A three-dimensional time-domain finite-element formulation for periodic structures

A formulation is presented for a three-dimensional time-domain finite-element method that can be used to analyze the scattering of a plane wave obliquely incident on a (doubly) infinite periodic structure using one unit cell. A broadband frequency response can be obtained in a single execution. The specifics of the method are shown for scattering problems, but it should be straightforward to extend it to radiation problems. The method solves for a transformed field variable (instead of solving directly for the electric field) in order to easily enable periodic boundary conditions in the time domain. The accuracy and stability of the method is demonstrated by a series of examples where the new formulation is compared with reference solutions. Accurate results are obtained when the excitation (frequency range) and the geometry are such that no higher order propagating Floquet modes are present. The accuracy is degraded in the presence of higher order propagating modes due to the rather simple absorbing boundary condition that is used with the present formulation. The method is found to be stable even for angles of incidence close to grazing.