Shape and topology optimization in fluids using a phase field approach and an application in structural optimization

We consider the problem of shape and topology optimization in fluid mechanics with a general objective functional. A phase field approach is introduced and discussed in terms of well-posedness and first order necessary optimality conditions. The state constraints are either given by the Stokes or the stationary Navier-Stokes equations. We find that minimizers of the diffuse interface setting have a converging subsequence as the interface thickness tends to zero. If this sequence fulfills a certain convergence rate or the total potential power is minimized in a Stokes flow, we obtain that the limit element is a minimizer of the sharp interface formulation. Additionally, we can derive in both, the Stokes and stationary Navier-Stokes setting, optimality conditions of the sharp interface model which can be verified to be the limit of corresponding optimality systems of the phase field model. Finally, we also apply this approach to structural optimization, where we want to find the optimal material distribution of two given elastic materials for a general objective functional. Using the techniques developed before, we can derive convergence results of a phase field approach similar to the fluid mechanical setting and discuss both the diffuse and the sharp interface formulation with regard to well-posedness and necessary optimality conditions