  The adjoint method allows efficient calculation of the gradient with respect
to the design variables of a topology optimization problem. This method is
almost exclusively used in combination with traditional
Finite-Element-Analysis, whereas Fourier-based solvers have recently shown
large efficiency gains for homogenization problems. In this paper, we derive
the discrete adjoint method for Fourier-based solvers that employ compatibility
projection. We demonstrate the method on the optimization of composite
materials and auxetic metamaterials, where void regions are modelled with zero
stiffness.
