Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

Banks, Harvey Thomas; Bokil, Vrushali A.; Cioranescu, Doina; Gibson, Nathan L.; Griso, Georges; Miara, Bernadette

doi:10.21236/ada440029

Cited by 4 publications

(2 citation statements)

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“…For general time-dependent solutions, it was proved in many studies by the two-scale homogenization approach that the macroscopic Maxwell equations can be strongly different from the microscopic ones: instantaneous material laws turn into constitutive laws with memory, see [9,[11][12][13][14][15][16][17]. A more general case has been considered in [18] with polarization of composite ingredients being not instantaneous but obeying the Debye or Lorenz polarization laws with relaxation. The complexity of the macroscopic constitutive laws is discussed in [19].…”

Section: Introductionmentioning

confidence: 99%

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

Amirat

Shelukhin

2016

Math Methods in App Sciences

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Communicated by H. AmmariWe 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. the variables with the hat symbol .O/ denote reference values and the variables with the prime superscript denote dimensionless values, respectively. Let O " and O be the reference values of " and : " D O "" 0 , D O 0 . Remind that both " and are dimensionless parameters in the Gaussian system of units. We choose O J D O O E then, in the dimensionless variables, equation (1) becomes curl 0

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Section: Introductionmentioning

confidence: 99%

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

Amirat

Shelukhin

2016

Math Methods in App Sciences

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“…Homogenization of two-scale Maxwell type equation is treated in Bensoussan et al [15], Sanchez-Palencia [88] and Jikov et al [62] using two-scale asymptotic expansion and compensated compactness theory. The two-scale convergence method (see [79,5]) is employed for Maxwell equation in Wellander and Kristensson [98] and Amirat and Shelukhin [8]; see also the references [18,14] where the related unfolding method is employed. In the contexts of metamaterials, Kohn and Shipman [63] and Bouchitté and Schweizer [24] employed two-scale convergence to study effective properties of Maxwell equations in media with high contrast coefficients.…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

Multiscale maxwell equations : homogenization and high dimensional finite element method

Chu¹

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Solving multiscale partial differential equations is exceedingly complex. Traditional methods have to use a mesh size of at most the order of the smallest scale to produce accurate approximations. Due to the importance of these problems in practical applications, there have been intensive efforts to study multiscale partial differential equations theoretically, and to develop feasible numerical methods to solve them. Although there has been a huge literature on the subject, there are still many open challenging problems that need theoretical study and effective solution methods.The thesis studies multiscale Maxwell equations. Comparing to other types of equations, such as multiscale elliptic equations and multiscale elasticity equations, multiscale Maxwell equations have attracted far less attention, although they arise from many important applications.Homogenization of multiscale Maxwell equation possesses features that are not seen in elliptic problems due to the fact that in a domain D ⊂ R d , unlike H 1 0 (D), the space H 0 (curl , D) is not compactly embedded in L 2 (D) d . Homogenization of multiscale Maxwell equations can be significantly more complicated than homogenization of multiscale elliptic equations in many circumstances.The thesis contributes rigorous study of mathematical homogenization of multiscale Maxwell equations. It includes new homogenization errors when the solution to the homogenized equation possesses low regularity. Such errors (even for elliptic problems) have not been deduced in the literature. We derive the error for twoscale Maxwell equations, but the procedure works the same way for other types of equations such as two-scale elliptic problems, and two-scale elasticity problems. The thesis develops the sparse tensor finite element approach, using edge finite elements, for solving the high dimensional multiscale homogenized Maxwell equations. It obtains the solution to the homogenized equation, which describes the solution to the multiscale equation macroscopically, and the scale interacting (corrector) terms, which encode the microscopic information, at the same time. The method achieves essentially optimal complexity. From the finite element solutions, we construct a numerical corrector for the solution of the multiscale problem, with an explicit error in terms of the homogenization error and the finite element error in the two-scale cases.In Chapter 2, we consider the time independent multiscale Maxwell equation which is of the same type as that considered in Bensoussan et al. [15], but in the multiscale setting, in a domain D ⊂ R d (d = 2, 3). We use multiscale convergence viii Abstract (see [79,7]) to derive the multiscale homogenized equation, from which the homogenized equation is obtained. In the two-scale case, when the solution of the homogenized equation u 0 ∈ H 1 (curl , D), we derive the standard O(ε 1/2 ) homogenization error, where ε is the microscopic scale. However, in polygonal domains which are of interest in finite element discretization, u 0 only belongs to a we...

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Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

et al. 2006

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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.

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Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

Cited by 4 publications

References 16 publications

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

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

Multiscale maxwell equations : homogenization and high dimensional finite element method

Homogenization of Periodically Varying Coefficients in Electromagnetic Materials

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