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Topology optimization is a mathematical tool for the computational design of structures,
particularly valuable in the early stages of the design process. Gradient-based
optimization methods, combined with finite element analysis, form the foundation of
an iterative procedure where the design is continuously updated and evaluated. To expand
the applicability of topology optimization, this thesis explores a range of design
problems involving path-dependent and non-linear material behavior, including hyperelasticity,
transient thermo-elasticity, and elasto-plasticity.

The thesis begins with an introduction to topology optimization, followed by formulations,
models, and numerical procedures for handling non-linear and path-dependent
responses. Thermodynamical principles are presented and linked to constitutive modeling,
followed by the balance laws and non-linear solution strategies employed in
the optimization frameworks. Topology optimization procedures for both single- and
multi-material structures are developed, along with material interpolation schemes
tailored to each class of materials. Finally, path-dependent sensitivity analysis is derived
and discussed in the context of transient and elasto-plastic problems.

Five appended papers provide detailed studies of specific applications. Paper A introduces
a framework for stress-constrained topology optimization of finite strain hyperelastic
structures under non-proportional loading, showing that different load trajectories
leading to the same terminal state can result in distinct optimal designs. Paper
B investigates non-linear transient thermo-elasticity in the design of multi-material
thermal actuators, where the material distribution is influenced by the operating time.
Papers C–E address the design of 3D elasto-plastic structures for tailored mechanical response,
plastic work maximization, self-locking behavior, and stiffness optimization.