynomial extensions for polynomial-degree-robust potential and flux reconstruction in three space dimensions, preprint (2017), https://arxiv.org/abs/1701.02161.

• [23]

G. N. Gatica, A. Marquez and W. Rudolph, A priori and a posteriori error analyses of augmented twofold saddle point formulations for nonlinear elasticity problems, Comput. Methods Appl. Mech. Engrg. 264 (2013), no. 1, 23–48.

• [24]

G. N. Gatica and E. P. Stephan, A mixed-FEM formulation for nonlinear incompressible elasticity in the plane, Numer. Methods Partial Differential Equations 18 (2002), no. 1, 105–128.

• [25]

A. Hannukainen, R. Stenberg and M. Vohralík, A unified framework for a posteriori error estimation for the Stokes problem, Numer. Math. 122 (2012), 725–769.

• [26]

D. S. Hughes and J. L. Kelly, Second-order elastic deformation of solids, Phys. Rev. 92 (1953), 1145–1149.

• [27]

K.-Y. Kim, Guaranteed a posteriori error estimator for mixed finite element methods of linear elasticity with weak stress symmetry, SIAM J. Numer. Anal. 48 (2011), 2364–2385.

• [28]

P. Ladevèze, Comparaison de modèles de milieux continus, Ph.D. thesis, Université Pierre et Marie Curie (Paris 6), 1975. Google Scholar

• [29]

P. Ladevèze and D. Leguillon, Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal. 20 (1983), 485–509.

• [30]

P. Ladevèze, J. P. Pelle and P. Rougeot, Error estimation and mesh optimization for classical finite elements, Engrg. Comp. 8 (1991), no. 1, 69–80.

• [31]

A. F. D. Loula and J. N. C. Guerreiro, Finite element analysis of nonlinear creeping flows, Comput. Methods Appl. Mech. Engrg. 79 (1990), no. 1, 87–109.

• [32]

R. Luce and B. I. Wohlmuth, A local a posteriori error estimator based on equilibrated fluxes, SIAM J. Numer. Anal. 42 (2004), 1394–1414.

• [33]

J. Nec̆as, Introduction to the Theory of Nonlinear Elliptic Equations, John Wiley & Sons, Chichester, 1986. Google Scholar

• [34]

S. Nicaise, K. Witowski and B. Wohlmuth, An a posteriori error estimator for the Lamé equation based on $H\left(\mathrm{div}\right)$-conforming stress approximations, IMA J. Numer. Anal. 28 (2008), 331–353. Google Scholar

• [35]

S. Ohnimus, E. Stein and E. Walhorn, Local error estimates of FEM for displacements and stresses in linear elasticity by solving local Neumann problems, Internat. J. Numer. Methods Engrg. 52 (2001), 727–746.

• [36]

D. A. D. Pietro and J. Droniou, A hybrid high-order method for Leray–Lions elliptic equations on general meshes, Math. Comp. 86 (2017), no. 307, 2159–2191. Google Scholar

• [37]

D. A. D. Pietro and J. Droniou, ${W}^{s,p}$-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a hybrid high-order discretisation of Leray–Lions problems, Math. Models Methods Appl. Sci. 27 (2017), no. 5, 879–908.

• [38]

M. Pitteri and G. Zanotto, Continuum Models for Phase Transitions and Twinning in Crystals, Chapman & Hall/CRC, Boca Raton, 2002. Google Scholar

• [39]

W. Prager and J. L. Synge, Approximations in elasticity based on the concept of function space, Quart. Appl. Math. 5 (1947), 241–269.

• [40]

S. I. Repin, A Posteriori Estimates for Partial Differential Equations, Radon Ser. Comput. Appl. Math. 4, Walter de Gruyter, Berlin, 2008. Google Scholar

• [41]

R. Riedlbeck, D. A. Di Pietro and A. Ern, Equilibrated stress reconstructions for linear elasticity problems with application to a posteriori error analysis, Finite Volumes for Complex Applications VIII – Methods and Theoretical Aspects, Springer Proc. Math. Stat. 199, Springer, Cham (2017), 293–301. Google Scholar

• [42]

R. Riedlbeck, D. A. Di Pietro, A. Ern, S. Granet and K. Kazymyrenko, Stress and flux reconstruction in Biot’s poro-elasticity problem with application to a posteriori analysis, Comput. Math. Appl. 73 (2017), no. 7, 1593–1610.

• [43]

D. Sandri, Sur l’approximation numérique des écoulements quasi-Newtoniens dont la viscosité suit la loi puissance ou la loi de Carreau, RAIRO Model. Math. Anal. Numer. 27 (1993), no. 2, 131–155.

• [44]

L. R. G. Treloar, The Physics of Rubber Elasticity, Oxford University Press, USA, 1975. Google Scholar

• [45]

R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester, 1996. Google Scholar

• [46]

R. Verfürth, A review of a posteriori error estimation techniques for elasticity problems, Comput. Methods Appl. Mech. Engrg. 176 (1999), 419–440.

• [47]

M. Vogelius, An analysis of the p-version of the finite element method for nearly incompressible materials. Uniformly valid, optimal error estimates, Numer. Math. 41 (1983), 39–53.

• [48]

M. Vohralík, On the discrete Poincaré–Friedrichs inequalities for nonconforming approximations of the Sobolev space ${H}^{1}$, Numer. Funct. Anal. Optim. 26 (2005), no. 7–8, 925–952. Google Scholar

• [49]

M. Vohralík, Unified primal formulation-based a priori and a posteriori error analysis of mixed finite element methods, Math. Comp. 79 (2010), 2001–2032.

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,

Export Citation

© 2018 Walter de Gruyter GmbH, Berlin/Boston.