Dali Zhang

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 (Pade) 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.

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

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 Pade 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.

Padé approximations for identification of air bubble volume from temperature- or frequency-dependent permittivity of a two-component mixture

The article presents a numerical method developed for identification of information about structural parameters of a two-component mixture from effective complex permittivity measurements. The identification is based on the reconstruction of the spectral function in the analytic Stieltjes representation of the effective permittivity using Padé approximation. The spectral function contains all information about the microgeometry of the mixture, it is used to calculate volume fractions of the components in the mixture. Padé approximation is derived from a constrained minimization problem. Numerical results of recovering volume fraction of air in mixtures of air prolate and oblate spheroidal inclusions in water and in ethanol show good agreement of theoretical and predicted values. The proposed method can be used for estimating volume fractions and other structural parameters using the effective complex permittivity of two-component composite materials.

Coupling of the effective properties of a random mixture through the reconstructed spectral representation

The spectral representation of the effective complex permittivity of a two-component composite medium is used to develop an approach to coupling of various effective properties of a random mixture. The spectral function contains all information about the microstructure, hence providing a coupling link between the various effective properties of the same composite material. It is demonstrated that the representation can be reconstructed from measurements of one effective property and used then to evaluate other properties of the same material. The reconstruction problem is very ill-posed and requires regularization. Several numerical examples of reconstruction of the spectral function from broadband measurements of the effective complex permittivity and the measurements of the effective thermal conductivity are shown. The approach can be used for indirect estimation of the thermal conductivity (or other properties) of the medium from broadband measurements of the effective complex permittivity.

Padé approximations in inverse homogenization and numerical simulation of electromagnetic fields in composites

The paper considers rational Padé approximation of the spectral function of composites with fine microstructure and discusses its use in characterization of the microgeometry of composite materials and in numerical simulation of time-domain electromagnetic fields in composites. It is assumed that the scale of the structure is much smaller than the smallest wavelength of the applied field. We use Stieltjes representation of the effective complex permittivity of the composite and derive its Padé approximation. The spectral function in this representation contains all information about the microgeometry of the mixture. Having reconstructed the Padé approximation, we recover information about the composite structure. The resulting time-domain equations governing the electromagnetic fields are of convolution type. We use rational Padé approximation to derive equations for internal variables for time-domain simulation. We show that electromagnetic fields computed using such internal variables, correspond to the fields in S-equivalent composite structures.