Fully Adaptive and Integrated Numerical Methods for the Simulation and Control of Variable Density Multiphase Flows Governed by Diffuse Interface Models

The present work is concerned with the simulation and optimal control of two-phase flows. We provide stable time discretization schemes for the simulation based on both, smooth and non-smooth free energy densities, which we combine with a practical, reliable and efficient adaptive mesh refinement concept for the spatial variables. Furthermore, we consider optimal control problems for two-phase flows and, among other things, derive first order optimality conditions. In the presence of smooth free energies we encounter classical Karush-Kuhn-Tucker (KKT) conditions, while in the case of non-smooth free energies we can prove C(larke)-stationarity. Moreover, we propose a dual weighted residual concept for spatial mesh adaptivity which is based on the newly derived stationarity conditions. We also address future research directions, including closed-loop control concepts and model order reduction techniques for simulation and control of variable density multiphase flows.

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