Semi-Smooth Newton Methods on Shape Spaces

本建议的目的是建立一种新的方法来分析和解决形状空间中受变分不等式（VI）约束的计算形状优化问题。与没有明确依赖于给定域的经典VI相比，VI约束形状优化问题特别具有挑战性，主要有两个原因：首先，需要在固有的非线性、非凸和无限维形状空间中操作。其次，对于任意的形状泛函，不能指望依赖于vi的解的形状导数的存在或以线性映射的形式得到形状导数，这意味着不能引入伴随状态，因此，不使用任何正则化技术就不能直接解决问题。在SPP项目P20“优化形状流形上的变分不等式”中，提供了受障碍问题约束的形状优化问题的体积形状导数的理论结果，并制定了一种有效的优化算法。本提案旨在将在另一个项目中建立的方法扩展到一般的，在经典意义上的非形状可微VI约束问题。该建议的主要思想是考虑牛顿形状导数而不是经典形状导数，以便制定一阶必要最优性条件。建立牛顿形导数格式是本项目分析和数值研究的指导原则。更准确地说，所得到的方案能够分析和计算处理由经典意义上的非形状可微的vi约束的形状优化问题，因此这些问题可以在没有任何正则化技术的情况下处理和解决，通常只导致近似的形状解。此外，该方案还为形状空间上的半光滑牛顿方法等高阶优化方法的形成打开了大门。除了建立牛顿形状导数格式外，本项目的进一步目标是研究VIs形状优化领域的适当形状空间公式，解决方案的存在性和适定性，包括形状空间中的平稳概念，形状空间上的半光滑牛顿方法，网格无关算法方法，不确定性的鲁棒处理以及应用问题的解决方法，例如来自（热）力学领域。此外，形状空间方法及其新颖的牛顿形状导数格式为与其他项目合作解决基于形状的问题公式提供了基础。

The aim of this proposal is to set up a novel approach for investigating analytically and solving computationally shape optimization problems constrained by variational inequalities (VI) in shape spaces. In contrast to classical VIs, where no explicit dependence on the domain is given, VI constrained shape optimization problems are in particular highly challenging because of two main reasons: Firstly, one needs to operate in inherently non-linear, non-convex and infinite-dimensional shape spaces. Secondly, one cannot expect for an arbitrary shape functional depending on solutions to VIs the existence of the shape derivative or to obtain the shape derivative as a linear mapping, which imply that the adjoint state cannot be introduced and, thus, the problem cannot be solved directly without any regularization techniques. Within project P20 'Optimizing variational inequalities on shape manifolds' in an SPP, theoretical results on volumetric shape derivatives for shape optimization problems constrained by the obstacle problem are provided and an efficient optimization algorithm is formulated. This proposal aims at extending the approaches established within another project to general, in the classical sense non-shape differentiable VI constrained problems. The main idea of this proposal is to consider Newton-shape derivatives instead of classical shape derivatives in order to formulate first-order necessary optimality conditions. Setting up a Newton-shape derivative scheme is the guiding principle for the analytical and numerical investigations within this project. More precisely, the resulting scheme enables the analytical and computational treatment of shape optimization problems constrained by VIs which are non-shape differentiable in the classical sense such that these can handled and solved without any regularization techniques leading often only to approximated shape solutions. Moreover, such a scheme opens the door for formulating higher order optimization methods like semi-smooth Newton methods on shapes spaces. Besides setting up a Newton shape derivative scheme, further goals of this project are investigations in the area of shape optimization for VIs regarding appropriate shape space formulations, existence and well-posedness of solutions including stationary concepts in shape spaces, semi-smooth Newton methods on shape spaces, mesh independent algorithmic approaches, robust treatment of uncertainties and solution approaches to application problems like, e.g. from the field of (thermo-)mechanics. Besides that, the shape space approach together with its novel Newton shape derivative scheme provides a basis for cooperation with other projects addressing shape based problem formulations.