Computing dispersion relations with modal analysis methods

Dispersion diagrams depicting dispersion relations are usually computed from a periodic structure single unit cell using either analytical methods, such as the plane wave expansion (PWE) method, or numerical methods, such as the wave finite element (WFE) method, by applying Bloch-Floquet periodic boundary conditions. However, the latter is still not available in most commercial FE codes. In this work, we propose a method to compute the dispersion diagram using standard modal analysis tools available in FE commercial codes currently used for structural analysis. We show that if a metastructure consisting of a few cells is "periodized" by "wrapping around" the degrees of freedom at its ends, its natural frequencies will be points on the dispersion curve pass bands. The wavenumbers corresponding to these natural frequencies can be obtained from the corresponding vibration modes using Prony's method. These natural frequencies are real for Hermitian structures and complex for non-Hermitian structures. Besides providing a useful tool for wave dispersion analysis using standard FE codes, the proposed method provides interesting insights into the dynamics of periodic structures. The proposed method is applied to simple passive and active periodic rod structures.