Complex band structures of two dimensional phononic crystals: Analysis by the finite element method

Abstract

In this work, the calculation of complex band structures of two-dimensional bulk phononic crystals (2DPCs) is discussed by the finite element method. A modification of the classical ω(k) approach—calculating the unknown frequencies for a real wave number—is modified to a k(ω) solution, which allows the evaluation of complex wave numbers for real frequencies. The dispersion relation of a 2DPC in a square lattice is presented and it is shown that the problem reduces to a polynomial eigenvalue problem with quadratic and quartic eigenvalue problems in the Γ−X−M−Γ directions. The developed method is applied for solid-vacuum PCs made of isotropic materials consisting cylindrical holes. Complex dispersion diagrams are calculated with various Poisson's ratios and the mode shapes of the propagating and the evanescent modes are presented. The significance of the complex bands is discussed.