Fluid dynamic shape optimization with phase fields and Lipschitz methods

• Applicants: Professor Dr. Michael Hinze; Dr. Christian Kahle
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 543959359

This project develops, analyzes and numerically implements combined pha-sefield and sharp interface techniques in pde-constrained shape optimizati-on, 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 simultaneously optimize the topology of the fluid domain. Then, on the other hand a novel, mesh-quality conser-vingW1,∞−sharp interface approach is used to sharpen out the shapes ob-tained 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. In addition to the development of numerical algorithms and an interface that intertwines both shape optimization approaches, the underlying shape optimization problems and algorithms are analyzed numerically and convergence is investigated.

• DFG Programme: Research Grants