Numerical tools for efficient simulations of wave propagation in damped periodic structures

Periodic structures exhibit very specific properties in terms of wave propagation. In this paper, some numerical tools for dispersion analysis of periodic structures are presented, with a focus on the ability of the methods to deal with the dissipative behavior of the systems. The classical Floquet-Bloch approach is first presented, as a reference. This technique uses proper boundary conditions on the unit cell, it is quite simple to implement but dealing with damping is not easy for 2D or 3D cases. Secondly, the ?Shifted-Cell Operator? technique is described. It consists in a reformulation of the PDE problem by ?shifting? in terms of wave number the space derivatives appearing in the mechanical behavior operator inside the cell, while imposing continuity boundary conditions on the borders of the domain. Damping effects can be introduced in the system and a quadratic eigenvalue problem yields to the dispersion properties of the periodic structure. Both approaches are first analyzed in terms of mesh convergence, and then a focus is proposed on tools for the post-processing of dispersion diagrams in damped configurations. The ?group velocity? of each branch in the diagram is defined including damping effects, and it is shown that it constitutes a pertinent indicator for the branches tracking from one computational point to another.