Anisotropic mesh adaptation for quasistatic crack propagation in brittle materials

The numerical simulation of quasistatic fracturing of brittle material, where no predefined crack path is imposed, is a challenging problem. In particular, we deal with the Francfort-Marigo model which requires the minimization of the well-known nonconvex and nonsmooth Mumford-Shah functional. To deal with a smoother problem which eases the minimization process of the energy, we consider the Γ-approximation via the Ambrosio-Tortorelli variational model. Moreover, from a modeling viewpoint, we induce the quasistatic crack evolution via an applied displacement slowly changing in time. It is well established that the numerical simulation of crack propagation is often biased by an inappropriate numerical discretization, leading to nonphysical results. In particular, a non-optimal or a fixed computational mesh can affect the fracture evolution, driving it along non-realistic directions. This issue can be successfully tackled by resorting to anisotropic adapted meshes, which have been employed to model phenomena exhibiting strongly directional features. To drive mesh adaptation, we devise an a-posteriori anisotropic error estimator of the residual associated with the gradient of the energy functional. Through anisotropic meshes we are able to locate, size and orient the mesh elements in order to follow the intrinsic directionalities of the crack. Thanks to this grid adaptation process, we obtain solutions reliable from a physical viewpoint and with a relatively small computational cost. In this talk, we describe the Francfort-Marigo model and its Γ-approximation in case of plane and antiplane displacements. Then, we focus on the anisotropic mesh adaptation strategy. We finally assess the reliability, robustness, and efficiency of the proposed approach on some benchmark tests.