An FFT-based approach for Bloch wave analysis: application to polycrystals

Abstract

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.

FFT based approaches in micromechanics: fundamentals, methods and applications

“…A second group of works studying problems which does not fulfill the strict RVE periodicity have been recently introduced in FFT frameworks Zhou and Bhattacharya (2021), Segurado and Lebensohn (2021). These studies rely on the use Bloch boundary conditions, that were introduced by Bloch (Bloch 1929) to describe the quantum wave functions of electrons in a crystal lattice.…”

“…The problem was exploited in Zhou and Bhattacharya (2021) in an static setting, to find bifurcations in the deformation of a nonlinear periodic medium. In Segurado and Lebensohn (2021), Bloch conditions were used to study acoustic wave propagation in elastic heterogeneous materials. In particular, an FFT based method was proposed to obtain the dispersion diagram of a periodic heterogeneous material.…”