Optimizing Variational Inequalities on Shape Manifolds

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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.