Derivative recovery and a posteriori error estimate for extended finite elements

Abstract

The goal of this work is to devise a simple and effective local a posteriori error estimate for partition of unity enriched finite element methods such as the extended finite element method (XFEM). 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 obtained 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. The XMLS shape functions are 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 one of the superconvergent patch recovery technique of Zienkiewicz and Zhu, which is only C0. Parametric studies of the general performance of the estimator are also carried out.

The extended/generalized finite element method: An overview of the method and its applications

“…For this purpose, error estimators will need to be developed. Bordas and Duflot [18] have developed error estimators based on L 2 projections.…”

Scale selection in nonlinear fracture mechanics of heterogeneous materials

“…The classical SPR technique is used. Our experience of SPR-C and SPR-CX [54][55][56][57] compared to other non-equilibrated recovery techniques in the presence of discontinuities and singularities [58][59][60] shows that the benefits of enforcing consistency, local equilibrium, and boundary conditions is expensive and relatively complex to implement given the observed benefits. We decided to keep the recovery procedure simple and focused our efforts on the modelling error evaluation because whilst advanced techniques such as SPR-C and SPR-CX do decrease the error level and improve the effectivity of the error indicator and the convergence of the approximate error to the exact error, it is always possible, even without satisfying equilibrium, consistency and boundary conditions constraints, to obtain the same error level (with more refined meshes) [55].…”

A recovery-explicit error estimator in energy norm for linear elasticity

“…For instance, reference [38] indicates: "Unfortunately, for an implicit mesh it would be very difficult to implement such a superconvergent recovery scheme of the stress field for elements that intersect the boundary". However in the XFEM framework, where the mesh is independent of the crack, efficient recovery techniques have been already proposed based on the Moving Least Squares (MLS) technique [39,40,41,42,43] and some on the SPR technique [30,34], which introduce worthy improvements to the solution, especially along the boundaries, even in elements trimmed by the crack. In this work we will use the SPR-C technique to obtain an improved stress field from the FE stress field, being this last field rather inaccurate in the case of bilinear elements.…”