  We propose a new discrete FFT-based method for computational homogenization
of micromechanics on a regular grid that is simple, fast and robust. The
discretization scheme is based on a tetrahedral stencil that displays three
crucial properties. First, and most importantly, the Fourier representation of
the associated Green operator is defined for any finite q-vector generated by
the periodic boundary conditions and that does not belong to the Reciprocal
Lattice of the discrete grids. As shown in the paper, this property guaranties
that, for any elastic contrats, even infinite, mechanical equilibrium is always
mathematically stable, i.e. free of any unphysical patterns, such as
oscillations, ringing or checkerboarding, a property which is not shared by the
original Moulinec-Suquet method \cite{moulinec1994fast,moulinec1998numerical}
nor by the rotated scheme proposed by Willot \cite{willot2015fourier}. Second,
the components of tensorial quantities are all defined on the same location,
which permits the use of any elastic anisotropy and any spatial variation of
the material fields. Third, convergence to equilibrium using the simplest
iterative scheme, the "basic scheme", is fast and the number of iterates
stabilizes at high contrasts, so that infinite contrast is obtained without
additional computational cost.
