Improved implementation and robustness study of the X‐FEM for stress analysis around cracks

Numerical crack propagation schemes were augmented in an elegant manner by the X‐FEM method. The use of special tip enrichment functions, as well as a discontinuous function along the sides of the crack allows one to do a complete crack analysis virtually without modifying the underlying mesh, which is of industrial interest, especially when a numerical model for crack propagation is desired. This paper improves the implementation of the X‐FEM method for stress analysis around cracks in three ways. First, the enrichment strategy is revisited. The conventional approach uses a ‘topological’ enrichment (only the elements touching the front are enriched). We suggest a ‘geometrical’ enrichment in which a given domain size is enriched. The improvements obtained with this enrichment are discussed. Second, the conditioning of the X‐FEM both for topological and geometrical enrichments is studied. A preconditioner is introduced so that ‘off the shelf’ iterative solver packages can be used and perform as well on X‐FEM matrices as on standard FEM matrices. The preconditioner uses a local (nodal) Cholesky based decomposition. Third, the numerical integration scheme to build the X‐FEM stiffness matrix is dramatically improved for tip enrichment functions by the use of an ad hoc integration scheme. A 2D benchmark problem is designed to show the improvements and the robustness. Copyright © 2005 John Wiley & Sons, Ltd.