Fluid dynamic shape optimization with phase fields and Lipschitz methods

This project aims at developing, analyzing and numerically implementing combined phasefield and

sharp interface techniques in pde-constrained shape optimization, where the application focuses on

shape optimization in fluid flow. This will be achieved by utilizing on the one hand a phase-field ansatz

with non-smooth homogeneous free energy densities, which yields robust numerical approximations

on grids which are well aligned to the shape, and, by a Darcy-type approach also allows to simul-

taneously optimize the topology of the fluid domain. Then, on the other hand a novel, mesh-quality

conserving W ^1,∞ −sharp interface approach is used to sharpen out the shapes obtained by the diffuse

interface method. With this approach we expect to overcome well known shortcomings in numerical

shape optimization, where many methods suffer from grid degeneration, in particular when the initial

guess for the optimization algorithm is very different from the final solution.