  A method based on the Fast Fourier Transform is proposed to obtain the
dispersion relation of acoustic waves in heterogeneous periodic media with
arbitrary microstructures. The microstructure is explicitly considered using a
voxelized Representative Volume Element (RVE). The dispersion diagram is
obtained solving an eigenvalue problem for Bloch waves in Fourier space. To
this aim, two linear operators representing stiffness and mass are defined
through the use of differential operators in Fourier space. The smallest
eigenvalues are obtained using the implicitly restarted Lanczos and the
subspace iteration methods, and the required inverse of the stiffness operator
is done using the conjugate gradient with a preconditioner. The method is used
to study the propagation of acoustic waves in elastic polycrystals, showing the
strong effect of crystal anistropy and polycrystaline texture on the
propagation. It is shown that the method combines the simplicity of classical
Fourier series analysis with the versatility of Finite Elements to account for
complex geometries proving an efficient and general approach which allows the
use of large RVEs in 3D.
