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Goal functional evaluations and adaptive finite elements for phase-field fracture/multiphysics problems

Currently, fracture propagation and multiphysics problems are major topics in applied mathematics and engineering. It seems to turn out that one of the most promising methods is based on a variational setting (proposed by Francfort and Marigo) and more specifically on a thermodynamically consistent phase-field model. Here a smoothed indicator function determines the crack location and is characterized through a model regularization parameter. In addition, modeling assumes that the fracture can never heal, which is imposed through a temporal constraint, leading to a variational inequality system.

In this talk, this basic fracture model is augmented with several hints and discussions of serious challenges in developing state-of-the-art numerical methods. Key aspects are robust and efficient algorithms for imposing the previously mentioned crack irreversibility constraint, treatment of the indefinite Jacobian matrix in terms of a monolithic error-oriented Newton approach, computational analysis of the interplay of model and discretization parameters, and finally coupling to other multiphysics problems such as fluid-filled fractures in porous media, fluid-structure interaction, and steps towards high performance computing for tackling practical field problems.

Our focus in these developments will be on goal functional evaluations, i.e., certain quantities of interest. This aim is accomplished with adaptive finite elements and more specifically, on the one hand, with predictor-corrector mesh adaptivity and, on the other hand, using partition-of-unity dual-weighted residual a posteriori error estimation. Together with high performance computing, mesh adaptivity allows for systematic investigations of the previously mentioned problem settings.

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