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Strict error bounds for linear and nonlinear solid mechanicsproblems using a patch-based flux-free method

Published online by Cambridge University Press:  20 December 2010

Régis Cottereau ,
Ludovic Chamoin  and
Pedro Díez
Régis Cottereau*
Affiliation:
Laboratoire MSSMat, École Centrale Paris, CNRS, Grande voie des vignes, 92295 Châtenay-Malabry, France
Ludovic Chamoin
Affiliation:
LMT-Cachan, ENS Cachan, CNRS, Paris 6 University, 61 avenue du Président Wilson, 94230 Cachan, France
Pedro Díez
Affiliation:
Laboratori de Càlcul Numèric, Universitat Politècnica de Catalunya, Jordi Girona 1-3, 08034 Barcelona, Spain
*
a Corresponding author:pedro.diez@upc.edu
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Abstract

We discuss, in this paper, a common flux-free method for the computation of strict errorbounds for linear and nonlinear finite-element computations. In the linear case, the errorbounds are on the energy norm of the error, while, in the nonlinear case, the concept oferror in constitutive relation is used. In both cases, the error bounds are strict in thesense that they refer to the exact solution of the continuous equations, rather than tosome FE computation over a refined mesh. For both linear and nonlinear solid mechanics,this method is based on the computation of a statically admissible stress field, which isperformed as a series of local problems on patches of elements. There is no requirement tosolve a previous problem of flux equilibration globally, as happens with othermethods.

Keywords

Verificationa posteriori error estimationerror boundsresidual-based estimatorsflux-free error estimator
Type
Research Article
Information
Mechanics & Industry , Volume 11 , Issue 3-4: Giens 2009 , May 2010 , pp. 249 - 254
Copyright
© AFM, EDP Sciences 2010

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References

Références

M. Ainsworth, J.T. Oden, A posteriori error estimation in finite element analysis,Pure and Applied Mathematics, Wiley-Interscience, 2000
Wiberg, N.E., Díez, P., Adaptive modeling and simulation, Comput. Methods Appl. Mech. Eng. 195 (2006) 205–480 CrossRefGoogle Scholar
P. Ladevèze, J.-P. Pelle, Mastering calculations in linear and nonlinear mechanics, Mechanical Engineering, Springer, 2005
Ladevèze, P., Upper error bound on calculated outputs of interest for linear and nonlinear structural problems, C. R. Mécanique 334 (2006) 399–407 CrossRefGoogle Scholar
Chamoin, L., Ladevèze, P., Bounds on history-dependent or independent local quantities in viscoelasticity problems solved by approximate methods, Int. J. Numer. Meth. Eng. 71 (2007) 1387–1411 CrossRefGoogle Scholar
Ladevèze, P., Chamoin, L., Florentin, E., A new non-intrusive technique for the construction of admissible stress fields in model verification, Comput. Methods Appl. Mech. Eng. 199 (2010) 766–777 CrossRefGoogle Scholar
Zienkiewicz, O.C., Zhu, J.Z., A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Numer. Meth. Eng. 24 (1987) 337–357 CrossRefGoogle Scholar
Babuška, I., Rheinboldt, W.C., Error estimates for adaptive finite element computations, SIAM J. Numer. Anal. 15 (1978) 736–755 CrossRefGoogle Scholar
Ladevèze, P., Leguillon, D., Error estimate procedure in the finite element method and applications, SIAM J. Numer. Anal. 20 (1983) 485–509 CrossRefGoogle Scholar
Zhu, J.Z.,A posteriori error estimation – the relationship between different procedures, Comput. Methods Appl. Mech. Eng. 150 (1997) 411–422 CrossRefGoogle Scholar
Choi, H.-W., Paraschivoiu, M., Adaptive computations of a posteriori finite element output bounds: a comparison of the “hybrid-flux” approach and the “flux-free” approach, Comput. Methods Appl. Mech. Eng. 193 (2004) 4001–4033 CrossRefGoogle Scholar
Parés, N., Díez, P., Huerta, A., Subdomain-based flux-free a posteriori error estimators, Comput. Methods Appl. Mech. Eng. 195 (2006) 297–323 CrossRefGoogle Scholar
Carstensen, C., Funken, S.A., Fully reliable localized error control in the FEM, SIAM J. Sci. Comput. 21 (1999-2000) 1465–1484 CrossRefGoogle Scholar
Machiels, L., Maday, Y., Patera, A.T., A “flux-free” nodal Neumann subproblem approach to output bounds for partial differential equations, C.-R. Acad. Sci. Ser. I – Math. 330 (2000) 249–254 Google Scholar
Morin, P., Nochetto, R.H., Siebert, K.G., Local problems on stars: a posteriori error estimators, convergence, and performance, Math. Comp. 72 (2003) 1067–1097 CrossRefGoogle Scholar
Prudhomme, S., Nobile, F., Chamoin, L., Oden, J.T., Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems, Numer. Methods Partial Differ. Eqs. 20 (2004) 165–192 CrossRefGoogle Scholar
J.P. Moitinho de Almeida, E.A.W. Maunder, Recovery of equilibrium on star patches using a partition of unity technique, Int. J. Numer. Meth. Eng. 2008, submitted
Sauer-Budge, A.M., Bonet, J., Huerta, A., Peraire, J., Computing bounds for linear functionals of exact weak solutions to Poisson’s equation, SIAM J. Numer. Anal. 42 (2004) 1610–1630 CrossRefGoogle Scholar
Parés, N., Bonet, J., Huerta, A., Peraire, J., The computation of bounds for linear-functional outputs of weak solutions to the two-dimensional elasticity equations, Comput. Methods Appl. Mech. Eng. 195 (2006) 406–429 CrossRefGoogle Scholar
Parés, N., Díez, P., Huerta, A., Bounds of functional outputs for parabolic problems. Part I: Exact bounds of the discontinuous Galerkin time discretization, Comput. Methods Appl. Mech. Eng. 197 (2008) 1641–1660 CrossRefGoogle Scholar
Parés, N., Díez, P., Huerta, A., Bounds of functional outputs for parabolic problems. Part II: Bounds of the exact solution, Comput. Methods Appl. Mech. Eng. 197 (2008) 1661–1679 CrossRefGoogle Scholar
Cottereau, R., Díez, P., Huerta, A., Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method, Comp. Mech. 44 (2009) 533–547 CrossRefGoogle Scholar
Becker, R., Rannacher, R., An optimal control approach to shape a posteriori error estimation in finite element methods, Acta Numerica 10 (2001) 1–102 CrossRefGoogle Scholar
Prudhomme, S., Oden, J.T., On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors, Comput. Methods Appl. Mech. Eng. 176 (1999) 313–331 CrossRefGoogle Scholar
J.J. Moreau, Fonctionnelles convexes, Cours du Collège de France, 1966
Paraschivoiu, M., Peraire, J., Patera, A.T., A posteriori finite element bounds for linear-functional outputs of elliptic partial differential equations, Comput. Methods Appl. Mech. Eng. 150 (1997) 23–50 CrossRefGoogle Scholar