Anisotropic mesh adaptation for the generalized Ambrosio-Tortorelli functional with application to brittle fracture

Quasi-static crack propagation in brittle materials is modeled via the Ambrosio-Tortorelli approximation. The crack is modeled by a smooth phase-field, defined on the whole computational domain. Since the crack is confined to a thin layer, the employment of anisotropic adapted grids is shown to be a really effective tool in containing computational costs. We extend the anisotropic error analysis, applied to the classical Ambrosio-Tortorelli approximation by Artina et al., to the generalized Ambrosio-Tortorelli functional, where a unified framework for several elasticity laws is dealt with as well as a non-convex fracture energy can be accommodated. After deriving an anisotropic a posteriori error estimator, we devise an algorithm which alternates optimization and mesh adaptation. Both anti-plane and plane-strain configurations are numerically checked.