P. Wörle:

"Shape Optimization for a Variational Inequality: Comparison of a Regularized and a Unregularized Approach";

Supervisor: J. Schöberl, K. Sturm; Institut für Analysis und Scientific Computing, 2022; final examination: 2022-01-16.

The aim of shape optimization is to optimize a cost function which is in some way dependent on a set $\Omega$. In this thesis the cost function is given by the squared $L^2$-Norm of $y_\Omega- \overline{y}$ over a domain $D$ for some given target function $\overline{y}$. Here $y_\Omega$ is the solution of a variational inequality depending on $\Omega$ which in addition solves an obstacle problem. The dependence of the variational inequality on $\Omega$ usually is such that the variational inequality is solved only on $\Omega$. In contrast in this thesis the right-hand side of the variational inequality is dependent of $\Omega$. This problem has been solved in different ways in the literature. Two such approaches are compared with the focus lying on the differences in the numericals solutions of the optimization problem.In the first approach the variational inequality is regularized and a shape derivative formula for the regularized problem is calculated. The formula is tested numerically on a concrete example and compared with the results from Netgen/NGSolve, a software that solves finite-element-problems developed at TU Wien. In the second approach the unregularized problem is solved directly and a shape derivative formula is stated. The numerical calculation of this shape derivative is more complex. A new algorithm is developed which allows the calculation under additional assumptions on the problem. For this algorithm convergence is shown. The numerical results of both approaches are compared and a third approach is shown, which combines techniques of the two former ones.

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