A non-smooth phase-field approach to shape optimization with instationary fluid flow

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This project aims at introducing, analyzing and numerically realizing well-posed formulations of shape optimization problems with instationary fluid flow. This will be achieved by utilizing a phase-field ansatz and associated non-smooth homogeneous free energy densities, which yield robust numerical approximations and, for the sake of efficient computations, admit several levels of model reduction ranging from adaptive meshing to proper orthogonal decomposition. Moreover, a Darcy-type approach allows to simultaneously optimize the topology of the fluid domain. We consider the following research objectives, which in all items will be carried in a strongly intertwined effort between the Berlin and the Hamburg group:(a) Investigation of the continuity and differentiability properties of our optimization framework;(b) Development and analysis of fully integrated adaptive numerical methods for adaptive mesh refinement for the efficient discretization of phase-field based shape and topology optimization with instationary fluid flow;(c) Analysis of the optimization problem, in particular of the a posteriori error estimator for the phase field model in the zero-limit of the phase field paramete. (d) Derivation of dual-weighted residual based error estimates for adaptive mesh refinement for the efficient discretization of phase-field based shape and topology optimization with instationary fluid flow;(e) Development, implementation and analysis of algorithmic solution frameworks for phase-field based shape and topology optimization with instationary fluid flow;(f) Development, implementation and analysis of reduced order models for phase-field based shape and topology optimization with instationary fluid flow.Here the Berlin group takes the lead in (a)-(b), and the Hamburg group in (e)-(f). The package in (c)-(d) will strongly benefit from the previous collaborations of the two groups within two other SPPs.