A new numerical simulation method using the finite element methodology (FEM) is presented to study electromagnetic scattering due to an arbitrarily shaped material body doped randomly with thin and short metallic wires. The FEM approach described in many standard text books is appropriately modified to account for the presence of thin and short metallic wires distributed randomly inside an arbitrarily shaped material body. Using this modified FEM approach, the electromagnetic scattering due to cylindrical, spherical material body doped randomly with thin metallic wires is studied

Deshpande, Manohar D.

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https://ntrs.nasa.gov/search.jsp?R=20050182129 2019-08-29T20:14:19+00:00ZStudy of Electromagnetic Scattering From Material Object Doped Randomly With Thin Metallic 
Wires Using Finite Element Method 
Manohar D. Deshpande, NASA Langley Research Center, Hampton, VA., USA
Abstract
A new numerical simulation method using the finite element methodology (FEM) is presented to study
electromagnetic scattering due to an arbitrarily shaped material body doped randomly with thin and short
metallic wires.   The FEM approach described in many standard text books [1,2] is appropriately modified
to account for the presence of thin and short metallic wires distributed randomly inside an arbitrarily
shaped material body.   Using this modified FEM approach, the electromagnetic scattering due to cylindri-
cal, spherical material body doped randomly with thin metallic wires is studied.   
I. Introduction
Electromagnetic (EM) scattering properties such as mono-static/bistatic radar cross section of a
homogeneous material object can be controlled or modified by embedding metallic/non-metallic inclu-
sions in the object. These inclusions can be in the form of short metallic wires, small thin metallic plates, or
small metallic particles of various shapes [3]. The EM scattering properties can be controlled by selecting
proper size, shape of these inclusions. Recent advances in fabrication technology have allowed for the
inclusion of nano-scale metallic wires in these objects. However, current fabrication techniques used have
no control over the spacing and arrangement of these small sized wires in a binding medium. As a result,
these short wires are arranged in an arbitrary fashion. It is important and economically advantageous to
know the EM scattering properties of these objects populated randomly with these wires/particles prior to
its actual fabrication. In this work we describe a modified FEM approach to determine the EM scattering
properties of randomly dispersed thin and short metallic wires in an arbitrarily shaped material object. 
II. Numerical Modeling
In this section, first, we present a brief outline of the FEM approach to estimate the EM scattering from an
arbitrarily shaped material object without any metallic wires or other inclusions. Then, we explain the steps
to modify this simple FEM approach to account for presence of randomly populated thin and short metallic
wires in the object.   Figure 1 shows an arbitrary shaped material object of permittivity  and permeability
 populated randomly with small/nano size metallic wires. To estimate the EM scattering from the object
shown in Figure 1 without any metallic wires, the entire scattering volume is split into two regions by
enclosing the object by a fictitious surface  as shown in Figure 1. The electric field inside the surface
 satisfy the vector wave equation 
                                             (1)
Following the usual steps involved in the FEM formulation, the equation (1) can be reduced to [4]
          (2)
where  is a vector testing function and  is the unit outward normal to the surface . If you introduce
the surface current  as an additional unknown, then the right hand side of equation (2) can be simplified
to
εr
µr
S1
S1
1 µr⁄( ) E
I
∇×∇× k0
2
εrE
I
– 0=
1 µr⁄( ) T∇× E
I
∇×• k0
2
εrE
I
T•–( ) vd
V
∫ T nˆ 1 µr⁄( ) EI∇×( )×• s1d
S1
∫–=
T nˆ S1
J
                        
The equation (2) can then be written as
                (3)
 In equation (3), in addition to the unknown electric field, the surface current  is introduced as an extra
unknown and to generate additional equations, continuity of tangential electric field across the fictitious
surface boundary can be used. Hence, 
                             (4)
where  and  are the vector potential functions for the region outside the surface  and  is the 
incident field. To facilitate the solution of equations (3) and (4) the region inside the fictitious surface  is 
discretized into tetrahedron and the electric field over each tetrahedron can be expressed as                        
(5)
where  and  are, respectively, the unknown amplitude of electric field and the vector basis
function associated with the  edge of  tetrahedron. The surface currents  and  required for the
calculations of vector potentials can also be expressed in terms of vector basis functions  as
                                       ,                         (6)
T nˆ 1 µr⁄( ) E
I
∇×( )×• s1d
S1
∫– jωµ0 T nˆ H×• s1d
S1
∫ jωµ0 T J• s1d
S1
∫= =
Arbitrary Shaped
 Material Object
Fictitious Surface
Enclosing 3_D Object
Thin, Short 
Metallic Wires
Figure1 Geometry of arbitrary shaped material object embedded randomly with
              thin and short metallic wires.
S1
1 µr⁄( ) T∇× E
I
∇×• k0
2
εrE
I
T•–( ) vd
V
∫ jωµ0 T J• s1d
S1
∫– 0=
J
E
S1
 –
F∇×– jωµ0A– 1 jωε0( ) A∇•∇⁄ Einc+ +=
A F S1 Einc
S1
E x y z, ,( ) bmn Wm
n
x y z, ,( )
m 1=
6
∑
n 1=
N
∑=
bm
n Wm
n
x y z, ,( )
m
th
n
th J M
W
J Iinˆ Wi×
i 1=
3
∑= M IiWi nˆ×
i 1=
3
∑=
Using the expressions (5), (6), and the Method of Moments (MoM), the equations (3) and (4) are converted
into set of simultaneous equations: 
                                              (7)
If  are the number of tetrahedron edges and  are the edges lying on the surface , then the coef-
ficient matrix is of the order .which can be solved for  and . From the
knowledge of the surface currents, the EM scattering properties of an arbitrarily shaped object are esti-
mated.   
The simulation technique presented so far estimates the EM scattering from a homogenous material
object. To estimate the EM scattering due to an arbitrarily shaped body populated randomly with short thin
wires we follow the procedure described below. Let us assume that when the object is embedded with thin
wires, a wire occupies one of the positions of edges out of total  edges. To populate randomly the
homogeneous object with  number of wires, we generate  random numbers between
. The edges corresponding to these random numbers are assumed to be occupied by thin metal-
lic wires. Consequently,  coefficients on these edges will be zero. The modified matrix equation for the
object randomly populated with thin metallic wires can be simply obtained by eliminating rows and col-
umns corresponding to those edges where metallic wires are assumed to be present. Hence the matrix
equation (6) when applied to the object loaded with thin metallic wires gets modified to
                                             (8)
 where the  and  are the matrices obtained from  and  by eliminating the rows and
columns corresponding to the edges where metallic wires are assumed to be present. 
III. Numerical Results and Discussion:
For a numerical experiment we consider a material sphere of radius  with permittivity
, permeability  illuminated by a plane EM wave. To estimate bistatic
radar cross section of the material sphere we assume that the sphere is illuminated by a plane wave with
incident angle , . To facilitate the bistatic RCS calculation the sphere is discretized
using the COSMOS/Geostar as is shown in Figure 2(a). Using the procedure described above the bistatic
RCS of the material sphere is calculated and presented in Figure 2(b). The number of tetrahedron used to
discretized the sphere were 123 resulting in  and . The numerical data presented
in Figure 2(b) are confirmed with the other published results.       A good agreement between the results
obtained using the present procedure and other published results confirms the validity of the present for-
mulation. 
Now to estimate the bistatic RCS for the material sphere doped with  number of very thin metallic
wires,    random numbers uniformly distributed between 1 and  are generated. If  is
the percentage of doping, then the number wires to be used for doping can be calculated from 
S11 S12
S21 S22
b
I
0
v
=
Ned Napt S1
Ned Napt+( ) Ned Napt+( )× bmn Ii
Ned
Nwire Nwire
1 N∼ ed{ }
bm
n
Sred b vred=
Sred vred S v
k0a 1.0=
εr 4.0 j0.0–= µr 1.0 j0.0–=
θin 180
o
= φin 0o=
Ned 238= Napt 132=
Nwire
Nwire Ned 238= Pd
. For the present example, with    and  choices the number of
wires used for doping were .   The edges, corresponding to these 12 random numbers (uni-
formly distributed between 1 and 238) are assumed to be thin metallic wires. The doped material sphere
with the randomly oriented wires is shown in Figure 3(a). Following the FEM procedure described above,
the bistatic RCS for the new structure   is estimated and is shown in Figures 3(b). Various numerical exam-
ples were simulated for varying concentrations of wires, their orientations and placing. These results will
be discussed at the time of presentation.
IV Conclusions
A new and modified finite element methodology (FEM) has been successfully developed to study
electromagnetic scattering from a arbitrarily shaped material object populated randomly with thin metallic
wires. Bistatic RCS of material objects in spherical, cylindrical, and cubical shapes randomly populated
with thin metallic wires have been studied. 
V References
[1] John L. Volakis, et al., "Finite Element Method for   Electromagnetics," IEEE Press, , New York, 1998.
[2] J. Jin, “ The Finite Element Method in Electromagnetics,”  John Wiley & Sons, Inc., New York, 1993
[3] J. K. Abraham, et. al., “Tailoring the dielectric properties of meta materials, ” 2003 IEEE MTT-S
Digest, pp.   1837-1840.
[4] M. D. Deshpande, et al., " A new approach to estimate complex permittivity of dielectric materials at
microwave frequencies using waveguide measurements," IEEE Trans. Microwave and Theory Tech-
niques, Vol. 5,  No. 3 , pp. 359-366, March 1997.
Figure 2(a) Material sphere (k0a 1= )discretized using tetrahedron elements
T h e t a ( D e g r e e s )
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0- 4 0
- 3 5
- 3 0
- 2 5
- 2 0
- 1 5
- 1 0
- 5
0σ
λ2
----- dB EθEφ
Figure 2(b) Bistatic RCS of material sphere shown
                   in Figure 2(a)
T h e t a ( D e g r e e s )
0 2 0 4 0 6 0 8 0 1 0 0 1 2 0 1 4 0 1 6 0 1 8 0- 4 0
- 3 5
- 3 0
- 2 5
- 2 0
- 1 5
- 1 0
- 5
0
populated randomly with thin metallic wires
Figure 3(a) Material sphere ( k0a 1=
                   in Figure 3(a)
Figure 3(b) Bistatic RCS of material sphere shown
)
EθEθ
EφEφσ
λ2
----- dB
Nwire
Pd
100
-------- Ned×= Pd 5= Ned 238=
Nwire 12=


Study of Electromagnetic Scattering From Material Object Doped Randomly With Thin Metallic Wires Using Finite Element Method

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