Wave directionality in periodic lattice structures

Abstract

We discuss the wave directionality for in-plane and out-of-plane waves in three-dimensional selfsimilar and non-self-similar square periodic lattices. The lattices studied are square lattice, square in the square hierarchical lattice, and hexagon in a square hierarchical lattice. Finite Element Analysis is used to calculate dispersion relations and directionality properties for these lattice structures following a 25/25 grid covering the complete Brillouin zone. The investigation is performed by combining the Floquet-Bloch theory of periodic structures with the commercial FE solver, Comsol Multiphysics. A method to distinguish between the in-plane and out-of-plane wave modes is explained in detail. Multiple surfaces are selected to show that the dispersion surfaces overlap in frequency and beaming direction; thus, requiring an efficient way to distinguish the inplane and out-of-plane modes. The effects of the hierarchy on the wave beaming in the chosen lattices are studied. The in-plane waves are observed to beam mostly along 0° and 90° directions in the square lattice, while a more equal spread is observed for the hierarchical lattices. Significant beaming is observed in the hexagon in the square lattice for the out-of-plane waves. The effect of hierarchy on the generation of bandgaps is studied. Compared with a non-hierarchical square lattice, the hierarchical lattice structures display significantly lowered group velocity magnitudes. The non-self-similar hierarchical lattice was found to provide bandgaps at lower frequencies and slower group velocities as compared to the self-similar hierarchical lattice.