In this talk we study a Dirichlet control problem for the Poisson equation, where the control
is assumed to be piecewise constant function which is allowed to take $M \ge 2$ different values. The
space of admissible Dirichlet controls is non-convex and therefore standard derivatives in Banach spaces are not applicable. Furthermore piecewise constant functions do not belong to $H^{1/2}$ standard
extension techniques to consider the weak solution of the Dirichlet problem do not apply. Therefore
we resort to the notion of very weak solutions of the state equation in $L^p$ spaces. We then study the
differentiability of the shape-to-state operator of this problem and derive the first order necessary
optimality conditions using the topological state derivative. In fact we prove the existence of the
weak topological state derivative for a multimaterial shape functional which is then
expressed via an adjoint variable. The topological derivative resembles formulas found for derivative
in the more standard Dirichlet control problems. In the final part of the paper we show how to apply
a multimaterial levelset algorithm within the finite element software NGSolve and present several
numerical examples in dimension three.
| Subject : | : | oral |
| Topics | : | 7-6 - Shape and topology optimization |
| PDF version | : |  PDF version |
