Homogenization of the Maxwell Equations: Case II. Nonlinear Conductivity

Wellander, Niklas

doi:10.1023/a:1021797505024

Cited by 27 publications

(22 citation statements)

References 11 publications

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“…This regularization corresponds to the averaging done in the definition of the inverse two-scale transform defined in [23]. It is also in line with the regularization used in the corrector results in [34] and [35].…”

Section: Remark 12mentioning

confidence: 74%

“…Wellander / Periodic homogenization in Fourier space 35 The solution of (28) provides an orthonormal basis {φ l (η, ·)} for L 2 (T n ) which is used to solve the original problem by means of Floquet-Bloch waves (see [11], Theorem 1.1). Obviously, by rescaling, η = εξ, y = x/ε, and dividing by ε 2 we get the corresponding eigenvalue problem at the ε scale, i.e.…”

Section: N Wellander / Periodic Homogenization In Fourier Spacementioning

confidence: 99%

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The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space

Wellander

2009

Asymptotic Analysis

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A two-scale Fourier transform for periodic homogenization in Fourier space is introduced. The transform connects the various existing techniques for periodic homogenization, i.e., two-scale convergence, periodic unfolding and the Floquet-Bloch expansion approach to homogenization. It turns out that the two-scale compactness results are easily obtained by the use of the two-scale Fourier transform. Moreover, the Floquet-Bloch eigenvalue problems for differential operators is recovered in a natural and straight forward way by the use of this transform. The transform is generalized to the (N + 1)-scale case. N. Wellander / Periodic homogenization in Fourier spaceNguetseng [24] and further developed in [1,2] and many other papers thereafter, e.g. in [19] and [20]. The Floquet-Bloch expansion [6,16] and the corresponding Bloch-wave homogenization method [3,5,[9][10][11] is a high-frequency method which recently has been further developed. It can be used to find the usual homogenization results, but more importantly, it provides dispersion relations for wave propagation in periodic structures [25], e.g. for photonic band gap structures. In [4] a two-scale transform was defined and used to homogenize fluid flow in porous media, see also [7,8,17] and [23] (in [8] and [17] it is called periodic unfolding). The two-scale transform simplifies the existing two-scale convergence proofs. The idea with the two-scale transform is to map bounded sequences of functions defined on L 2 (Ω) to sequences defined in the product space L 2 (Ω × T n ) and then taking the weak limit in L 2 (Ω × T n ). It gives the same limit as in the two-scale convergence method but when proving corrector results we do not have to assume any regularity of the solution to the cell problem, which was noted also in [9] using Bloch analysis. In [18] a similar approach was used to map nonperiodic coefficients to functions defined on Ω × T n .In this paper we give an explicit example of a two-scale transform operator by means of a two-scale Fourier transform, presented earlier in [36] and [37]. It turns out that the standard two-scale convergence results can be obtained at once in the frequency space, as well as we recover the classical Floquet-Bloch eigenvalue problems by the use of this transform.The paper is organized in the following way. In Section 2 we give some basic definitions, mainly to do with two-scale convergence. In Section 3 we define and explore the two-scale Fourier transform and its application to homogenization of PDEs. In Section 4 we define the two-scale transform operator by use of the two-scale Fourier transform. It is characterized and applied to the standard periodic elliptic PDE, proving convergence and correctors. Section 5 is devoted to the Floquet-Bloch homogenization method. We demonstrate how the corresponding eigenvalue problems are obtained by the use of the two-scale Fourier transform. In Section 6 we generalize to the (N + 1)-scale Fourier transform. Finally, in Section 7 we close the paper by some concluding remarks. Prelimi...

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Section: Remark 12mentioning

confidence: 74%

Section: N Wellander / Periodic Homogenization In Fourier Spacementioning

confidence: 99%

The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space

Wellander

2009

Asymptotic Analysis

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“…The rigor of the approach has been justified with extensive mathematical theories including two-scale expansions, G-convergence, compensated compactness, and two-scale convergence [26,37,[45][46][47][48][49][50][51].…”

Section: Homogenization Theorymentioning

confidence: 99%

“…It was demonstrated that the macroscopic Maxwell equations can have significant differences when compared to their microscopic counterpart turning instantaneous material laws into constitutive laws with memory [26,30,47,[52][53][54]. A more general case was considered in [48] where polarization of the composite ingredient obeyed the Debye or Lorenz polarization laws with relaxation.…”

Section: Homogenization Theorymentioning

confidence: 99%

Inverse Synthesis of Electromagnetic Materials Using Homogenization Based Topology Optimization

El-Kahlout¹,

Kızıltaş²

2011

PIER

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Abstract-Recent studies on artificial materials demonstrate that substantial improvements in electromagnetic response can be attained by combining different materials subject to desired metrics. However, the perfect material combination is unique and extremely difficult to determine without automated synthesis schemes. In this paper, we develop a versatile approach to design the microstructure of periodic materials with prescribed dielectric and magnetic material tensors. The proposed framework is based on a robust material model and generalized inverse synthesis tool relying on topology optimization. The former is derived using homogenization theory and asymptotic expansion applied to Maxwell equations and can characterize the effects of anisotropy and loss of materials with periodic unit cells of arbitrary geometries and multi-phases much smaller than the wavelength. Resulting Partial Differential Equation (PDE) is solved numerically using Finite Element Analysis (FEA) and is validated with results in literature. The material model proves to be fast and numerically stable even with complex inclusions. The topology optimization problem is applied for the first time towards designing the unit cell topology of periodic electromagnetic materials from scratch with desired dielectric and magnetic tensors using offthe-shelf materials, i.e., readily available constituents obtained from El-Kahlout and Kiziltas isotropic ceramic powders. The proposed framework's capability is demonstrated with five design examples. Design with anisotropic permittivity is also fabricated. Results show that the framework is capable of designing, in an automated fashion, non-intuitive material compositions from scratch with desired electromagnetic properties.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

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

Shelukhin¹,

Terent’ev²

2009

PIER B

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Abstract-We evaluate effective dielectric permittivity and electric conductivity for water-saturated rocks based on a realistic model of a representative cell of the pore space which has periodical structure. We have applied the method of two-scale homogenization of the Maxwell equations, which results in up-scaling coupled equations at the microscale to equations valid at the macroscale. We have analyzed the interfacial Maxwell-Wagner dispersion effect and the Archie law as well.

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Homogenization of the Maxwell Equations: Case II. Nonlinear Conductivity

Cited by 27 publications

References 11 publications

The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space

The two-scale Fourier transform approach to homogenization; periodic homogenization in Fourier space

Inverse Synthesis of Electromagnetic Materials Using Homogenization Based Topology Optimization

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

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