Overview of MM and UTD Methods at The Ohio State University
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Overview of MM and UTD Methods at The
Ohio State University
EDWARD H. NEWMAN, FELLOW, IEEE, AND RONALD J. MARHEFKA, SENIOR MEMBER, IEEE
Techniques appropriate for low and high frequency numerical A. Overview of the M e t h o d of Moments
evaluation of the scattered fields from complex shapes are dis-
cussed. Object size in terms of wavelengths dictates the type of A basic problem of electromagnetic scattering i s that of
solution to be used. For targets in and just above resonance, the a k n o w n impressed field incident u p o n a m e d i u m which
method of moments (MM) is used. The limitation in this case is
is free space except for a scatterer or inhomogeneitywhich
that the scatterer cannot be too large in terms of the wavelength.
For radar targets in the microwave band, the uniform geometrical i s composed of some combination of perfect conductors
theory of diffraction (UTD) and modifications thereof form the basis and dielectridferrite material. The first step in the MM solu-
for the computation. 'This has the basic limitation that the scatter- t i o n t o this problem is t o use the equivalence theorems [2]
ing mechanisms need to be known and included in the model. t o replace the perfectly conducting surfaces by equivalent
Examples are given for both frequency regimes.
electric surface currents, and the dielectric and/or ferrite
material by equivalent volume electric and/or magnetic cur-
I. INTRODUCTION rents. I n the equivalent problem, the scattered fields are the
free space fields of the equivalent currents. Finding these
The analysis of complex scattering structures can be
accomplished using various numerical methods. Two com- currents i s the main task of the MM solution.
plementary techniques that have found great success in The next step in the MM solution i s t o obtain a set of cou-
many classes of electromagnetic applications are the pled integral equations for the equivalent currents. I n our
method of moments (MM) and the uniform geometrical w o r k these equations are statements of the boundary con-
theory of difraction (UTD). The Ohio State University dition that the total electric field tangential t o the surface
ElectroScience Laboratory (OSU-ESL) has been researching of the perfect conductor must vanish, and that the total
the theories and applying t h e m in the form of user-oriented electric and magnetic fields in the dielectriclferrite body
satisfy the volume equivalence theorems.
computer codes for many years. This paper reviews some
of the effort directed toward the practicular set of problems The final step i n the MM solution i s t o solve the integral
useful for electromagnetic scattering from complex bodies. equations for the equivalent currents. The equivalent cur-
In addition, it i s centered around w o r k specifically used i n rents are expanded in terms of a finite set of N basis or
the computer codes mentioned in this paper. M o r e research expansion functions. The N u n k n o w n coefficients in the
is ongoing in these areas at OSU-ESL and elsewhere. expansion are found by enforcing N weighted averages of
the integral equations t o be valid. This transforms the inte-
gral equations into a system of N simultaneous linear alge-
I I. METHOD
LOW-FREQUENCY OF MOMENTS
TECHNIQUES
braic equations which can be compactly written in matrix
This section will present an overview of recently devel- form as
oped low-frequency m e t h o d of moments (MM) [I] tech- [ZII = v. (1)
niques at The O h i o State University ElectroScience Labo-
ratory. These include MM solutions for the modeling of Here [Z] is the order N impedance matrix, V i s the N ele-
arbitrary 3D perfectly conducting surfaces, T M scattering ment voltage vector, and I i s the current or solution vector
from a general 2D cylinder, and scattering from material whose N elements contain the coefficients in the original
coated edges. expansion for the equivalent currents. Once [Z] and Vare
known, /can be found using standard techniques of matrix
algebra, thus providing an approximation t o the equivalent
Manuscript received M a y l , 1988;revised November 4,1988. This currents.
work was supported by numerous U.S. government agencies A n advantage o f MM solutions i s that they are capable
including the JointService Electronics Program (Contract N 0 0 0 1 4 - of producing very accurate results. The coupled integral
78-C-0049between The Ohio State University Research Foundation equations are essentially exact, and the MM provides a
and the Program.
The authors are with the Department of Electrical Engineering,
direct numerical solution of these equations. No a priori
The Ohio State University, Columbus, OH 43212, USA. knowledge of the form of the solution i s needed. All phe-
I E E E Log Number 8927975. nomena of the problem, i.e., surface waves, creeping waves,
0 1989 IEEE
0018-9219/89/0500-0700$01.00
700 PROCEEDINGS OF THE IEEE, VOL. 77, NO. 5, M A Y 1989edge conditions, edge or corner diff ractions, etc., are inher-
ent in the integral equations, and will be automatically
included in a well-formulated MM solution. As will be illus-
trated later, a second advantageof MM solutions i s that they
are i n practice applicable t o geometrically complex scat-
terers.
The main limitation of MM solutions is a result of the fact
that the number of terms, N, which must be retained i n the
expansions for the equivalent currents is proportional t o
the electrical size of the body. Thus, as the frequency is
increased N must increase, and at some point the computer
storage or CPU time required t o set u p and solve (1)becomes
prohibitive. For this reason, MM solutionsareoften referred
to as l o w frequency solutions, applicable when the scat- Fig. 2. A polygonal plate is segmented into quadrilateral
terer i s not too large electrically. surface patch basis functions.
B. Polygonal Plate Modeling the solution, or if computations are desired at different fre-
About 13 years ago, an effort was begun t o develop a sur- quencies.
face patch MM technique [3] which w o u l d be able t o model Clearly, the surface patch modeling technique i s of lim-
an almost arbitrary perfectly conducting body. Figure 1 ited use if there is n o reasonable way for a user t o tile a c o m -
shows a surface patch model for a very simple surface, i.e., plicated surface. To alleviate this problem w e employ a
a rectangular plate. The surface is tiled or segmented into polygonal plate model of a complicated surface. The insert
a number of rectangular patches, typically between 0.1 X in Fig. 3 shows a 15-plate model of the Concorde aircraft.
and 0.25 X o n a side. Adjacent pairs of these patches form To describe this surface the user need only describe the
a piecewise sinusoidal, rectangular, surface patch, dipole coordinates of the corners of the 15 polygonal plates. The
basis function, and are indicated i n Fig. 1 by the arrows. code (not the user) then segments each plate into the sur-
face patch basis functions as illustrated i n Figs. 1 and 2. Fur-
ther, the code checks t o see which plates intersect, and
isin k(lll-h)
wrln k h
IO d B / DIV
N O R M A L I Z E D TO 1.342dB/f “,“,‘,”,”,‘,“,’,ED
h h
e =goo
Fig. 1. A rectangular plate is segmented into rectangular
r-4
surface patch basis functions.
Note that there are t w o orthogonal arrays of these basis
functions t o model an arbitrary vector current density o n
the plate. Also, the basis functions are placed on the plate
such that there i s continuityof the longitudinal component
of current (and thus n o line charges o n the plate), the nor-
mal componentofcurrentvanishesattheedgesof the plate,
and the tangential component of current is finite at the
edges o f the plate. Figure 1 also illustrates the shape and
functional form for a typical rectangular basis function. As
illustrated i n Fig. 2, a polygonal plate is i n general seg-
mented i n t o piecewise sinusoidal, quadrilateral, surface
patch, dipole basis functions.
Prior t o analyzing the scattering from a given perfectly
conducting surface, it i s necessary t o tile or segment that
surface into the patch basis functions. For a surface as com-
plicated as an aircraft, an automobile, or a satellite, requir-
ing several hundred patches, this i s an almost intractable TAIL
task if it must be done by hand. Also, a n e w segmentation
Fig. 3. A comparison of computed and measured RCS of
of the surface i s necessary if one wishes t o increase or the Concorde aircraft in the azimuth plane and for hori-
decrease the number of patches t o test the convergence of zontal polarization.
NEWMAN AND MARHEFKA: MM AND UTD METHODS AT OHIO STATE UNIVERSITY 701automatically places overlap surface patch basis functions
at the plate/plate junctions t o insure continuity of the nor-
mal component of current at the junctions. The advantages
of the polygonal plate modeling technique are as follows.
A complicated shape can often be described in terms LOSSY
of a relatively few plates which can be defined by a INHOMOGENEOUS
user without the help of a sophisticated graphics
package. The user is completely divorced from the
details of the tiling of the surface.
From the user’s standpoint, the description of the Fig. 4. Geometry for T M scattering by a general cylinder
composed of perfectly conducting and sheet impedance, as
body i s frequency independent, since the code auto- well as dielectriciferrite cylinders of arbitrary cross section.
matically segments the plates into more (fewer)
patches as the frequency is increased (decreased).
The density of patches o n a given plate can be volumecurrents.Thesefour current distributions are found
adjusted, independently from the density o n other as the MM solution of four coupled integral equations [15].
plates, by simply changing o n e number i n an input As an example of the use and accuracy of the general cyl-
file. inder model, Fig. 5 shows the MM solution for the bistatic
The “Electromagnetic Surface Patch (ESP) Code” [4]-[Ill
MATERIAL COATED ELLIPTIC CYLINDER
i s a user-oriented computer code which implements the
polygonal plate and surface patch MM solution, together a,= 0.4 m
with the ability t o model t h i n wires [12], [13]. Recently, the
ability t o treat t h i n dielectric plates has been added t o the a,= 0.5 m
code. ESP can treat several plates which intersect along a b 2 = 0.361 m
common edge, and can also treat wire t o plate junctions
provided they are at least 0.1 X from the edge of a plate. MM pr=2 . 0
Excitation can be either by a planewave for scattering prob- -EIGENFUNCTION f = 300 Mhz
lems or by a voltage generator in the wire for antenna prob-
lems. ESP can compute most of the quantities of engi-
neering interest including current distributions, and “C
antenna input impedance, efficiency, and mutual coupling.
ESP can compute the amplitude and phase of near or far
zone radiated and scattered fields for all possible polar-
izations of the fields, including cross polarizations and the
radial component i n the near zone. ESP also incorporates
an impedance matrix interpolation scheme which permits
wideband data t o be obtained with reduced CPU time [14].
At a frequency at which the length of the Concorde i s
about 1.8 A, Fig. 3 shows acomparison of t h e c o m p u t e d and
measured (on a scale model of the Concorde) radar cross
section (RCS) of the Concorde in the azimuth plane and for
horizontal polarization. Although the polygonal plate
model of the Concorde may appear crude, reasonable
results are obtained because at l o w frequencies the cur- PHI (DEG)
rents and the RCS are more dependent u p o n theoverall size Fig. 5 . A comparison of the bistatic echo width of a coated
and shape of the body than o n electrically small details of elliptic cylinder computed with the general cylinder MM
the structure. code and an eigenfunction solution.
C. General 2 0 Cylinder
echo w i d t h of a perfectly conducting coated elliptic cyl-
Figure4 illustrates the p r o b l e m o f T M scattering byagen- inder computed by the MM and by an approximate eigen-
era1 cylinder composed of perfectly conducting, inhomo- function solution developed by Richmond [16]. Note that
geneous and lossy dielectridferrite, and sheet impedance the outer surface of the conductor and coating are both
cylinders of essentially arbitrary cross section. The various elliptic, and thus the coating i s of nonuniform thickness.
cylinders may or may not contact each other. A sheet Other recently developed solutions for scattering by
impedance, of which a resistive sheet is a special case, is material-coated perfectly conducting cylinders include
a model for an electrically t h i n dielectric sheet. As described eigenfunction solutions for material-coated elliptic [I61 and
i n Section 11-A, w e employ a straightforward MM solution parabolic [I71 cylinders, in which the coefficients in the
for the general cylinder i n which the perfectly conducting eigenfunction expansion must be determined by the MM.
and sheet impedance cylinders are replaced by 2 polarized Also, Richmond has developed a physical basis function
electric surface currents, the dielectric cylinders are MM solution which he used to analyze the scattering from
replaced by 2 polarized electric volume currents, and the a thin dielectric strip using only three basis functions,
ferrite cylinders are replaced by 9 and p polarized magnetic regardless of the w i d t h of the strip [18].
702 PROCEEDINGS OF THE IEEE, VOL. 77, NO. 5, M A Y 1989D. Material-Coated Edges shown i s a comparison of the computed and measured echo
w i d t h from 8 t o 16 GHz. Below 11 GHz the echo w i d t h is
O n e method for controlling the scattering from an edge
essentially constant, and then at 11 GHz it begins t o oscil-
is t o coat or otherwise place lossy material in the vicinity
late. The explanation for this behavior is that at about 11
of the edge. In order t o study this problem w e have devel-
GHz the first T M surface wave m o d e begins t o propagate
oped MM solutions for T M or TE scattering by a arbitrarily
in the dielectric covered ground plane. It i s interesting that
shaped lossy and inhomogeneous dielectriclferrite cylin-
the presence of the surface wave i s automatically included
der i n the vicinity of a perfectly conducting edge. As illus-
i n the MM solution of the integral equation.
trated i n Fig. 6, the edge is modeled by a perfectly con-
ducting half plane.
Ill. HIGH FREQUENCYTECHNIQUES
This section discusses the high-frequency modeling effort
using the uniform geometrical theory of diffraction (UTD)
at The Ohio State University ElectroScience Laboratory.
First, an overview of the latest theoretical developments
will be discussed. Next, the application of these techniques
t o near zone (nonparallel ray) and far zone (parallel ray)
W modeling will be outlined.
Fig. 6 . Geometry for a T M or TE scattering by a dielectric/
ferrite cylinder in the presence of a perfectly conducting
half plane. A. Overview of the Uniform Geometrical Theory of
Diffraction
The uniform geometrical theory of diffraction (UTD) i s a
I n principle, the material-coated edge of Fig. 6 could be
high-frequency asymptotic solution that follows a scatter-
modeled using the general cylinder code described above.
ing center interpretation of the returns from complex
However,in practicean infinitenumberof u n k n o w n s w o u l d
shapes. It i s based o n t h e w o r k of Keller [23] w h o d e v e l o p e d
be required o n the half plane, and thus the solution w o u l d
the geometrical theory of diffraction, which i s a ray-based
be impractical. Instead, w e employ an MMlGreen’s func-
solution that is a correction t o geometrical optics. Recent
tion solution[l9]forthecoated e d g e i n w h i c h thedielectric1
developments strive to provide uniform solutions such that
ferrite volume i s replaced by equivalent electric and mag-
the total field i s continuous and bounded. This i s accom-
netic volume polarization currents, while the presence of
plished with the use of transition functions that generally
the half plane i s exactlyaccounted for by including the half-
are unity, b u t at shadow boundaries and caustic regions
plane Green’s function in the kernel of the integral equa-
cause the diffraction coefficients t o go t o their proper lim-
tions [20], [21]. The advantage of this technique i s that the
its.
only unknowns in the MM solution are the equivalent cur-
Examples of UTD solutions that are based o n canonical
rents representing the material cylinder. The coated half-
geometries are the uniform diffraction by a wedge [24] and
plane solution has been generalized t o a parabolic cylinder
the uniform diffraction b y a c u r v e d surface[25]. M a n y other
which can model a thick edge [22].
structural features d o not have nice exact solutions. For
The insert in Fig. 7 shows a T M wave w i t h edge o n inci-
those types of problems, various methods of approximat-
dence on a t h i n dielectric slab o n t o p of a half plane. Also
ing the solutions are used. These heuristic methods of
determining new, or modifying existing, diffraction coef-
cy = 2.6 0.538cm ficientsoften producevery useful practical solutions.These
solutions have t o be carefully studied t o gauge the range
of their o p t i m u m usability.
An example of a heuristically derived diffraction coeffi-
HALF -PLANE
- .-l 5 r cient is the scattering from a corner (vertex). A solution that
-
a
works well in the near zone has been formulated from
equivalent currents [26]. This is a nonparallel ray solution.
E -20 It does, however, provide approximate results in the far
w zone. A solution based on extensions of m o d e r n equivalent
I currents concept [27] can be formulated which is a parallel
\
m -25 ray solution and does give a physical optics result near spec-
U
Y
ular [28].
I The double diffraction between edges can contribute in
-30 certain regions of a scattering pattern. O n e example i s in
s the plane of a knife-edged plate for vertical (perpendicular
0 t o the plate) polarization. I t i s the double terms that begin
3 -35 t o provide the null that i s expected in the planeof the plate.
W
A newly developed two-dimensional solution [29] thatworks
-401 1 1 1 11
. l I I ’I J everywhere for the parallel-ray problem gives uniform
8 IO 12 14 16 results. For the other polarization, the so-called edge wave
FREQ. ( G H z ) diffraction (not the same as Ufimtsev’s fringe wave) is an
Fig. 7. A comparison of the computed edge on echo width important contributor in low-level regions, especially in the
for a dielectric slab on the surface of a half plane. plane of the plate [30].
NEWMAN AND MARHEFKA: MM AND UTD METHODS AT OHIO STATE UNIVERSITY 703Dielectric coated surfaces are also of interest i n UTD. An The NEC-BSC version 3 includes at the present time the
approximate specular solution that i s of practical interest following UTD field terms. The direct field from the trans-
has been proposed [31] for a flat plate. I n addition, w o r k i s mitter t o receiver can be included. The fields reflected from
being done o n improving this solution and incorporating a plate, diffracted by a plate edge, diffracted by a plate cor-
surface waves [32]. Dielectric curved surface solutions are ner, doubly reflected between plates, and the fields
also being investigated [33]. reflected-diffracted and diffracted-reflected between
The advantage of UTD i s that it can handle large-sized plates and edges and corners can be included also. In addi-
scattering objects in an efficient manner. The larger the tion, the triply reflected, reflected-reflected-diffracted,
object in terms of a wavelength, in principle, the better the reflected-diff racted-reflected, diff racted-reflected-
solution. Complex objects can be systematically built u p reflected, and the double diffracted fields are under devel-
using the major local scattering centers based o n the dif- opment.
fraction coefficients like the ones discussed here. Higher For curved surfaces the reflected field from the curved
order interactions between various scattering centers can surfaces and the flat end caps are provided. The creeping
be included by combining the individual mechanisms (at waves for the elliptic cylinders are included and the creep-
least outside the transition regions). Numerical efficiency ing waves for the other curved surfaces are under devel-
results from the assumption that the important scattering opment. The diffracted fields from the curved surface and
centers can be found and that the ray paths can be quickly end cap junctions are included. The curved edge caustic
traced [34]. I t i s not necessary t o integrate the surface cur- corrections are presently not included. In version 2, the
rents o n large bodies. double interaction between parallel curved surfaces was
The disadvantages of UTD hinge o n the fact that it is included for some limited cases. Plate-to-curved-surface
assumed a priori that the most important mechanisms are interaction was included in version 1 for limited cases [40]
known. If some terms are left out o n purpose or by over- and will eventually be incorporated into version 3.
sight, the accuracy of the total solution may diverge. Usu- I n the near zone situation, the geometrical optics needs
ally this occurs only over a small region of a pattern and in t o be included even f o r t h e flat plate. I n the limit as the range
most instances a b o u n d o n the accuracy is k n o w n by the gets large for a flat straight-sided plate, the reflected field
level of the discontinuity that results. As more terms are and the edge-diffracted field start t o get canceled by the
added t o the solution, the more time it takes to trace and corner-diffracted fields. The residual amount, in principle,
shadow the rays. I n addition, the accuracy of the solutions goes t o approximately the physical optics result in the spec-
depend o n knowing the diffraction coefficients. For engi- ular region. The present near zone corner diffraction coef-
neering purposes, these are usually k n o w n except in rather ficient formulation approximately achieves this, with
small regions for higher order terms. improvements being investigated.
Disadvantages of doing the far zone scattering problem
B. Near Zone Modeling using the nonparallel ray approach i s primarily numerical.
For many years the OSU-ESL has been designing and W h e n distances get large, the numerical accuracy o n most
applying UTD t o practical problems. The UTD concept has machines will eventually not be able t o correctly calculate
been embodied i n many user-oriented computer codes. I n the fields.
the area of near zone (nonparallel ray) target modeling, a As an example of the near zone scattering prediction
computer code entitled the NEC-basic scattering code capabilities of the NEC-BSC [37], the geometry i n Fig. 8 i s
(NEC-BSC) i s being developed. I t can be used t o model
complex scattering structures w i t h basic shapes. For ver-
sion 2, the shapes are multiple flat plates and finite elliptic
cylinders [35], [36]. Version 3, which i s under development
[37], also allows the useof multiple sectioned cone frustums
and finite composite ellipsoids [38], [39]. The code can be
used for the prediction of radiation patterns of antennas in
the nearzoneof complex structures with theobserver being
either in the near or far zones. The source can be fixed or Fig. 8. Cone cylinder geometry.
moving. The near zone of the object is defined as the region
within 2D2/hoftheobject, w h e r e D i s itsoverall dimension. used. The source and receiver are 864 in. away from the
The NEC-BSC i s not an antenna code, that is, it does not object. The calculated result i n Fig. 9 can be compared w i t h
determine information such as current distributions or the measured result taken at Pacific Missile Test Center in
radiation resistance of the antennas. I t determines the scat- Fig. 10 at 10 GHz for horizontal polarization. Note that the
teringfrom astructure produced by an antennawith agiven specular scattering from the side of the cylinder i s much
current distribution or pattern factor. It can interface with broader and lower than w o u l d be expected if the source
a method of moments code t o provide the needed infor- and receiver were truly in the far zone of the cylinder. The
mation. Coupling between antennas can be found using code predicts this near zone effect very well.
the vector effective height of the receiver. The coupling
between antennas i s the spacial coupling only. It can be
C. Far Zone Modeling
used t o approximately represent the situation in a spherical
range. It i s necessary t o d o the same sort of normalization A computer code based o n parallel ray theories can be
procedure that needs t o be d o n e in the measurement sit- written. This avoids the normalization and large range
uation. For example, a sphere can be used t o calibrate the numerical questions discussed above. I n addition, some o f
scattered fields because its value i s known. the theories can be formulated in the far zone in a simpler
704 PROCEEDINGS OF THE IEEE, VOL. 77, NO. 5, MAY 1989approximate morecomplexones. The basic shapes are mul-
2 tiple-sided flat plates, multiple-section cone frustums, and
finite-composite section ellipsoids at the present time.
The RCS-BSC will include some plate-to-plate interac-
tions. In preparation of understanding what needs t o be
included in these situations in a UTD sense, a two-dimen-
sional code has been developed [44]. This code includes u p
to and including all third order interactions between two
plates. In addition, some fourth order interaction terms
have been included. This code is ideal for analyzing
dihedral configurations. It i s necessary t o include only the
diffraction from the edges and their reflection in the other
-40t ' I
plate. In addition, double diffraction between edges w i t h
possible intervening reflections have been included. The
newly developed far zone double diffraction coefficients
-SOL I I I t I [29]are used. With this formulation for double, it i s not nec-
180 I20 60 0 essary to use false edges, or imposed edges as some people
TAIL ANGLE NOSE call them. As an example, a 9-in. two-dimensional dihedral
Fig. 9. UTD calculated near zone backscatter pattern from has been analyzed in the E-plane using UTD as shown in
a cone-cylinder geometry at 864-in. range, horizontal polar- Fig. 11 and MM as shown in Fig. 12 at 10 GHz. Note that the
ization, and 10 GHr.
two agree very well even in the low-level regions.
The interaction between plates and curved surfaces can
also be handled in a similar way as in the near zone [45], [46].
: 0 2
0
+-I /--
-5
-
m
D
cn
-20
U
(r
-40 0
5 -20
W
-30- 1
-60
180 90 0
TAIL ANGLE NOSE -401 I I I I 1 I
0 90 180
Fig. 10. Measured backscatter pattern from a cone-cylin- ANGLE
der geometry at 864-in. range, horizontal polarization, and
10 CHz. Fig. 11. Backscatter from a two-dimensional 9-in. dihedral
at 10 GHz in the E-plane calculated using UTD.
manner. It is still necessary to provide transition functions
that correct the ray picture in some cases. For example, the
scattering from a circular rim can be found in closed form
[41] for the bistatic scattering case. In addition, the bistatic
scattering in the specular region of the side of a cone frus-
t u m can be found [42]. With these techniques the bistatic
scattering from a cylinder or cone frustum of any size can
be calculated by just summing the scattering from u p t o
four points.
The corner diffraction coefficient can also be taken care
of using modern equivalent current concepts [27], [28]. With
this technique the bistatic scattering from a multiple-sided
flat plate can be found by summing terms from only the
corners of the plate or complex structure made u p o f plate
facets.
A computer code called the radar cross section-basic 0 90 180
scattering code (RCS-BSC) is being developed [43], which ANGLE
will provide the far zone scattering from complex struc- Fig. 12. Backscatter from a two-dimensional 9-in. dihedral
tures using these techniques. It uses basic shapes t o at 10 CHz in the E-plane calculated using MM.
N E W M A N A N D MARHEFKA: Mh4 A N D UTD M E T H O D S A T OHIO STATE UNIVERSITY 705As in the case of plate-to-plate interaction, these terms can IO
be formulated in a numerically efficient way without mul-
tiple integrations being needed.
As an example of h o w UTD can be used t o analyze the 0
far zone scattering from a model of an aircraft, a Boeing 737
has been analyzed using the techniques discussed above.
The computer model i s shown i n Fig. 13. It i s composed of
- 10
- 20
- 30
- 50
')Ir 1 1 ' ' 1 ' 1 ' I1
0 90 180
TAIL ANGLE NOSE
Fig. 14. First order UTD backscattered result for horizontal
polarization in the azimuth plane of a 1/20 scale model of
a Boeing 737 without engines at 10 CHz.
Fig. 13. Computer model of Boeing 737 aircraft.
a composite cone frustum model for the fuselage and
engines. The wings and stabilizers are modeled as flat plates 5
m
with thick edges. Obviously, this is not an exact match with V
the 1/20 scale model tested against measurements, let alone v,
0
the real aircraft. The important thing here, however, is t o a
be able t o m o d e l the salient features of the aircraft as they
pertain t o the most important scattering centers for this
case. The pattern cut of interest is in the plane of the wings,
that is, the azimuth cut.
The UTD fields from the m o d e l are calculated using only
first order terms without the engines included. The results
are compared w i t h measurements made o n the OSU-ESL
-50
0
w u 90 180
compact range at 10 GHz for backscatter. The 1/20 scale TAIL ANGLE NOSE
model of the aircraft measured w o u l d not allow the engines Fig. 15. Measured backscattered result for horizontal
t o be removed so absorber was placed over the front and polarization in the azimuth plane of a 1/20 scale model of
back faces. The results are i n dB relative t o a square meter a Boeing 737 without engines at 10 CHz.
at the 1/20 scale model size at 10 GHz. The calculated result
for the horizontal polarization i s shown i n Fig. 14, and the
measured result in Fig. 15. higher order terms. The computer time t o calculate the
I n general the agreement is verygood. The fuselage return models without the engines was just under 8 m i n on a VAX
i s even close though the calculated results are somewhat 11l780.
narrower as anticipated because of the flat-sided repre- In addition t o the RCS-BSC code mentioned above, a
sentation of the curved surface. The major differences in good approximation t o the backscattered signal can be
the peaks associated w i t h the sp,ecular from the leading obtained by an envelope description of the scattering of
edge of the wings and stabilizers are d u e t o t w o possible complex structure by combining the backscattered signal
mechanisms. First, the rounded edge model used could be of basic shapes forming the structure [47]. Analytic expres-
improved, since the radii of curvature of the edge i s small sions for the envelope o f the backscattered fields o f some
i n terms of a wavelength i n this case. In addition, it can be basic shapes, such as finite circular cylinder, circular cone,
shown that theedgewaveson the leading edgeof thewings cone frustum, hemisphere, circular disk, and a straight edge
can strongly affect these peaks. This i s not a low cross-sec- have been developed. This type of technique is very useful
tion aircraft, so the first order solution used here may be for initial design w o r k and for very large structures where
determined t o be sufficient. If a different aircraft had been the side lobe details w o u l d need t o be sampled at very small
studied, it might be necessary t o include more of these angular increments t o be assured of obtaining the peaks.
706 PROCEEDINGS OF THE IEEE, VOL. 77, NO. 5 , M A Y 1989REFERENCES GTD analysis of the diffraction ot electromagnetic wave5 by
a smooth convex surface," / € E € Trans. Antennas Propagat.,
[I] R. F. Harrington, Field Computation by Moment Methods. vol. AP-28, No. 5, pp, 631-642, Sept. 1980.
New York: Macmillan, 1968. [26] F. A. Sikta, W. D. Burnside, T. T. Chu, and 1. Peters, Jr., "First
[2] R. F. Harrington, Time-Harmonic Electromagnetic Fie/ds. order equivalent current and corner diffraction scattering
New York: McGraw-Hill, 1961, secs. 3-5, 3-11. from flat plate structures," / € E € Trans. Antennas Propagat.,
[3] N. N. Wang, J. H. Richmond, and M. C. Gilreath, "Sinusoidal vol. AP-31, NO. 4, pp. 584-589, July 1983.
reaction formulation for radiation and scattering from con- [27] A. Michaeli, "Elimination of infinities in equivalent edge cur-
ducting surfaces," /€€€ Trans. Antennas Propagat., vol. AP-23, rents, Part l: Fringe current components," /€€€ Trans. Anfen-
pp. 376-382, May 1975. nas Propagat., vol. AP-34, No. 7, pp. 912-918, July 1986.
[4] E. H. Newman and R. L. Dilsavor, "A user's manual for the [28] T. Brinkley, "Far Zone Bistatic Scattering from a Flat Plate,"
electromagnetic surface patch code: ESP version 111," The M.Sc. Thesis, Spring 1988,The Ohio State University, Depart-
Ohio State University, ElectroScience Laboratory, Report ment of Electrical Engineering.
716148, prepared under Grant NSG 1613 with the National [29] R. Tiberio, G. Manara, G. Pelosi, and R. G. Kouyoumjian,
Aeronautics and Space Admin., May 1987. "High-frequency diff raction by a double wedge," presented
[5] E. H. Newman and D. M. Pozar, "Electromagnetic modelling at the IEEE AP-S International Symposium, Vancouver Can-
of composite wire and surface geometries," /€€E Trans. ada, June 17-21, 1985.
Antennas Propagat., vol. AP-26, pp. 784-789, Nov. 1978. [30] F. Sikta, "UTD Analysis of Electromagnetic Scattering by Flat
[6] J. H. Richmond, D. M. Pozar, and E. H. Newman, "Rigorous Plate Structures," Ph.D. Diss., Dec. 1981,The Ohio State Uni-
near-zone field expressions for rectangular sinusoidal sur- versity, Department of Electrical Engineering.
face monopole," /€€€ Trans. Antennas Propagat., vol. AP-26, [31] W. D. Burnside and K. W. Burgener, "High frequency scat-
pp. 509-510, May 1978. tering by a thin lossless dielectric slab," / € € € Trans. Antennas
[;7 E. H. Newman and D. M. Pozar, "Considerations for efficient Propagat., vol. AP-31, No. 1, pp. 104-110, Jan. 1983.
wirelsurface modeling," / € E € Trans. Antennas Propagat., vol. [32] R. G. Rojas-Teran, "A Uniform GTD Analysis of the EM Dif-
AP-28, pp. 121-125, Jan. 1980. fraction by a Thin DielectriciFerrite Half-Plane and Related
[8] E. H. Newman and P. Tulyathan, "A surface patch model for Configurations," Ph.D. Diss., Winter 1985, The Ohio State
polygonal plates," / € E € Trans. Antennas Propagat., vol. AP-30, University, Department of Electrical Engineering.
pp. 588-593, July 1982. [33] H. T. Kim, "High-Frequency Analysis of EM Scattering from
[9] E. H. Newman and M. R. Schrote, "On the current distri- a Circular Conducting Cylinder with DielectriciFerrite Coat-
bution for open surfaces," / € € E Trans. Antennas Propagat., vol. ing," Ph.D. Diss., Spring 1986, The Ohio State University,
AP-31, pp. 515-518, May 1983. Department of Electrical Engineering.
[IO] E. H . Newman, P. Alexandroupoulos, and E. K. Walton, [34] "The Modern Geometrical Theory of Diffraction, Volumes
"Polygonal plate modeling of realistic structures," /€€€Trans. 1-3," CTD Short Course Notes, 1980, The Ohio State Uni-
Antennas Propagat., vol. AP-32, pp. 742-747, July 1984. versity ElectroScience Laboratory, Department of Electrical
[ I l l E. H. Newman, "The equivalent separation(s) for the self- Engineering.
impedance of thin strips," /€€€ Trans. Antennas Propagat., vol. [35] R. 1. Marhefka and W. D. Burnside, "Numerical Electromag-
AP-35, pp. 110-113, Jan. 1987. netic Code-Basic Scattering Code, NEC-BSC (Version 2), Part
[I21 J. H. Richmond, "Radiation and scattering by thin-wire struc- I: User's Manual," Tech. Rep. 712242-14, Dec. 1982, The Ohio
tures in a homogeneous conducting medium," / € € E Trans. State University ElectroScience Laboratory, Department of
Antennas Propagat., vol. AP-22, p. 365, Mar. 1974. Electrical Engineering; prepared under Contract No. N00123-
[I31 Y. T. Lin and 1. H. Richmond, "EM modeling of aircraft at low 79-C-1469 for Naval Regional Contracting Office.
frequencies," / E € € Trans. Antennas Propagat., vol. AP-23, pp. [36] R. J. Marhefka,"Numerical Electromagnetic Code-Basic Scat-
53-56, Jan. 1975. tering Code, NEC-BSC (Version 21, Part II: Code Manual,"
[I41 E. H. Newman, "Generation of wideband data from the Tech. Rep. 712242-15, Dec. 1982, The Ohio State University
method of moments by interpolating the impedance matrix," ElectroScience Laboratory, Department of Electrical Engi-
/ € € E Trans. Antennas Propagat., vol. 36, pp. 1820-1824, Dec. neering; prepared under Contract No. N00123-79-C-1469 for
1988. Naval Regional Contracting Office.
[I51 M. Kragalott, "The Method of Moments Solution to Trans- [37] R. J. Marhefka,"Numerical Electromagnetic Code-Basic Scat-
verse Magnetic Scattering by a General Cylinder," M.Sc. The- tering Code, NEC-BSC (Version 31, User's Manua1,"Tech. Rep.
sis,TheOhioStateUniv.,Columbus,Dept.of Elec. Engr.,Aug. 718422-3,1989, under preparation, The Ohio State University
1988. ElectroScience Laboratory, Department of Electrical Engi-
[I61 J.H. Richmond, "Scattering by a conducting elliptic cylinder neering; prepared under Contract No. N60530-85-C-0249for
with dielectric coating," Radioscience, vol. 23, pp. 1061-1066, Naval Weapons Center.
Nov./Dec. 1988. (381 R. J. Marhefka and J. H. Choi, "Bistatic Scattering Analysis of
[1;7 E. H. Newman, "Scattering bya material coated paraboliccyl- an Ellipsoid," Applied Computational Electromagnetics Soci-
inder," / € E € Trans. Antennas Propagat., accepted for publi- ety, in Proc. 3rd Annual Review Conf., Monterey, California,
cat ion. March 24-26, 1987.
[I81 J. H. Richmond, "Scattering by thin dielectric strips," /E€€ 1391 J. H. Choi, "Bistatic Scattering Analysis of an Ellipsoid," M.Sc.
Trans. Antennas Propagat., vol. AP-33, pp. 64-68, Ian. 1985. Thesis, Autumn 1986,TheOhio State University, Department
[I91 E. H. Newman, "An overview of the hybrid MMIGreen's func- of Electrical Engineering.
tion method in electromagnetics," Proc. / € € E , vol. 76, pp. 270- [40] R. 1. Marhefka, "Analysis of Aircraft Wing-Mounted Antenna
282, Mar. 1988. Patterns," Report 2902-25, June 1976, The Ohio State Uni-
[20] E. H. Newman, "TM scattering by a dielectric cylinder in the versity ElectroScience Laboratory, Department of Electrical
presence of a half-plane," / € € E Trans. Antennas Propagat., vol. Engineering; prepared under Grant No. NGL 36-008-138 for
AP-33, pp. 773-782, July1985. National Aeronautics and Space Administration.
[21] E. H. Newman, "TM and TE scattering by a dielectriciferrite 1411 K. C. Chiang, "Bistatic Scattering from a Finite Circular Cyl-
cylinder in the presence of a half-plane," / E € € Trans. Anten- inder," M.Sc. Thesis, Winter 1984, The Ohio State University,
nas Propagat., vol. AP-34, pp. 804-812, June 1986. Department of Electrical Engineering.
[22] E. H. Newman and J. L. Blanchard, "TM scattering by an [42] W. Ebihara, "UTD Bistatic Scattering from a Cone Frustum,"
impedance sheet extension of a parabolic cylinder," / € € E M.Sc. Thesis, June 1986, The Ohio State University, Depart-
Trans. Antennas Propagat., vol. AP-36, pp. 527-534, Apr. 1988. ment of Electrical Engineering.
[23] J. B. Keller, "Geometrical theory of diffraction," 1. Opt. Soc. [43] R. J . Marhefka, "Radar Cross Section-Basic Scattering Code,
Am., vol. 52, no. 2, pp. 116-130, 1962. RCS-BSC (Version2), User'sManua1,"Tech. Rep. 718295,1988,
[24] R. G. Kouyoumjian and P. H. Pathak, "A uniform geometrical under preparation, The Ohio State University ElectroScience
theory of diffraction for an edge in a perfectly conducting Laboratory, Department of Electrical Engineering; prepared
surface," Proc. / € € E , vol. 62, pp. 1448-1461, Nov. 1974. under Contract No. F33615-86-K-I023for Wright Patterson Air
[25] P. H. Pathak, W. D. Burnside, and R. J.Marhefka, "A uniform Force Base.
NEWMAN A N D MARHEFKA: MM A N D U T D METHODS AT O H I O STATE UNIVERSITY 707[44] K. R. Aberegg, "UTD Interaction Between Plates for Back- Dr. Newman is a member of Commission B of URSI, and is a past
scatter," M.Sc. Thesis, December 1985, The Ohio State Uni- chairmanoftheColumbus sectionsofthe IEEEAntennasand Prop-
versity, Department of Electrical Engineering. agation and Microwave Theory and Techniques Societies.
[45] J. L. Volakis, W. D. Burnside, and L. Peters, Jr., "Electromag-
netic Scattering from Appendageson a Smooth Surface," / € € €
Trans. Antennas Propagat., vol. AP-33, No. 6, pp. 736-743, June Ronald Joseph Marhefka (Senior Member,
1985. IEEE) was born in Cleveland, OH, on June
[46] R. H. Campbell, "Electromagnetic Backscatter from a Flat Plate 2,1947. He received the B.S.E.E. degree from
in the Vicinity of a General Curved Surface," M.Sc. Thesis, Ohio State University, Athens, in 1969, and
Dec. 1984, The Ohio State University, Department of Elec- the M.S. and Ph.D. degrees in electrical
trical Engineering. engineering from Ohio State University,
[47] C. W. I . Pistorius, "An Envelope Description of the Back- Columbus, in 1971 and 1976, respectively.
scattered Power Patterns of Conducting Bodies," M.Sc. The- Since 1969, he has been with the Ohio
sis, Winter 1984, The Ohio State University, Department of State University ElectroScience Laboratory.
Electrical Engineering. His research interests are in the areas of
developing and applying high-frequency
asymptotic solutions such as the Uniform Geometrical Theory of
Diffraction, hybrid solutions, and other scattering techniques. He
has applied these methods to numerous practical antenna and
scattering problems, including airborne, spacecraft, and ship-
board antenna analysis and radar cross-section prediction.
Edward H. Newman (Fellow, IEEE) was born He i s the author of the user-oriented computer code, the NEC-
in Cleveland, OH, on July 9, 1946. He Basic Scattering Code, which has been distributed to over 100 U.S.
received the B.S.E.E, M.S., and Ph.D. government agencies and companies as well as a number of for-
degrees in electrical engineering from Ohio eign countries. He i s also the author of codes for the analysis of
State University in 1969, 1970, and 1974, radar cross section and contributed to other user-oriented codes
respectively. such as the Aircraft Code. In 1975, he coauthored a paper that won
Since 1974 he has been a member of the the IEEE Antennaand Propagations Society's Best Application Paper
ElectroScience Laboratory at Ohio State and one that won the R. W. P. King Award. In addition, he has been
University. His primary research interest is a lecturer at numerous GTD short courses.
in the development of method of moments Dr.MarhefkaisamemberofTau BetaPi,EtaKappaNu, Phi Kappa
techniaues for the analvsis of general Phi, and Sigma Xi. He has served as SecretaryiTreasurer, ViceChair-
antenna or scattering problems, and he is the piimary author of man, and Chairman of the Columbus joint chapter of the IEEE
the "Electromagnetic Surface Patch Code" (ESP). Other research Antenna and Propagation and Microwave Theory and Techniques
interests include electrically small antennas, arrays, printed-circuit Societies during 1977-1980, respectively. He i s an associate editor
antennas, antennas in inhomogeneous media, and scattering from of the IEEE TRANSACTIONS ON ANTENNAS A Y D PROPAGATION and also an
material coated edges. He has published over 35 journal articles associate editor of the Applied Computational Electromagnetics
in these areas. Society Journal and Newsletter.
708 PROCEEDINGS OF THE IEEE. VOL. 77, NO. 5, M A Y 1989You can also read
