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**Improved implementation and robustness study of the X-FEM for stress analysis around cracks.**
*(English)*
Zbl 1122.74499

Summary: 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 ‘geomatrical’ 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.

### Keywords:

X-FEM; convergence rate; integration scheme; preconditioner \(J\)-integral; crack propagation
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\textit{E. Béchet} et al., Int. J. Numer. Methods Eng. 64, No. 8, 1033--1056 (2005; Zbl 1122.74499)

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