In this paper, we introduce an implementation of the extended finite element method for fracture problems within the finite element software ABAQUS TM . User subroutine (UEL) in Abaqus is used to enable the incorporation of extended finite element capabilities. We provide details on the data input format together with the proposed user element subroutine, which constitutes the core of the finite element analysis; however, pre-processing tools that are necessary for an X-FEM implementation, but not directly related to Abaqus, are not provided. In addition to problems in linear elastic fracture mechanics, non-linear frictional contact analyses are also realized. Several numerical examples in fracture mechanics are presented to demonstrate the benefits of the proposed implementation.
SUMMARYA new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method (XFEM) is presented. The procedure is an adaptation of the superconvergent patch recovery (SPR) technique for the XFEM framework. It is based on three fundamental aspects: (a) the use of a singular +smooth stress field decomposition technique involving the use of different recovery methods for each field: standard SPR for the smooth field and reconstruction of the recovered singular field using the stress intensity factor K for the singular field; (b) direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement; and (c) assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack. The method was validated by testing it on problems with an exact solution in mode I, mode II, and mixed mode and on a problem without analytical solution. The results obtained showed the accuracy of the proposed error estimator.
SUMMARYThe superconvergent patch recovery (SPR) technique is widely used in the evaluation of a recovered stress field r * from the finite element solution r fe . Several modifications of the original SPR technique have been proposed. A new improvement of the SPR technique, called SPR-C technique (Constrained SPR), is presented in this paper. This new technique proposes the use of the appropriate constraint equations in order to obtain stress interpolation polynomials in the patch r * p that locally satisfy the equations that should be satisfied by the exact solution. As a result the evaluated expressions for r * p will satisfy the internal equilibrium and compatibility equations in the whole patch and the boundary equilibrium equation at least in vertex boundary nodes and, under certain circumstances, along the whole boundary of the patch coinciding with the boundary of the domain. The results show that the use of this technique considerably improves the accuracy of the recovered stress field r * and therefore the local effectivity of the ZZ error estimator.
SUMMARYThe application of the extended finite element method (XFEM) to fracture mechanics problems enables one to obtain accurate solutions more efficiently than with the standard finite element method. A component can be modelled without the need to build a mesh that matches the crack geometry, and thus remeshing as the crack grows is unnecessary. In the XFEM approach, the interpolation on certain elements is enriched with functions that make it feasible to represent the crack tip asymptotic displacement fields by using a local partition of unity method. However, the enrichment is only partial in the blending elements connecting the enriched zone with the rest of the mesh, and consequently pathological terms appear in the interpolation, which lead to increased error. In this study we propose enhancing the blending elements by adding hierarchical shape functions where appropriate; this permits compensating for the unwanted terms in the interpolation. This technique is an extension of the study of Chessa et al. (Int. J. Numer. Meth. Engng. 2003; 57:1015-1038) to fracture mechanics problems. The numerical results show that the proposed enhancement always results in greater accuracy. Moreover, enhancing the blending elements makes it possible to recover the convergence rate that is decreased when the degrees of freedom gathering technique is used to improve the condition number of the stiffness matrix.
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