Optimizing Variational Inequalities on Shape Manifolds

优化形状流形上的变分不等式

• 批准号: 314066674
• 负责人: Professor Dr. Volker Schulz
• 金额: --
• 依托单位: Arbeitsgruppe Optimierung bei partiellen Differentialgleichungen
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/314066674?language=en
• 关键词: Optimizing; Variational; Inequalities; Shape; Manifolds

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.

形状优化在许多应用领域中与变分不等式类似具有重要意义。然而，形状优化问题的约束条件组成的变分不等式尚未考虑在现有的文献。该建议旨在一种新的方法，形状优化问题的形状流形和所产生的框架，从无限维黎曼几何，这是最近开发的申请人。这种方法使形状优化与矢量束中的最优控制问题的理论联系成为可能，这将是本项目中分析和数值研究的指导原则。因此，该项目的目标是在变分不等式的形状优化领域的调查，关于适当的黎曼形状流形配方，解的存在性和适定性，形状向量束的半光滑牛顿方法，网格独立的算法方法，不确定性的鲁棒处理和（热）力学领域应用问题的解决方案。除此之外，形状流形方法及其新颖的形状度量，增强了离散化和算法的鲁棒性，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。