# Numerics and optimisation: non-smooth variational problems for modelling superconductivity and friction

Below a critical temperature, electrical resistance vanishes in superconducting materials. Electricity is transported virtually without losses.

Disc brakes, such as those used in bicycles, generate the braking action through friction between the brake pads and the brake disc. This friction depends on the surface structure of the brake pads.

With the exception of the present report, these two topics are unlikely to be discussed in the same context.

But mathematical modelling of the physical processes of superconductivity and frictional contact leads to similar mathematical problems. Both are the focus of the DFG Priority Programme ‘Non-Smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization’. Professor Irwin Yousept’s and Professor Gerhard Starke’s research groups are participating in the priority programme throughout its entire period from October 2016 until October 2022. The mathematical similarity between the questions lies in the term ‘non-smooth’.

The relevant process variables – current density in the case of superconductivity, tension in the brake pad in the case of friction – are not clearly influenced by the created fields in all parameter areas. Instead, they have a ‘kink’ in the most interesting area: when reaching the critical current density and during the transition from adhesion to sliding, respectively.

In the mathematical description of the processes, these ‘kinks’ are variational inequalities. In the past decades, a comprehensive solution theory and numerical methods have been developed to construct approximations of such variational inequalities efficiently. Adaptive mesh refinements based on error estimators are an important component of the solution strategy, as they ensure that the dimension of discretised problems does not become excessive. The strategy further involves suitable iterative processes for approximating a solution to the discretised problems, which are still highly non-linear and non-smooth. Complementarity conditions incorporating Lagrange multipliers play a role in both sub-aspects.

The established methods for variational inequalities are not directly applicable to the problems our sub-projects seek to solve, however. We are working with hyperbolic evolution variational inequalities (in the superconductivity problem) and quasi-variational inequalities (in the friction problem). In the first case, time-dependent irregularities in current densities and singularities in the electromagnetic fields may occur; in the second case, the variation formulation itself depends on the solution. Both projects yielded enough research topics for one doctoral dissertation each, followed by plenty of post-doctoral research.