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Civil-Comp Proceedings
ISSN 1759-3433
CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY
Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 212

A Recovery Error Estimator for Singular Problems Using Singular+Smooth Field Splitting

J.J. Ródenas, E. Giner, J.E. Tarancón and O.A. González

Research Centre on Vehicles Technology, Department of Mechanical and Materials Engineering, Polytechnic University of Valencia, Spain

Full Bibliographic Reference for this paper
J.J. Ródenas, E. Giner, J.E. Tarancón, O.A. González, "A Recovery Error Estimator for Singular Problems Using Singular+Smooth Field Splitting", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 212, 2006. doi:10.4203/ccp.84.212
Keywords: error estimation, stress recovery, superconvergent patch recovery, generalised stress intensity factor, equivalent domain integral, finite element method, singular stress fields.

Summary
When applied to smooth solution problems, recovery-type error estimators, especially those based on the use of the superconvergent patch recovery (SPR) technique [1], have been shown to provide very accurate results. However, the efficiency of the error estimators decreases in singular problems due to their special nature.

In this paper, a new recovery-type error estimator for singular problems in linear elasticity, called the SPR-C-GSIF technique, has been developed. The idea behind the new recovery technique is to obtain the recovered stresses as the contribution of a singular recovered stress field and a smooth recovered stress field . The recovered singular stress field is reconstructed once the generalised stress intensity factor K (GSIF) is extracted from the standard finite element solution. The value of K, which characterizes the singular field, is then substituted into the singular analytical expressions for the stresses to have a better estimate of the singular part of the stress field. For the smooth recovered stress field, an enhanced SPR technique [2] which ensures the exact satisfaction of the equilibrium and compatibility equations of the recovered stresses in the patch is used. The stresses are directly evaluated at the integration points using a 'conjoint polynomial' enhancement to account for the different values of stresses obtained at each integration point from different patches.

In order to obtain an accurate estimation of the GSIF, a domain integral approach has been used [3]. The use of domain integrals for characterizing elastic singular problems is very efficient, accurate and easy to implement as a post-processing part of a finite element analysis.

The test problem used in the numerical examples is a portion of an infinite L-shaped domain which has been loaded with pure mode I stresses, see Figure 1. Since the effectivity index is a measure of the ratio of the estimated error to the exact error , Figure 2 compares the global effectivity indices obtained using the proposed method with the results obtained with the standard SPR technique. A clear improvement can be observed. It has also been checked that the improvement is remarkable at the local level.

Figure 1: L-shaped domain mode I. Geometric model.
Figure 2: Improvement of global effectivity index for quadratic elements

References
1
O.C. Zienkiewicz, J.Z. Zhu, "The Superconvergent Patch Recovery and a-Posteriori Error Estimates. Part I: The Recovery Technique", International Journal for Numerical Methods in Engineering, 33, 1331-1364, 1992. doi:10.1002/nme.1620330702
2
J.J. Ródenas, M. Tur, F.J. Fuenmayor, A. Vercher, "Improvement of the superconvergent patch recovery technique by the use of constraint equations: the SPR-C technique", Submitted to International Journal for Numerical Methods in Engineering, 2005. doi:10.1002/nme.1903
3
B.A. Szabó, I. Babuška, "Finite Element Analysis", John Wiley & Sons, Inc., New York, 1991.

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