Project Details
Projekt
Semi-Smooth Newton Methods on Shape Spaces
Subject Area
Mathematics
Term
from 2019 to 2023
Project identifier
Deutsche Forschungsgemeinschaft (DFG) - Project number 423771068
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.
DFG Programme
Priority Programmes
