This talk is concerned with the optimal control of two immiscible fluids. For the mathematical formulation we use a coupled Cahn-Hilliard/Navier-Stokes system which involves a variational inequality of 4th order. We discuss the differentiability properties of the control-to-state operator and the corresponding stationarity concepts for the control problem. We present strong stationarity conditions and provide a numerical solution algorithm based on a bundle-free implicit programming approach which terminates at an at least C-stationary point which, in the best case, is even strongly stationary
joint work with Winnifried Wollner
An optimal control problem for a time-discrete fracture propagation process is considered. The nonlinear fracture model is treated once as a linearized one, while the original nonlinear model is dealt with afterwards. The discretization of the problem in both cases is done using a conforming finite element method. Regarding the linearized case, a priori error estimates for the control, state, and adjoint variables will be derived. The discretized nonlinear fracture model will be analyzed as well, which leads to a quantitative error estimate, while we avoid unrealistic regularity assumptions.
joint work with Christian Kahle, Jens-Uwe Repke
In lab-on-a-chip devices liquid drops move over solid surfaces. A specific contact angle distribution can be utilized to influence both speed and path of each drop. In previous works, we improved and implemented a phase field model for energy-consistent simulations of moving contact lines. Building on this, we present first results for the optimal control of droplets on solid surfaces. The static contact angle distribution acts as the control variable and the phase field model is used to calculate the drop dynamics. As optimization problems, desired drop shapes and positions are considered.