  Progresses in additive manufacturing technologies allow the realization of
finely graded microstructured materials with tunable mechanical properties.
This paves the way to a wealth of innovative applications, calling for the
combined design of the macroscopic mechanical piece and its underlying
microstructure. In this context, we investigate a topology optimization problem
for an elastic medium featuring a periodic microstructure. The optimization
problem is variationally formulated as a bilevel minimization of phase-field
type. By resorting to Gamma-convergence techniques, we characterize the
homogenized problem and investigate the corresponding sharp-interface limit.
First-order optimality conditions are derived, both at the homogenized
phase-field and at the sharp-interface level.
