Optimizing Variational Inequalities on Shape Manifolds

形状优化在许多应用领域具有重要意义，类似于变分不等式。然而，在现有的文献中，对由变分不等式组成的约束条件下的形状优化问题还没有很多的研究。这一建议旨在根据形状流形和由无限维黎曼几何得到的框架来解决形状优化问题的新方法，该框架是申请人最近开发的。这种方法使形状优化与向量丛中的最优控制问题在理论上联系起来，这将是本项目分析和数值研究的指导原则。因此，本项目的目标是研究关于适当的黎曼形状流形的变分不等式的形状优化，解的存在性和适定性，形状向量丛上的半光滑牛顿方法，网格无关的算法方法，不确定性的稳健处理和(热)力学领域的应用问题的求解方法。此外，形状流形方法及其新的形状度量增强了离散化和算法的健壮性，为与其他解决基于形状的问题公式的项目合作提供了基础。

Shape optimization is of importance in many fields of applications similarly to variational inequalities. However, shape optimization problems with constraints consisting of variational inequalities have not yet been considered much in the existing literature. This proposal aims at a novel approach to shape optimization problems in terms of shape manifolds and the resulting framework from infinite dimensional Riemannian geometry, which has been developed recently by the applicant. This approach enables a theoretical connection of shape optimization with optimal control problems in vector bundles, which will be the guiding principle for the analytical and numerical investigations within this project. Thus, the goals of this project are investigations in the area of shape optimization for variational inequalities regarding appropriate Riemannian shape manifold formulations, existence and well-posedness of solutions, semi-smoth Newton methods on shape vector bundles, mesh independent algorithmic approaches, robust treatment of uncertanties and solution approaches to application problems from the field of (thermo-)mechanics. Besides that, the shape manifold approach together with its novel shape metrics enhancing discretization and algorithmic robustness provides a basis for cooperation with other projects addressing shape based problem formulations.