Topology optimization of non-linear, dissipative structures and materials (English)

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Topology optimization is a mathematical tool for finding optimal distributions of material phases within a design domain. It is commonly used in early design stages to generate conceptual structural layouts. Ultimately, designs generated by topology optimization consist of distinct material phases with enhanced performance. Most research on topology optimization methods focuses on linear elastic mechanical problems with the objective to maximize stiffness. Whereas linear problems are well understood, methods for solving geometrically and/or material non-linear, transient, path-dependent problems are less so. This thesis consists of a general introductory part and six appended papers in which topology optimization frameworks that account for non-linear response are presented. The introductory part gives an overview of topology optimization methods, motivates optimization for non-linear problems and details the specific solution methods for the non-linear, transient and path-dependent frameworks on which the included papers are based on.All papers but Paper E present non-linear optimization frameworks that employ hyperelastic material models and are based on finite strain theory. In Papers A and B, topology optimization methods for multiple material phases and for tangent stiffness maximization, respectively, are established. It is shown that an arbitrary number of phases can be included in the optimization and that the optimized distribution of the phases depends on the load magnitude. It is also shown that the definition of the stiffness measure used in topology optimization of non-linear elastic structures is of great importance. In Papers C, D and F, inelastic constitutive models are included in the optimization frameworks. Rate-dependent, i.e. viscous, inertial and finite strain effects are combined with topology optimization to generate maximum energy absorbing structures. Both macroscopic structures and microstructural designs are presented, wherein plastic work is optimized. It is shown that micro and macro response can be tailored for a specific load magnitude, loading rate and load path. The developed scheme in Paper F enables optimization of structures for maximum stiffness while constraining the maximum specific plastic work. Since significant plasticity occurs at design domain boundaries, a regularization technique for explicit control of boundary effects in topology optimization problems, presented in Paper E, is utilized. This optimization approach shows that by constraining the specific plastic work, stiff designs with smooth stress distributions are generated.