This project is concerned with the analysis, simulation and optimization of rate-independent systems with non-convex energy. Models of this type arise in a variety of applications in solid mechanics, such as for example the brittle damage of a workpiece under the influence of external loads. Rate-independent systems governed by non-convex energies provide a bunch of mathematical challenges: there exists a variety of different solution concepts and, in neither of these concepts, the solutions are in general unique. Moreover, solutions frequently provide a substantial lack of regularity, in particular jumps in time may occur. All this is caused by a complex interplay of a non-smooth dissipation and a non-convex energy.

Within this project, we intend to analyze rate-independent systems and optimal control thereof when equipped with the concept of so-called balanced viscosity solutions. This concept is based on a viscous regularization and a subsequent passage to the limit in the viscosity parameter. Besides the existence of optimal controls, we are also interested in viscous approximations of these optimal control problems, which offers a possibility to solve these optimization problems numerically.

Our vision is to provide a general, analytically sound and numerically robust framework for the simulation and optimization of rate-independent systems with non-convex energies. Moreover, our techniques and methods shall be practically verified on the basis of real-world applications such as damage evolution in brittle materials.