Within this proposal, we consider the numerical approximation and solution of control problems governed by a quasi-static brittle fracture propagation model. As a central modeling component, a phase-field formulation for the fracture formation and propagation is considered.

The fracture propagation problem itself can be formulated as a minimization problem with inequality constraints, imposed by multiple relevant side conditions, such as irreversibility of the fracture-growth or non-selfpenetration of the material across the fracture surface. These lead to variational inequalities as first order necessary conditions. Consequently, optimization problems for the control of the fracture process give rise to a mathematical program with complementarity constraints (MPCC) in function spaces.

Within this project, we intend to focus on mathematical challenges, that are also motivated by applications, such as control of the coefficients of the variational inequality, or nonsmooth and/or nonconvex cost functionals in the outer optimization, such as, e.g., maximizing the released energy of the fracture. We will develop first and second order optimality conditions for the resulting MPCC as well as other obstacle-like formulations. Additionally, we will consider the discretization by finite elements and show the convergence of the discrete approximations to the continuous limit. These findings will be substantiated with prototype numerical tests.