Elie Chahine

Spider XFEM: an extended finite element variant for partially unknown crack-tip displacement

In this paper, we introduce a new variant of the extended finite element method (Xfem) allowing an optimal convergence rate when the asymptotic displacement is partially unknown at the crack tip. This variant consists in the addition of an adapted discretization of the asymptotic displacement. We give a mathematical result of quasi-optimal a priori error estimate which allows to analyze the potentialities of the method. Some computational tests are provided and a comparison is made with the classical Xfem.

Study of Some Optimal XFEM Type Methods

The XFEM method in fracture mechanics is revisited. A first improvement is considered using an enlarged fixed enrichment subdomain around the crack tip and a bonding condition for the corresponding degrees of freedom. An efficient numerical integration rule is introduced for the nonsmooth enrichment functions. The lack of accuracy due to the transition layer between the enrichment aera and the rest of the domain leads to consider a pointwise matching condition at the boundary of the subdomain. An optimal numerical rate of convergence is then obtained using such a nonconformal method.

Some improvements of Xfem for cracked domains

The XFEM method for fracture mechanics is revisited. A first improvement is considered using an enlarged fixed enriched subdomain around the crack tip and a bonding condition for the corresponding degrees of freedom. An efficient numerical integration rule is introduced for the nonsmooth enrichment functions. The lack of accuracy due to the transition layer between the enrichment area and the rest of the domain leads to consider a pointwise matching condition at the boundary of the subdomain. An optimal rate of convergence is then obtained, numerically and theoretically, even for high degree polynomial approximation.