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对允许形状的类别进行限制。在这个项目中，我们致力于研究带凸约束的形状优化问题的离散化和迭代求解。这种约束导致了意想不到的数学困难和现象。首先，需要适当的凸性离散概念来防止数值方法的锁定效应；其次，最优凸形通常是非光滑的，这需要仔细的收敛分析。相关的应用涉及由偏微分方程式定义的约束，范围从具有对称性破缺的最佳绝缘模型，以及具有低流动阻力或最大扭转刚度的物体的设计，到确定特殊形状，例如几何上具有恒定宽度的物体。该项目的目标是开发和分析可靠和有效地计算最佳凸形的数值方法，并在科学和几何应用中确定最佳形状。特别是发展了离散的凸性概念，形状导数的适当表示，网格正则性和保凸性的识别，以及离散凸集的紧性性质。

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.