Stieltjes representation of the 3D Bruggeman effective medium and Padé approximation

• Published in: Applied mathematics and computation 2011-05, Vol.217 (17), p.7092-7107
• Main Authors: Zhang, Dali, Cherkaev, Elena, Lamoureux, Michael P.
• Format: Article
• Language: English
• Subjects: Approximation; Approximations and expansions; Bruggeman self-consistent effective medium approximation; Complex permittivity; Density; Effective complex permittivity; Exact sciences and technology; Homogenizing; Mathematical analysis; Mathematical models; Mathematics; Numerical analysis; Numerical analysis in abstract spaces; Numerical analysis. Scientific computation; Padé approximation; Partial differential equations; Real functions; Representations; Sciences and techniques of general use; Spectra; Spectral density function; Stieltjes representation; Three dimensional; Two-phase composite
• ISSN: 0096-3003; 1873-5649
• DOI: 10.1016/j.amc.2011.01.020

The paper deals with Bruggeman effective medium approximation (EMA) which is often used to model effective complex permittivity of a two-phase composite. We derive the Stieltjes integral representation of the 3D Bruggeman effective medium and use constrained Padé approximation method introduced in [39] to numerically reconstruct the spectral density function in this representation from the effective complex permittivity known in a range of frequencies. The problem of reconstruction of the Stieltjes integral representation arises in inverse homogenization problem where information about the spectral function recovered from the effective properties of the composite, is used to characterize its geometric structure. We present two different proofs of the Stieltjes analytical representation for the effective complex permittivity in the 3D Bruggeman effective medium model: one proof is based on direct calculation, the other one is the derivation of the representation using Stieltjes inversion formula. We show that the continuous spectral density in the integral representation for the Bruggeman EMA model can be efficiently approximated by a rational function. A rational approximation of the spectral density is obtained from the solution of a constrained minimization problem followed by the partial fractions decomposition. We show results of numerical rational approximation of Bruggeman continuous spectral density and use these results for estimation of fractions of components in a composite from simulated effective permittivity of the medium. The volume fractions of the constituents in the composite calculated from the recovered spectral function show good agreement between theoretical and predicted values.