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.

提出了一种基于快速傅立叶变换的方法来获得声波在具有任意微结构的非均质周期介质中的色散关系。使用体素化的代表性体积元素 (RVE) 明确考虑微观结构。色散图是通过求解傅立叶空间中布洛赫波的特征值问题获得的。为此，通过在傅立叶空间中使用微分算子定义了两个代表刚度和质量的线性算子。最小特征值是使用隐式重新启动的 Lanczos 和子空间迭代方法获得的，所需的刚度算子的逆是使用带有预处理器的共轭梯度完成的。该方法用于研究声波在弹性多晶中的传播，显示出晶体各向异性和多晶纹理对传播的强烈影响。结果表明，该方法将经典傅立叶级数分析的简单性与有限元的多功能性相结合，以解决复杂的几何形状，证明是一种有效且通用的方法，允许在 3D 中使用大型 RVE。