Simulation and Optimization of Rate-Independent Systems with Non-Convex Energies

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The project is concerned with the analysis, simulation, and optimization of rate-independent systems with non-convex energies. Various systems in continuum mechanics behave approximately rate-independent. A prominent example is the damage evolution in brittle materials. The associated models are given in form of evolutionary variational inequalities based on a positive homogeneous and convex dissipation functional and an energy functional. If the latter is convex, too, then solutions are typically unique and time-continuous. In contrast to that, rate-independent systems with non-convex energies, such as damage models, lead to several mathematical challenges: in general, solutions are not continuous in time, and there are multiple solution concepts with different characterizations of the discontinuities in time. Moreover, solutions are often not unique, even when restricted to a specific notion of solution. Therefore, the numerical simulation and all the more the optimization of rate-independent systems with non-convex energies is a challenging task.The project will face these issues based on the notion of Balanced-Viscosity-(BV)-solutions, which was recently developed to enable a more precise characterization of the discontinuities in time. Existence of BV-solutions can be shown by viscous regularization accompanied by a limit analysis for vanishing viscosity. So far however, it is mostly assumed for this purpose that the data are smooth in time. The project is divided into three branches: The first branch includes an extension of the existence theory to non-smooth data and a refined characterization of the analytical properties of the solution set, like for instance compactness properties. Both aspects will be of major importance for the other two branches. The second one is concerned with the development, analysis, and implementation of efficient and robust numerical methods to approximate BV-solutions. Furthermore, in the third branch, suitable optimization problems shall be formulated and analyzed.For the numerical approximation of BV-solutions, we apply time-incremental local minimization methods. While first numerical results look promising, a rigorous convergence analysis is so far only known for rather restrictive assumptions that are for instance not fulfilled in case of damage. Moreover, time-adaptive methods have not been investigated so far for this type of problems. With regard to optimization, even less is known. This in particular concerns the approximation of optimal solutions via viscous regularization, which also offers an opportunity to solve the optimization problems. With the proposed project, we focus on these open questions and aim to establish an analytically sound framework for the simulation and optimization of rate-independent systems with non-convex energies. Damage processes in brittle materials will serve as prototypical application therefor.