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容许形状的类别进行限制。在本计画中，我们主要研究具凸性约束的形状最佳化问题的离散化与迭代解法。这种限制导致了意想不到的数学困难和现象。首先，需要适当的离散概念的凸性，以防止锁定效应的数值方法，第二，最佳凸形状通常是非光滑的，这需要仔细的收敛性分析。相关的应用涉及由偏微分方程定义的约束，范围从对称性中断的最佳绝缘模型，以及具有低流动阻力或最大扭转刚度的机构的设计，到确定特殊形状，例如几何形状中恒定宽度的机构。该项目的目标是开发和分析数值方法，用于可靠和有效地计算最佳凸形，并确定科学和几何应用中的最佳形状。特别是发展的离散概念的凸性，适当的表示形式的衍生物，识别网格的正则性和保凸性的同构和紧凑性的离散凸集。

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.