Electromagnetic scattering is the effect caused by EM waves such as light or radio waves hitting an object. The waves will then be scattered and the scattered field contains useful information about the object, see    . Electromagnetic scattering happens in many situations, for example, sun light scattered by atmosphere, radio waves scattered by buildings or planes, and so on. The study of EM wave scattering is of great interest and importance since it helps advance many different fields ranging from Medical Technology to Computer Engineering, Geophysics, Photonics, and Military Technology, see   . Unfortunately, most wave equations cannot be solved analytically to get an exact solution. Therefore, numerical methods are sought to approximate the solution asymptotically. Unlike solving scalar wave scattering problem   , solving EM wave scattering is much more complicated and computationally expensive due to the vector nature of EM waves.
In  and  , a theory for solving electromagnetic wave scattering problem by many small perfectly conducting and impedance bodies was developed. In   , numerical methods for solving EM wave scattering by one and many small perfectly conducting bodies are presented. In this paper, we focus on EM wave scattering by many small impedance bodies. A numerical method for solving this problem, based on the above theory, is described and tested. The problem is solved under the assumptions that the characteristic size a of the bodies is much
smaller than the distance d between neighboring bodies, where
, and this distance d is much less than the wave length , where k is the wave number. The distribution of these small bodies is assumed to follow this law
in which is an arbitrary open subset of the domain that contains all the small bodies, is the number of the small bodies in , and N is the distribution function of the bodies
The boundary impedance of the bodies is of the form , where h is a continuous function such that , and . The function h and constant can be chosen as desired.
To make the paper self-contained, the theory of EM wave scattering by one and many small impedance bodies is given in Sections 2 and 3. In Section 4, a numerical method for solving the EM scattering problem is presented. The solution of this problem is computed asymptotically and error analysis of the asymptotic solutions is also provided.
2. Electromagnetic Wave Scattering by One Small Impedance Body
Let D be a bounded domain of one small body, a be its radius, and S be its smooth boundary, . Assume that the dielectric permittivity and magnetic permittivity are constants. Let E and H denote the electric field and magnetic field, respectively. is the incident field and is the scattered field. The electromagnetic wave scattering by one small impedance body problem can be stated as follows
where is the frequency, is the wave number, , is the wave length, is the boundary impedance of the body, and is a unit vector that indicates the direction of the incident wave . This incident wave satisfies the relation . The scattered field satisfies the radiation condition (8). Here, N is the outward pointing unit normal to the surface S.
It is known from  that problem (3)-(8) has a unique solution and its solution is of the form
where is the incident plane wave defined in (7) and J is an unknown pseudovector. J is assumed to be tangential to S and can be found from the impedance boundary condition (5). Here E is a vector in and is a pseudovector.
Once we have E, from (3) H can be found by the formula
The asymptotic formula of E when the radius a of the body D tends to zero is
where , and the point is an arbitrary point inside the small body D, see  . So, instead of finding J to get E, we can just find one pseudovector Q
The analytical formula for Q is derived in  which can be summed up in the following theorem.
Theorem 1. One has
and is the surface area of S.
Here, correspond to , and coordinates in , is a 3 × 3 identity matrix, and is the j-th component of the outer unit normal vector to the surface S.
3. Electromagnetic Wave Scattering by Many Small Impedance Bodies
Now, consider a domain containing M small bodies , , and are their corresponding smooth boundaries. Let and D' be the complement of D in . We assume that . We also assume that the dielectric permittivity and magnetic permittivity are constants. Let E and H denote the electric field and magnetic field, respectively. is the incident field and v is the scattered field. The electromagnetic wave scattering by many small impedance bodies problem involves solving the following system
where v satisfies the radiation condition (8), is the frequency,
is the wave number, , , is a unit vector that
indicates the direction of the incident wave , and is the boundary impedance of the body . These ’s are given by the following formula
where is a continuous function in a bounded domain ,
The distribution of small bodies , , in satisfies the following assumption
where is the number of small bodies in , is an arbitrary open subset of ,
and is the parameter from (20).
From (15) and (16) we have
if . Once we have E, then H can be found from this relation
From (26) and (25), one can get (16). Thus, we need to find only E which satisfies the boundary condition (17). It was proved in  that under the assumptions (21), the problem (15)-(18) has a unique solution and its solution is of the form
When , the asymptotic solution for the electric field is given by
Therefore, instead of finding , we can just find . The analytic formula for is derived in  by using formula (13) and replacing in this formula by the effective field acting on the m-th body
The effective field acting on the m-th body is defined as
and , is a point in . When , the effective field is asymptotically equal to the field in (29) as proved in  .
From (20), (30), and (31), one gets
where c is a positive constant depending on the shape of the body , , , and
Here, correspond to , and coordinates in , is a 3 × 3 identity matrix, and , , is the j-th component of the outer unit normal vector to the surface .
4. Numerical Method for Solving EM Scattering Problem by Many Small Impedance Bodies
Our goal is to find in (32). Take curl of (32), set , and let , we have
where , see  . Solving this linear system gives us the curl of , for .
This linear system can be solved directly using Gaussian elimination method. Then E can be computed as follows
Matrix is of size 3x3 and can be computed as follows
Let then one can rewrite (34) as
where by the subscripts the corresponding coordinates are denoted, e.g. , , and
in which .
5. Error Analysis
The error of the method presented in Section 4 can be estimated as follows. From the solution E of the electromagnetic scattering problem by many small bodies given in (27)
we can rewrite it as
Comparing this with the asymptotic formula for E when given in (29)
we have the error of this asymptotic formula is
Thus, to reduce the error, one needs to reduce the quantity .
To illustrate the idea of the method described in 4, consider a domain as a unit cube placed in the first octant such that the origin is one of its vertex. This domain contains M small bodies. The small bodies are particles which are distributed uniformly in the unit cube. The following physical parameters are used to solve the problem
・ Speed of wave, .
・ Frequency, .
・ Wave number, .
・ Constant .
・ Volume of the domain that contains all the particles, .
・ Direction of plane wave, .
・ Vector .
・ Function where M is the total number of particles and a is the radius of one particle.
・ Function .
・ Function .
・ The distance between two neighboring particles, cm, where b is the number of particles on a side of the cube.
・ Vector : .
The radius a of the particles is chosen variously so that it satisfies and dissatisfies the assumption . The numerical solution to EM wave scattering problem by many small impedance bodies is computed for and 1000 particles.
Table 1 and Figure 1 show the results of solving the electromagnetic wave scattering problem with particles, the distance between neighboring particles is cm, and with different radius a of particles. When the radius of particles decreases from cm to cm, the error of the asymptotic solution decreases rapidly from about to . Note that when cm, , which does not satisfy the assumption . When , the error of the solution is always less than .
Table 2 and Figure 2 show the results of solving the problem with particles, when the distance between neighboring particles is cm, and with different radius a. From Table 2, one can see that the error of the asymptotic solution is also very small when , less than , but greater than the previous case when for corresponding a. This time, the error is also decreasing rather quickly when the radius of the particles decreases from cm to cm. The small error when guarantees that the asymptotic formula (31) for the solution E is well applicable under this assumption.
Table 1. Error of the asymptotic solution E when and cm.
Table 2. Error of the asymptotic solution E when and cm.
Figure 1. Error of the asymptotic solution E when and cm.
Figure 2. Error of the asymptotic solution E when and cm.
In this paper, we present a numerical method for solving EM wave scattering problem by many small impedance bodies. For illustration, numerical experiments are also provided. The solution to the EM wave scattering problem can be computed numerically and asymptotically using the described method, and the result is highly accurate if the assumption is satisfied.
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