Homogenization of Maxwell's Equations in Lossy Biperiodic Metamaterials

A novel homogenization technique is proposed for computing the quasi-static effective parameters of the lossy bi-periodic artificial structure materials. This technique is based on the Floquet's Theorem which allows reducing the studied domain to the elementary cell with pseudo-periodic conditions on the lateral sides. The studied domain is extended by adding a vacuum layer in order to impose correctly the Silver-Müler absorbing boundary condition. This homogenization technique is a numerical method using the Finite Element Method and based on the evaluation of the macroscopic fields by averaging the local fields on the elementary cell. The effective constitutive parameters are obtained from the macroscopic fields and inductions. The numerical validation of this approach is presented in 2D and 3D by computing the effective conductivities for square cylinders and cubes suspended in a host isotropic medium. The obtained results are compared to our previous approach based on Unfolding Method and Finite Element Method (UFEM). On the one hand this technique can be applied to homogenize any bi-periodic metamaterial with elementary cells having inclusions of arbitrary geometry and on the other hand it takes into account the effect of the inclusions shape on the effective parameters.