This talk was part of the Workshop on "New perspective on Shape and Topology Optimization" held at the ESI December 11 — 15, 2023.

I will present two applications of the phase-field approach to Topology Optimization. First, I will focus on optimal material distribution for a time-evolving elastoplastic medium subjected to kinematic hardening. I will show the existence of optimal configurations and compute their first-order optimality conditions. The latter is obtained by subsequent approximations, where a time discretization and a regularization of the flow rule are considered. In the second part of the talk, I will present a phase-field model for the optimal distribution of periodic microstructures in an elastic material. Besides showing existence of solutions and optimality conditions, I will discuss the convergence to a sharp-interface and homogenized problem, obtained in the limit as the phase-field parameters and the microstructures periodicity tend to 0.