  Computational micromechanics and homogenization require the solution of the
mechanical equilibrium of a periodic cell that comprises a (generally complex)
microstructure. Techniques that apply the Fast Fourier Transform have attracted
much attention as they outperform other methods in terms of speed and memory
footprint. Moreover, the Fast Fourier Transform is a natural companion of
pixel-based digital images which often serve as input. In its original form,
one of the biggest challenges for the method is the treatment of
(geometrically) non-linear problems, partially due to the need for a uniform
linear reference problem. In a geometrically linear setting, the problem has
recently been treated in a variational form resulting in an unconditionally
stable scheme that combines Newton iterations with an iterative linear solver,
and therefore exhibits robust and quadratic convergence behavior. Through this
approach, well-known key ingredients were recovered in terms of discretization,
numerical quadrature, consistent linearization of the material model, and the
iterative solution of the resulting linear system. As a result, the extension
to finite strains, using arbitrary constitutive models, is at hand. Because of
the application of the Fast Fourier Transform, the implementation is
substantially easier than that of other (Finite Element) methods. Both claims
are demonstrated in this paper and substantiated with a simple code in Python
of just 59 lines (without comments). The aim is to render the method
transparent and accessible, whereby researchers that are new to this method
should be able to implement it efficiently. The potential of this method is
demonstrated using two examples, each with a different material model.
