Fast SVD-Based Linear Elastic Eigenvalue Problem Solver for Band Structures of 3D Phononic Crystals

In this article, a Fast Linear Elastic Eigenvalue Problem Solver (FLEEPS) is developed to calculate the band structures of three-dimensional (3D) isotropic phononic crystals (PnCs). In brief, FLEEPS solves in linear time complexity the smallest few eigenvalues and associated eigenvectors of the linear elastic eigenvalue problem originating from the finite difference discretization of the frequency-domain linear elastic wave equation. Notably, FLEEPS employs the weighted singular value decomposition based preconditioner to greatly improve the convergence rate of the conjugate gradient iteration, and uses the fast Fourier transform algorithm to accelerate this preconditioner times a vector, based on the structured decomposition of the dense unitary factor T of this preconditioner. Band structure calculations of several 3D isotropic PnCs are presented to showcase the capabilities of FLEEPS. The preliminary MATLAB implementation of FLEEPS is available at https://github.com/FAME-GPU/FLEEPS-MATLAB.