Approximation of non-smooth optimal convex shapes with applications in optimal insulation and minimal resistance

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Devising and analyzing numerical methods for shape optimization problems typically requires the restriction of the class of admissible shapes. In this project we aim at investigating the discretization and iterative solution of shape optimization problems with convexity constraint. This constraint leads to unexpected mathematical difficulties and phenomena. First, appropriate discrete notions of convexity are required to prevent locking effects of numerical methods, and second, optimal convex shapes are typically nonsmooth which necessitates a careful convergence analysis. Related applications involve constraints defined by partial differential equations and range from models for optimal insulation with breaks of symmetry, and the design of bodies with low flow resistance or maximal torsion stiffness, to the determination of special shapes such as bodies of constant width in geometry. The goal of the project is to develop and analyze numerical methods for the reliable and efficient computation of optimal convex shapes and to identify optimal shapes in scientific and geometric applications. Particular aspects are the development of discrete notions of convexity, appropriate representations of shape derivatives, identification of mesh regularity and convexity preserving diffeomorphisms and compactness properties of discrete convex sets.