Open Access

Tanja Lochner

• In this thesis, we first consider the periodic homogenization of the linearized elasticity equation with slip-displacement conditions of a two-scale composite of two solids. The jump conditions are motivated by fibre-reinforced materials, which are often modelled by perfect bonding between fibre and matrix, which may not be true in practice. We are interested in the impact of the interface jumps in displacement on the (upscaled) partial differential equations and distinguish three different cases. While one material is connected, the other one is either disconnected, globally connected or unidirectionally connected. In all three cases, we show the existence and uniqueness of the solution and prove some general compactness and convergence results, whereby we apply the method of periodic unfolding. In the end, we derive the homogenized problem. In the second part of the thesis, we combine the methods of homogenization and parameter identification. We consider the homogenized linearIn this thesis, we first consider the periodic homogenization of the linearized elasticity equation with slip-displacement conditions of a two-scale composite of two solids. The jump conditions are motivated by fibre-reinforced materials, which are often modelled by perfect bonding between fibre and matrix, which may not be true in practice. We are interested in the impact of the interface jumps in displacement on the (upscaled) partial differential equations and distinguish three different cases. While one material is connected, the other one is either disconnected, globally connected or unidirectionally connected. In all three cases, we show the existence and uniqueness of the solution and prove some general compactness and convergence results, whereby we apply the method of periodic unfolding. In the end, we derive the homogenized problem. In the second part of the thesis, we combine the methods of homogenization and parameter identification. We consider the homogenized linear elasticity problem, whereby we assume perfect bonding on the interface of the two-scale composite, and want to deduce from measurements of the deformation on the boundary of a body the structure of the periodicity cell, which can be parametrized by finite real vector. After proving some general properties of the homogenized tensor, which describes the stiffness of the homogenized material, we show that there exists at least one solution of the minimization problem, which minimizes the L^2-difference of the measured deformation and the computed deformation for some given structure of the periodicity cell. Using shape optimization, in particular the Lagrangian method of Céa, we derive the Gâteaux derivative of the homogenized tensor, which we need to compute the Gâteaux derivative of the target functional. Finally, we use these results to apply gradient-based algorithms for some numerical simulations in the steady-state and time-dependent case.…