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Beschreibung:
We develop and analyze a numerical two-scale optimization scheme for structural shape optimization. Whenever cost functionals, such as the elastic energy, are subject to minimization, while meeting a constraint on the available material, the emergence of microstructures can be observed. The costs are constantly reduced by fine-scale structures on a decreasing length scale δ. For an optimum, a generalized notion is required in the limiting case δ → 0, where materials are macroscopically characterized by an effective density θ ∈ [0, 1] and an effective behavior only. These are governed by the concrete realization of the underlying microstructure and there are provable optimal, yet nonunique, constructions. Being mostly of theoretical interest, they lack a physical counterpart, which could serve as guidance for manufacturing real materials. The goal of this work is to establish a flexible algorithmic method, that can assess the approximation quality of alternative, simple microstructure models. A numerical simulation and optimization can only be computationally feasible when distinct length scales are treated in a coupled but separated manner. This work builds upon the intuitive idea of pointwise probing the reaction of the underlying microstructure. A linearization of the macroscopic displacement serves as an affine-periodic boundary condition for a locally attached cell problem, whose solution is used as the microscopic correction. The intuitive approach is rigorously derived and linked to well-known concepts of homogenization, two-scale convergence and the heterogeneous multiscale method (HMM). This allows to adopt results on the approximation quality of the two-scale approach and its discretization. The deformation of the elastic bodies is modeled via the partial differential equations of linearized elasticity. Within the microscopic cells, the shapes of perforations, parametrized by few degrees of freedom, will be subject to optimization. For this purpose, the shape derivative will be computed analytically in ...