Convergence properties and flat bands in platonic crystal band structures using the multipole formulation

We present converged band diagrams for Bloch-Floquet bending waves in a thin elastic plate containing a square array of circular perforations, using the multipole formulation developed by Movchan et al. (2007) and applied in the situation where the perforations are no longer considered to be small in comparison with the lattice pitch. We give tables of converged frequencies and the number of multipoles necessary to achieve them, for a range of radii of the perforations for both clamped-edge and free-edge boundary conditions. We find that the larger filling fraction leads to extremely flat bands within the band structure; this can be explained by considering the energy of the vibrational modes. We derive the energy balance relation as well as convenient expressions for the group velocity of eigenmodes, which reveal the interplay between the Helmholtz and the modified Helmholtz components of the eigenfield. © 2010 Taylor & Francis