• Funding: DFG project in the DFG Priority Programme 1962 Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization
• Duration: 01.10.2019 - 30.09.2022
• Project management:Sören Bartels and Gerd Wachsmuth

Project description

The development and analysis of numerical methods for shape optimisation problems often requires the restriction to certain classes of admissible domains. In this project we will consider shape optimisation problems where the objective functional is minimised only over the set of convex domains. The focus is on the discretisation and iterative solution of these problems. Surprisingly, this convexity restriction leads to various mathematical phenomena and difficulties. On the one hand, suitable discrete formulations of convexity are required, otherwise locking may occur. On the other hand, the optimal convex regions are typically not smooth and therefore a careful convergence analysis is required. The project also includes applications with partial differential equations, as well as models for optimal isolation, the design of bodies with low flow resistance or maximum torsional stiffness and also the finding of special convex bodies, for example bodies of constant width. The aim of the project is to develop and analyse numerical methods for the reliable and efficient calculation of optimal convex bodies. Furthermore, we will identify optimal solids in other areas of science and geometry.