Optimal Control of Variational Inequalities of the Second Kind with Application to Yield Stress Fluids

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The project is concerned with the optimal control of variational inequalities of the second kind involving a non-differentiable norm of first derivatives as characteristic nonsmooth feature. Such kind of variational inequalities arise in the modeling of yield stress fluids as for instance Bingham fluids. Within this project we focus on Bingham fluids in their stationary form. The simulation of such nonsmooth fluid flows is frequently based on regularization techniques and the same holds for their optimization. In order to avoid an associated regularization error, the aim of our project is to optimally control yield stress fluid flows with minimal, respectively, without any regularization.The project addresses theoretical and numerical aspects. First we investigate the limited differentiability properties of the solution operator associated with the variational inequality. Based on these findings, necessary and sufficient optimality conditions are derived in the analytical part of the project. Concerning the numerical component of the project, the differentiability results are used to design a mesh-independent trust-region algorithm in function space for the solution of the optimal control problem. This algorithm will be realized with the in-house flow simulation package FEATFLOW in order to solve application-driven benchmark problems.