Anisotropic Mesh Adaptation for Crack Detection In Brittle Materials

The quasi-static brittle fracture model proposed by G. Francfort and J.-J. Marigo can be $\Gamma$-approximated at each time evolution step by the Ambrosio--Tortorelli functional. In this paper, we focus on a modification of this functional which includes additional constraints via penalty terms to enforce the irreversibility of the fracture as well as the applied displacement field. Second, we build on this variational model an adapted discretization to numerically compute the time-evolving minimizing solution. We present the derivation of a novel a posteriori error estimator driving the anisotropic adaptive procedure. The main properties of these automatically generated meshes are to be very fine and strongly anisotropic in a very thin neighborhood of the crack, but only far away from the crack tip, while they show a highly isotropic behavior in a neighborhood of the crack tip instead. As a consequence of these properties, the resulting discretizations follow very closely the propagation of the fracture,...

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