## On solution of contact shape optimization problems with Coulomb friction and a solution-dependent friction coefficient

### Petr Beremlijski, J. Haslinger, J. Outrata a R. Pathó

We are dealing with a discretized problem of shape optimization of 2D and 3D elastic body in unilateral contact, where the coefficient of friction is assumed to depend on the unknown solution. Mathematical modeling of the Coulomb friction problem leads to a system of finite-dimensional implicit variational inequalities parametrized by the so-called control variable, that describes the shape of the elastic body. It has been shown that if the coefficient of friction is Lipschitz and sufficiently small, then the discrete state problems are uniquely solvable for all admissible values of the control variable (the admissible set is assumed to be compact), and the state variables are Lipschitzian functions of the control variable.

The shape optimization problem belongs to a class of so-called mathematical programs with equilibrium constraints (MPECs). The uniqueness of the equilibria for fixed control variables enables us to apply the so-called implicit programming approach. Its main idea consists in minimization of a nonsmooth composite function generated by the objective and the (single-valued) control-state mapping. In our problem, the control-state mapping is much more complicated than in most MPECs solved so far in the literature, and the generalization of the relevant results is by no means straightforward. For the solution of this nonsmooth problem we use our Matlab implementation of bundle trust method proposed by Schramm and Zowe. In each step of the iteration process, we must be able to find the solution of the state problem (contact problem with Coulomb friction) and to compute one arbitrary Clarke subgradient.

To get subgradient information needed in the used numerical method we use the differential calculus of Mordukhovich. For solving a state problem we use the method of successive approximations. Each iterative step of the method requires us to solve the contact problem with given friction. As a result, we obtain a convex quadratic programming problem with a convex separable nonlinear inequality and linear equality constraints. For the solution of such problems we use a combination of inexact augmented Lagrangians in combination with active set based algorithms.

Numerical examples illustrate the efficiency and reliability of the suggested approach.