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1722
IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 36. NO 12, DECEMBER 1988
Generalized Formulations for Electromagnetic
Scattering from Perfectly Conducting and
Homogeneous Material Bodies-Theory
and Numerical Solution
Abstract-Generalized E-field form ulatio n for three-dimen sional scat-
tering from perfectly conducting bodies and generalized coupled operator
equat ions for three-dimens ional scat ter ing f rom mater ial bodies are
introduced. The suggested approach is to use a fict i t ious electric current
flowing
on
a mathematical surface enclosed inside the body to simulate
the scattered field and , in the material case, to use in addit ion a fict i t ious
electric current flowing on a mathemat ical surface enclosing the body to
simulate the field inside the body. Applica tion of th e respective bou ndar y
condi t ions l eads to operator equat ions to be solved for the unknown
ficti t ious currents which fac il i tate the fields in the various regions thro ugh
the magnetic vector potential integral . Th e existence and uniqueness of
the so lu t ion are d i scussed . These al t ernat ive operator equat ions are
solvable via the method of moments. In particular, impulsive expansion
funct ions for the currents in conjun ct ion wi th a poin t -matching tes t ing
procedure can be used wi thout degrading th e capabi l ity of th e numerical
solution to yield accurately near-zone and surface fields. The numerical
solution is simple to execute, in most cases rapidly converging, and is
general in that bodies of smooth but o therwise arb i t rary surface, both
lossless and lossy, can be handled effectively. Boundary condit ion checks
to see the degree to which the requi red b oundary condi t ions are sat i sf ied
at any set of poin t s
on
the body surface are eas ily made for val idat ing the
solution. Finally, results are given and compared with available analytic
solu t ions , which demonst rate the very good accuracy
of
t h e m o m en t
procedure.
I . INTRODUCTION
HREE-DIMENSIONAL problems of electromagnetic
T
cattering by perfectly conducting and material bodies
have been a sub ject of intense investigation an d research to the
electromagnetic community for many years. The study of
electromagnetic scattering is not solely of academic interest,
but of practical importance as well in many application areas.
These efforts have led to a development of a large number of
analysis tools and modeling technique s for quantitative evalua-
tion of electromagnetic scattering by various objects. Among
these methods, surface integral equation formulations are
probably the most suitable ones for numerical solutions. The
general procedure is to reduce the three-dimensional problem
to two dimensions by cast ing the problem in terms of unknow n
functions defined on the surface of the body rather than in
terms of unknown volume functions. In considering scattering
from a conducting body
(Fig.
l ) ,
he problem is formulated in
Manuscript received September 17, 1986; revised September 25, 1987.
The autho rs are with the Departm ent of Electrical Engineering, Technion-
IEEE Log
Number 8823639.
Israel Institute of Technology, Technion City, Haifa, 32000 Israel.
J
Mm
1-
I
perfect ly conducting
closed surface S
Fig. 1. General problem
of
scattering by perfectly conducting body
terms of the yet to be determined surface curre nt J, induced on
the conducting body surface S. This can be done in two
alternative ways discussed both by Poggio and Miller in [11.
One formulation, known as the E-field integral equation, is
derived by setting the component tangential to
S
of the sum of
the incident electric field an d the electric field due to J,, both
calculated with the condu cting body absent, equal to zero on S.
The other formulation, known as the H-field integral equation,
is derived by setting the comp onent tangential to S of the sum
of the incident magnetic field and the m agnetic field due t o J,,
both calculated with the conducting body absent, equal to zero
just inside
S.
In considering scattering from a homogeneous
material body (Fig. 2 ) , he problem can be formulated in terms
of yet to be determined equivalent electric and magnetic
currents
J,,
M, over the body surface S Application of
bound ary conditions leads to a set of four integral equations to
be satisfied. Linear combinations of these four equations leads
to a coupled pair of integral equations to be solved. One choice
of combination constants gives the formulation described by
Poggio and Miller [ l j . Another choice of combination
constants gives the formulation obtained by Muller
[ 2 ] .
If
either the E-field or the H-field integral equation for the
conducting body case were solved exactly, we w ould have the
true solution. Similarly, if the coupled pair of integral
equations for the material body case were solved exactly, we
would have the true solution. To obtain approximate solutions,
these equations are reduced to m atrix equations via the method
of moments [3j. The solution of the matrix equations is then
0018-926X/88/1200-1722 01.00
988 I E E E
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LEVIATAN
et al. : GENERALIZED FORMULATIONS
FOR
ELECTROMAGNETIC SCATTERING
1723
Fig.
2 .
General problem of scattering by homogeneous material body
carried out in the computer by inversion
or
elimination, and
sometimes by i terative techniques. O nce the unknow n surface
current
J,
in the conducting body case and the unknown
equivalent surface currents J,, M, in the material body c ase are
determined, the analysis of these scattering problems is
completed as the fields and the field-related parameters may
then be calculated in a straightforward manner.
In this paper, we introduce alternative formulations for
scattering from perfectly conducting and homogeneous mate-
rial bodies with smooth surfaces. The novel idea is to
formulate the problem in term s of unknown functions that for a
conducting body are all defined on a mathematical surface
enclosed in the body, and for material body are partially
defined on a mathematical surface enclosed in the body and
partially on a mathematical surface enclosing the body.
Specifically, in considering scattering from a perfectly con-
ducting body, we simulate an equivalence for the region
exterior to the body by m eans of a fictitious elec tric curren t
J,,
flowing on a smooth mathematical surface
SI
enclosed in
S.
This current is assumed to radiate in free space. The operator
equation for
J,,
is then formally derived by setting the
component tangential to
S
of the sum of the incident electric
field and the electric field due to J,,, both calculated with the
body absent, equal to zero on S . Similarly, in considering
scattering from a material body, we simulate an equivalence
for the region exterior to S by means of current J,, as we d o in
the conducting-body case and, in addition, simulate an
equivalence for the region interior to the body by means of a
fictitious equivalent current J,, flowing on a smooth mathe-
matical surface
S ,
enclosing
S .
This current is assumed to
radiate in an unbounded space fi l led with the medium
composing the object . The operator equations for
J,,
and J,,
are then formally derived by an enforcement of the boundary
condition, namely, the continuity of the tangential components
of the electric and magnetic fields across S , which leads to a
set of two operator e quations to be satisfied by the fields in the
two simulated equivalent situations. It should be pointed out
that it is certainly not claimed that exact solutions to the
suggested operator equations are guaranteed for any selection
of S, and
S,.
The existence of an exact solution is intimately
related t o the analytic continuability of the scattered fields
toward the interior region and of the internal field toward the
exterior region. Exact solutions, if they exist, are actually
equivalent currents, which produce the true fields in the
respective regions. The existence question will be addressed
further in Section
111.
To solve the proposed operator equations, we can apply a
method of moments numerical solution. Being numerical, our
solution will never be exact whether a mathematically admis-
sable solution exists
or
not. Our objective is thus to match the
boundary condit ion to some desired com putational accuracy
and this can be effected for certain cho ices of
Si
and S, even if
the existence of the solution cannot be guaranteed from a
strictly mathem atical point of view . In pa rticular, the choice of
impulsive sources as expansion functions for the unknown
currents is well-suited. Good resu lts can be obtained using an
expansion of impulsive currents that l ie a distance away from
the surface because the fields these currents generate on the
surface constitute a basis of smooth field functions. Being
smooth field functions, they are suitable for representing
smooth quanti ties on the boundary and a re l ikely to render the
final solution ac curate, not only in the far zone but in the near
zone and on the surface as well. The notable advantage of the
displaced implusive curren ts is that they not only yield sm ooth
field functions on the surface but also enable us to determine
the fields anywhere analytically. T he quality that the fields are
known anywhere analytically is appealing as one can save
laborious surface current integrations when calculating the
fields at the various stages of the solution. Note that there are
quite a few field calculations involved. First, when construct-
ing the generalized impedance matrix. Second, when testing
the solution by checking the degree to which the required
boundary conditions are satisfied over a denser set of points on
the boundary. Third, when computing field-related quantities
of interest after the solution has been established. Further-
more, since we are actually using a basis of smooth field
functions for representing fields on the boundary surface, a
simple point-matching procedu re can be conveniently adopted
for testing. Notice that the attractive combination of an
impulsive current expansion and a point-matching testing
procedu re which rende rs the solution trivial canno t be success-
fully applied to the standard surface formulations. First,
impulsive currents on the surface would inherently yield poor
surface and near-zone field approximations. Second, even if
one is interested merely in far-field quantities, one should
refrain from testing procedu res, such as point matching, which
give empha sis to surface quantities but rather resort to a testing
procedure that will average out the inaccuracies.
As already stated, our objective is to match the boundary
condit ions over the surface to some desired accuracy. In our
solution, however, we force the boundary condit ions to be
obeyed only at a finite number of points on the boundary
surface. Surely, the field between the match points might
happen to be quite different from what is required by the
boundary conditions. Therefore, the convergence of the solu-
tion must be validated from a chec k on how well the boundary
condit ions are matched between the points. A question that
naturally arises is the relationship between the error in the
boundary condition match and the errors in the exterior
scattered field and in the interior field. We are not aware of a
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LEVIATAN et al. GENERALIZED FORMULATIONS FOR ELECTROMAGNETIC SCATTERING
1725
free space
(po, o)
and excited by impressed sources
(Ji, M i ) .
Harmonic eJwtime dependence is assumed and suppressed.
The geometry of the problem is shown in Fig. 2. The
permeability of the material composing the body is
p
and its
permittivity is E . Both
p
and E are considered complex to
account for dissipation. T he total field on and exterior to S is
the sum of the incident field
Elnc,
i ')and the field
(ES,
H s )
scattered from the material body. The field inside the material
body is denoted by
E,
H) .
We primarily seek the scattered
field in the region exterior to the obstacle as well as the field
inside the obstacle.
The proposed m ethod for solving the considered problem is
to set up two simulated equivalent situations, one for the
region exterior to S and the other for the region interior to S ,
by means of two electric surface current distributions. In the
simulated equivalence for the exterior region, we employ the
model used in the metallic case and shown in Fig.
3 .
The
scatterer is thus replaced by free space with yet unknown
fictitious surface current distribution
J,;
flowing on a smooth
mathematical surface Si enclosed in S . We denote the total
field on and exterior to S in the simulated equivalence for the
region exterior to S shown in Fig. 3by Einc+ E(J,,), Hinc+
H(J.$J),
where
(E(J,;), H(J,;))
is the field due to
J,;,
calculated
in free space. This total field is simulating the total field
(Einc
+ E , HI '
+
H S )
present in the region exterior to
S
in the
original situation shown in Fig. 2 . Similarly, in the simulated
equivalence for the interior region, shown in Fig.
4 ,
the
impressed sources are removed, the exterior region is filled
with homogeneous material indentical
to
that composing the
object, and yet unknown fictitious surface current J,, is
distributed on a smooth mathematical surface S , enclosing S .
This current, radiating in an unbounded homogeneous medium
of constitutive parameters p and e , is assumed to simulate the
electromagnetic field inside the material body. W e denote the
interior to S in the simulated equivalence for the region
interior to S shown in Fig. 4by (E(J,,), H(J,,)). This field is
simulating the field (E, H ) present inside the body in the
original situation shown in Fig.
2.
The question as to the existence of current J,; and J,, for
arbitrarily selected surfaces S; and S, which exactly produce
the scattered field (E,, H ) on and exterior to S and the
field
(E,
H ) inside
S ,
respectively, will be discussed later.
Like in the metallic case, it is impossible to guarantee the
existence of such currents, in general, from a strictly
mathematical point of view. Again, we will assume f or present
purposes that for the considered choices of inner and outer
surfaces S; and So there exist current distributions J,; and J,,
which produce the true fields in the respective regions. Note
that in this event the currents JSiand J,, are in fact equivalent
currents.
Hence, for such inner and outer surfaces S I and S o , he two
simulated equivalent situations can be pieced together by
enforcing the simulated fields in the two regions to obey the
continuity conditions for the tangential components across the
material boundary S . This leads to the o perator equations:
iix [E(JSi)-E(J,,)] = x E' ' on S
(2)
~ X [ H ( J , ~ ) - H ( J , , ) ] = i i x H I n C n S
3 )
unbounded homog eneous space
P
, * )
mathemat ical closed
surface So
/
mathemat ical closed
sur f ace
S
Fig. 4.
Simulated equivalence for region interior
to
S .
where
ii
is a unit vector outward normal to
S .
Clearly, if
2 )
and 3 ) are satisfied, then by uniqueness [9, sec.
3-31,
the
electromagnetic fields
(E(J,,), H(J,,))
in the region exterior to
S
and (E(J,,), H(J,,)) in the region interior to
S
will be
exactly equal to the true scattered and total fields in these
respective regions. Equations
(2)
and (3) thus constitute a
generalized set of coupled op erator equation for the problem
of Fig. 2in which
Elnc,
Inc) s known and J,, and J,,, for
given
SI
and
S o ,
respectively, are the unknowns to be
determined. On ce these currents are found, the analysis of the
problem is completed as field and field related quantities can
be readily calculated.
111. EXISTENCE
In this section, we first examine the strictly mathematical
requirements that guarantee the existence of current distribu-
tions J,, and J,, on arbitrarily selected S , and So , which
produce the true fields in their respective regions.
As discussed by Millar
[lo],
the exterior scattered field in
both the metallic and material cases can be continued
analytically into the region interior to S provided that in this
process no singularity of the exterior scattered field is crossed.
Note that the scattered field must have singularities within (or
on) S for otherwise it would vanish identically
[
1 1 1 . The
problem of locating singularities of the exterior scattered field
has been studied by Millar
[121.
The location of the singulari-
ties depends on the form of the scatterer surface and the
smoothness of the incident field on this surface. Therefore,
each problem would dem and detailed consideration on its own
merits. Similarly, the interior field in the m aterial case can be
continued analytically into the region exterior to
S
provided
that in this process no singularity of this field is crossed. The
internal field must also have singularities outside (or on) S for
otherwise it would vanish indentically. Again, the location of
these singularities depends on the form of the scatterer surface
and the smoothness of the incident field on this surface and
each problem would d emand detailed consideration on its own
merits. Notice that the continuation of the scattered field
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IEEE TRANSACTIONS ON ANlENN AS AND PROPAGATION, VOL.
36,
NO.
12.
DECEMBER 1988
toward the region interior to
S
is effected in free space with the
scatterer absent. Similarly, the continuation of the internal
field toward the region exterior to S is effected in homogene-
ous unbounded space filled with the same material composing
the scatterer. Also, each analytic continuation, if exists, is, of
course, unique.
As an example, consider the simple two-dimensional
scattering problem of a curr ent filament situated at distance p
from the axis of a perfectly conducting cylinder of circular
cross section of radius a (a
< p ) . The solution to this
problem is analytically derivable and can be found in [9, sec .
5-91. Applying large order asymptotic expansions for Bessel
and Hankel functions, it can be readily shown that the series
[9, eq.
(5-120)]
representing the scattered field on and exterior
to the cylinder p
a
is also uniformly convergent in the
interior annular region
a 2 / p
< p
6) [lo]. For this case, the expansion in
terms of Mathieu functions conve rges uniformly on and within
the cylinder as far as the interfocal segment. Thus excluding
the interfocal segmen t, the analytic continuation of the exterior
scattered field is valid everywhere throughout the interior
region.
We now assume, without further comment, that the scat-
tered field (ES, Hs) can be analytically continued to some
extent into the region inside S and, in the material case, that
the internal field
(E, H)
can be analytically continued to some
extent into the region exterior to S. Notice again that the
continuation of the scattered field towa rd the region interior to
S is effected in free space (with the sca tterer absent) and that of
the internal field is effected in homog eneou s unbounded space
filled with the same material composing the scatterer. We
denote the former by (E Sp, Hp)and the latter by (EP,
HP)
as
shown, respectively in Figs. 5and
6.
Now given a smooth
mathematical surface SI inside S lying within the region
through which the continuation of the scattered field (ESPP,
Hsp)
is valid an d enclosing the re gion containing the singulari-
ties, we can use the equivalence principle [9, sec. 3-51 to set up
an equivalent situation. A pertaining illustration is shown in
Fig. 5 . Let the original scattered field (ES, HS) exist on and
external to S, the analytic continuation of the scattered field
(ESPp, p)exist between S and SI, and let the sou rce-free field,
denoted by (ES,H),having a tangential electric field ove r
SI
equal to that of ESP, exist internal to SI . To suppor t these
fields, there must be a surface equivalent current J,, over S, to
account for the discontinuity in the magnetic field across
SI .
This current is
(4)
where
fi,
is a unit vector outward normal to
SI.
From the
uniqueness theorem [9, sec.
3-31, we know that the field
interior to SI produced by J,, radiating in free space will be
(E,H),
between
SI
and
S
the field will be ESP,H), and
farther out , exterior to S, the field will be (ES, H). In a
J,,
=
fi,
X (HSP
-
HS
on
SI
u n b o u n d e d h o m o g e n e o u s s p a c e
P o C O )
ECE ,C.
??I
m a th e m a t i ca l c l o se d
surface S
m a th e m a t i ca l c l o se d
surface
S ,
Fig. 5 .
Analytic continuation
of
scattered field towards region interior to
S.
u n b o u n d e d h o m o g e n e o us sp a ce
( P *
( N '
+
N )/2 field points
r ; ,m = 1 , 2 , - . . , N S o n S . T h e r e s u l t i s
[ z ~ I & =
B
(24)
where
L L J
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LEVIATAN er al . . G EN ER A LI ZED FO R M U LA T I O N S FO R ELEC TR O M A G N ETI C SC A TTERI N G
1729
In
(24), [Z,] i s a 4N by 2(N' + N ) generalized impedance
matrix, is a 2( N' + N )-element generalized unknow n
current column vector, and
is a 4Ns-element generalized
voltage source column vector. In (25), the matrices [Z& ],
p ,
q =
1 ,
2, are precisely the matrices introduced in (19) and
defined in detail thereafter. Specifically, each
[Z ,q]
enotes
an Ns by NI matrix whose (m, n ) element is the
iirn
component of the electric field at observation point rk on S
due to a current element I l b of unit moment
Zl;, =
1).
Similarly, each [ 2 ] , p , q = 1, 2, denoted an Ns by N
matrix whose (m, n) element is the negative of the i:,,
component of the electric field at observation point r;, on S
due to a current element II;, of unit moment (Zlf, = 1).
Further, each [ZLp,], , q
=
1, 2, denotes an S y
N'
matrix
whose (m, n) element is the ii,, component of the magnetic
field at observation point
r;,
on
S
due to a current element
ll;,
of unit moment
Z/q,l
=
I). Similary,
[Z','pq],
,
q
=
1, 2 ,
denotes an N by
NI'
matrix whose (m,
n)
element is the
negative of the i;, component of the magnetic field at
observation point r; on S due to a current element llf,, of unit
moment (I/;,
=
1). In (26), the vectors ,q
=
1 ,
2 ,
are
precisely the vectors introduced in (20) and defined in detail
thereafter. Specifically, each T; enotes an -element colum n
vector whose nth element is
I/:,.
Similarly, each F: q
=
1,
2, denotes an NI'-element colum n vector whose nth elemen t is
I/;, , . Finally, in (27), the vectors Vep , = 1, 2, are precisely
the vectors introduced in (21) and defined in detail thereafter.
Specifically, each
vep
enotes the Ns-element colum n vector
whose mth element is the negative of the i;,,, component of
E' ' at observation point rk on S . Similarly, each
Fhp,
p =
1 ,
2, denotes an Ns-element column vector whose mth element is
the negative of the
iirn
omponent
HI '
at observation point
rk
on S .
Having formulated the matrix equation
(lo),
the unknown
current vector can be found in a simple manner. If the
boundary condition is imposed at
Ns = 1
2(N'
+ NI')
points
on S , then the solution will be, in analogy to (22),
If, on the other hand, the boundary condition is forced at N
>
1/2(N1 +
N )
points on S , then the solution will be, in
analogy to
(23),
This completes the solution of matrix equation (24). Once the
unknown current is derived, either from (28) or (29), one can
readily proceed to evaluate an approximate scattered field
(E ', H ')
in the exterior region, an approximate field
(E ,
HI') in the interior region, and, of course, any other field-
related quantity of interest.
VI.
NUMERICALESULTS
A
versatile computer program has been developed using the
formulation of the preceding se ction. To check the accuracy of
the suggested method, we consider a conducting sphere, a
conducting cylindrical rod with rounded ends whose axis of
symmetry coincides with the z axis, and a dielectric sphere
illuminated by an incident plane wave of unit magnitude.
E' '
= U, exp ( - j k , z )
(30)
1
70
HI '=
uy xp (
- j k ,z )
(31)
propagating in the z direction. For the spherical cases, the
exact solution can be foun d in
[9,
sec. 6-91. For the conducting
cylindrical rod with rounded ends case, a numerical solution is
available in [141. Some computational results obtained with
the program are given in this section and compared with the
available solutions. To limit the data displayed, respresenta-
tive results will be shown, without loss of generality, in the
principal xz plane.
The location of the sources may affect the rate of conver-
gence. Based
on
previous studies
[4]-[7],
one would expect
the numerical results to converge faster to a sufficiently
accurate value when the sources are situated on surfaces
concentric with S and of figure similar to
S .
For the spheres,
the mathematical surfaces S , and So are thus taken to be
spherical surfaces of radii r' and rI1, respectively, concentric
with S . The current elements are evenly spaced along the
latitudinal and longitudinal lines of the respective spherical
surfaces. The match points are also evenly spaced on S . It was
found that for a sphere of radius rs selections of
r
between
0.2rs and 0.W and of rl 'greater than 1 5rs have a comparable
rate of convergence. In contrast, the rate of convergence
deteriorates when the inner sources approach either the sphere
center or the sphere surface, and when the outer sources
approach the sphere surface. For the numerical examples, the
inner source points are located on a spherical surface of radius
rl =
0.2rS and the outer source points are located on a
spherical surface of radius
rl' =
2.0rs.
In
line with the above
criteria, for a conducting cylindrical rod with rounded ends of
diameter
d'
and length
I
the mathematical surface
S,
is taken
to be a finite hemisphere-capped cylinder of diameter 0.2d'
and length 1 - 0.8d (i.e., the distances between the centers of
hemispherical caps of Si nd of S are equal). Furthermore,
with regards to the option of imposing the boundary condition
in the least square error sense, it was empirically found that,
although in some cases one can achieve the same accuracy
using fewer sources, thereby gaining the advantage of invert-
ing smaller matrices, in general this option is redundant. In the
metallic case, we thus take an equal number of inner source
points
N'
and match points
Ns.
In the material case, w e take
an equal number of inner source points NI , outer source points
N , and match points
Ns.
This common num ber is denoted for
convenience by N. ote that in the material case the value of
N
represents twice as many sources as in the metallic case. It
should be remarked that the sources do not have to be split
equally between the inner and outer regions. Other combina-
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.......
N = 16
.25
N: 6
_ _ _
\
0
45 90
135 I80
e ( d e v p s )
Fig.
7.
Plots
of
boundary condition error AE versus 8 in x z plane
for
metallic sphere
of
radius
r s =
0.2X, for various numbers of sources and
match points
N .
t ions can be used and may even yield a more rapidly
converging solution. Clearly, in any event, o ne should test the
solution by increasing the number of sources and match points
and verify the fulfillment of the boundary conditions between
the match points. Furthermore, for any desired quantity of
interest , one should examine the numerical convergence by
comparing the results for an increasing number of sources and
match points. If the compu ted results are sufficiently close, the
solution can be taken as satisfactory.
A . Perfectly
Conducting
Sphere
Results for the problem of plane wave scattering by a
perfectly conducting sphere are shown in Figs.
7-10.
The
conducting sph ere in Figs. 7-9 is of radius rs = 0.2X, where X
is
the wavelength in free space. In F ig. 10,a larger conducting
sphere of radius rs = 1 .OX is examined.
First, we study the convergence of the boundary condition
error AE,, defined by
(f ix
E '+ ElnC / n S
IEincI
E , = (32)
This quantity reveals how well the boundary condition is
satisfied between the match points. Plots of AI? as a function
of the polar an gle
8
in the xz plane for various values of the
parameter
N
re presented in Fig.
7 .
Cases considered are
N
= 16, 25, and 36. T he boundary condit ion error, which by (1)
is zero at the match points, increases smoothly and reaches a
maximum between the points. A s the number of sources and
match points increases, the m aximum of
AEbc
on the surface
falls sharply. No te that even forN s small as 36 the maximum
is smaller than 0.3 5 percent. Th is nature of convergenc e has
been observed in other cases involving spheres
of
other radii
.......
N = 4
~ N.16
N.36
_ _ -
**** exac t
0
45
90 135
180
e d e g m s )
Fig. 8.
Plots of surface current Je on a metallic sphere of radius r s = 0.2X
versus
8
in
xz
plane, for various numbers
of
sources and match points
N.
......
N.8
N.16
_ .
.36
****
e x o c i
o m i
I
I
154
00
0 4 5 90 135 180
9 d e g r e e s )
Fig. 9.
Plots of scattering cross section
U
versus
8
in
x z
plane for various
numbers
of
source s and match points
N or
case
of
metallic spher e of radius
r s =
0 . 2 h
and when the sources were distributed differently. Thus, by
forcing (1) to be obeyed at a sufficiently dense set of points on
S ,
(1)
can be satisfied within a low error between the match
points as well, and consequently fields and field-related
quantities of interest can be approximated to a satisfactory
degree of accuracy.
Fig. 8shows plots
of
the
8
componen t of the surface induced
current J given by
J = i x (H '+ HInc)
(33)
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GENERALIZED FORMULATIONS FOR ELECTROMAGNETIC SCATTERING
1731
160
120
x
\ 0
40
0
N.25
N
=49
__
N.100
**** e x o c t
.
.
.
_ _ - _
0
45
D O 1:15 1 8 0
0 (dryrce.)
Fig.
10.
Plots of scattering cross section U versus 6 in xz plane for various
numbers
of
sourc es and match points Nfor case
of
metallic sphere of radius
rs = 1 O X
versus 8 in the xz plane for various values of N . Here, the
interval from
0
to 90 on 8 is in the shadow region of S
while 90 to 180 interval is in the lit portion of
S .
Note
that for N = 36 the results are in excellent agree men t with the
exact eigenvalue solution [9, sec. 6-91. It should be remarked
that the accuracy obtained here is equivalent to that obtained
by Rao et al. [15] using 96 t r iangular patches to model the
sphere.
Next, we compute the scattering cross section defined by
IE
( E i n c
U =
lim 4ar2
7- m
(34)
where r is the radial distance from th e origin. Plots of U versus
8
in the
xz
plane for various values of N are shown in Fig. 9
and compared with the exact solution. Again, very good
agreement with the exact solution is seen for N = 36.
Finally, we co mpu te the scattering cross section (34) for the
larger sphere. Plots of
U
versus 8 in the xz plane for various
values of N are depicted in Fig. 10 and compared with the
exact solution. Of course, we expect that more sources wil l
now be required to render the solution accurate. Here the
results converge to the exact solution for
N
not larger than
100.
B. Perfectly Conducting Cylindrical Rod with
Hemispherical Caps at the Ends
Results for the problem of plane wave scattering by a
perfectly conducting cylindrical rod with hemispherical caps at
the ends are shown in Fig. 11. The rod is of diameter
d' =
0.4X and length I =
1.5h.
Plots of
U
as a function of 8 in the
xz
plane for various values of
N
are dipicted in F ig. 11and
compared with Andreasen's numerical result [141. Observe
0 D O
0 7 5
0 60
4
\ o 45
0 JC
0 1:
0 O
N.65
N.96
=192
* *
*
Andreasen
. ___.
..
_ _ _ - -
0
45 00 135 180
9
(dcyrerr)
Fig.
1 1
Plots
of
scattering cross section
U
versus 0 in xz plane for various
numbers of sources and match points for case of perfectly conducting
cylindrical rod with rounded ends
of
diameter
d
=
0.41
and length I =
1.51.
that for N = 192 our result is in excellent agreement with
Andreasen's result. Note also that even for N as small as 96,
our result is quite close to that of Andreasen.
C. Dielectric Sphere
Results for the problem of plane wave scattering by a
dielectric sphere are shown in Figs.
12-14. T he sphericai
scatterer considered is of radius rs
=
0.2X. The sphere is of
permeability
p
=
po
and permittivity E = 3eO.
In a manner analogous to that presented in the metallic case,
we first carry out a study of the con verge nce of the boundary
condition erro rs
AE
and AHk defined by
Ifi
x (E5'+E' '-E )I on S
IE' 'I
l fi x (H + HI '- HI1) on S
(H' ')
AEb, = (35)
AHbc= (36)
Plots of U and AHk as a function of 8 in the xz plane for
various values of parameter
N
are presented in Fig. 12.Cases
considered are N
=
25, 36 , and 49. The boundary condition
er rors
AE,
and
AHbc,
which by (2) and (3) are zero at the
match points, increase smoothly and reach a maximum
between the points. As the numb er of sources and match points
increases, the maxima of AE, and AH, on the surface fall
sharply. Note that even
for N
as small as 49, the maxima of
AEk
and
AH,
are sm aller than 0.7 percent.
To give some addit ional information on the conv ergence as
the number of expansion functions and match points is
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1732
4
3
L
>
I
0
........
N: 25
~ N = 3 6
N=
49
;
_ _ _ _
, '
IEEE TRANSACTIONS ON ANTENN AS AND PROPAGATION. VOL
36, NO. 12, DECEMBER 1988
0 45 90 I35 I80
O( d e g i r e \ )
(b )
Fig. 12.
(a)
Plots of
boundary condition error
AEh
versus 8 in
xz
plane for
dielectric sphere of radius rs = 0.2X and pe rmittivity t =
3to,
for various
numbers
of
sources and match points
N.
b) Plots
of
boundary condition
error AH h versus
0
in xz plane for dielectric sphere of radius r = 0.2X and
permittivity E = 3t0, for various numb ers of sources and match points N.
increased, we investigate the convergence of the approximate
scattered field to the exact solution on the surface of the
sphere. For this purpose, we define the scattered field erro rs
A E and A H as follows:
(ESr-EZxactIn S
IEinCI
E = (37)
R
I
. 3 1
w 4
- I
........
N =
25
__ N.36
N =
49
8
6
-
X 4
a
2
0
0 45 90 135 180
degrr6i)
( b )
Fig. 13. (a)
Plots
of scattered field error
A E
on boundary versus
8
in xz
plane for dielectric sphere
of
radius
r s
= 0.21 and perm ittivity
t
= 3to ,
for
various numbers
of
sources and match points N. b) Plots
of
scattered field
e r ro r A H on boundary versus 0 in xz plane for dielectric sphere
of
radius
rs
=
0.2X
and perm ittivity t = 3to,
for
various numbers
of
source s and r.atch
points N.
(HSr-H&actI
n S
IHinCI
H = (38)
where
(E ,xact,
,x,c,) enote the exact values of the scattered
field obtained using the result of
[9,
sec. 6-91.
Plots
of A
E
and
A H as a function
of 19
in the xz plane are depicted in Fig. 13.
I ,
Note the convergence of the fields as the number
N
ncreases.
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LEVIATAN et al. : GENERALIZED FORMULATIONS FOR ELECTROMAGNETIC
SCATTERING
1733
0
16
0 0 8 1
o o o ,
, ,
, ,
4 5 9 0 135 180
8
deg iee3 )
The operator equations are solvable by the method of
moments. In particular , an impulsive expansion for represent-
ing the unknown current can be used. The noticeable
advantage of the impulsive expansion is that it enables
us
to
determine the fields anywhere in space analytically, thereby
rendering the quite a few field calculations involved in the
various stages of the solution trivial. At the same time, this
expansion yields accurate results not only in the far-zone but in
the near-zone and
on
the surface a s well because the displaced
impulsive sources generate smooth fields on the surface
suitable for representing smooth quantities on the surface.
Finally, s ince we are using, though indirectly, a smooth
expansion for the fields
on
the surface, a point-matching
procedure can be selected for testing.
Th e proposed method has already been applied successfully
to two-dimensional waveguide
[4]-[6]
and free-space
[7]
scatter ing problems,
where the various fields have been
approximated using filamentary currents situated on suitably
selected surfaces. The numerical procedure was found to be
simple to apply, of wide range of applicability, and rapidly
converging. An application of this method to three-dimen-
Fig. 14. Plots
of
scattering cross section U versus
e
in xz plane
for
various
sional scattering problems where the various fields are
approximated using impulsive
current
elements has been
presented in this paper. The numerical solution for three-
numbers
of
sources and match points
N or
case of dielectric sphere of
radius rs = 0 2X and permittivity E
=
3to.
A similar convergence to the exact solution is also found for
the scattered field outsid e the sphere and for the field inside the
sphere. These plots will not be shown here.
Finally, the scattering cross section given by (34) is
computed for the dielectric sphere case. Plots of
U
versus
0
in
the xz plane for various values of N are shown in Fig. 14 and
compared with the exact solution. Very good agreement with
the exact solution is seen for N =
36.
VII . CONCLUSION
Generali zed formulations for three-dimensional problem s of
scattering by perfectly conducting and homogeneous material
bodies have been proposed. The innovative approach is to use
the field of a fictitious electric current flowing o n a mathe mati-
cal surface enclosed within the body to simulate the exterior
scattered field. In the material body case, the field of an
additional fictitious electric current flowing on a mathematical
surface enclosing the body is used to simulate the internal
field. Application of the boundary conditions at the body
surface leads to alternative operator equations to b e solved for
the unknown currents which facilitate the fields in the various
regions via the magnetic vector potential integral.
Attention has been paid
to the questions of existence and
uniqueness of the solution. It is found that from a strictly
mathematical point of view, one cannot, in general, guarantee
the existence of current distributions, for arbitrary selected
surfaces inside and outside the body, that will produce the true
fields in the respective regions. The existence of an exact
solution is intimately related to the analytic continuability of
the scattered field towards the interior region and the
internal field, in the material body case, towards the exterior
region. Exact solutions, if they exist, are actually equivalent
currents which produce the true fields in the respective
regions.
dimensional problems is also simple to execute, rapidly
convergin g, and general in that bodies of smooth but otherwise
arbitrary surface both lossless and lossy can be handled
effectively. It should be clear that it is almost impossible to
state a rule of thumb as to the choice of source location and
number.
In
any case, one shou ld test the solution by examining
both the degree to which the boundary conditions are satisfied
over a denser set of points on the boundary and the numerical
convergence of the considered quantity as the number of
sources and match points is increased. Som e choices of source
location may speed up the convergence, but even for choices
less than optimal the solution will usually converge to the
appropriate limiting value without seriously taxing the com-
puting system. In any case, it is immediately known if the
results are inaccurate by using the boundary condition check.
As presented here, the formulation deals exclusively with a
single scatterer. The extension of this formulation to encom-
pass the multiscatterer case, say Mo ne s, is s traightforward. In
this case, the exterior scattered field is simulated by the field
of M sets of sources, each situated inside its corresponding
body, while the field inside each of the material bodies is
simula ted, as before, by the field of an appropriate set situated
outside the body. Boundary conditions are then sim ultaneously
applied at selected points
on
the M surfaces.
The suggested technique is mainly applicable to scatterers
with smooth surface. A lack of smoothness can cause the
method to fail since the fields generated on the surface by the
impulsive current elements that lie a distance away from the
surface are smooth, an d clearly a sum of such sm ooth fields is
not best suited for representing fields near edges where the
fields are singular . This deficiency can in pr inciple be
overcome by incorporating, in addition to the impulsive
current elements, surface currents capable of representing the
correct edge singularity in subdomains near the edges.
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IEEE TRANSAC TIONS ON ANTENNAS AND PROPAGAT ION, VOL. 36, NO. 12, DECEMBER 1988
Howev er, it might even be possible
to
use impulsive Sources
near
the
edges to the edge behavior to accuracy that
may be sufficient for engineering need s. These appro aches are
[9] R.
F.
Harrington,
Time-Harmonic Electromagnetic Fields.
New
York: McGraw-Hill , 1961.
R. F. Millar, Rayleigh hypothesis in scattering problem,
Electron.
Lett.,
vol. 5 ,
pp.
416-418, Aug. 1969.
[ IO ]
currently under investigation by us.
REFERENCES
111
I21
131
A . J. Poggio and E.
K .
Miller, Integral equation solutions of three-
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Computer Techniques
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Electromagnetics,
R. Mittra, Ed. Oxford, England: Pergamon,
1973, ch. 4.
C. Muller,
Foundations
of
the Mathematical Theory
of
Electro-
magnetic Waves.
R. F. Harrington,
Field Computation by Moment Methods.
New
York: Macmillan, 1968.
New York: Springer Verlag, 1969.
[ I ]
[
121
R. C ourant and-D. Hilbert ,
Methods
of
Mathematical Physics,
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2 .
R. F. Millar, The Rayleigh hypothesis and singularities of solutions to
the Helmholtz equation,
Bull. Radio Elec. Eng. Div. Nut. Res.
COumnC. Ca n. .
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DD.
23-27. Aor. 1970.
New York: Interscience, 196 2, pp. 317-318.
..
J. R. Mautz and R.
F.
Harrington, H-field, and E-field, and
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Elek. Ubertragung.,
vol. 32, pp. 157-164, Ap r. 1978 .
M . G . Andreasen, Scattering from bodies of revolution, IEEE
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vol. AP-13, pp. 303-310, Mar. 1965.
S . M. Rao, D. R. Wilton, and A. W. Glisson, Electromagnetic
scattering by surfaces
of
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IEEE Trans. Antennas
Propagat.,
vol. AP- 30, pp. 409-418, M ay 1982.
[4]
Y. Leviatan, P. G. Li, A. T. Adams, and J . Perini, Single-post
inductive obstacle in rectangular waveguide, IEEE Trans. Micro-
wave Theory Tech.,
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806-811, Oct. 1983.
..
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Y. Leviatan and G. S . Sheaffer, Analysis of inductive dielectric posts
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IEEE Trans. Microwave Theory Tech.,
vol. MTT-35, pp. 48-59, Jan. 1987.
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dielectric cylinders using a multifilament current model,
IEEE
Trans. Antennas Propagat.,
vol. AP-35, pp. 1119-1 127, Oct. 1987.
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matching versus integral equation scattering solutions for a perfectly
conducting bod y,
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pp. 857-865, July 1986.
[6]
Amir
Boag,
for a photograph and biography please see page 1127 of the
October 1987 issue of this TRANSACTIONS,
71
[8]
Alona b a g ,
for
a photograph and biography please see Page 1607 of the
November 1988 issue
of
this TRANSACTIONS.