  An FFT framework which preserves a good numerical performance in the case of
domains with large regions of empty space is proposed and analyzed for its
application to lattice based materials. Two spectral solvers specially suited
to resolve problem containing phases with zero stiffness are considered (1) a
Galerkin approach combined with the MINRES linear solver and a discrete
differentiation rule and (2) a modification of a displacement FFT solver which
penalizes the indetermination of strains in the empty regions, leading to a
fully determined equation. The solvers are combined with several approaches to
smooth out the lattice surface, based on modifying the actual stiffness of the
voxels not fully embedded in the lattice or empty space. The accuracy of the
resulting approaches is assessed for an octet-lattice by comparison with FEM
solutions for different relative densities and discretization levels. It is
shown that the adapted Galerkin approach combined with a Voigt surface
smoothening was the best FFT framework considering accuracy, numerical
efficiency and h-convergence. With respect to numerical efficiency it was
observed that FFT becomes competitive compared to FEM for cells with relative
densities above 7%. Finally, to show the real potential of the approaches
presented, the FFT frameworks are used to simulate the behavior of a printed
lattice by using direct 3D tomographic data as input. The approaches proposed
include explicitly in the simulation the actual surface roughness and internal
porosity resulting from the fabrication process. The simulations allowed to
quantify the reduction of the lattice stiffness \rev{as well as to resolve the
stress localization of 50% near large pores.
