Selection of Material Parameters in a Chaotic Small Particle Embedded Medium for Wave Scattering Design

Abstract

Solution to the problem of electromagnetic (EM) wave scattering on a set of small size impedance particles of arbitrary shape with the chaotic rule of their distribution is sought for by the asymptotic approach. The particles are distributed in a homogeneous volume with the constant material parameters. Solution to the problem is derived under the condition that the characteristic size of particles tends to zero; besides, the quantity of particles approaches to infinity at a specific principle. The solving procedure is reduced to derivation of an explicit form of solution that avoids the need to solve the governing integral equation, which is used to determine the fields in the particle’s surfaces. This allows to keep out of integration of the derivatives of Green function, which are presented in a kernel of the derived integral equation. The practical importance of approach consists of creating the media or materials with the close to desired inhomogeneous value of the effective refractive index or magnetic permeability. The explicit analytical relations are reduced for the above physical parameters, and they are verified by computations. It is substantiated that the chaotic distribution of particles in the initial medium makes possible to obtain more contrast material parameters comparing with the regular distribution of particles.

1. Introduction

The EM wave scattering attracts the attention of many researchers owing its application in the various fields of antennas design, microelectronics, communications, radioastronomy, power engineering, avionics, environmental observation, etc. As s rule, this phenomenon describes the behavior of the EM field components in the presence of complex scattering objects that cause the change of the initial incident field in a great extent. The solving the EM wave scattering problems started from the consideration of the simple form of scatterers with canonical geometrical shapes. The definitive research in this relation was pioneering work of Mie [1], where the solution of the EM wave scattering problem on the metallic sphere was obtained. The forthcoming investigations were focused on the coordinate bodies with the different physical properties [2, 3, 4, 5, 6] and the scatterers of arbitrary shape [7, 8, 9, 10]. In fact, the application of the analytical approach for the bodies of arbitrary shapes is impossible; therefore, the numerical methods must be applied.

In relation to this, one can attract attention to the article [11], in which author used the method of moments to solve the integral equation for the electrical component of EM field. The models on a planar triangular patch are used for modeling the non-coordinate surfaces of arbitrary shapes. The advantages of such models have been specified at the early research by Sankar and Tong [12] and by Wang [13].

In addition to the above, the asymptotic approach is a perspective method, which allows to get the analytical form of solution to the problem of scattering on a set of small bodies [14]. The convenience of this approach is the possibility to get a close approximate solution, which is applicable to many engineering tasks [15].

The asymptotic solution of integral equation for the case of scattering by an impedance convex cylinder at the grazing incidence has been studied in [16]. The proposed method joins a combination of an asymptotic field’s expansion based on the theory of boundary layer with the use of the boundary and matching conditions. The paper [17] was one of the first works, in which the usual boundary integral equation method and asymptotic approach were combined for solving the 2D and 3D problems of diffraction. An effective method for the asymptotic phase front determination, based on the moment’s method, was proposed in [18, 19].

In this chapter, we study the problem of scattering the EM waves on a set of arbitrary small impedance particles, which are placed in a homogeneous medium, which is characterized by constant permittivity ${\epsilon }_0>0$_, and permeability ${\mu }_0>0$ was considered in [20]. For solving the problem of scattering the EM waves on a set of chaotically placed particles, we improve the idea proposed there. The particles are embedded in some limited domain D. As a result, such a domain obtains new physical properties. Particularly, it can have the effective spatially inhomogeneous permeability $\mu \left(x\right)$, which can be vary either by the boundary impedance of embedded particles or by their density distribution.

Thus, even though the original domain has a constant permeability ${\mu }_0$, the new domain becomes an inhomogeneous permeability $\mu \left(x\right)$. This value has an analytical representation in terms of the distribution density of particles and the boundary impedance of particles.

The assumptions from [21], applied to the study of scattering the EM waves on a set of small size spherical particles with the impedance-type boundary conditions, are used. Simultaneously, we generalize the approach [22], where a single body of arbitrary shape was considered, to the case of set of the bodies. Mainly, the chapter is focused on the numerical results aimed to show that the chaotic embedding the small particles has advantages from both the reason, such as the better characteristics of the convergence of the proposed successive approximation method for solving the auxiliary system of linear algebraic equations (SLAE) and ability to get the characteristics of material parameters of resulting inhomogeneous medium differ more on the characteristics of homogeneous medium that in the case of regular distribution of particles. For example, the above was demonstrated for the EM wave scattering in [23] and for light scattering in [24].

The chapter is organized as follows. In Section 2, we get short information about the physics of problem. Section 3 proposes the mathematical description of problems, and the presented formulas make easy the understanding the matter despite the fact that many of them were given in the previous publications of authors. The solving procedure is explained in Section 4. Section 5 contains the results of numerical modeling that demonstrate the advantages of the chaotic distribution of particles in contrast to their regular distribution. In Section 6, we discuss the advantages of the proposed distribution of particles and emphasize the practical importance of the approach proposed. The conclusions in Section 7 finish the research with announce some forthcoming investigations.

2. The physics of problem

The discussion about the physical peculiarities of problem one can find in [25, 26, 27]. We explain here the main physical parameters of the problem. Let the boundary impedance of the small particles be

$${\zeta }_m\left(x_m\right)=h\left(x_m\right)/b^{\nu },$$E1

where x_m$\in$D_m is a point in the m-th particle, and the function h(x_m) is a continuous and diminishes out the domain D. Form (1) of the boundary impedance is correct in the physical sense because its module can be large and arbitrary, and the condition Reh(x_m)>0 is true; the number $\nu \in$ (0, 1], we can prescribe arbitrarily.

The value of b is the particle’s maximum characteristic size, that is.

$$b=0.5\underset{1\le m\le M}{max}\left(\text{diam}{D}_m\right).$$E2

The small size of subdomain D_m is understood as $\mathit{kb}\ll 1$, and k is the wavenumber.

We accept that the boundary conditions in the surface S_m of D_m are given as

$${\mathbf{E}}_{\mathit{tm}}=-{\zeta }_m\left({\mathbf{H}}_t\times \mathbf{N}\right),$$E3

where “$\times$” marks the cross product of vectors. The vector functions E_t (H_t) characterize the tangential values of E(H) at S_m, and N is the unit outer normal to S_m.

Crucial in our approach is the function p(x), characterizing the particle’s density in the arbitrary subdomain Δ of whole domain $D$. We assume that the embedded particles number N(Δ) in Δ is

$$\mathrm{N}\left(\Delta \right)=\left[1+o\left(1\right)\right]\left(1/b^{2-\nu }\right)\underset{\Delta }{\int }p\left(x\right)\mathit{dx},$$E4

and $b\to 0$, $p\left(x\right)\ge 0$ is a continuous arbitrary function, which is zero outside $D$, where the particles are embedded. Both the functions $h\left(x\right)$ and $p\left(x\right)$ can be given that results in a desired magnetic permeability and a refractive index in $D$.

3. Mathematical description of problem

We assume that a set of bodies $D_m$, where $1\le m\le M$, is embedded in a domain with the constant values of ${\epsilon }_0$,${\epsilon }_0$, and $k^2={\omega }^2{\epsilon }_0{\mu }_0$; $\omega$ is the frequency.

The problem of EM wave scattering is reduced to determine the vectors $\mathbf{E}$ and $\mathbf{H}$, satisfying the set of Maxwell equations

$$\Delta \times \mathbf{E}=i{\mathit{\omega \mu }}_0\mathbf{H},\Delta \times \mathbf{H}=-i{\mathit{\omega \epsilon }}_0\mathbf{E}$$E5

in the domain $D=R^3\backslash \bigcup _{m=1}^M{D}_m$

We give the EM field components as

$$\mathbf{E}={\mathbf{E}}_i+{\mathbf{E}}_s,\mathbf{H}={\mathbf{H}}_i+{\mathbf{H}}_s$$E6

the vectors ${\mathbf{E}}_i,{\mathbf{H}}_i$ stand for the incident fields satisfying (5) in whole space $R^3\backslash D$; components $E_s$ and $H_s$ define the scattered field.

The impedance boundary conditions we give as

$$\left(\mathbf{N}\times \left(\mathbf{E}\times \mathbf{N}\right)\right)={\zeta }_m\left(\mathbf{N}\times \mathbf{H}\right),$$E7

on $S_m$ at $1\le m\le M$. Function ${\zeta }_m\left(x_m\right)$ is the boundary impedance, and $\mathbf{N}$ is a unit normal pointing out $D_m$.

Usually, the impedance boundary conditions are

$${\mathbf{E}}_t={\zeta }_m\left({\mathbf{H}}_t\times {\mathbf{N}}_i\right),$$E8

where ${\mathbf{N}}_i$ is a unit normal to $S_m$ directed in $D_m$, and the tangential part of $\mathbf{E}$ is

$${\mathbf{E}}_t=\left(\mathbf{N}\times \mathbf{E}\right).$$E9

If we prescribe ${\mathbf{E}}_t$ as

$${\mathbf{E}}_t=\mathbf{E}-\mathbf{N}\cdot \left(\mathbf{E}\times \mathbf{N}\right)=\left(\mathbf{N}\times \left(\mathbf{E}\times \mathbf{N}\right)\right),$$E10

and use the equalities $\left(\left[\mathbf{N}\times \left(\mathbf{H}\times \mathbf{N}\right)\right]\times \mathbf{N}\right)=\left(\mathbf{H}\times \mathbf{N}\right)$, $\left(\mathbf{N}\times \mathbf{N}\right)=0$, where “.” denotes the dot product of vectors, then we obtain the impedance boundary condition (7). We suppose that the boundary impedance ${\zeta }_m$ is constant and that $Re\left({\zeta }_m\right)\ge 0$.

We consider the case of a plane incident wave, namely

$${\mathbf{E}}_i=\beta e^{\mathit{ik}\left(\alpha \cdot \mathbf{x}\right)},$$E11

$\beta$ is a constant vector, and unit vector $\alpha \in S^2$, where $S^2$ is a unit sphere in whole space ${\mathrm{R}}^3$. By this, both the vectors $\alpha$ and $\beta$ satisfy t the condition $\left(\alpha \cdot \beta \right)=0$. Besides, the vectors ${\mathbf{E}}_s$ and ${\mathbf{H}}_s$ must satisfy the Silver-Müller condition of radiation

$${\mathbf{E}}_s-\sqrt{\frac{{\mu }_0}{{\epsilon }_0}}\left({\mathbf{H}}_s\times \mathbf{N}\right)\simeq o\left(1\right),$$E12

$${\mathbf{E}}_s-\sqrt{\frac{{\mu }_0}{{\epsilon }_0}}\left({\mathbf{H}}_s\times \mathbf{N}\right)\simeq o\left(1\right),$$E13

If to use the relation

$$\mathbf{H}=\frac{\nabla \times \mathbf{E}}{i{\mathit{\omega \mu }}_0},$$E14

we obtain

$$\nabla \times \nabla \times \mathbf{E}-k^2\mathbf{E}=0$$E15

in the domain D.

Thus, the problem (5)–(7) is reduced to the determination of $\mathbf{E}\left(x\right)$ by (15), which meets the conditions (10) and (12), (13).

4. Solution methods

4.1 Asymptotic representation of the field

The solution to problem of the EM wave scattering on the set of uniformly distributed small impedance particles is given in the explicit form in [28]. This solution satisfies the Helmholtz equation at the arbitrary continuous integrand function $P_m\left(t\right)$ because the incident field ${\mathbf{E}}_i$ satisfies this equation as well.

Let the boundary impedance ${\zeta }_m$ of $S_m$ is (1). The quantity $\mathrm{N}\left(\Delta \right)$ of the particles embedded in an arbitrary subdomain $\Delta \in D_m$ is (4).

Alternatively, solution of [28] can be presented as

$$\mathbf{E}\left(x\right)={\mathbf{E}}_i\left(x\right)+\sum _{m=1}^M\left[{\nabla }_{x}g\left(x,x_m\right),Q_m\right]+\sum _{m=1}^M\nabla \times \underset{S_m}{\int }\left(g\left(x,y\right)-g\left(x,x_m\right)\right)P_m\left(y\right)\mathit{dy},$$E16

where

$$Q_m=\underset{S_m}{\int }{P}_m\left(y\right)\mathit{dy}.$$E17

Subscript “x” at operator $\nabla$ in (16) marks that differentiation is taken to argument x. If relation $P_m\left(y\right)=O\left(b^{-\nu }\right)$ is correct that follows from the estimate of terms corresponding to integral equation, which is reduced for $P_m\left(t\right)$ in [25], we have $Q_m=O\left(b^{2-\nu }\right)$.

Using Theorem 1, given in [28], we present (16) in form

$$\mathbf{E}\left(x\right)={\mathbf{E}}_i\left(x\right)+\sum _{m=1}^M\left[{\nabla }_{x}g\left(x,x_m\right),Q_m\right]+o\left(1\right).$$E18

When | x — x_m | is the same order of smallness than $b$, then term at $m=j$ in (16) can be eliminated by the asymptotic (effective) field definition [25]. In the terms of asymptotic, the previous formula is

$$\mathbf{E}\left(x\right)\simeq {\mathbf{E}}_i\left(x\right)+\sum _{m=1,m\ne j}^M\left[{\nabla }_{x}g\left(x,x_m\right),Q_m\right],$$E19

The last formula has a practical significance because it allows us to exclude the term with $m=j$ in the SLAE for the determination of the function vectors $\mathbf{U}$ (see Eq. (24) below). One can show if $b\to 0$, then sum in (21) transforms to the respective integral. The last property gives the possibility to reduce the Fredholm integral equation with respect to the vector $\mathbf{E}\left(x\right)$ (see [21]).

$$\mathbf{E}\left(x\right)={\mathbf{E}}_i\left(x\right)+\frac{\eta }{i{\mathit{\omega \mu }}_0}\nabla \times \underset{D}{\int }g\left(x,y\right)h\left(y\right)p\left(y\right)\mathbf{A}\left(\nabla \times \mathbf{E}\left(y\right)\right)\mathit{dy},$$E20

the parameter $\eta$ defines the shape of particle, the functions $h\left(y\right)$, $p\left(y\right)$ are defined in (1) and (4). The operator $\mathbf{A}=\mathbf{I}-\mathbf{B}$, $\mathbf{I}$ is $3\times 3$ identity matrix; the numbers $b_{\mathit{ij}},i,j=1,2,3$of $3\times 3$ matrix $\mathbf{B}$ are defined as in [28]

where $\mid S_m\mid$ is the $m-$th particle area. Eq. (20) has a great importance because it allows to reduce the closed formula for the permeability of the new inhomogeneous material.

4.2 The explicit formula for vector $\mathbf{E}$

Now, we illustrate the method how to determine the $Q_m$, which is applied for search the components of electrical field $\mathbf{E}$. We define the auxiliary vector functions

$${\mathbf{U}}_m=\left(\nabla \times \mathbf{E}\right)\left(x_m\right).$$E21

If they are found, we can calculate the vectors

$${\mathbf{Q}}_m=\frac{{\zeta }_m{\eta }_m}{i{\mathit{\omega \mu }}_0}{b}^{2-\nu }h\left(x_m\right)\mathbf{A}\left({\mathbf{U}}_m\right),$$E22

where number ${\eta }_m$ defines the shape of particle. For example, this number is $4\pi$ for a sphere, and for an ellipsoid with semiaxes $a,b,c$ is

$${\eta }_m=4\pi \sqrt[p]{\frac{t_1^p+t_1^p{t}_2^p+t_2^p}{3}},$$E23

where $b=t_{1}a,c=t_{2}a$ and $p=1.6075$. The field $\mathbf{E}$ is found using (18), provided vectors ${\mathbf{Q}}_m$ are determined. In order to reduce SLAE for determination of ${\mathbf{U}}_m$ we act by the operator $\nabla$ on (18), and replacing the sum $\sum _{m=1}^M$ with sum $\sum _{m=1,m\ne j}^M$according to the definition of an effective field, we have SLAE

$${\mathbf{U}}_j={\mathbf{U}}_{\mathit{ij}}+\frac{{\zeta }_m{\eta }_m}{i{\mathit{\omega \mu }}_0}{b}^{2-\nu }\sum _{m\ne j,m=1}^{M}h\left(x_m\right){\nabla }_x\left[{\nabla }_{x}g\left(x_j,x_m\right)\times \mathbf{A}\left({\mathbf{U}}_m\right)\right]$$E24

at $1\le j\le M$. By this, the vectors ${\mathbf{U}}_{\mathit{ij}}$ and ${\mathbf{U}}_j$ are sought as

$${\mathbf{U}}_{\mathit{ij}}=\nabla \times {\mathbf{E}}_i\left(x_j\right),{\mathbf{U}}_j=\nabla \times \mathbf{E}\left(x_j\right).$$E25

If vector functions ${\mathbf{U}}_j$ are known, one can to find vector $\mathbf{E}\left(x\right)$ by (18) in space ${\mathrm{R}}^3$.

4.3 Material parameters of inhomogeneous medium

Using Eq. (20), we can reduce an explicit form of both the magnetic permeability and refractive index of new inhomogeneous material. If to act the operator $\nabla \times \nabla \times$ on (20), we get

$$\left[\nabla \times \left[\nabla \times E\left(x\right)\right]\right]=k^{2}E\left(x\right)-\frac{\eta }{i{\mathit{\omega \mu }}_0}\nabla \times \left(h\left(x\right)p\left(x\right)A\left(\nabla \times E\left(x\right)\right)\right).$$E26

Here, we refer to the known formula $\nabla \cdot \nabla \times =0$, the relation

$${\nabla }^{2}g\left(x,y\right)-k^{2}g\left(x,y\right)=\delta \left(x-y\right)$$

and take into account that function $h\left(x\right)$ is a scalar one. From (22), we derive

$$\begin{array}{l}\left[\nabla \times \left[\nabla \times E\left(x\right)\right]\right]=\frac{k^2}{1+\frac{\eta }{i{\mathit{\omega \mu }}_0}A\left(h\left(x\right)p\left(x\right)\right)}-\frac{\eta }{i{\mathit{\omega \mu }}_0}\frac{\nabla \left(h\left(x\right)p\left(x\right)\times \left(\nabla \times E\left(x\right)\right)\right)}{\frac{k^2}{1+\frac{\eta }{i{\mathit{\omega \mu }}_0}A\left(h\left(x\right)p\left(x\right)\right)}},\\ \end{array}$$E27

where

$$K^2\left(x\right)=\frac{k^2}{\frac{\eta }{i{\mathit{\omega \mu }}_0}A\left(h\left(x\right)p\left(x\right)\right)}$$E28

On the other hand, we get

$$\nabla \times \nabla \times E\left(x\right)=K^2\left(x\right)E\left(x\right)+\left(\frac{\nabla \mu \left(x\right)}{{\mu }_0}\times (\nabla \times E\left(x\right)\right)$$E29

if to use the relations$\nabla \times$ to $\nabla \times E=\mathit{i\omega \mu }\left(x\right)H$, $\nabla \times E=\mathit{i\omega \mu }\left(x\right)H$, and $k^2\left(x\right)={\omega }^2\epsilon \left(x\right)\mu \left(x\right)$.

If to compare Eqs. (27) and (29), one can see that the last term in Eq. (27) arises by virtue of the new permeability $\mu \left(x\right)$. This $\mu \left(x\right)$ is result of the excitation of surface current at $S_m$. Let us demonstrate the recipe to reduce an explicit formula for $\mu \left(x\right)$. To this end, we introduce the function

$$\mathrm{\Psi }\left(x\right)=1-\frac{\eta }{i{\mathit{\omega \mu }}_0}A\left(h\left(x\right)p\left(x\right)\right).$$E30

If $\epsilon \left(x\right)={\epsilon }_r{\epsilon }_0=\text{const}$, where ${\epsilon }_r$ is the relative permeability of domain, where the particles are embedded, we have

$$\mu \left(x\right)={\mu }_0/\mathrm{\Psi }\left(x\right),$$E31

that’s why, $K^2\left(x\right)={\omega }^2\mathit{\epsilon \mu }\left(x\right)$.

Hence, a new physical result follows from Eq. (27), namely the new inhomogeneous domain $D$ is characterized by the variable permeability

$$\mu \left(x\right)={\mu }_0/\mathrm{\Psi }\left(x\right)=\frac{{\mu }_0}{1-\frac{\eta }{i{\mathit{\omega \mu }}_0}A\left(h\left(x\right)p\left(x\right)\right)},$$E32

it becomes a new wavenumber

$$K^2\left(x\right)={\omega }^2\mathit{\epsilon \mu }\left(x\right)$$E33

with refractive index $n=\sqrt{{\epsilon }_r\mu \left(x\right)}$.

The ${\epsilon }_0$ is substantiated by $\epsilon$if a material with relative permittivity ${\epsilon }_r$ is studied, $\epsilon ={\epsilon }_0{\epsilon }_r$. The elements of matrix $A$ are replaced by some numbers corresponding to the shape of particle. For example, this number is equal to $2/3\pi$ for sphere, it is determined by the values of semiaxes $a,b$, and $c$ for ellipsoid, and it is calculated numerically for other bodies.

5. Numerical examples

5.1 The exactness of computations

The numerical data below illustrate the analysis of error for the asymptotic representation (18) and the dependence of convergence of the iterative method while solving SLAE (24) at the different parameters b, d, M, and $\omega$ of the problem studied. The computations testify the applicability of asymptotic method.

Such physical parameters are used in study:

• Permittivity of vacuum ${\epsilon }_0$= 8.854188 × 10⁻¹² F/m.

• Permeability of vacuum ${\mu }_0$= 1.256637 × 10⁻⁶ H/m.

• Vector characterizing the plane wave direction $\alpha$= (0, 0, 1).

• The vector $\beta$= (0, 1, 0), (the condition $\left(\alpha \times \beta \right)=0$ is true).

• The constant v = 0.9.

• The distance between adjacent particles $d=O\left(b\right)$.

• Function p(x) = Mb^2−v/|D|, where M is quantity of particles in D, and |D| is its volume.

• Vector ${\left.{\mathbf{U}}_i\left(x_m\right)=\nabla \times \beta e^{1\mathit{k\alpha }\cdot x}\right|}_{x=x_m}$.

The rest of the problem parameters b, M,$\omega$, and function $h\left(x\right)$ are variable. The chaotic distribution of particles is defined by the standard random function available in many programming languages; such a recipe of particle distribution was used in [29]. The area of particle distribution is defined by the parameters of domain $D$.

Figure 1 shows the typical value of vector $\mathbf{E}\left(x\right)$ ($x-$component amplitude) in the forward direction. The results are presented for the domain $D$ in the form of cube with the sides equal to 5 cm, and the characteristic size $b$ of particle is 0.05 mm, therefore the constraint $d>10b$ [22], which is necessary for the convergence of the iterative process (24) is satisfied. The distance in Figure 1 along the $x$ and $y$ axes is given in cm, and this testify that at the given frequency $\omega$ equal to 188.4 MHz (wavenumber $k=0.3{\mathrm{m}}^{-1}$) the characteristic of radiation is poorly directed. The average distance $d$ for such size of $D$ is equal to 0.5 cm, therefore the requirement $\mathit{kb}\ll 1$ [21] is met as well. One can show that the amplitude is “periodical,” and it diminishes if distance from the center increases. The amplitudes of the $E_y$, $E_z$ behave similar. They become smooth distribution if to provide with a quite high accuracy.

Figure 1.

The $E_x$ amplitude in the forward direction.

The operator $\nabla$ acts on the functions ${\mathbf{U}}_m$ in the right-hand part of Eq. (24), and this results in the presence of all components $U_x$, $U_y$, and $U_z$ here. Therefore, an iterative procedure is needed to determine the solution of (26), which affects the $\mathbf{E}\left(x\right)$ determination accuracy by virtue of (19) and (24). In contrast to [28], where the relative error RE for $\mathbf{E}\left(x\right)$ was determined as the error of the asymptotic solution, the relative error is defined here through the error of solution to SLAE (24) because derivation of the difference of Green’s function and estimation of the values $Q_m$ in the representation of the solution is difficult for the case of chaotic distribution of particles. We define the relative error of solution to SLAE (24) as

$$\mathit{RE}=\mid \frac{\Vert U_{n+1}\Vert -\Vert U_n\Vert }{\Vert U_{n+1}\Vert }\mid ,$$E34

where sign $\Vert \cdot \Vert$ defines norm of the respective vectors. Subscript $\left\{x,y,z\right\}$ is omitted here in the designation of the respective components of $\mathbf{U}$; the subscripts $\left\{n,n+1\right\}$ define the number of steps in the iterative procedure. Because the operations determination of $\mathbf{E}\left(x\right)$ through the components of $\mathbf{U}$ in the formulas (19) and (22) are linear ones, the errors of $\mathbf{U}$ and $\mathbf{E}\left(x\right)$ are the same order. The given considerations allow us to draw a conclusion about the correctness of definition of the relative error for $\mathbf{E}\left(x\right)$by the proposed method.

The attained error for $E_y$ versus the distance $d$ between particles is shown in Figure 2 (lower line) at the fixed $b$. The upper blue line characterizes the value for the uniform placement of particles in a similar domain $D$. In fact, the error depends on $d$ slightly, where for the uniform distribution this dependence is significant.

Figure 2.

The relative error RE of $E_y$ component versus the distance $d$ between the particles.

Figure 3 shows the $E_y$ error dependence on size $b$ at the fixed $d$. The average distance between particles is equal about o.05 cm, and values of $b$ is given in mm. The error for both the chaotic and uniform particle distributions is close, and difference grows when $b$ increases. In Figure 4, the dependence of errors on $\omega$ is shown at the fixed $b$ and $d$, namely $b=0.05\mathrm{mm}$, and $d=0.5\mathrm{mm}$. The error close to the previous case, by this it decreases when $\omega$ grows, and the error for a chaotic distribution is smaller.

Figure 3.

The relative error RE of $E_y$ component versus the size $b$ of particle.

Figure 4.

The relative error RE of $E_y$ component versus the frequency $\omega$.

Figures 5 and 6 demonstrate study of the error’s characteristics in $D$ for particles of different sizes.

Figure 5.

The amplitude of $E_y$ component for the different points in $D$ versus size b of particle.

Figure 6.

The amplitude of $E_y$ component for the different points in $D$ versus the frequency $\omega$.

In Figure 5, the minimal, maximal, and the error in the central point of $D$ are shown at the different $b$, and the values of $b$ are given in mm. One can see that first two errors are lower for chaotically placed particles, and the errors for central particle depend on its size. Similar data are got for the frequency $\omega$ dependence (Figure 6), and the difference is that the errors decrease if $\omega$ grows.

It is turned out that the characteristics of convergence at solving the SLAE (24) depend on the recipe of arranging the particles in the chaos ensembles. In Figures 7–9, the comparative characteristics of convergence are shown and compared for four recipes of arranging the particles, namely the regular, chaotic general, chaotic layered (the particles are arranged in one layer, and several such layers form $D$), and chaotic layered with the normal distribution. There is shown the relative error of component $E_y$, which is measured in percents. The analysis of results testifies that the quality of convergence is the best for the case of the latter distribution. In Figure 7, the dependence of error is shown at the different distances $d$ between particles. One can see that the regular (uniform) distribution. If minimal distance grows, the error approaches for all considered recipes.

Figure 7.

The relative error RE of $E_y$ component versus the distance $d$ between the particles at four recipes of distribution of particles.

Figure 8.

The relative error RE of $E_y$ component versus the size $b$ of particles at four recipes of distribution of the particles.

Figure 9.

The relative error RE of $E_y$ component versus the frequency $\omega$ at four recipes of distribution of the particles.

The dependence of error on size of particles is shown in Figure 8 at the fixed $d=0.5\mathrm{cm}$. One can see that error in this case practically does not depend on the recipe of particle distribution; it grows for the all cases when size $b$increases. The above testifies that the relation between $b$ and $d$$d\approx 10b$ is optimal for the considered geometry of $D$.

The different characteristics of convergence are observed at the study of convergence characteristics depending on the frequency $\omega$. In Figure 9, this dependence is shown at fixed $d=0.5\mathrm{mm}$ and $b=0.05\mathrm{mm}$. One can observe the error is most sensitive at the regular distribution of particles, and the maximal value of error is equal to 1.8% at 120 MHz, but the error for this distribution decreases in contrast to the two of them, for which errors grow. This testify that the error is sensitive to the recipe of distribution in a great extent.

5.2 The magnetic permeability of inhomogeneous material

In Figure 10, the dependence of magnetic permeability $\mu \left(x\right)$ is shown depending on the size $b$ of particle at several fixed $d$ ($d$ is measured in mm as well). The value of $\mu \left(x\right)$ is normalized on the value ${\mu }_0$ ($4\pi \times {10}^{-7}$ H/m, namely permeability of free space). The values are shown for such relation of parameters $b$ and $d$, which ensures the convergence of iterative procedure of solving SLAE (24). This does not contradict the physical nature because the properties of material in $D$ at the increase of $d$ approach to the properties of initial homogeneous material.

Figure 10.

Values $\mu$ versus size $b$ of the particles.

The permeability $\mu \left(x\right)$ of the resulting inhomogeneous medium depends on the function $h\left(x\right)$ defining the surface impedance of particles as well. In Figure 11, the dependence of $\mu \left(x\right)$ on the imaginary part $Im\left(h\left(x\right)\right)$ of this function is shown at the fixed $b=0.05\mathrm{mm}$; the values of $d$ are given also in mm. Similarly, to the previous example, the results are presented for such relation $Im\left(h\left(x\right)\right)$ and $d$, which ensures the convergence of iterative procedure of solving SLAE (24). Other characteristics of inhomogeneous domain $D$ with embedded particles change when the function $h\left(x\right)$ varies. The amplitude $E$ of scattered EM field is one of such characteristics. The dependence of $E_x$ on the radius a of particles is shown in Figure 12. One can see that the amplitude increases when a grows. The values of wavenumber $k=1.0{\mathrm{mm}}^{‐1}$, therefore $b$ and $d$ are measured in mm.

Figure 11.

The values $\mu$ versus function $\mid Imh\left(x\right)\mid$.

Figure 12.

The values of $max\mid E_x\mid$ versus radius $a$ of the particles.

The proposed approach allows creating the materials with the piecewise constant $\mu \left(x\right)$. Such values can be designed either by embedding the different number$M_m$ of particles in the subdomains ${\mathrm{\Delta }}_m$ of the domain $D$ or by change of function $h\left(x_m\right)$ in respective subdomains. Both recipes have their own advantages that depend on the physical and geometrical parameters of domain $D$. Sometimes, it is necessary to have a constant $\mu \left(x\right)$along some direction (for example, a constant along the axes $y$, $z$, and a piecewise continuous along the axis $x$).

In Figure 13, the piecewise constant distribution of $\mu \left(x\right)$ is shown for the case of four subdomains of $D$. The number of particles $M_p=11\times 5\times 5$ is the same in all domains; the functions $h\left(x_m\right)$ are: $h_1=-7i$, $h_2=-9i$, $h_3=-11i$, and $h_4=-13i$ (i.e. $h\left(x_m\right)$ is piecewise constant totally).

Figure 13.

The piecewise constant $\mu \left(x\right)$ for the different $h\left(x\right)$ in the separate subdomains ${\mathrm{\Delta }}_p$.

The results in Figure 14 demonstrate creating a material with the specific effective permeability $\mu \left(x\right)$. The initial domain $D$ is split into four subdomains ${\mathrm{\Delta }}_p$, the number of particles in such subdomain is equal to 275, 225, 165, and 99, frequency $\omega$ is 50GHz. The forms of particles are ellipsoids of rotation with the semiaxis $b=0.0001\mathrm{m}$ and the relation between semiaxes $a=0.8b$ and $c=0.6b$. The rule of distribution of the particles is such that the subdomains ${\mathrm{\Delta }}_p$ are cubes with side 0.016 m. This allows us to design the piecewise constant value of $\mu \left(x\right)$ [26]. Additionally, the function $h\left(x\right)$, defining the boundary impedance $\zeta$, changes specially in separate subdomain. The proposed recipe allows us to design the $\mu \left(x\right)$, which grows by the quadratic weighed rate from the center of $D$ to its periphery. The obtained results confirm the possibility of creating the magnetic materials with the specific peculiarities.

Figure 14.

Specific distribution of $\mu \left(x\right)$ at the different numbers of particles in subdomains and various $h\left(x\right)$_.

6. Discussions

The main advantage of the approach proposed is that the semi-analytical solution to the EM wave scattering problem enables to derive the close formula for the material parameters of the resulting inhomogeneous medium. The main computational difficulty caused by the chaotic distribution of particles was eliminated owing the modification of the algorithm for solving the SLAE with respect to the auxiliary vector function, which enables to get the form of closed solution to the initial scattering problem.

The comparison of the characteristics of convergence with the case of regular distribution of particles demonstrated the advantages of the iterative procedure in comparison with the latter case. The investigation of dependence of error on the distance between particles, their size, and frequency of radiation elucidated the more sensitive of them, which provides the best characteristics of convergence.

The novelty of chapter is that the advantages of the chaotic distribution of particles allows to get more effective characteristics of the iterative procedure for solving the scattering problem, as well as wider range of the material parameters, unlike the case of regular distribution of particles.

7. Conclusions

We have successfully applied the asymptotic approach to solve the EM wave scattering problem on a set of chaotically placed particles of small size with the impedance boundary conditions. The obtained semi-analytical solution was reduced to solving the SLAE for some auxiliary functions, which allowed us to get the close formula for the EM field electrical component. This SLAE was solved effectively by the method of successive approximations. The numerical data testify that to get the exactness equal to several dozens of percents, and the number of iterations should be about 10–15 ones. The numerical results confirm the analytical conclusion about the ability to create the material with the space inhomogeneous magnetic permeability either by embedding into a homogeneous material a set of small particles with the boundary impedance chosen appropriate or changing the geometrical characteristics of the particles including their distribution density or size. The perspective of approach is generalization on the case of conductive media.

Conflict of interest

Authors declare no conflict of interest.