[en] Numerical crack propagation schemes were augmented in an elegant manner by the X-FEM method applied to fracture mechanics. 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 an evident industrial interest. The conventional approach for crack tip enrichment (described in [2,3]) is that only a specific layer of elements are enriched around the crack tip. We show that this “topological” approach does not yield an increase of the order of the asymptotic convergence rate when compared to unenriched finite elements, as when the crack is part of the mesh. It rather modifies the proportionality factor of the asymptotic convergence rate. In this study, we propose another enrichment scheme which yields a convergence rate that appears to be close to that of regular finite elements used when the solution field does not show singularities. The enriched basis in X-FEM degrades the rigidity and mass matrices condition numbers (the mass matrix typically appears in case of time dependent problems such as wave propagation in cracked bodies). To recover the condition number of non enriched matrices, we introduce a preconditioning strategy which acts block-wise on the set of enriched degrees of freedom associated to each node. This strategy uses a local (nodal) Cholesky based decomposition. Another issue is brought by the integration scheme used to build the matrices. The nature of the asymptotic functions are such that any Gauss-Legendre based integration scheme will only poorly converge with respect of the order of the quadrature. We propose a modified integration scheme to handle that issue. We apply the new technique developed to the estimation of stress intensity factors along the crack front of 3D cracks and use these SIFs for crack propagation using a Paris type fatigue law.
