  This paper applies the Recursive Projection Method (RPM) to the problem of
finding the effective mechanical response of a periodic heterogeneous solid.
Previous works apply the Fast Fourier Transform (FFT) in combination with
various fixed-point methods to solve the problem on the periodic unit cell.
These have proven extremely powerful in a range of problems ranging from
image-based modeling to dislocation plasticity. However, the fixed-point
iterations can converge very slowly, or not at all, if the elastic properties
have high contrast, such as in the case of voids. The paper examines the
reasons for slow, or lack of convergence, in terms of a variational
perspective. In particular, when the material contains regions with zero or
very small stiffness, there is lack of uniqueness, and the energy landscape has
flat or shallow directions. Therefore, in this work, the fixed-point iteration
is replaced by the RPM iteration. The RPM uses the fixed-point iteration to
adaptively identify the subspace on which fixed-point iterations are unstable,
and performs Newton iterations only on the unstable subspace, while fixed-point
iterations are performed on the complementary stable subspace. This combination
of efficient fixed-point iterations where possible, and expensive but
well-convergent Newton iterations where required, is shown to lead to robust
and efficient convergence of the method. In particular, RPM-FFT converges well
for a wide range of choices of the reference medium, while usual fixed-point
iterations are usually sensitive to this choice.
