Keywords: stress intensity factors, error estimation, strict bounds, extended finite element method.

In the field of mechanical engineering, the finite element method is now very largely used. Since this method leads to an altered solution of the reference problem, the question of the quality of the calculations have always been a top research issue. The early works provide an a posteriori error estimation in the global energy norm. One can distinguish three strategies: the constitutive relationship error [1], the error based on the residuals of the equilibrium equations [2], the error based on smoothing techniques [3]. The efficiency of these estimators have been proved. However, they do not allow the engineer to acces the local quality of the solution. Dimensioning criterions require quantities of interest like the displacement, the stress at a point, the stress intensity factors (SIF) etc. More recent research tasks deal with local error estimators derived from the previous global estimators: [4,5,6,7,8].

In this paper we are interested in the control of the SIF for a mixed mode problem. A few authors tried to estimate the quality of the SIF: [8]. In [9,10] error estimation for the J-integral are proposed, in reference [11] guaranted bounds are established. Here we propose an a posteriori error estimator to control the calculation of the SIF by a finite element method which leads to strict and accurate bounds that are easy to compute.

In the first time, we consider a standard finite element model in plane linear elasticity with a static loading. The SIF extraction technique is based on a linear functional of the displacement solution, it involves an area integral on a patch of elements. The method to estimate the error on the SIF uses the concepts of the error in a constitutive relation and, the local error estimation developed at the LMT Cachan [1]. The implementation of this method requires the resolution of the adjoint problem which depends on the extractor operator. The dual problem is solved by the finite element method too. Then, the product of the global errors of the two problems provide bounds for and . Numerical results for problems with reference solutions demonstrate the accuracy of the method.

In the second time, we consider the extension of the method to problems solved by the XFEM [12] based on the partition of unity method [13] .

1

P. Ladevèze, J.-P. Pelle, "Mastering calculations in linear and non linear mechanics", Mechanical engineering series, Springer, 2005.

2

I. Babuska, T. Strouboulis, "The finite element method and its reliability", Oxford University Press, Oxford, 2001.

3

O.C. Zienckiewicz, J.Z. Zhu, "A simple error estimator and adaptative procedure for practical engineering analysis", Int. J. Num. Meth. Eng, vol. 24, 337-357,1987. doi:10.1002/nme.1620240206

4

R. Rannacher, F.T. Stuttmeier , "A feedback approach to error control in finite element methods: application to linear elasticity", Computational mechanics, vol. 19, 434-446, 1997. doi:10.1007/s004660050191

5

J. Peraire, A. Patera, "Bounds for linear-functional outputs of coercive partial differential equations: local error indicators adaptive refinement", In P. Ladevèze and J.T. Oden, editors, Advances in Adaptive Computational Methods, pages 199-216. Elsevier,1998. doi:10.1016/S0922-5382(98)80011-1

6

P. Ladevèze, Ph. Rougeot, P. Blanchard, J.P. Moreau, "Local error estimators for finite element linear analysis", Comp. Methods Appl. Mech. Engrg., vol. 176, 231-246, 1999. doi:10.1016/S0045-7825(98)00339-9

7

S. Prudhomme, J.T. Oden, "On goal-oriented error estimation for elliptic problems: application to the control of pointwise error", Comp. Methods Appl. Mech. Engrg. vol., 176, 313-331, 1999. doi:10.1016/S0045-7825(98)00343-0

8

T. Strouboulis, I. Babuska, D.K. Datta, K. Copps, S.K. Gangaraj, "A posteriori estimation and adaptive control of the error in the quantity of interest. Part I: A posteriori estimation of the error in the Von Mises stress and the stress intensity factors", Comp. Methods Appl. Mech. Engrg., vol. 181, 261-294, 2000. doi:10.1016/S0045-7825(99)00077-8

9

P. Heintz, K. Samuelson, "On adaptative strategies and error control in fracture mechanics", Computer and Structures. vol. 82 485-497, 2004. doi:10.1016/j.compstruc.2003.10.013

10

M. Rüter, E. Stein, "Goal oriented a posteriori error estimates in linear elastic fracture mechanics", Comp. Methods Appl. Mech. Engrg. vol. 195 251-278, 2006. doi:10.1016/j.cma.2004.05.032

11

Z.C. Xuan, N. Parés, J. Peraire, "Computing upper and lower bounds for the J-integral in two dimensional linear elasticity", Comp. Methods Appl. Mech. Engrg. vol. 195 430-443, 2006. doi:10.1016/j.cma.2004.12.031

12

T.Belytschko, T. Black, "Elastic crack growth in finite elements with minimal remeshing", Int. J. Num. Meth. Eng, vol. 45, 601-620,1999. doi:10.1002/(SICI)1097-0207(19990620)45:5<601::AID-NME598>3.0.CO;2-S

13

J.M. Melenk, I. Babuska, "The partition of unity finite element method: Basic theory and applications", Comp. Methods Appl. Mech. Engrg, vol. 139, 289-314, 1996. doi:10.1016/S0045-7825(96)01087-0

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