Abstract
In the context of global/goal-oriented error estimation applied to computational mechanics, the need to obtain reliable and guaranteed bounds on the discretization error has motivated the use of residual error estimators. These estimators require the construction of admissible stress fields verifying the equilibrium exactly. This article focuses on a recent method, based on a flux-equilibration procedure and called the element equilibration + star-patch technique (EESPT), that provides for such stress fields. The standard version relies on a strong prolongation condition in order to calculate equilibrated tractions along finite element boundaries. Here, we propose an enhanced version, which is based on a weak prolongation condition resulting in a local minimization of the complementary energy and leads to optimal tractions in selected regions. Geometric and error estimate criteria are introduced to select the relevant zones for optimizing the tractions. We demonstrate how this optimization procedure is important and relevant to produce sharper estimators at affordable computational cost, especially when the error estimate criterion is used. Two- and three-dimensional numerical experiments demonstrate the efficiency of the improved technique.
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References
Ainsworth M, Oden J (1993) A priori error estimators for second-order elliptic systems. Part 2: An optimal order process for calculating self-equilibrating fluxes. Comput Math Appl 26: 75–87
Babuška I, Rheinboldt W (1978) A posteriori error estimates for the finite element method. Int J Numer Meth Eng 12: 1597
Babuška I, Strouboulis T (2001) The finite element method and its reliability. Oxford University Press, Oxford
Babuška I, Strouboulis T, Upadhyay C, Gangaraj S, Copps K (1994) Validation of a posteriori error estimators by numerical approach. Int J Numer Methods Eng 37: 1073–1123
Becker R, Rannacher R (2001) An optimal control approach to a posteriori error estimation in finite element methods. In: Isereles A (eds) Acta Numerica, vol 10. Cambridge University Press, Cambridge, pp 1–102
Carstensen C, Funken S (2000) Fully reliable localized error control in the fem. SIAM J Sci Comput 21(4): 1465–1484
Chamoin L, Ladevèze P (2007) Bounds on history-dependent or independent local quantities in viscoelasticity problems solved by approximate methods. Int J Numer Methods Eng 71(12): 1387–1411
Chamoin L, Ladevèze P (2008) A non-intrusive method for the calculation of strict and efficient bounds of calculated outputs of interest in linear viscoelasticity problems. Comput Methods Appl Mech Eng 197(9–12): 994–1014
Cirak F, Ramm E (1998) A posteriori error estimation and adaptivity for linear elasticity using the reciprocal theorem. Comput Methods Appl Mech Eng 156: 351–362
Coorevits P, Dumeau J, Pelle J (1999) Control of analyses with isoparametric elements in both 2d and 3d. Int J Numer Methods Eng 46: 157–176
Cottereau R, Díez P, Huerta A (2009) Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method. Comput Mech 44(4): 533–547
de Almeida JM, Maunder E (2009) Recovery of equilibrium on star patches using a partition of unity technique. Int J Numer Meth Eng 79(12): 1493–1516
Florentin E, Gallimard L, Pelle J (2002) Evaluation of the local quality of stresses in 3d finite element analysis. Comput Methods Appl Mech Eng 191: 4441–4457
Ladevèze P (1975) Comparaison de modèles de milieux continus. Ph.D. thesis, Universite P. et M. Curie (in French)
Ladevèze (2008) Strict upper error bounds on computed outputs of interest in computational structural mechanics. Comput Mech 42(2):271–286. http://dx.doi.org/10.1007/s00466-007-0201-y
Ladevèze P, Chamoin L (2010) Calculation of strict error bounds for finite element approximations of non-linear pointwise quantities of interest. Int J Numer Methods Eng 84(13): 1638–1664
Ladevèze P, Leguillon D (1983) Error estimate procedure in the finite element method and application. SIAM J Numer Anal 20(3): 485–509
Ladevèze P, Maunder E (1996) A general method for recovering equilibrating element tractions. Comput Methods Appl Mech Eng 137: 111–151
Ladevèze P, Pelle J (2004) Mastering calculations in linear and nonlinear mechanics. Springer, New York
Ladevèze P, Rougeot P (1997) New advances on a posteriori error on constitutive relation in finite element analysis. Comput Methods Appl Mech Eng 150: 239–249
Ladevèze P, Marin P, Pelle J, Gastine J (1992) Accuracy and optimal meshes in finite element computation for nearly incompressible materials. Comput Methods Appl Mech Eng 94(3): 303–315
Ladevèze P, Rougeot P, Blanchard P, Moreau J (1999) Local error estimators for finite element linear analysis. Comput Methods Appl Mech Eng 176(1–4): 231–246
Ladevèze P, Chamoin L, Florentin E (2010) A new non-intrusive technique for the construction of admissible stress fields in model verification. Comput Methods Appl Mech Eng 199:766–777. doi:10.1016/j.cma.2009.11.007. http://www.sciencedirect.com/science/article/B6V29-4XPYXDV-1/2/174b4df4415326ba82032f6679df331479df3314
Machiels L, Maday Y, Patera A (2000) A flux-free nodal Neumann subproblem approach to output bounds for partial differential equations. Comptes Rendus Académie des Sciences - Mécanique, Paris 2000 330(1): 249–254
Morin P, Nochetto R, Siebert K (2003) Local problems on stars: a posteriori error estimators, convergence, and performance. Math Comput 72(243): 1067–1097
Panetier J, Ladevèze P, Chamoin L (2010) Strict and effective bounds in goal-oriented error estimation applied to fracture mechanics problems solved with the xfem. Int J Numer Methods Eng 81(6): 671–700
Paraschivoiu M, Peraire J, Patera A (1997) A posteriori finite element bounds for linear functional outputs of elliptic partial differential equations. Comput Methods Appl Mech Eng 150: 351–362
Parés N, Díez P, Huerta A (2006) Subdomain-based flux-free a posteriori error estimators. Comput Methods Appl Mech Eng 195(4–6): 297–323
Parés N, Santos H, Díez P (2009) Guaranteed energy error bounds for the poisson equation using a flux-free approach: solving the local problems in subdomains. Int J Numer Methods Eng 79(10): 1203–1244
Parret-Fréaud A, Rey C, Gosselet P, Feyel F (2010) Fast estimation of discretization error for fe problems solved by domain decomposition. Comput Methods Appl Mech Eng 199: 3315–3323
Peraire J, Patera A (1998) Bounds for linear-functional outputs of coercive partial differential equations: local indicators and adaptive refinement. In: Ladevèze P, Oden J (eds) Advances in adaptive computational methods in mechanics. Studies in Applied Mechanics, vol 47. Elsevier, pp 199–216. doi:10.1016/S0922-5382(98)80011-1. http://www.sciencedirect.com/science/article/B8GXV-4NR6463-C/2/599f64fa26ef69780b6aaafdd2d0b920aafdd2d0b920
Pled F, Chamoin L, Ladevèze P (2011) On the techniques for constructing admissible stress fields in model verification: Performances on engineering examples. Int J Numer Methods Eng. doi:10.1002/nme.3180
Prager W, Synge J (1947) Approximation in elasticity based on the concept of functions spaces. Q Appl Math 5: 261–269
Prudhomme S, Oden J (1999) On goal-oriented error estimation for elliptic problems: application to the control of pointwise errors. Comput Methods Appl Mech Eng 176(1–4):313–331. doi:10.1016/S0045-7825(98)00343-0. http://www.sciencedirect.com/science/article/B6V29-40B2XYV-M/2/1ef91c8610e64eb8e9beb2caf6b6585e85e
Prudhomme S, Nobile F, Chamoin L, Oden J (2004) Analysis of a subdomain-based error estimator for finite element approximations of elliptic problems. Numer Methods Partial Differ Equ 20(2): 165–192
Ramm E, Rank E, Rannacher R, Schweizerhof K, Stein E, Wendland W, Wittum G, Wriggers P, Wunderlich W (2003) Error-controlled adaptive finite elements in solid mechanics. Wiley, New York
Rannacher R, Suttmeier F (1997) A feedback approach to error control in finite element methods: application to linear elasticity. Comput Mech 19: 434–446
Strouboulis T, Babuška I, Datta D, Copps K, Gangaraj S (2000) A posteriori estimation and adaptive control of the error in the quantity of interest—part 1: a posteriori estimation of the error in the von Mises stress and the stress intensity factors. Comput Methods Appl Mech Eng 181: 261–294
Verfürth R (1996) A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, Stuttgart
Wiberg N, Díez P (2006) Special issue. Comput Methods Appl Mech Eng 195: 4–6
Zienkiewicz O, Zhu J (1987) A simple error estimator and adaptive procedure for practical engineering analysis. Int J Numer Methods Eng 24: 337–357
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Pled, F., Chamoin, L. & Ladevèze, P. An enhanced method with local energy minimization for the robust a posteriori construction of equilibrated stress fields in finite element analyses. Comput Mech 49, 357–378 (2012). https://doi.org/10.1007/s00466-011-0645-y
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DOI: https://doi.org/10.1007/s00466-011-0645-y