In this paper, a fully numerical homogenization technique is presented for the retrieval of the effective constitutive parameters of a periodic composite medium. Based on the eigensolution returned by the utilized field-flux FEM formulation, a proper field averaging process of the field components is proposed over specific paths of the periodic unit cell of an arbitrary structure. Subsequently, the constitutive relations for every eigenmode supported by the medium are derived, forming a linear system of equations. The solution of this system returns a set of fifteen effective constitutive parameters, including the magnetoelectric coefficients, which account for possible cross-polarization effects, causing bianisotropy.
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The option of averaging the normally continuous field components by integrating on the volume of a unit cell may appear to be in contrast with the other proposed options, where the integrating paths lay on the boundaries of the unit cell. Comparing this situation to consideration of atomic scales, under the assumption that the wavelength is much larger than the unit cell’s dimensions, there is practically no information relative to the field variation very close to the atoms or molecules. However, in the case of a computational solution of a metamaterial composite medium, provided that the information of the field distributions is available in detail (the field distribution is known inside the unit cell, and, especially, on and around the scatterers), we make use of this intrinsic detail by integrating the field component of interest on the whole volume of the meta-atom (unit cell). The results presented in this section exhibit values very close relative to the integration on the alternative paths, indicating that there is no violation of any physical law.
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