Kurzfassung

This project aims at developing, analyzing and numerically implementing combined phasefield andsharp interface techniques in pde-constrained shape optimization, where the application focuses onshape optimization in fluid flow. This will be achieved by utilizing on the one hand a phase-field ansatzwith non-smooth homogeneous free energy densities, which yields robust numerical approximationson grids which are well aligned to the shape, and, by a Darcy-type approach also allows to...This project aims at developing, analyzing and numerically implementing combined phasefield andsharp interface techniques in pde-constrained shape optimization, where the application focuses onshape optimization in fluid flow. This will be achieved by utilizing on the one hand a phase-field ansatzwith non-smooth homogeneous free energy densities, which yields robust numerical approximationson grids which are well aligned to the shape, and, by a Darcy-type approach also allows to simultaneously optimize the topology of the fluid domain. Then, on the other hand a novel, mesh-qualityconserving W 1,∞ −sharp interface approach is used to sharpen out the shapes obtained by the diffuseinterface method. With this approach we expect to overcome well known shortcomings in numericalshape optimization, where many methods suffer from grid degeneration, in particular when the initialguess for the optimization algorithm is very different from the final solution.

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