Reconstruction of spectral function from effective permittivity of a composite material using rational function approximations

The paper deals with the problem of reconstruction of microstructural information from known effective complex permittivity of a composite material. A numerical method for recovering geometric information from measurements of frequency dependent effective complex permittivity is developed based on Stieltjes analytic representation of the effective permittivity tensor of a two-component mixture. We derive the Stieltjes representation for the effective permittivity of the medium using the eigenfunction expansion of the solution of a boundary-value problem. The spectral function in this representation contains all information about the microgeometry of the mixture. A discrete approximation of the spectral measure is derived from a rational (Padé) approximation followed by its partial fractions decomposition. The approach is based on the least squares minimization with regularization constraints provided by the spectral properties of the operator. The method is applied to calculation of volume fractions of the components in a mixture of two materials in a Bruggeman effective medium analytic model which has a continuous spectral density and to analytical models of two-phase composites with coated cylindrical and ellipsoidal inclusions. The numerical results of reconstruction of spectral measure for a mixture of silver and silicon dioxide and a composite of magnesium and magnesium fluoride show good agreement between theoretical and predicted values. The approach is applicable to geological materials, biocomposites, porous media, etc.