Homogenization of the Maxwell Equations: Case II. Nonlinear Conductivity

The Maxwell equations with uniformly monotone nonlinear electric conductivity in a heterogeneous medium, which may be non-periodic, are homogenized by two-scale convergence. We introduce a new set of function spaces appropriate for the nonlinear Maxwell system. New compactness results, of two-scale type, are proved for these function spaces. We prove existence of a unique solution for the heterogeneous system as well as for the homogenized system. We also prove that the solutions of the heterogeneous system converge weakly to the solution of the homogenized system. Furthermore, we prove corrector results, important for numerical implementations.     

Homogenization of the stationary periodic Maxwell system in the case of constant permeability

The homogenization problem in the small period limit for the stationary periodic Maxwell system in ℝ³ is considered. It is assumed that the permittivity η^ε(x)=η(ε⁻x), ε > 0, is a rapidly oscillating positive matrix function and the permeability μ₀ is a constant positive matrix. For all four physical fields (the electric and magnetic field intensities, the electric displacement field, and the magnetic flux density), we obtain uniform approximations in the L ₂(ℝ³)-norm with order-sharp remainder estimates. __________ Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 41, No. 2, pp. 3–23, 2007 Original Russian Text Copyright ? by M. Sh. Birman and T. A. Suslina Dedicated to the memory of the great mathematician Mark Grigor’evich Krein Supported by RFBR grants no. 05-01-01076-a, 05-01-02944-YaF-a.     

Homogenization of parabolic equations an alternative approach and some corrector-type results

We extend and complete some quite recent results by Nguetseng [Ngu1] and Allaire [All3] concerning two-scale convergence. In particular, a compactness result for a certain class of parameterdependent functions is proved and applied to perform an alternative homogenization procedure for linear parabolic equations with coefficients oscillating in both their space and time variables. For different speeds of oscillation in the time variable, this results in three cases. Further, we prove some corrector-type results and benefit from some interpolation properties of Sobolev spaces to identify regularity assumptions strong enough for such results to hold.This research was supported by The Swedish Research Council for the Engineering Sciences (TFR), The Swedish National Board for Industrial and Technological Development (NUTEK), and The Country of Jämtland Research Foundation     

Homogenization of time harmonic Maxwell equations: the effect of interfacial currents

We consider the Maxwell equations for a composite material consisting of two phases and enjoying a periodical structure in the presence of a time‐harmonic current source. We perform the two‐scale homogenization taking into account both the interfacial layer thickness and the complex conductivity of the interfacial layer. We introduce a variational formulation of the differential system equipped with boundary and interfacial conditions. We show the unique solvability of the variational problem. Then, we analyze the low frequency case, high and very high frequency cases, with different strength of the interfacial currents. We find the macroscopic equations and determine the effective constant matrices such as the magnetic permeability, dielectric permittivity, and electric conductivity. The effective matrices depend strongly on the frequency of the current source; the dielectric permittivity and electric conductivity also depend on the strength of the interfacial currents. Copyright © 2016 John Wiley & Sons, Ltd.     

A Comparison of homogenization, Hashin-Shtrikman bounds and the Halpin-Tsai Equations

In this paper we study a unidirectional and elastic fiber composite. We use the homogenization method to obtain numerical results of the plane strain bulk modulus and the transverse shear modulus. The results are compared with the Hashin-Shtrikman bounds and are found to be close to the lower bounds in both cases. This indicates that the lower bounds might be used as a first approximation of the plane strain bulk modulus and the transverse shear modulus. We also point out the connection with the Hashin-Shtrikman bounds and the Halpin-Tsai equations. Optimal bounds on the fitting parameters in the Halpin-Tsai equations have been formulated.