Semi-Smooth Newton Methods on Shape Spaces


The main aim of this project is to set up an approach for investigating analytically and solving computationally shape optimization problems constrained by variational inequalities (VI) in shape spaces. Shape optimization problem constraints in the form of VIs are challenging, since classical constraint qualifications for deriving Lagrange multipliers generically fail. In this project, we 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. 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 in 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.


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