Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

Abstract

In this paper we employ the periodic unfolding method for simulating the electromagnetic field in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the effective parameters for a Debye dielectric medium in the cases of circular and square microstructures in two dimensions. We assume that the composite materials are quasi-static in nature, i.e., the wavelength of the electromagnetic field is much larger than the relevant dimensions of the microstructure. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) 10. SPONSOR/MONITOR'S ACRONYM(S) SPONSOR/MONITOR'S REPORT NUMBER(S) DISTRIBUTION/AVAILABILITY STATEMENTApproved for public release; distribution unlimited SUPPLEMENTARY NOTESThe original document contains color images. ABSTRACTIn this paper we employ the periodic unfolding method for simulating the electromagnetic eld in a composite material exhibiting heterogeneous microstructures which are described by spatially periodic parameters. We consider cell problems to calculate the e ective parameters for a Debye dielectric medium in the cases of circular and square microstructures in two dimensions. We assume that the composite materials are quasi-static in nature, i.e., the wavelength of the electromagnetic eld is much larger than the relevant dimensions of the microstructure.

Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

“…The details of the method highlighted here may be found in [2]. A similar approach known as the recursive convolution (RC) method has been used to compute the discrete convolution terms that appear in Maxwell's equations [16,19].…”

“…Results of numerical experiments involving square microstructures can be found in [2]. In all of these computations we have used a conjugate gradient method to solve the resulting linear systems that arise after discretizing our model in space and time.…”

A note on two scale compactness for $p = 1$

“…The latter proof is the simplest and most intuitive, due to the fact that the periodic unfolding method reduces two-scale convergence in L p (Ω) to standard weak L p convergence in Ω×Y (where Y is the period of the oscillations) of the unfolded functions, thus allowing us to replace rapidly oscillating test functions with non-oscillatory test functions. This method has been used in many contexts, including electromagnetism, homogenization in a domain with oscillating boundaries, and thin junctions in linear elasticity [3,4,8,11,12,14,15,17].…”

Frequency Dispersion of Dielectric Permittivity and Electric Conductivity of Rocks via Two-Scale Homogenization of the Maxwell Equations

“…It was proved in many studies that the macroscopic Maxwell equations can be strongly different from the microscopic ones: instantaneous material laws turn into constitutive laws with memory [13,[17][18][19][20][21][22][23][24][25]. More general case has been considered in [26], with polarization of composite ingredients being not instantaneous but obeying the Debye or Lorenz polarization laws with relaxation. Complexity of the macroscopic constitutive laws is discussed in [27,28].…”

“…Numerical evaluation of effective permittivity and effective conductivity on the basis of the two-scale homogenization theory was performed in many publications including [31] for the time-harmonic Maxwell equations (see [26]). The main result of [31] is a successful testing of the numerical algorithm at a fixed low frequency both by comparison of the calculated effective conductivity with those predicted by the Maxwell-Garnett approach [1] and by comparison with an exact electric field related to a specific boundary value problem for the Maxwell equations for the case when inclusions are less conductive than the host medium.…”