PD Dr. Stephan Schmidt, Trier University

Title: Shape Hessians for a Wide Class of PDE Constrained Problems

Abstract:

The aim of this talk is to provide an overview of recent developments in Newton-type methods for shape optimization. We adopt a holistic perspective, starting from first-order boundary representations of shape derivatives and proceeding to the construction of full shape Hessians based on material derivatives. In particular, the volumetric form of the shape Hessian, expressed via material derivatives, is especially well suited for finite element implementations. This formulation enables the computation of Newton directions for PDE-constrained shape optimization in a fully coupled manner within mixed finite element spaces.

The talk follows a progressive structure: it begins with simulation problems, then moves to shape optimization without PDE constraints—such as those arising in the formation of capillary bridges—and finally addresses the PDE-constrained case, including both inverse problems and applications in computational fluid dynamics (CFD).