  Most of the FFT methods available for homogenization of the mechanical
response use the strain/deformation gradient as unknown, imposing their
compatibility using Green's functions or projection operators. This implies the
allocation of redundant information and, when the method is based in solving a
linear equation, the rank-deficiency of the resulting system. In this work we
propose a fast, robust and memory-efficient FFT homogenization framework in
which the displacement field on the Fourier space is the unknown: the
displacement based FFT (DBFFT) algorithm. The framework allows any general
non-linear constitutive behavior for the phases and direct strain, stress and
mixed control of the macroscopic load. In the linear case, the method results
in a linear system defined in terms of linear operators in the Fourier space
and that does not require a reference medium. The system has an associated full
rank Hermitian matrix and can be solved using iterative Krylov solvers and
allows the use of preconditioners. A preconditioner is proposed to improve the
efficiency of the system resolution. Finally, some numerical examples including
elastic, hyperelastic and viscoplastic materials are solved to check the
accuracy and efficiency of the method. The computational cost reduction respect
the Galerkin-FFT was around 30%.
