Error estimation for crack simulations using the XFEM

Abstract

SUMMARY The extended finite element method (XFEM) is by now well‐established for crack calculations in linear elastic fracture mechanics. An advantage of this method is its discretization independence for crack simulations. Nevertheless, discretization errors occur when using the XFEM. In this paper, a simple recovery based error estimator for the discretization error in XFEM‐calculations for cracks is presented. The method is based on the Zienkiewicz and Zhu error estimator. Enhanced smoothed stresses incorporating the discontinuities and singularities because of the cracks are recovered to enable the error estimation for arbitrary distributed cracks. This approach also allows the consideration of materials with generally inelastic behaviour. The enhanced stresses are computed by means of a least square fit problem. To assess the quality of the error estimator, global norms and the effectivity index for the global energy norm for examples with known analytical solutions are presented. Copyright © 2012 John Wiley & Sons, Ltd.

3-D local mesh refinement XFEM with variable-node hexahedron elements for extraction of stress intensity factors of straight and curved planar cracks

“…Basically, the enhanced smoothed stresses are recovered by projecting the element stresses onto the nodes, and by interpolating the nodal stresses with the same ansatz functions that are being used for calculating the displacements. In order to accurately reflect stress discontinuity along crack face as well as stress singularity at crack tip, Prange et al [3] adopted the asymptotic stress fields in linear elastic fracture mechanics as the crack-tip branch enrichment functions for the smoothed stresses. The approximation of smoothed stresses can be written as (19) ( ) 00000 00 0 0 0 00 000 000 00 0000 0 00000 …”

“…Furthermore, another important problem that is often encountered in the vicinity of crack, particularly near crack-tip, is the high gradients [1][2][3]. Modeling the high gradients using mesh-based methods usually requires a fine mesh in the vicinity of crack.…”

“…The elements to be refined have been detected by a posteriori error estimation algorithm. The adaptive procedure using a posteriori error estimation in terms of the XFEM is adopted from the work done by Prange et al [3]. The Zienkiewicz and Zhu error estimator [39] is used and that is based on a stress smoothing technique.…”

“…An error indicator applied to subsequently refined meshes is gained with a relative error, and every element with a relative error exceeds a given specified value of tolerance error is then refined with a set of subdivision elements. For further information, interested readers can refer to [2,3,18].…”

3D Ductile crack propagation with the XFEM

“…For the observed jump in the history data field the modified Heaviside enrichment H(X) is used. In vicinity of the crack front the front enrichments f j (X) originally applied for a stress smoothing procedure in [10] ar employed. After successfully projecting the history data onto the nodes they are interpolated to the new integration points.The Projection of the old degrees of freedom is done with the same enrichment functions as in the XFEM procedure.…”