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PROCEEDINGS OF THE FIFTEENTH UK CONFERENCE OF THE ASSOCIATION OF COMPUTATIONAL MECHANICS IN ENGINEERING
Edited by: B.H.V. Topping
First Attempts at A Posteriori Error Estimation in eXtended Finite Element Methods
S. Bordas1 and M. Duflot2
1Civil Engineering Department, University of Glasgow, United Kingdom
S. Bordas, M. Duflot, "First Attempts at A Posteriori Error Estimation in eXtended Finite Element Methods", in B.H.V. Topping, (Editor), "Proceedings of the Fifteenth UK Conference of the Association of Computational Mechanics in Engineering", Civil-Comp Press, Stirlingshire, UK, Paper 49, 2007. doi:10.4203/ccp.85.49
Keywords: XFEM, a posteriori error estimation, derivative recovery, extended moving least squares, extended global recovery, strain/stress smoothing.
We present and compare in this paper two derivative recovery and a posteriori error estimation techniques for partition of unity enriched finite elements such as the extended finite element method (XFEM).
In the work presented, we fashion and test two simple and effective local a posteriori error estimators (or indicators), specially adapted to partition of unity enriched finite element methods such as the extended finite element method (XFEM) [1,2]. The first estimator is based on extended moving least squares (XMLS) smoothing, the second on an extended global recovery technique, that we coin XGR, both are of the Zienkiewicz and Zhu type .
The XMLS estimation, based on an idea of Tabbara et al.  is explained first. In each element, the local estimator is the L2 norm of the difference between the raw XFEM strain field and an enhanced strain field computed by extended moving least squares (XMLS) derivative recovery computed from the raw nodal XFEM displacements. The XMLS construction is taylored to the nature of the solution. The technique is applied to linear elastic fracture mechanics, in which near-tip asymptotic functions are added to the MLS basis. This basis is constructed from weight functions following the diffraction criterion to represent the discontinuity. The result is a very smooth enhanced strain solution including the singularity at the crack tip. Results are shown for two and three dimensional linear elastic fracture mechanics problems in mode I and mixed mode. The effectivity index of the estimator is always close to 1. and improves upon mesh refinement. It is also shown that for the linear elastic fracture mechanics problems treated, the proposed estimator outperforms the Zienkiewicz and Zhu estimator of , which is expected, since the latter provides only C0 continuity of the recovered strains. Parametric studies of the general performance of the estimator are also carried out.
The XGR method, extending the idea of Zienkiewicz and Zhu  to enriched finite elements is as follows. The enhanced strain field is constructed as a linear combination of the finite element shape functions used for the displacement approximation, enriched with functions derived from the gradients of the near-tip fields. The coefficients of this linear aproximation are obtained by a global minimization of the difference between the raw solution and this enhanced solution. In all examples treated, the XGR method yields a recovered strain field which is not as smooth as the XMLS, but is computationally cheaper. As in the XMLS, the estimator always yields an effectivity which is close to 1. and improves upon mesh refinement.
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