Complexity of band structures: Finite element calculation of complex band structures for one and two dimensional phononic crystals

The calculation of complex band structures for surface gratings (one dimensional phononic crystals) and for two dimensional phononic crystals is discussed in the presented paper. We show an extension of the finite element method and the semi-analytical finite element method based on the dynamic condensation to achieve this aim. Complex band structures are particularly important for surface gratings, since the folded surface waves become evanescent beyond the sound cone. This behavior and the complex interconnections between the real propagating modes are discussed. The presence of an evanescent mode within the complete stop band is shown for surface gratings. This describes the spatial decay of the elastic waves inside the stop band. The presented method is extended for two dimensional phononic crystals with square lattices whereby the different orientations of the wave vector lead to polynomial eigenvalue problems including quadratic and quartic eigenvalue problems.