Strict and effective bounds in goal-oriented error estimation applied to fracture mechanics problems solved with the XFEM
Résumé
In this work, we analyze a method that leads to strict and high-quality local error bounds in the context of fracture mechanics. We investigate in particular the capability of this method to evaluate the discretization error for quantities of interest computed using the extended finite element method (XFEM). The goal-oriented error estimation method we are focusing on uses the concept of constitutive relation error along with classical extraction techniques. The main innovation in this paper resides in the methodology employed to construct admissible fields in the XFEM framework, which involves enrichments with singular and level set basis functions. We show that this construction can be performed through a generalization of the classical procedure used for the standard finite element method. Thus, the resulting goal-oriented error estimation method leads to relevant and very accurate information on quantities of interest that are specific to fracture mechanics, such as mixed-mode stress intensity factors. The technical aspects and the effectiveness of the method are illustrated through two-dimensional numerical examples.