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

设计和分析形状优化问题的数值方法通常需要限制允许形状的类别。在这个项目中，我们的目的是研究具有凸性约束的形状优化问题的离散化和迭代解。这种约束导致了意想不到的数学困难和现象。首先，需要适当的离散凸性概念来防止数值方法的锁定效应，其次，最优凸形状通常是非光滑的，这需要仔细的收敛分析。相关的应用包括由偏微分方程定义的约束，从具有对称断裂的最佳绝缘模型，以及具有低流动阻力或最大扭转刚度的体的设计，到确定特殊形状（例如几何中的恒宽体）的范围。该项目的目标是开发和分析可靠和有效计算最佳凸形状的数值方法，并确定科学和几何应用中的最佳形状。特别的方面是发展离散凸性的概念，适当的形状导数的表示，网格规则和保持凸性的微分同态的识别和离散凸集的紧性。

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.