Optimal Control of Static Contact in Finite Strain Elasticity

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Static contact problems in the regime of finite strain elasticity are an important class of mechanical problems with nonlinear, non-smooth behaviour. Finite strains occur if the considered materials are soft, for example for rubber or biological soft tissue. Aim of his project is the development and analysis of algorithms for the optimal control of these problems.We construct and analyse a regularization and homotopy approach, where the non-penetration condition of the contact problem and the local injectivity condition of elasticity are relaxed. Properties of the regularized problems and convergence of the homotopy are studied. The resulting regularized optimal control problems will still be nonlinear and non-convex and will be solved by a function space oriented composite step method. This method will exploit problem structure of finite strain elasticity, in particular the variational structure of elasticity and the group structure of deformations. To finally approximate solutions of the original problem, we will develop and analyse an affine invariant path-following method that is tailored for this class of problems.