Inverse homogenization with diagonal Padé approximants

The paper formulates inverse homogenization problem as a problem of recovery of Markov function using diagonal Pade approximants. Inverse homogenization or de-homogenization problem is a problem of deriving information about the micro- geometry of composite material from its effective properties. The approach is based on reconstruction of the spectral measure in the analytic Stieltjes representation of the effective tensor of two-component composite. This representation relates the n-point correlation functions of the microstructure to the moments of the spectral measure, which contains all information about the microgeometry. The problem of identification of the spectral function from effective measurements in an interval of frequency has a unique solution. The problem is formulated as an optimization problem which results in diagonal Padea p- proximation and exact formulas for the moments of the measure. The reconstructed spectral function can be used to evaluate geometric parameters of the structure and to compute other effective parameters of the same composite; this gives solution to the problem of coupling of different effective properties of a two-component composite material with random microstructure. 1 Uniqueness of reconstruction of Stieltjes representation of the effective property The structural information is contained in the spectral measure μ in the Stieltjes representation of the effective complex permittivity � ∗ developed in (1, 8, 6) in the course of computing bounds for the effective permittivity � ∗ of an arbitrary two- component mixture. The spectral measure or its moments can be reconstructed from effective measurements and used to characterize parameters of the microstructure (7, 5, 10) or to estimate other effective properties of the same material (3, 4). It is shown in (3) that the measure μ can be uniquely recovered from the effective complex permittivity given on an arc C⊂ C in the complex plane. The problem is reduced to inverse potential problem and solved using regularized method in (3, 4). A concept of S − equivalency of structures is introduced (3) for micro-geometriesof composites which cannot be distinguished by effective measurements; it is shown that the S−equivalent structures correspond to the same spectral functions. Numerical algorithm based on (p, q)-Pade approximations is constructed in (10). (2) develops application to viscoelastic torsion problem and evaluation of volume fractions of materials in the composite from torsion experiment. We considera stationaryrandomfine-scalemixtureof two materialswith properties � 1 and � 2, and introducea characteristic function χ of the region O1 occupied by the first material for a realization η ∈ Ω ,w hereΩ is the set of all realizations of the random medium, χ(x, η )=1 if x ∈O 1 and χ(x, η )=0 otherwise. The complex permittivity of the medium is modeled by a (spatially) stationary random field � (x, η), x ∈ R d and η ∈ Ω, � (x, η )= � 1χ(x, η )+ � 2(1 − χ(x, η)). The stationary random fields E(x, η) and D(x, η), are related by D(x, η )= � (x, η)E(x, η) and satisfy the equations ∇· D =0 , ∇× E = 0, � E(x, η)� = ek with ek being a unit vector in the k th direction, for some k =1 ,...,d ,w here �·� is ensemble average over Ω or spatial average over all of R d . The effective tensor � ∗ is defined as coefficient of proportionality between the averaged fields: � D� = � ∗ � E� . The inverse homogenization problem (3) is a problem of characterization of function χ from the effective property � ∗ :