Abstract
 Top of page
 Abstract
 1. Introduction
 2. Formulation of the Problem
 3. Concept of TwoScaled Convergence
 4. Local Problem
 5. Comparison With Classical Mixture Formulae
 6. Conclusions
 Acknowledgments
 References
 Supporting Information
[1] This paper contains an overview of the homogenization of anisotropic materials at fixed frequency using the concept of twoscaled convergence. The homogenized electric and magnetic parameters, the relative permittivity, and the relative permeability are found by suitable averages of the solution of a local problem in the unit cell. A comparison between the exact homogenization method presented in this paper and the traditional mixture formulae, which are based on physical arguments, is made.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Formulation of the Problem
 3. Concept of TwoScaled Convergence
 4. Local Problem
 5. Comparison With Classical Mixture Formulae
 6. Conclusions
 Acknowledgments
 References
 Supporting Information
[2] Many engineering problems contain materials which are heterogeneous on a length scale (microscopic) that is small compared with the typical length scales (macroscopic) of the problem. Nevertheless, the underlying heterogeneous microscopic structure affects the macroscopic properties of the material. Accurate methods to model these microscopic effects are therefore important to develop.
[3] In electromagnetic problems, when the wavelength is long compared to the periodicity of the microstructure, the homogenized values of the electric or magnetic material parameters are pertinent macroscopic quantities. Several homogenization procedures have been suggested in the literature. Some of these are based upon physical arguments, e.g., the Maxwell Garnett formula, the Böttcher mixture rule or Bruggeman formula, and the coherent potential (CP) formula [Sihvola, 1999]. They have proven useful in many situations, e.g., low volume fraction of homogeneous spherical or ellipsoidal inclusions in a homogeneous host material, but they fail if the volume fraction is too high or if the inclusions are not spheres or ellipsoids. Then a more accurate homogenization procedure has to be used, which includes all contributions of the interaction between the inclusions.
[4] In this paper we review a general mathematical procedure to obtain the homogenized (or effective) material parameters, which includes all interactions effects between the inclusions. Specifically, we employ recent advances in the mathematics of twoscale convergence, which was introduced in 1989 by Nguetseng [1989]. Specifically, we review some of the more important results of this convergence concept, but for the mathematical details we refer to the existing literature on this subject.
[5] A typical homogenization situation is depicted in Figure 1, where we, in several steps, shrink the length scale ε, which is the periodicity of the material. The twoscaled convergence predicts the limit of this process and gives a procedure of how to compute the effective material parameters of the material with microstructure. The results are not restricted to low volume fractions of the inclusions or to a twophase composition of materials. In fact, the results are quite general and can be applied to a large variety of engineering situations.
[6] The results obtained using twoscale convergence are selfcontained in contrast to similar approaches that assume an asymptotic expansion of the solution in terms of the microscopic scale [e.g., see SanchezPalencia, 1980]. The theory of twoscale convergence does not rely on such an expansion. The periodic variations in the material parameters generate the same type of variations in the twoscale converged solution, which characterizes the microscopic solution completely.
[7] This paper is organized in the following way. In section 2 the basic assumptions of the homogenization problem are stated, and in section 3 the fundamental properties of the twoscaled convergence are reviewed. The local problem is stated in section 4, and a comparison between the classical mixture formulae and the exact homogenization procedure is presented in section 5. Some conclusions of the theory are discussed in section 6.
2. Formulation of the Problem
 Top of page
 Abstract
 1. Introduction
 2. Formulation of the Problem
 3. Concept of TwoScaled Convergence
 4. Local Problem
 5. Comparison With Classical Mixture Formulae
 6. Conclusions
 Acknowledgments
 References
 Supporting Information
[9] In Ω, the electromagnetic fields satisfy the Maxwell equations. The field solutions depend on the size of the Y^{ε} cells, and therefore all fields are indexed by the periodicity ε. The Maxwell equations in the presence of a source term J^{ε} are
and
The boundary conditions on ∂Ω are traditionally taken as
which also is adopted in this paper. Other boundary values, such as the penetrable case, are also possible to analyze. In the penetrable case the excitation is exterior to the domain Ω. This problem is more complex, and the reader is referred to the literature for the solution of this problem [Wellander and Kristensson, 2002]. To simplify this review, we omit the discussion of the sources below.
[10] We now state the general assumption made on the relative permittivity and the relative permeability used to describe the material in Ω. The material can be inhomogeneous, and the relative electric permittivity dyadic ϵ(x) and the relative magnetic permeability dyadic μ(x) are threedimensional complexvalued dyadics. They also satisfy a coercivity and a boundedness condition (C_{1,2} > 0)
where ϵ† denotes the Hermitian of the dyadic ϵ. The first condition states that the medium is passive, i.e., it is dissipative. The second one is a condition that guarantees boundedness of the parameters. Moreover, the quantities are assumed to be Y periodic, i.e., ϵ(y + ê_{k}) = ϵ(y) for every value of the local variable y ∈ ^{3} and for every k = 1, 2, 3. Similar result holds for the relative permeability μ. Due to the dyadic characters of the relative permittivity and the relative permeability, inhomogeneous anisotropic situations can be analyzed.
[11] Since the medium is Y^{ε} periodic, we scale the arguments in ϵ and μ with the periodicity ε. In fact, the constitutive relations adopted in this paper are
This implies that the scaled material parameters used in (1) and (2) are Y^{ε} periodic, i.e.,
for all x ∈ Ω, k = 1, 2, 3, since ϵ is Y periodic.
3. Concept of TwoScaled Convergence
 Top of page
 Abstract
 1. Introduction
 2. Formulation of the Problem
 3. Concept of TwoScaled Convergence
 4. Local Problem
 5. Comparison With Classical Mixture Formulae
 6. Conclusions
 Acknowledgments
 References
 Supporting Information
[12] The basic mathematical tool in the analysis of the homogenized or effective material parameters is the concept of twoscaled convergence. In this section we review some of the properties of this convergence principle. Only the basic properties needed for the results presented in this paper are stated, and the reader who wants to study this convergence concept in more detail is referred to the literature [Allaire, 1992; Bossavit, 1995; Nguetseng, 1989; Holmbom, 1997; Wellander, 1998; Svanstedt and Wellander, 2001; Wellander, 2001, 2002; Cioranescu and Donato, 1999].
[13] The concept of twoscaled convergence was introduced by Nguetseng [1989] and has later been analyzed further in the mathematical literature [see, e.g., Allaire, 1992; Bossavit, 1995; Holmbom, 1997; Wellander, 1998; Svanstedt and Wellander, 2001; Wellander, 2001, 2002; Cioranescu and Donato, 1999]. A sequence {u^{ε}} in L^{2}(Ω)^{3} twoscale converges to u_{0} ∈ L^{2}(Ω × Y)^{3} if
[Nguetseng, 1989] for every smooth vector field ϕ(x, y), which is Y periodic in y. The volume measure of ^{3} is denoted dυ_{x} or dυ_{y}, depending on the integration variable. We denote twoscale convergence by u^{ε}u_{0}. Allaire [1992] proved that it is possible to extend the set of test functions to the set of admissible function, i.e., all functions ϕ(x, y) (Y periodic in y, but not necessarily smooth in x and y) such that
[14] We easily see that if u^{ε}(x) u_{0}(x, y) then u^{ε}(x) converges weakly to the mean of u_{0} over the unit cell Y. To see this, take a test function ϕ ∈ (L^{2}(Ω))^{3} that is independent of y and the definition implies
where the mean over the unit cell Y is denoted
The average over the unit cell (Y cell) is denoted
and Y is the volume of the unit cell (in our case we have scaled so that Y = 1). Therefore u^{ε}(x) u_{0}(x, y) implies that u^{ε}(x) ⇀ u(x) weakly in (L^{2}(Ω))^{3}.
[15] It is also not hard to prove that if u^{ε}(x) u(x) strongly in (L^{2}(Ω))^{3} (norm convergence in (L^{2}(Ω))^{3}), then it also twoscale converges to the same limit. The converse is, however, not true. Moreover, if a(x, y) is a smooth Y periodic function in y, then a^{ε}(x) = a(x, x/ε) twoscale converges to a(x, y). In fact, every a(x, y) ∈ L^{2}(Ω × Y) is obtained as a twoscale limit of a function a^{ε}(x) in L^{2}(Ω) [Allaire, 1992]. We can conclude that the convergence in the twoscaled sense is more general than both the usual strong and weak convergence concepts.
[16] The two usual concepts of convergence, strong (norm convergence) and weak convergence, are too coarse instruments to preserve the information of the field at the microscale. The concept of twoscale convergence, however, preserves some of the relevant information of field at the microscale. This is proved to be an ideal tool in the homogenization procedure of materials with microscopic scale [see, e.g., Allaire, 1992; Bossavit, 1995; Nguetseng, 1989; Holmbom, 1997; Wellander, 1998; Svanstedt and Wellander, 2001; Wellander, 2001, 2002; Wellander and Kristensson, 2002].
[17] The existence of a twoscale limit of a family of fields E^{ε} and H^{ε}, such as the solutions to the Maxwell equations in (1) and (2), as ε 0, is guaranteed provided a uniform bound can be established; that is, we need uniform bounds on the fields of the following form (a priori estimates):
for all ε where
The constants C_{i}, i = 1, 2, 3, 4, do not depend on the parameter ε, but only on the domain Ω and the material parameters in Ω. These a priori estimates are fundamental in the development of the homogenization procedure [Allaire, 1992; Nguetseng, 1989]. For the boundary conditions in (3) such an a priori estimate can be established [Wellander, 1998]. The a priori estimates for other appropriate boundary conditions, such as the penetrable case, can also be obtained [see Wellander and Kristensson, 2002], but the analysis is more complex.
4. Local Problem
 Top of page
 Abstract
 1. Introduction
 2. Formulation of the Problem
 3. Concept of TwoScaled Convergence
 4. Local Problem
 5. Comparison With Classical Mixture Formulae
 6. Conclusions
 Acknowledgments
 References
 Supporting Information
[19] The a priori estimates in (5) suffice to guarantee that there are subsequences of {E^{ε}} and {H^{ε}}, so that twoscale converge [Nguetseng, 1989; Wellander, 1998; Svanstedt and Wellander, 2001], i.e.,
and
where the fields, E(x) and H(x), are the averaged fields over the unit cell Y, i.e.,
and the scalar fields Φ_{1} and Ψ_{1}, and the vector fields E_{0}, H_{0}, E_{1} and H_{1} denote the limit functions. Notice that the first terms in the twoscaled convergence in (6), E and H, are independent of the local variable y, and the second terms are gradients w.r.t. the local variable y. The twoscaled limits contain information about the field on the microscopic level, and, moreover, the fields E^{ε} and H^{ε} do not converge strongly to E and H, respectively, i.e., lim_{ε0} ∥E^{ε}(x) − E(x)∥ ≠ 0 [Allaire, 1992]. However, we have
[Wellander and Kristensson, 2002]. This is the so called corrector result.
[20] The unknown functions Φ_{1} and Ψ_{1}, which contain the information of the behavior of the fields on the microscale, are found by employing the Maxwell equations (1) and (2) and the constitutive relations (4). A separation of variables arguments implies that these functions can be obtained as the solution of a local problem. The result is
[Allaire, 1992; Nguetseng, 1989; Wellander, 1998], where
The functions (y) and (y) are determined by the solution of the local problem in the unit cell Y (i = 1, 2, 3)
[Allaire, 1992; Nguetseng, 1989] or in a weak formulation
for all Y periodic test function ω.
[21] The local problem in (7) is an electrostatic problem in the unit cell Y. These equations are solved for the unknowns (y) and (y) in the unit cell Y in each unit direction ê_{i}, i = 1, 2, 3, with periodic boundary conditions on ∂Y. From these solutions we then find the effective relative permittivity and permeability dyadics in terms of the following averages:
[Allaire, 1992; Nguetseng, 1989] where I_{3} is the identity dyadic in ^{3}. This procedure provides a constructive algorithm to find the homogenized parameters; solve (7) and compute the averages in (8). We also see that in general the homogenized parameters ϵ^{h} and μ^{h} are anisotropic dyadics, even though the material parameters ϵ(y) and μ(y) inside the unit cell are isotropic, i.e., proportional to the identity dyadic I_{3}, ϵ(y) = ϵ(y)I_{3} and μ(y) = μ(y)I_{3}. Notice that the homogenized parameters in (8) are constant dyadics in Ω. It can be proven that if ϵ is a symmetric dyadic (reciprocal material), then ε^{h} is also a symmetric dyadic [Wellander and Kristensson, 2002]. Similar results hold for the homogenized permeability dyadic μ^{h}.
[22] Moreover, it is possible to prove [Wellander and Kristensson, 2002] that the averaged fields, E(x) and H(x), satisfy the Maxwell equations with the homogenized relative permittivity and permeability ϵ^{h} and μ^{h} given by (8), i.e.,
and
[23] The local problem, (7), is solved by a suitable numerical approach, e.g., FEM. We illustrate the homogenization procedure by an example that occurs in radome applications. In radomes, the matrix material is usually reinforced by glass fiber to give the radome mechanical structure. The periodicity of the glass fiber is in general small compared to the wavelength of the application. Therefore a homogenization of the complex material gives material parameters that accurately model the interaction between the material and the electromagnetic fields. The electromagnetic parameters of this homogenization are uniaxial. A numerical simulation in two dimensions of the local field in the unit cell Y of a twoshaft weave is illustrated in Figure 3. The glass fiber weave is depicted in the bottom left figure, and the model of this glass fiber is illustrated in the lower right figure. The upper figure illustrates the solution of the local problem in the unit cell. The intensity of the local field is shown in a horizontal cross section of the unit cell.
5. Comparison With Classical Mixture Formulae
 Top of page
 Abstract
 1. Introduction
 2. Formulation of the Problem
 3. Concept of TwoScaled Convergence
 4. Local Problem
 5. Comparison With Classical Mixture Formulae
 6. Conclusions
 Acknowledgments
 References
 Supporting Information
[24] The classical mixture formulae are well known to the electrical engineers. They are derived under the assumption that all scattering effects can be ignored [Sihvola, 1999]. This implies that they give an approximate value of the material parameters of the homogenized material, which is most accurate at low volume fractions. This is in contrast to the exact method described in this paper, where all scattering effects are included and which gives an accurate value of the homogenized material parameters for all volume fractions. Moreover, the classical mixture formulae apply only to certain generic form of the inclusions, while the exact homogenization procedure has very small limitations in its range of applications, both to geometry and to material components. To illustrate the differences and the similarities between these two ways of calculations, we make a comparison between the procedures in this section. All the details of the analysis are given by Kristensson [2002].
[25] The classical mixture formulae apply to homogeneous, spherical, or ellipsoidal inclusions in a homogeneous, isotropic, or anisotropic background material [Sihvola, 1999]. Therefore let the material consist of two homogeneous materials; an isotropic background material with relative permittivity ϵ_{b} and periodically arranged spherical inclusions with relative permittivity ϵ_{i}. We denote the radius of the sphere by a, and all materials are nonmagnetic, μ = 1. The volume fraction of the inclusions is then f = 4πa^{3}/3.
[26] Several of the effective relative permittivity expressions of a mixture of homogeneous spherical inclusions are represented in the formula [Sihvola, 1999]
where ϵ_{h} is the effective relative permittivity of the mixture. The integer ν represents different mixture formulas, e.g., ν = 0 gives the Maxwell Garnett formula, ν = 2 gives the Böttcher mixture rule or Bruggeman formula, and ν = 3 gives the coherent potential (CP) formula. The Maxwell Garnett formula is explicitly given by
[28] Some of these classical mixture formulae are derived for a random distribution of spheres, but they are nevertheless often applied to a regular lattice problem. All classical mixture formulae have their domain of validity for small volume fractions f. The differences between the mixture formulae are best seen from the power series expansion in f, i.e.,
The coefficient α = 3ϵ_{b}(ϵ_{i} − ϵ_{b})/(ϵ_{i} + 2ϵ_{b}) is present in all these formulae, and this contribution represents the dipole contribution. The β coefficients for the mixture formulas are all different, and they are given in Table 1 [Sihvola, 1999].
Table 1. Different Coefficients in an Expansion of ϵ^{h} = ϵ_{b} + αf + βf^{2} for Small Volume Fractions f and Different Mixture Formulae^{a}ν  β 


0  α^{2}/(3ϵ_{b}) 
2  α^{2}ϵ_{i}/(ϵ_{b}(ϵ_{i} + 2ϵ_{b})) 
3  α^{2}(4ϵ_{i} − ϵ_{b})/(3ϵ_{b}(ϵ_{i} + 2ϵ_{b})) 
Rayleigh  α^{2}/(3ϵ_{b}) 
[29] The heterogenous material with spherical inclusions has been solved with the complete homogenization procedure that is reviewed in this paper. The results and the details of the analysis are given by Kristensson [2002]. From this complete solution it is possible to extract a small volume fraction expansion. The analysis is rather complex, and we refer the interested reader to this reference for details. It is possible to explicitly write down the complete solution of the local problem, but for comparative reasons the low volume fraction result is most illustrative. The result for small radii a of the sphere is
[Kristensson, 2002] where
where the sums S_{1}, S_{2}, and S_{3} explicitly are
Note that the first term in this exact homogenization procedure is the Maxwell Garnett formula in (9), and, moreover, that the Rayleigh formulae is correct if powers of order fifteen and higher in the radius are ignored. All other mixture formulae do not give the correct behavior beyond the dipole term. Therefore, for this configuration, we see that the Maxwell Garnett (and Rayleigh) formula is the most reliable one.