Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System with Variable Fluid Densities

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In this project we study optimal and closed-loop control problems for variable density multiphase flows governed by diffuse interface models with nonsmooth energies. The resulting problems fall into the realm of mathematical programs with equilibrium constraints (MPECs) in function space. Moreover, model order reduction (MOR) techniques for nonsmooth Cahn-Hilliard Navier-Stokes systems and their application in control of multiphase flows are considered. We intend to (a) study differential stability properties of the solution operator of the coupled Cahn-Hilliard Navier-Stokes system and derive so-called strong stationarity conditions for the optimal control problem;(b) develop, analyse and implement bundle-free implicit programming techniques for the numerical solution of the reduced control problem;(c) develop and analyse fully integrated adaptive numerical methods for the control of multiphase flow problems with variable fluid densities which guarantee a locally refined resolution of the local processes at the interface;(d) derive dual-weighted residual based error estimates for adaptive mesh refinement for the efficient discretization of the non-smooth Cahn-Hilliard Navier-Stokes system;(e) develop, implement and analyse reduced order models for multiphase flows with variable densities governed by diffuse interface models with nonsmooth energies.(f) develop, implement and analyze model-predictive feedback control strategies for multiphase flows governed by variable density diffuse interface models with nonsmooth energies.