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' Michigan State
University

 

 

 

 

This is to certify that the

thesis entitled

A Combination of Rao-Wilton-Glisson and
Asymptotic Phase Basis Functions to Solve the

Electric and Magneti<brgsieelltgd£t§tegral Equations

John Robert Gulick

has been accepted towards fulfillment
of the requirements for

14.3. degree in Electrigal
Engineering

944‘. [0%

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Date Wrdn lb; 200‘

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6/01 cJCIRC/DetoDuest-pts

A COMBINATION OF RAO-WILTON-GLISSON AND ASYMPTOTIC PHASE
BASIS FUNCTIONS TO SOLVE THE ELECTRIC AND MAGNETIC FIELD
INTEGRAL EQUATIONS
By

John Robert Gulick

AN ABSTRACT OF A THESIS
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
MASTER OF SCIENCE
Department of Electrical and Computer Engineering
2001

Professor Leo C. Kempel

ABSTRACT
A COMBINATION OF RAO-WILTON-GLISSON AND ASYMPTOTIC PHASE
BASIS FUNCTIONS TO SOLVE THE ELECTRIC AND MAGNETIC FIELD
INTEGRAL EQUATIONS
By

John Robert Gulick

Using the method Of moments to solve the electric and magnetic field integral equations
for the currents on a PEC surface requires a large number of unknowns tO capture the
current’s rapid spatial variation across the surface. Rao-Wilton—Glissr‘m (RWG) vector
basis functions [I] have been successfully used for the past twenty years [1. 2. 3....].
Unfortunately, the required number of unknowns is on the order of 100 per square
wavelength making electrically large problems impractical. For large smooth Objects, the
rapid spatial variation in the current is due to phase variations rather than magnitude
variations. Thus, using asymptotic phase (AP) basis functions can drastically reduce the
number Of unknowns [3] for large, smooth metallic bodies. The AP basis function
incorporates the anticipated phase, hence represents a more efficient basis function for a
large class of problems. However, using RWG basis functions for monostatic calculations
is more efficient since the matrix entries need not be recomputed for each new incidence
angle, as is the case for an AP expansion. One can combine the methods; selecting RWG
or AP basis functions for a given geometry based on an element’s location within the
geometry. This allows the relaxation Of mesh density in smooth flat regions not near the
discontinuities resulting in a significant reduction of unknowns. This research shows that
combining functions is highly efficient and the effectiveness Of this method depends on

the geometry Of application.

TO Aubrey Lynn.
Integrity, Ambition, Knowledge, Communication. Excellence. Service. Love.

"Science serves humanity only when it is joined to conscience."
-Pope John Paul II, Padua. Italy, I992, to International Conference on Spacc Research

iii

ACKNOWLEDGEMENTS

Several people have significantly impacted and/or assisted me in thisjourney. To them. I

owe deep gratitude, that which is insufficiently accomplished by mere prose.

Great appreciation is given to Pamela Haddad who provided her method of moments

code for this research.

Dennis Nyquist, Ed Rothwell, and David Infante have inspired my interest and career by
Opening my eyes to the intriguing mathematical world Ofelectromagnetics. Their thirst
for knowledge. integrity, and sense Of humor set a high standard for engineers.

mathematicians, professors. and professionals.

I thank Leo C. Kempel, whose vast knowledge and energy inspired and continually

supported me through graduate school, for his concem. humor. and high expectations.

Col (USAF Ret) Gale Larson has guided me as an Officer mentor. Showing me the value

Of depth and breadth Of knowledge. excellence, integrity, and service.

Without the continued love and support of my beautiful wife. Amanda, all potential

would forever remain unfulfilled.

TABLE OF CONTENTS

List of Tables .................................................................................................................... vii
List Of Figures .................................................................................................................. viii
Key tO Abbreviations .......................................................................................................... x
Introduction ......................................................................................................................... I
Chapter I: Integral Equations ............................................................................................. 2
1. Configuration .............................................................................................................. 2

2. Electric Field Integral Equation .................................................................................. 4

3. Magnetic Field Integral Equation ............................................................................... 6
4. Combined Field Integral Equation (C FIE) ................................................................. 7
5. Summary ..................................................................................................................... 8
Chapter 2: Method of Moments .......................................................................................... 9
1. Method Of Moments .................................................................................................... 9

2. Expansion and Testing Functions ............................................................................. IO

3. Electric Field Integral Equation Method Of Moments ............................................. I I

4. Magnetic Field Integral Equation Method of Moments ........................................... I7
Chapter 3: Basis Function Specification ........................................................................... 19
I. Sub-domain Vector Basis Functions ......................................................................... I9

2. Impedance Matrix ..................................................................................................... 2|

3. Singularities .............................................................................................................. 24

4. Coding ....................................................................................................................... 24
Chapter 4: Results ............................................................................................................. 25
1. Square Plate .............................................................................................................. 25

2. The Kite: Description ................................................................................................ 26

3. The Kite: Results ...................................................................................................... 29
Code Validation Case: .............................................................................................. 30
RCS Curve Comparison: ........................................................................................... 3I
Selection of the RWG Region Thickness: ................................................................ 34
Convergence: 35
Mesh Density Analysis: ............................................................................................ 36

4. Extension to the EMCC Mini—Arrow ........................................................................ 38
Chapter 5: Conclusion ....................................................................................................... 40
I. Summary ................................................................................................................... 40
Summary Of Results from Specific Geometries ........................................................ 41
Conclusion ................................................................................................................ 42

2. Challenges ................................................................................................................. 43

3. Future Work .............................................................................................................. 43
Appendix A: Fundamental Theory ................................................................................... 46
l. Maxwell's Equations ................................................................................................. 46

2. Boundary Conditions ................................................................................................ SI

3. Vector Potentials ....................................................................................................... 5|
Magnetic Vector Potential: ....................................................................................... 52
Electric Vector Potential: (source free space.) .......................................................... 53

4. Scalar Green's Function ............................................................................................ 53
Appendix B: Vector Identities .......................................................................................... 56
Appendix C: Inner Kite Vertices ...................................................................................... 57
Appendix D: Additional Kite Separation Comparisons .................................................... 58
Bibliography ..................................................................................................................... 61

vi

LIST OF TABLES

Table 4.1: Problem Size ..................................................................................................... 30

Table 4.2: Mesh Density Analysis ..................................................................................... 36

vii

LIST OF FIGURES

Figure 1.1: Configuration ................................................................................................. 3
Figure 2. I: Current on a 4x4 Wavelength Square Plate ................................................. IO
Figure 3.1: Sub-domain Parameters ................................................................................ 20
Figure 3.2: Local Position in Terms OfGlobal 23
Figure 4. I: Sample results for a 4Ax4A plate .................................................................. 25
Figure 4.2: Scattering by a IOAxlOA plate ...................................................................... 26
Figure 4.3: The Kite ........................................................................................................ 26
Figure 4.4: Waterline Cut ............................................................................................... 27
Figure 4.5: Over the Top Cut .......................................................................................... 27
Figure 4.6: Kite Mesh ..................................................................................................... 28
Figure 4.7: 5 GHz Kite RCS Plot, d = 1.5 cm, Density Factor = 12/3 ........................... 31
Figure 4.8: IO GHz Kite RCS Plot, Mesh Density Factor = 12/12 ................................. 32
Figure 4.9: IO GHz Kite, d = I.5cm, Mesh Density Factor = 12/3 ................................ 33
Figure 4.10: IOGhz Kite RCS Plot, d = 1.5 cm, Density Factor = 12/3 ......................... 33
Figure 4.1 1: IO GHZ Kite RCS Plots, d = lcm, Mesh Density Factor = 12/3 ................ 34
Figure 4.12: 10 GI-Iz Kite RCS. OTT-T Mixed Basis Functions ................................... 35
Figure 4.13: 10 GHz Kite, d = 1 cm. Condition Number vs. Incident Angle ................. 36
Figure 4.14: 12 GHZ Kite RCS, OTT-T, d = 1.25cm ..................................................... 37
Figure 4.15: Condition Number vs. Incident Angle ....................................................... 37
Figure 4.16: Mini-arrow scattering at 9 GHZ .................................................................. 38
Figure 5.1: Graded Mesh on Square Plate vs. Conesphere ............................................. 42

viii

Figure A. 1: Position Vectors ........................................................................................... 54

Figure D. I: 10 GHZ Kite RCS, AP Basis Functions ....................................................... 58
Figure D2: 10 GHZ Kite RCS, Mixed Basis Functions ................................................. 59
Figure D3: 10 GHz Kite RCS. RWG Basis Functions .................................................. 59

ix

KEY TO ABBREVIATIONS

AP — Asymptotic Phase

CFIE — Combined Field Integral Equation
EFIE — Electrical Field Integral Equation
EMCC — Electromagnetic Code Consortium
GHz — Gigahertz

IE — Integral Equation

MFIE — Magnetic Field hitegral Equation
MOM — Method of Moments

OTT-P — Over the top cut, phi polarization
OTT-T — Over the top cut, theta polarization
PEC — Perfect Electric Conductor

RC S — Radar Cross Section

RWG — RaO, Wilton, Glisson

WL-P — Waterline cut. phi polarization

INTRODUCTION

This thesis presents the theoretical development and numerical solution results for
implementing a method Of moments solution technique combining Rao-Wilton-Glisson
(RWG) basis functions [I] and Asymptotic (AP) basis functions [3] over the same
surface. With the intention Of broadening the potential audience beyond just the
electromagnetics community, included in Appendix A is background material
commencing from Maxwell’s independent equations. The theoretical development in
chapter one continues where Appendix A leaves Offand develops the familiar form of the
electric and magnetic field integral equations along with a standard method ofcombining
the equations. Chapter two discusses the method of moments technique and introduces
the implementation of combined RWG and AP basis and testing functions to derive the
impedance matrix for the electric field integral equation (EFIE) and magnetic field
integral equation (MFIE). In Chapter three, the basis functions are incorporated into the
EF IE matrix equation in a manner that can be solved computationally. Chapter 4 presents
computational results. comparison. and discussion for implementation of this technique
on two square plates, a kite geometry, and the Electromagnetic Code Consortium
(EMCC) mini-arrow. The final chapter summarizes the specific knowledge gained from

this research.

The research shows that the technique is a more efficient solution method than existing

methods for geometries with large. smooth, low curvature regions.

CHAPTER 1: INTEGRAL EQUATIONS

The first three chapters develop the theory for the problem this research addresses. After
discussing the configuration and general problem, chapter one introduces the electric and

magnetic field integral equations.

1. C onfiguration

The problem is a traditional scattering problem as shown in figure I. An arbitrary perfect
electrically conducting (PEC) surface is immersed in free Space. A known current source
creates impressed electric and magnetic fields (Ei and Hi). Due tO the necessity of

matching boundary conditions on the PEC surface, the impressed fields excite a surface

 

 

A
re)
Ei(r), Hi0.) PEC
44/
ES(r), Hs(r) b
Et(r), Ht(r)
A
Free Space

(3:3 0, FIZIIO. 020)

Figure 1.1: Configuration

current J, on the PEC Object. The surface current induces scattered electric and magnetic
fields (Es and H”), as required tO satisfy the boundary conditions. The total fields in the
free space region (Et and H') are the sum Of the impressed and scattered fields. It is the
total fields that must satisfy the boundary conditions on the surface of the PEC object.
The following fundamental equations are derived in Appendix A along with the complex
transform domain form of Maxwell's equations. Quantities E. H, A, J, and (I) have
suppressed position dependence r. Vector r’ denotes the source position while r denotes
the position vector to the Observation point. Note that the first five equations contain a

itr)!

suppressed e time dependence term.

V X E = —j(0flH ..Faraday‘s Law ( 1.1)
VXH =Ji +(O'+jw8)E ..Ampere‘s Law (1.2)
V-E=%p(r) ..Gauss' Law (1-3)
VoH = 0 ..Gauss’s Magnetic Law (1.4)
V-J = —ja)p(r) ..Continuity Equation (1.5)
(1),. = gV'A ..Lorentz gauge conditions (1.6)
k 2 = 602,118 ..wave number k in free space (1.7)
11H = VX A ..H in terms Of vector potential A (1.8)
E = —V(De — ij ..E in terms of vector potential A (1.9)
It X Et = 0 ..Bozmdary Conditions describing the (1.10)
fix Ht = Js tangential component ofthe total field. (1.1 I)
VZA + sz = --,UJi ..scalar Helmholtz equation (1.12)
g (r I I") = 17:79—ij ..unbound region Green's function for ( 1.12) (1.13)
A = flip-103g“ I r')ds' ..a solution to (1.12) in terms Of(1.l3) (1.14)

Given the impressed current, we know the impressed fields. By enforcing the boundary
conditions for the total fields, we can solve for the desired scattered fields. If we can
solve for the surface currents, we can then get A from (1.14) and find the scattered fields,
from (1.6), (1.8), and (1.9). Due tO the uniqueness theorem [4]. if we find a solution to
Maxwell’s equations by enforcing all the relevant boundary conditions. the solution is

unique.

2. Electric Field Integral Equation

Enforcing the boundary condition [ME = O at the surface ofthe PEC Object and using
subscript t to denote the tangential field components,
I _ I s _ i _ s
E,-E,+E,—O=>E,——E,, (1.15)

Writing —E: using (1.6), and (1.9),

I
k3 VVA]. (1.16)

I

 

 

Ej’(r)=l:—V(1(:)V-A)—ij = —jw(A +

l
Expanding A with (1.14) and substituting into (1.16),

1
k2

 

E:(r = r3.) = jwpUqJ(r')g(rg |r')ds'+ VV-Jl‘J(r')g(r_, |r')ds'] . (1,17)

I

In the second term of the integrand Of(1.l7), the divergence Operator may be taken inside
the integration since it Operates on Observation points while the integration is taken over
source points. The notation denoting tangential field components will be suppressed

unless required for clarity.

V'I.[J(r')g<rlr')]ds'=I.V-[J<r'>g<rlr'>ldv' (1.18)

Using the vector identity V-( WV) 2 wV-V + V-Vw on the integrand of the right hand
side, V-[J(r')g(r | r')] = g(r | r')V-J(r')+J(r')-Vg(r | r'). Since J(r') is a function of
primed coordinates and is a constant with respect to the unprimed coordinates. the
unprimed derivative of J(r') = 0 :> V-J(r') = 0. Due tO the symmetry ofthe Green's
function, Vg(r | r') = —V 'g(r | r'). Using the same vector identity as above,

—J(r')~V 'g(r | r') = —V '-[J(r')g(r | r')]+ g(r | r')V '-J(r') . For closed three—dimensional
bodies. if we split the volume into two surfaces. 5. and 53 along a cut c with contours C1

and C3 respectively and in Opposite directions, we can apply a two-dimensional version Of

the divergence theorem as in [20] on p17.

LV '-[J(r')g(r | r')]ds'
2 L] V 'o[J(r')g(r | r')]ds ' + L V '-[J(r')g(r | r')]ds'
= <13 fi,-J(r')g(r 1 r')dl + 43 fi,-J(r')g(r | r')a’l 11.19)
= 0.

For Open surfaces, this identity follows by straightforward application Of the divergence

theorem. Therefore,

V-J [J(r')g(r 1 r')]ds'= I g<r1r'>V'-J(r'>ds". (1.20)
S S
The electric field integral equation (EFIE) is by substitution Of( 1 .20) into (1.17),

l
k2

 

V130} | r')V'°J(r')a'S']. (1.21)

I

Ei<r = r.) = jwuflgatga. 1 r')ds'+

3. Magnetic Field Integral Equation

The tangential magnetic field is discontinuous by the amount ofcurrent density induced

on the surface ofthe PEC; fix H' = J, ,
J.(r'> = fi><[H"(r') + Him]. 11.22)
H5 may be written in terms OfA using (1.14),

H”(r)=ivXA=vxI J..(r')g(rlr')crv'. (1.23)
A

Due tO the physical nature Of the problem, we can safely assume uniform convergence of
the integrand. Thus, we can interchange the order Of integration and differentiation.

Then. rewriting (1.23) using the vector identity Vx(Vw) : wV x V — V x Vtt',

H‘(r) = I Vx[.1,(r')g(r 1 r')] ds'

: J;{g(r I r')VXJS(r') -J_,.(|")><Vg(r I r')}dS '. (1.24)

However. VXJ5(r') = 0 since the unprimed derivative ofa function of primed
coordinates is zero. From the symmetry ofthe Green's function. Vg(r | r') = —V'g(r I r') ,

(1.24) becomes,
mm = I,J...(r')><[V 'g<r 1 r')] ds '. (1.2s)

Inserting (1.25) into (1.22),

Jér) — rinr{;zx(.1,(r')><V'g(r I r') 48"} 11-26)

I'—)S

 

nXH'(r)=

where r —> 3* indicates that s is approached from the outside. Due to the discontinuity at

the surface. the integral should be taken in the principal value sense [5].

IWIT-(1’): ¥—§.fiXJ..(r')XV'g(r l r'M’S' (1.27)

Equation (1.27) is referred to as the magnetic field integral equation (MFIE), valid for

closed surfaces [9].

The BF IE and MF IE are actually integrodifferential equations since the unknown
quantity is in the integrand of a differential equation. However, they are commonly
referred to as integral equations. In general, they are both classified as inhomogeneous
F redholm equations where the EF IE is of the first kind and the MF IE is Of the second

kind. For more information on Classification, see [10].

4. Combined Field Integral Equation (CFIE)

Although either the MF IE or the EFIE is sufficient to finding the scattered fields. a
combination can also be used. Using only EF IE or M FIE leads to spurious resonances
for closed scattering bodies [5]. These are resolved using a method that introduces a
mixing constant OLE [0,1] to formulate the CFIE, providing stable, unique solutions for all

closed scatterers [1 1,12].

a[EFIE]+—%c—(oz —1)[MFIE] (1.28,
J .,

In this implementation, or = 1 reduces to a pure EF IE formulation while or = 0 reduces to

a pure MFIE formulation.

5. Summary

 

EFIE: E:(r>=./wu[l,~'<r')g1r. Ir')ds'+ ,. VI, gtr. 1r'>V new]

I

MFIE: I5><H'i(l‘) =l§§2- fiXJ.(r')><V'g(r I r')ds'

CFIE: a[EFlE]+%(a—l)[MF[E]
.1

None of these integral equations are easy to solve analytically except for a few special
circumstances. Numerical techniques such as the method of moments (MOM) can be
used to find solutions. When implementing the MOM, one chooses an expansion function
set that can accurately represent the anticipated unknown function while minimizing the
cost tO employ it. The method Of moments numerical technique applied to these integral

equations is the topic of the next chapter.

CHAPTER 2: METHOD OF MOMENTS

Numerical techniques, such as the method Of moments (MOM), can be used to find
solutions to the integral equations developed in the previous chapter. When
implementing the MOM, one chooses an expansion function set that can accurately
represent the anticipated unknown function while minimizing the cost to employ it. The
focal point of this research is on evaluating a potentially improved (good representation
ofthe solution with minimal employment cost) expansion function set fora particular
class of problems. This research considers combining two existing sub-domain
expansion functions, RaO‘Wilton—Glisson and Asymptotic Phase. within the same
problem. Chapter 2 discusses the method Of moments formulation using both types of

expansion functions and develops the impedance matrix for use in a numerical solution.

1. Method of Moments

For a surface, subdivided into a mesh of triangle elements, there are N edges. The basis
functions are associated with the edges of the mesh with support spanning two adjacent
triangles. The total surface current is formed by the superposition of the various basis
functions. Using the method of moments, we chose a sub-domain expansion function to
represent the current across each element. Since there are N edges, each edge has a
different expansion coefficient. We then choose a testing function. Using Galerkin's
method, the testing function is selected as the same as the basis function. We expand the
current with the expansion function, multiply each term in the integral equation by the

testing function, and integrate over the surface.

9

2. Expansion and Testing Functions

Using the method of moments, we approximate the current by a set of expansion
(equivalently basis) functions and "test" the integral equation with a testing function.
Rao, Wilton, and Glisson developed [1] a sub-domain basis function that can be used to
accurately approximate the current over a surface element and also serves as a testing
function. Let f(r) represent the RWG sub—domain basis function. Since that phase of the
incident field is known, we can assume the phase of the surface current should vary
spatially approximately as the incident phase varies in regions not near location of rapid
curvature change. This is one critical approximation made in the popular asymptotic
solution method, the physical optics (PO) method. [6] RWG basis functions, modified by
the incident phase term, are Asymptotic Phase (AP) basis functions [6] with

F(r) = f (r)e"""r . Thus, including the incident phase in the basis functions should allow
a reduction in the density ofthe mesh for regions away from discontinuities. Indeed this
concept is illustrated in figure 2.1 which compares the variation Of the real part of the x-
component of the current for standard RWG basis functions and AP basis functions for a

four wavelength square plate.

 

 

.n ‘ - ;

Rea. m1 R.{J,(.-)..~-w .}

4/lx4l plate.6'"C = 80",(1)inc = 0“
Figure 2.1: Current on a 4x4 Wavelength Square Plate

For testing functions, f(r), F*(r), and F(r) were considered. However. the lirst two lead
tO asymmetric matrices while the third choice yields a symmetric matrix as will be shown
in the next section. Thus. we will use F(r) for both the testing and expansion functions.
TO allow control over the testing and expansion function for each edge of the mesh
independently, we introduce a constant ,3” = 0 or B” = 1 such that the current expansion is
given by
)=2 J F Oz: Jf W” '
n 11 ° (2.1)
n: I

The subscript, n, denotes a specific edge Of the mesh. In this representation, [3” :
means AP function while [3" = 0 reduces tO the RWG basis function. For the impedance
matrix elements where the testing and expansion functions are the sanre. the
implementation Of the MOM is termed Galerkin's method. The impedance matrix is
derived by testing the EFIE and expanding the unknown current using our flexible (either

RWG or AP) basis and testing functions.

3. Electric Field Integral Equation Method Of Moments

     

EFIE: Ei(r)=./w# I,J(r')g(r.lr')dS"+ ‘,g(r.|r')V'-J(r')a’s” (2.2)

I
Assume the medium rs fiee space. Let the m indices denote the testingc edge and n denote

the expansion edge. Testing equation (2.2) withF F(r) requires pre- —multiplication by

"I

Fm(r) and integration over the domain Ofeach testing function.

IqF’"(r).Ei(r)dS = jwflIsIS.F...(|‘)°J(l")g(r, I r')a’s'ds

 

+

For notational convenience, define A, B, and C such that (2.3) has the form

()

A=jcouIB+£7CI

A = I F, (r)-E'(r)ds

II

B = I IF (r)-J(r')g(rs I r')ds 'ds

C = I Fm (r)-V I .. go; I r')V ’-J(r')ds 'ds
Manipulating C by first switching the order of integration.
C = I ..V'-J(r') I Fm(r)-Vg(rs I r')dsds'

then, using the vector id V-( WV) 2 onV + V-Vw as V-Vw : V-Vw— V-(Vw),

C = IS_V'-J(r')[I$V-[Fm(r)g(rs I r')]ds — IqV-Fm(r)g(rs Ir')dsIds

Using the same technique as in (1.19),
I,V-[F,.(r>g(r. 1 r')]ds -- I,fi,-F,,,(r)g(r. 1 rod! = 0 .
Therefore, C reduces to,
C = — I IV '-J(r') I V-Fm (r) g(rs I r')dsds',
Then, after switching order Of integration, we have

C = I ‘ I JV-Fm (r)V '-J(r') go; I r')ds 'ds.

12

(0
kafl J; Fm (r).VJ.S' g(rs I r')V 'oJ(rI)CIIS .618.

(2.3)

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

Introduce an incident electric field as

E' (r) = éeil’hwr) where
li' = -f' = —.i sine, cosrp, —j/sin 6, sin (I), — 2 c056,.
r = )Acx+,vy+éz
IE' or = —x sin 6,. cos (I), — y sin 9,. Sin 4),. — 2 cos 9,. and
e = O, cosa +951 sina.
Incorporate asymptotic phase (AP) expansion and testing functions as discussed in

section 2.2 using coefficient fl describing the amount Of phase. The testing function

I"

becomes,

at...
Fm(r)=fm(r)e I (A ), (2.11)

Expanding the current as the sum Of the currents over all N edges,

J (r') = ZJnfn (r,)e—m,,(i'-r'). (2.12)

Therefore A, B, and C (redefining C by pulling out a minus Sign) can be written so that

 

N I.
A : [wflXJn [:8— k3 C] WI‘ICI‘C

n=| o

A —"(k(,+[3,,, lf'or
A =Isfm(r)-ee /‘ ) d5 (2,13)
, 517". [3mr+B,,r' ,
B = I. I,,f.(r)f.<r >g(r. Ir')e ’ ‘ 'ds ds (2...)

C = J; Is" Cintegrand g(rs I r')dS 'dS (2.15)

where

13

Cimcgrand_ vl:fm(r)e-Wm(°r)]v i.[fn(rI)e—.//311(k'or')]. (2.16)

Using the product rule for differentiation,

V'Ifm (r)e—_113,,,(1'.rII = f... (r).VIe—/rr,,,(1l.rII+—_,113,(1.IV.fm (r).

Then, differentiating.
‘ij lil'r /' m .rsinOtcos¢,+I'Sin9,sin¢,+:cos9,
V6 I I : V6-13 ( . )

:15,,,(xsin6 cosq) + ysinO sinrp +zcosaie) I3 (1‘ ")

>113 tie-(”mfk‘tl

which means

V-f“m[m(r)e ”(i r)I=(V-f,,,(r)- ,,,(r) 1,3,,k) MM“). (2.17)

Thus, by substitution Of (2.17) mm (2.16),

 

(V-f. (r)—f.(r)-j/3,.I€’)(V'-f. (r ')—f. (r '>-jI3.I€')
Cirrtegrand : e.il;i.(fimr+l3nr') . (2. I 8)

For notational Simplicity, define

g1r1r’>=g(rIr')e‘t"'"‘*~"+i"). 12.19)

11.11.01.111.me yields
(V-f.(r)—f,.(r)-jfiml9)(V'-f.(r')—f.<r')-j/3,,I€’)
=V-f,.(r)(V'-f.(r'))—(fm(r)-113,.I€‘)V'-f,,(r')
—V-f,.<r>(f.<r'>-113.I€’)+(cow/3.12"Mam-113.12")
=V-fm<r)[V '-f.<r'>]—113,.I€’-f,.<r>[V '-f.<r '>]
—jl3.I€‘-f. (r '> [V-fmml— am. [1942, <r)][/€’-f.<r '1].
Thus,
‘Vof.(r>[V'-f,,(r')]
C: j j IgIrIr'y_IIIII.IIIIIII[VIIIIIIII may.
S —JfinkI'fn(r')[V°fm(r)i (2.20)
711.13. [IE’-fm(r>][I€’-f.(r ')]I

 

 

N
Combining A, B, C back into A = jwuzJ" [II—ALIC] results in
":1 ’1:

If," (r‘)-ée—I‘ ”III" II IIdSZJwfl;Jn Z157, (2.21)

where

15

If,,.(r)f,.(r ')

 

 

f—V-fm (r)[V antr 1)] \
Z'I‘III =1.l..g"<rlr'> +1 +ff3m'9’-fm<r>[V'-f,,<r'>] dsvds
k0 +113113’°f,.(r ') [V-fm(r)] (2.22)
(+£31.13. [k‘-f,,, (r)][k" .1” (r 3]

 

 

)1
is the EF 1E impedance matrix. For future reference, rewrite (2.21) and (2. 22) 1n a form

convenient for discussion;

.. N
I f (r)oée—I(I”I )" “215: 12011212, I LgIrIr'yjf’Fds ds (2.23)

 

 

where
(—V-fm (r) [V '-fn(r ')] )
.. 1 +j13m/2’.fm(r)[v'.fn(r')]
Z,I.I’I=fm(r)fn(r')+—7 . ~,-
k0 +JBnk 'fn(rI)[V'fm(r)] (2.24)
I+£3.13. [k’-f,,,<r)][k’-f.<r 1] j

and from (2.19),

g(r|r')= g(rlr')eI II '"III ’.

Note that if the testing and expansion functions are RWG, then B," and fin are zero. In

this case (2.23) and (2.24) reduce to the familiar [1, 2, 3. 8] expression,

16

.i fm (r)oée—"I”II°IdS
S

N ’f,.(r>f,.(r'>
= jquJn I I g(r | r') V-fm(r) [V '.r,,(r')] ds 'ds. (2.2s)
n=l _ k

L. () ..4

 

 

 

If either the testing or the expansion function is AP, one ofthe middle two terms in the
brackets of(2.24‘) are added. For matrix elements where both testing and expansion
functions are AP, we must compute all the terms in (2.24); the last three of which must be
computed at each incident angle. It is important to note that in all cases, the impedance

matrix is symmetric, thus we can use specialized solution methods.

4. Magnetic Field Integral Equation Method of Moments

A similar application of the method of moments can be applied to the magnetic field

integral. Recall the MFIE from equation (1.25),
" i __ Js(r) " V V! V d !
nXH (r)——2——-§3,n><J,(r)>< 8(l‘lr) S . (2.26)

The sI in equation (2.26) is a reminder that the MFIE is to be evaluated 8 > 0 distance
outside the surface, hence should be evaluated in the principal value sense. We test the

MFIE, as with the EFIE, with

Fm (r) : fm (r)e—IIIIIII(II.r)

and expanding the current as the sum of the currents over all N edges,

J (r') = fiJnfn (r')e""I”(II'II)

9

l7

yielding,

—j[3,,,( (k’o odSr)

I f (r)-n><HI(r')e
=12I r m-(r) 2%“ ,. )e_’II°(II”'IIII”II)d9

-i.fm<r>'fi><§..21.f.( (")><[V g(r|r')]e’ I W“ ds ds

N

:FAIFIE: ; :JZ Zita/FIE

m nm

([3,,,,,r+[3 r ’)
23:” =1 WI (”III a.
—fm (r)-fz><§ + f" (r')><V 'g(r | r')ds'

Again, we are using the notation from (2.19). Our first observation is the asymmetry of

MFIE

the MFIE impedance matrix. The Z is more difficult than the EFIE to evaluate due to

this asymmetry and performing the principal value sense integration.

l8

CHAPTER 3: BASIS FUNCTION SPECIFICATION

To implement the preceding impedance matrix in a numerical solver, we still need to
specify the spatial distribution of the basis functions, f. Additionally, we need to evaluate
the singular integrals when the source and observation elements coalesce. This chapter
uses the Rao-Wilton-Glisson [1] definition for sub-domain vector basis functions to write
the impedance matrix, in particular the electric field integral equation impedance matrix,
in a form appropriate for numerical implementation. We then discuss the self-cell

problem and its resolution for the case of flat triangles.

I. S ub-domain Vector Basis Functions

As described in Chapter 2, In and fm are standard RWG basis functions. Recall that for
asymptotic basis functions, we multiply the RWG function by the incident phase temt.
The final impedance matrix from Chapter 2 incorporates the combination of basis
functions using the parameters fl," and B". One can see by inspection that if both the
source and observation edge are RWG ([3,, = B," = 0), the impedance matrix reduces to the
familiar RWG MoM symmetric impedance matrix. As intended then, the phase factor is
separated from the RWG basis function in the impedance matrix equation. For numerical
implementation, at this point we need only substitute the RWG basis function definition

for f (r)and f"(r').

I"

19

As defined in [1], RWG sub-domain basis
functions for triangle patches effectively
model the current on a scatterer. These
basis functions describe the expanded
current only over two triangles sharing a

cormnon edge indicated by the expansion

 

and testing indices (n or m) and are zero 0 .
Figure 3.1: Subdon‘mn Parameters [l]
elsewhere, hence the term sub-domain.

Figure 3.1 illustrates the nature of these basis functions and the associated parameters for

the nth interior edge shared by triangles T,,+ and T,,‘. A point in either triangle is described
by either r from the global origin or pf (r) , a local position vector from emanating from
the free vertex (e.g. the vertex opposite the nth edge) ofthe triangle on which the point is
located. Notice the direction of pf (r) from the free vertex of T,,+ across the shared edge
towards the free vertex of T,,'. This establishes positive current reference direction from

th

the 7”,," to T,,'. As seen in figure 3.1, 1,, is the length of the n edge and A: is the area of

triangle Tut . The basis function f"(r) for the nth edge is then defined as

, forr in T*

I!

 

__ —np" (r) . _
fn(r)— 2A‘ , forr m T" (3.1)

H

 

0, elsewhere,

 

20

21551

where the surface divergence of fn(r) , proportional to the surface charge density

associated with the basis element, is given by

V-f"(r) =

 

H

—+, for r in 7::

II

H

—j, for r in T;

n

(3.2)

0, elsewhere.

Some properties that make the basis function f"(r) particularly suited to approximating

the surface current are detailed in [1] and stated here.

1. The current has no component normal to the boundary of the surface formed by

the triangle pair Tni', Tn”. Thus, no line charges exist along this boundary.

h . .
2. The current component normal to the nt edge 18 constant and continuous across

the edge. It is therefore implied that all edges of the triangle pair are free of line

charges.

3. Due to (3.2), the charge density is constant in each triangle and the total charge

associated with the pair is zero.

2. Impedance Matrix

We then substitute the definition for RWG basis function into the impedance matrix form

of the EF IE established previously (2.24). Recall

- :ior N ~ ~ ‘.
J. fm (r).ée—.l(ku+fim)l\ dS : ja)“ 2 JnJ‘ J'g(r I r')Z’f4”[;IEdS 'dS
S ”:1 S S

where

21

f —V-fm (r) [V at, (r )] \

E 1 +j/3mIE’-f,.(r)[V'-f,,<r'>]
Zuni] : fm (r)fn (r ') + —2 - Ai t
k0 +1ka ofn(r )[V-fm(r)]

K+fimfi.[19490)][19413.0'>] )

 

 

and

g(r 1 r') = ..+,.e‘j‘Re‘-’"'""~*'+“"">, R = |r - r'l .

Observe the form of the above equation is [Z] {J }={_V l, where [Z] is an NxN system and

N is the number of edges in the mesh. The tested left hand side of the EFIE becomes

A - ' k A?"- 1 A — ' k A”.
J fm (r).ee /( 0+I3m) rdS : 1M J‘T+ T— pi (r).ee ./( n+fim) rdS
S + .

2.4; (33)
Here we have introduced the sign carrying parameter
1, for r in T
t, = . (3.4)
-l, for r in TI'

and used i with the top sign for r in T,,' and the bottom sign for r in T,,‘. The right hand

side consists of a constant ( jam ) multiplying the unknown current expansion coefficient

EFIE
III)!

column vector (Jn) times a matrix Z . Matrix elements associated with the nlh row

and mth column are then given by

ZEFIE _ J'
nm T,:+

with

1 — 11,12 —./A7’-(B,..r+/3.r') ”EFIE .
T‘J ++T“ 4’” e e Z”’" dS dS (3-5)

I"

22

I; FIE
Z _

nm

In evaluation of anm ,it is useful to
. i .
recogmze that p, (r), 16 {mm} are expressed
. . i
1n global coordinates as p, (1') = i (l‘ — l',-)

where r,- is the global position vector to the

vertex opposite edge i on Tf' as shown in

t—V-fAr) [V '-f.<r '>] ‘
1 +mmléi-f.(r)[V'-f.(r')]
-Jf,.(r>f.(r')+—; . ~.
k. +1B.k~f.<r')[V-f,.<r)]
UM [k’of (r)][k’-f,, (r 5])

 

 

 

 

{ _l_mlm _l__nlll \
Am An—
. ,- lmlmp; r 1.1,,
+ +Jfimk .( ) .
1.1.97.0) 1.1.pf(r') 1 2’4»: An
: 2Ai . 2A1 +17 ~-llp:(r')tl
m n o +jBnkl= '7 '1 "1:1
2A;t A;
+fim fin kAi .lmlmpm (r) I’C‘i.lnlnp; Sr)
1 2A,; 2A; J

 

 

 

_z_,,,__l,,.l,,l,, + zmlmznl. _1+§3mfi°9i (r)+%l3n’;i‘P: (r')
-——p (“)M '+) ——

4A,:A: '" kSA,:A:+ +£3.13. 4pm. )][k"-p:(r')]

t I ll k2”: (”’93: (r')—4+j2flm 125p: (r)

_ mmnn

‘34]. A- A; +j2/3 k‘0p: (r ')+l3,.[3 [k"Pm(r)][k '93( )l .

 

Figure 3.2: Local Position in Terms of
Global Position Vectors

Figure 3.2.

23

3. Singularities

Equation (3.5) may be readily evaluated numerically unless the source and observation

— My '(er+[3nr')

edges are close. We resolve the singularity as in [7], with 6 ‘ suppressed
in the integrand of each term,

—'kR —'kR
.10 e]. _1

+ t e t i l t
l,:Pz-‘<">—R—ds = ,.p,- (UTdS

' (3.6)

+Ji(P'—P),

, 1
R s+(p—pi)J.Ti:Eds

where p,p',p,. are projections of position vectors r,r ', r, respectively onto triangle Tf .

The first integral is bounded, so it can be numerically integrated, while the second two

can be evaluated analytically. Similarly, the scalar integrals can be written

 

ex“ e‘-”"~R -1 1
L ds'=J+——ds'+J+—ds' (37)
T: R '1;- R T; R ' '

4. Coding

The code used for testing was primarily a modified version of the program, T riMom, by
Dr. Pamela Haddad [8]* implemented at Michigan State University. The modified code,
called HotPoppa, is a FORTAN based numerical solver. It allows for solution of the
EFIE using the basis function development included in the previous chapter. It was
additionally modified to import .grd mesh files created by SkyMesh2TM as well as SDRC

IDEAS Universal files (.unv).

*Dr. Pamela Haddad performed this work while on NSF Graduate Fellowship at the University of
Michigan. She is now a mentor of the technical staff at MIT’s Lincoln Laboratory.

24

CHAPTER 4: RESULTS

This chapter presents details regarding numerical solutions of the theory presented above.
Results for two different size square plates, a kite, and the EMCC mini-arrow are
discussed and compared. We shall analyze the kite in extensive detail as we explore
several implications of the mixed basis function method. Additionally, convergence
comparisons between various basis function implementations along with mesh density
analysis are presented for the kite. Radar cross section (RC S) values are used for

comparison as they indicate the scattering characteristics of the object.

1. Square Plate

Two square plates are initially considered showing the potential for mixed basis function
implementation to perform accurately as proposed. For a 4x4 wavelength plate, a tenth
of a wavelength edge length mesh leads to 2240 unknowns while the graded ( or non-
uniform) mesh reduces the number of unknowns to 1870. Figure 4.1 shows monostatic

RCS results and the graded an.

 

 

 

 

 

’0— Rain l n”: l
mesh for the 4x4 i§fifi
wavelength plate. We
observe excellent
agreement. The reference ' ' ,,
line shows results for tenth he 4 A —-i T

'n in in so

 

“Llr d-l L
ofa wavelength Sampling NIOIIOSlilllC Rt goffl-f/Sx-lk plate
V\ polarization

over the entire square using
Figure 4.1: Sample results for a 4M47t plate.[3]

25

 

 

 

 

 

only RWG basis and testing 66 . . L . so

 

 

 

 

 

 

 

 

 

 

 

— Relmence [ V v — Relerm I
0 EEEQPORWGI. 45. 9 EFE‘APiRWG
. — we EFIE—RWG
functions. EFIE-AP refers to ,0,
. . . e *35»
AP ba31s and testing functlons (E E
E 330»
8 ‘8
on the pictured graded mesh ‘r ‘25
20
while EFIE-RWG + AP ,5 . .
‘ Go in 4b 60 so 100 2i) 40 so so
shows the results for us1ng Themdmes, Mamas,
VV-polarization HH-polarization

RW funct'ons in the o t

G 1 u er Figure 4.2: Scattering by a 101x10). plate.[3]
region of the graded mesh and
AP functions in the inner region. For a 10x 10 wavelength plate, tenth of a wavelength

sampling results in a mesh of 14,600 edges while a graded mesh has only 4,792 edges

(67% reduction). Again, the results are very promising as we see in Figure 4.2.

2. The Kite: Description

The kite is an infinitesimally

thin PEC surface in free

 

space. (Figure 4.3) Its

length of 23.495 cm is

 

approximately eight

Figure 4.3: The Kite

wavelengths at ten GHz.

The kite is one face of the EMCC mini-arrow, an object studied in Section 4.4.

Additionally, we wish to define two ways of observing the kite for scattering solution
discussion. Described using standard spherical coordinate basis. the following “cuts”

describe the are of incident and observation angles.

26

l. Waterline cut (WL): Figure 4.4

Z
9:90",¢=0“—>180" /

 
  

2. Over-the-Top cut (OTT): Figure 4.5

X

f y
9 = —90" ——> 90“, ¢ = 0“ / /

Figure 4.4: WL Figure 4.5: OTT

Note that these two perspectives use full advantage of the symmetry of the kite. For each
perspective, we will consider theta and phi polarized incident waves. However, results
are not presented for the WL cut theta polarization since an incident wave thus oriented
induces no current. The three remaining cut/polarization combinations, over the top theta
pol , over the top cut phi pol and waterline cut phi pol, will be abbreviated OTT-T, OTT-

P, and WL-P respectively.

The kite presents an interesting geometry for method testing for several reasons.
1. It has features (relatively large, flat, and PEC) that take advantage of the
methodology.
2. It is small enough to achieve numerical results in a reasonable amount of time at
frequencies in the five to twelve GHz range.
3. It is the largest facet of the EMCC mini-arrow for which the community
maintains experimental data. We extend our study to the three-dimensional mini-

arrow in Section 4.4.

As with the square above, we break the kite into two regions, a proportional inner kite

and a border region. Refer again to Figure 4.3. The inner region is formed by taking the

27

intersection of lines parallel to the kite edges and separated by a constant distance d from
the edges. Due to geometry, and easily observed, the vertex points of the inner and outer
kites are greater than (1 apart. One can specify d to determine the inner kite (afier
describing the outer kite). Appendix E contains calculations used to derive the vertices of

the inner kite from the parameter d.

In general, we want to make d as small as possible because as we will see. a small (1
results in a smaller number of unknowns. However, to ensure accuracy, it is necessary to
keep d near half a wavelength to account for edge conditions not incorporated in the AP

basis functions. This is discussed flirther when we observe results.

Using Skymesh2TM to create a triangular mesh, we can
define the edge elements for use in our numerical solution.
The division of the kite into inner and outer regions
provides two essential levels of control. We can
independently control the mesh density in the two regions.
This allows us to maintain approximately ten elements per
wavelength in the border region and much less in the inner
region. Skymesh2TM accomplishes a gradient transition in
the inner region. Being able to control the mesh density in
the two regions enables us to take full advantage of the

different basis functions. The different regions in the

 

mesh allow us to spec1fy which type of ba31s funct1on Figure 41): 10 GHz Kite Mesh

28

(specifically, the value of [3) used for elements in each region. Therefore, as mentioned
in chapter 2, we can use a fine (ten element per wavelength) mesh with RWG basis
functions near discontinuities (edges in the ease of the kite) and AP basis functions in
sparsely meshed regions away from the discontinuities. Note that this method differs

from [3] where the basis function for an edge is determined by its length.

As we discuss kite results, we will use a mesh density factor fora given region or for the
kite as a whole. This number is related to the mesh creation and is proportional to the
number of elements per wavelength, but does not describe it directly. A factor of 12
roughly equates to a maximum element size of a tenth of a wavelength. Often the mesh
density factor is presented as a pair specifying the entire kite and is written outer/inner.
(e.g. A mesh density factor of 12/3 describes a fine outer mesh and a relatively sparse

inner mesh.)

3. The Kite: Results

Several mesh configurations were solved using a variety of basis function combinations
and slightly varying d-spacing for all three orientations (OTT-P, OTT-T, WL-P). A
frequency of 5 0112 is used for code validation; 10 0112 is presented for RC S curve
comparison while 12 6112 is used for a mesh density analysis. These selections are based
on the element size relative to the kite, relative to a wavelength, and the total number of

elements as it affects the run time.

29

Table 4.1 illustrates the size ofthe
problem for various configurations.
The problem quickly becomes quite
large with increasing frequency.
However, 12 GHz allowed for a larger
difference and variation control of the
inner mesh density lending itself to the

mesh density analysis.

Code Validation Case:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a 5 12717 1447 674 1060
b 1.5 5 3/12 1267 584 925
'c 10 12/12 5667 2734 4200
111.5 10 3/12 4523 2162 3342
le 1 10 3/12 3787 1794 2790
'f1.25 12 16/16 14294 6989 10641
'g1.25 12 14/14 10957 5340 8148
h1.25 12 12/12 8091 3926 6008
11.25 12 12/9 6787 3274 5030
j1.25 12 12/6 6027 2894 4460
11.25 12 12/2.8 5807 2784 4295
Table 4.1: Problem Size

Initially, 5 GHz results were studied, however, the edge number reduction was minimal

(1060 vs. 925). It was useful to see that using RWG in both regions, using AP in both

regions, and using a combination (RWG outer, AP inner) all produced similar RCS

values. The following results correspond to row (b) in table 4.1. The separation, d, is set

to 1.5 cm because it becomes impractical to make it any larger from a meshing standpoint

(the mesh becomes geometrically constricted). Setting (1 less than half a wavelength is

not detrimental to the solution in this case.

30

tie RC8 p101 V‘."aluhne. 1; U1 phi—P01
U r v T r v v

 

f

ml 1') i

33. q
1 l W-

. . x .

40 if

so» 1 ‘11 «

:11:
I ' f-‘1I‘E‘d
-70 > - - — WWI} 4

RES (tlBt-xn)

 

 

 

 

 

 

 

 

 

'8] A A A L A A A A
0 30 40 EU 80 1|_I';l 1 212' 1 40 1 60 183
Pin (dz-meets)
Kite RC8 Plot Over the Top Cut. Cut PhrPol P119 RPS Plot CM?! the T011 Cut, Cut Thetaapol

U 1 Y T T r v 1' r v T

 

 

RCS (strn)
PCS (dB-em)
£3

.50»

 

 

—— AP 10

., , ‘ ~70 A - Mued . ‘
- - -— awe ~ - — we: ..
_w A A A A A A A A A . A A A A A A .41

~80 430 - 40 730 U 20 40 60 8] .81] .11 | .. .10 . :1‘1 1] 20 JD [[1 5.1]
Theta (degrees) Theta (degrees)

Figure 4.7: 5 GHz Kite RCS Plot, (1 = 1.5 cm. Density Factor = 12/3

 

 

 

 

 

 

 

 

 

 

 

 

 

RCS Curve Comparison:

For a frequency of 10 GHz, three different meshes were considered. The first (c in table
4.1) has a mesh density factor of 12/ 12, approximately ten edges per wavelength across
both regions of the kite. RCS curves are shown in Figure 4.8. One can see that the three
curves in each figure (AP only, RWG only, and mixed with AP inner and RWG outer)
closely match each other. Some discrepancy (3dB max) is apparent in the OTT curves
(Figures 4.1] and 4.12) in the —60 to —40 and 40 to 60 degree theta range. However, we
can see that the mixed basis function method consistently matches AP only results for

this implementation.

31

F :29 PCS Plot Wat-3111119, 1"111 F'hif‘ol

0 fi Y T Y *Y Y

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

— 41:1
.201 _ _' _ 1; 3:3 1
l
- '7’“ t 1 A 1
E 1", 11811 ‘1.
..1—“.-‘ m . iv
v: .40 . , _ 1
- .13 .11
<92:- 5’3 , 1‘ fill! if 1 {1 1'11 i.
\K“; (i r f 2 I I A
' =9 . 1((11 {ii- i“- .
1' 0
4% W ‘ O . ?
ED r U a ‘ 4
. 6
.70 > 4
*1) - ‘A A A A A A A A
U .‘U 40 60 80 103 1 2'0 1 AU 1E0 1 80
Pm idegvees)
Kite PCS Plot Over the Top Cut, Cut PhrPol kite RES F'lot Over the Top Cut, fut Theta-Pol
0 1 U
{‘1
1L! 4 It] I! \
EU 10 f N
E m ”f 30
u! m
in m ‘9‘
TE 3
m '40 * tn 40 | W
L) U
rx 0: J?
-50 , .50 > t 'v
60» {I} » ll
AP
70 > . idl‘ed .1 " 7U
' — Riva"; l
a] A A A A A I A A A .m A A 1 l- A A
{$0 430 40 -2D 0 20 40 60 80 -80 4:30 "50 -Z‘U 0 2'0
Theta (degrees) Theta (dearer-21

Figure 4.8: 10 GHz Kite RCS Plot. Mesh Density Factor = 12/12

The second 10 GHz experiment has a mesh density factor of 12/3 where the inner kite is
sampled less. The separation d is 1.5 cm, or half a wavelength. In this case — (d) in table
4.1 — we observe a 20% reduction in unknowns. Figure 4.9 illustrates the element size
verses position in the mesh. Figure 10 shows the RC8 values for the various
perspectives. In the waterline cut, we begin to see discrepancies in the 145-160 degree
phi range where the RWG-only curve does not match the AP and mixed curves. The
same mismatch is visible in the OTT cuts across a large range of theta angles with almost
lOdB difference for certain angles. These mismatches are expected since RWG typically
requires a minimum sampling of ten elements per wavelength over the entire surface for

accurate results. Similar discrepancies exist for case (e) in table 4.1 where the border

32

region is shrunk to one cm (see figure 4.1 1). Notice that in case (e), the number of

unknowns is reduced 33.5% as compared to (c).

RCS (stm)

 

lOGHz
Kite

(iraded
Mesh

 

 

Edge
Length
Relative to
a
Wavelength

 

 

 

 

Figure 4.9: 10 GHz Kite, d = l.5em, Mesh Density Factor = 12/3

 

Kite PCS Plat Waterline. Cm Phrpul

RC8 (stm)

 

 

 

 

U 20 40 80 100

Ph1(degteesl

Kite RISE Plot Over the Top Cut, Cut Theta Pol ia 116 W133 P101 (wet the Top Cul. Cut PhtPot
n

 

 

 

01> .,
.10 .
~10
-2o
‘20
/ ' an
30 , 1 E
/ ' £
I 1 3 340
‘J ‘1 .‘ ‘ (n
1 . ' \, g
50 ' 11 'I 4 '50 _
1} l
1.
-60 i so
1 ~ 70
.70 ' P ‘
1
.m ‘

 

 

 

 

 

 

40 EU 83 83 50 JO

JO D 2U 10 U 20
Theta (degrees) Thela (degrees)

Figure 4.10: lOGhz Kite RCS Plot, d = 1.5 cm, Density Factor = 12/3

33

Vite PCS; Plot Waterline, Cut PhrPol

 

 

AP
' f‘diltéi'd
-20 r - - R“!

 

7~13

 

 

 

RC8 (stm)

     

 

 

 

213 40 60 Ea? 1m 1213 140 161] 1813
Phi (degrees)

kite PICS Plot Over the Top Cut. Cut Theta-Pol

 

 

 

 

 

 

PCS (stm)
RC8 (stm)

 

 

 

 

 

 

 

 

 

 

 

 

   

‘31

d.

83 ~60 -40 -20 O 20 4O 60 80
Theta (degrees)

Figure 4.11: 10 GHz Kite RCS Plots, (1 = lem, Mesh Density Factor = 12/3

Theta (degrees)

Particularly visible in Figure 4.11, we see that the mixed basis function RCS curve
closely matches the AP basis function curve even with d = 1 cm. The RWG curve does
not match because the sampling rate over most of the kite is less than ten elements per
wavelength. Remember that the mixed basis function method has a lower

implementation cost than using all AP basis functions.

Selection of the RWG Region Thickness:

If we view the above RCS data plotted differently, we can consider the implications of
changing d. Figure 4.12 shows RCS curves for mixed basis function implementation

with d = 1 cm, 1.5 cm, and for a uniform mesh (12/ 12). This figure is included here

34

 

because it is the most interesting of the , , “‘9 p.11... F"f" ""9?B‘S'5-‘3”T"fi'ap°‘

 

 

 

 

 

 

 

D
1 fl
orientations. The other eight orientations 401 [I T 3‘1"?“
- - — D '— l S
‘0’ A i
. . . q ,
are 1neluded in Appendix D. We can see a I” 7;;- 4 1" -.\~._ ,3 q‘ 35-.
g 1. \1 1:6 , :1 {‘5' r .1 ,::‘ '
. J 40 ,c 13' ~- 1 mat-1', w / l .
that as d is decreased, the RC8 values a ,1 l. 1" (,1 111 (r; .1 (.1 «19., / \1
.‘rr 1- 'I/u 1 i" . .1 1 .9
-- I; . 11 \-
' o [I .
. . . 11‘: V:
dev1ate from the htghest sampltng rate 993:..- '1‘:
.71'1 j. (‘1‘.
curve at certain theta angle ranges. Hence, ,0 ll . . . . . 1 . . if.
' 190 an - 40 .20 0 .13 40 to to
The-ta (degrees)
we experience the traditional trade-off of Figure “21 ‘0 (.in Kite RCS, OTT-T

Mixed Basis Fttnctions

number of unknowns verses quality of solution.

Convergence:

In numerical methods, rate of iterative solution convergence is a constant concern. Since
AP elements in the matrix require computation at every incident angle, we are naturally
concerned about the convergence of the matrix solution for each angle. Since we wanted
a convergence measure relative to an all-RWG basis function implementation for a given
mesh, we chose to observe the matrix condition number, the ratio of largest to smallest
eigenvalue. Figure 4.13 shows that although the condition number varies with angle,
there is no apparent correlation and the overall variation relative to RWG-only is
minimal. Therefore, we can conclude that additional matrix convergence issues do not

result from using combined basis functions.

35

'mle Cor-damn Number 19101 Waterime Cut Phi Pol r 1:.- i Uhd-llul' 115mm Plot [M1 tr..- 1”. 1'_ 1.! F». .m 11...” P01

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

351'; v . . . . r 3500 . T fir
AP p
i — 9W1} — F.»
"L-LIJ 25130 >
o‘ . .n . o n g .
*3 59001115511 11.111111511
E: 1w: » 3 1:00.
1111- > . 13301
5"? > . {our
1", A 4 A; A 1 1 A ‘ U A‘ L x - L L 4
11 ;-ji .m 1,1;1 an 1m 1m 1411 11.11 1113 '9“ "W "U 1” ‘-' ~'-' "J W ”0
Pm (degveez) 'i'wia dun-re;

Figure 4.l3: 10 GHz Kite. d = 1 cm. Condition Number vs. Incident Angle

Mesh Density Analysis:

We completed a mesh analysis to study two particular issues. First, how dense should the
mesh be to provide a converged RCS value? Second, how sparse can the inner mesh be?
Using 12 GHz, the number of unknowns necessary increases dramatically (33%) from the
1061-12 case (table 4.1). However, the increased frequency allows more flexibility for
this particular analysis. Half of a wavelength at 12 Ghz is approximately 1.25 cm so for
the 12 GHz experiments we use d = 1.25. We considered the phi angle of 63 degrees on

the OTT-T out since this was the region of maximum discrepancy among RCS curves.

Addressing the second question above first, we find that m-“

m 714... 14:1: .":. ‘_

due to physical meshing constraints, a 2.8 mesh density ng. 13.13 34144152347 i‘

RWG' 12.2.8 31.76 2647.6
Mix "14.14 41.40.977.43
Mix 12.1: -3352 515.26
MIX 12.2.21 -3122 3491.1
Table 4.2: Density Analysis
(9 = (13". 8 = 0“, Freq = IZGHZ

factor is the lowest possible. Below 2.8, the mesh has
the same number of unknowns. That is, the kite is

geometrically constricted below 2.8 mesh density factor.

36

To answer our first

Kite PCS Plot Over the Top Cut, Cut Theta-Pol

 

 

 

 

 

 

 

 

 

 

question, we again consider 0 ' Mixed12
10 - RWG 12
. . ' b O M x d 2.8 1
theta = 0 and phi = 63 Since - — Rivis 2.8
-20 1- (5120
this observation angle P b
E -30 - 4
w ‘0 Q &
‘D ’ 9
appears to have the largest 3% 40 _ l i9 9 is) i
g a I ' g..o. Q
RCS discrepancy. Using 60 - O . 1
- - o l
. . . 130 ~ 0 ‘ .
RWG basrs functions With a 9
-70 .-
density factor of 12/12, we
-:1| . 1 l 1 1 1 1 1 1
-80 -60 4:0 -20 o 20 40 so 80

observe from table 4.2 that

Theta (degrees)

Figure 4.14: 12 GHZ Kite RCS, OTT—T, d = 1.25cm

the RC8 value appears to

have converged since at a mesh density factor of 14/14 is very close to the value at 12/12.

Using mixed basis functions on a uniform mesh, the RC8 value does not converge until a

mesh density of 14/14. RCS values for 12/12 and 12/2.8 are compared for mixed vs.

RWG basis functions in figure 4.14. Interestingly, the discrepancy in the theta range of

discussion shows the RWG 12/ 12 value is most clearly matched by Mix 12/ 12 while

RWG 12/2.8 and Mix 12/2.8 match each

other, but not RWG 12/ 12. Aside from this

observation angle range, such discrepancy
does not arise (Appendix D). It appears
some other phenomenon may be affecting

our results in that observation angle range.

37

 

 

 

 

 

 

 

 

Vitr- C undltion #F‘lot Over the Top Cut, Cut Thetmpol
4000 Y 1' f 1 v T Y r
:‘\/"’
3503 » 1111‘. 111/ 1
l. ‘ ' k I
i' l /‘ rl i1 1' ii "
. .. 'u "4., \ .‘\ 1%}‘4
{10.1 1- ~./ ‘v . .t‘. “I. J ‘
11 1,1. .
t- o ooooooo o‘\./o F‘V/ a \ o 4
‘JUU ' .
t
g
“i 3003 P
E . Mixed 12
1:) ' ' _ H‘t‘nv'l} l?
150] 1 — Mired 2 13
- ewe 2 8
101]] r
SCENIFO Warm—«Y o -0 “0—th v
0 L 1 L l 4 1 J.

80 -60 «:0 1'1 0 20 40 60 80
That 5 (degrees)

Figure 4.152(‘ondition Number vs. Incident Angle

One can see in Figure 4.15 that the condition numbers are much larger for 12/ l 2 than for
12/2.8. This makes sense since the same problem is specified more finely in the 12/ 12

case. The angles studied are shown by the dots plotted in the RWG curve in Figure 4.15.

In the 12/2.8 mixed case, we see that we can maintain relatively accurate results
(especially for theta angles near 0) for the kite using this methodology to reduce the

number of unknowns.

4. Extension to the EMC C Mini-A rrow

The kite can be extended to similar shape, in particular, the three-dimensional mini-
arrow. The mini-arrow is formed by adding the point (8.935,0,3.932) to the kite and

extending a line from each kite vertex to this new point. In figure 4.16 a tenth of a

 

90' ..__
180 0‘ ' ”919-1- .L
2111
U +- 3......"""“”““”““ ' . i '. ' M 135': ~13». ¢¢ olarization
.. one.” polarization ,. -;:»~ I." P
,. . - _n

. ‘ ‘1‘-
i. ' ".1: ,
3': ."V l' ' Y 2‘3." 1 \ i
.43\ l i , I ! 45‘. ‘ K!
-50 ' I ' "r“? i
o .

0 to ten 150 am 250 sec .150 '1 f‘.‘ 1 .s ‘::u .211 .90 5:32‘ 1:552»
thtdegveos: 3': 1:21.] n.w.-.~:

Figure 4.16: Mini-arrow scattering at 9 GHz.[3]

38

wavelength sampled RWG only mesh (reference) is compared with a graded mesh using
the combined basis function methodology at 9 GHz. This demonstrates the methods
ability to simulate three-dimensional objects using CF 1E (alpha = .5) with mixed basis
functions. CF IE-AP refers to the implementation of RWG and AP basis functions. It
should be noted that UIUC [3] computed these results and duplication at M SU was
deemed unnecessary. Next, we consider the mini—arrow at 12 GHz to determine edge
reduction potential. However, due to the small surface area of each surface of the mini-
arrow, the mesh is geometrically restricted rather than current restricted. We do not
achieve a significant reduction in edge unknowns. Hence, the combined basis function

methodology is only beneficial for large smooth sections.

39

CHAPTER 5: CONCLUSION

We conclude with a surmnary of the research and the knowledge gained along with

noting some particular challenges and future work.

1. Summary

In the past, people have used Rao-Wilton-Glisson expansion and testing functions to
solve the PEC scattering integral equation problem. For accurate results, this method
requires element edge sizes at most a tenth of a wavelength or equivalently,
approximately one hundred element edges per square wavelength. Thus, increases in
frequency yield exponentially larger problems. Aberegg and Peterson [6] addressed the
issue by multiplying the RWG function by the phase term of the incident field (i.e.
asymptotic phase functions). While AP functions allow for less dense sampling in regions
where the surface current phase is not rapidly changing, they require computing each
matrix element at each angle of incidence, a costly disadvantage. Since AP functions still
require high sampling rates near discontinuities, using RWG functions in those regions
eliminates some of the added computation. Thus, the combination of RWG and AP basis
and expansion functions on the same surface has the advantages of both methods. This
research shows that such a combination of functions, appropriately used, does in fact
achieve benefits of both, faster computation time from RWG, less unknowns from AP.
We find that the gain is maximized for surfaces where the majority of the element edges
(not necessarily the majority of the surface) use RWG functions. We have also shown
that the advantages of the methodology are highly dependent on the physical

characteristics of the geometry. Unique contributions from this research include:

40

1. Impedance matrix form shown in equation (2.22).

2. Use of the traditional singularity extraction technique [21] with asymptotic phase
basis functions over flat triangles.

3. Convergence and mesh analysis for the kite geometry.

4. Investigation of region specific sampling with combined basis functions.

Summary of Results from Specific Geometries

The kite and square geometries show approximately 35% and 68% (respectively)
reduction in unknowns when discretized using a graded mesh and RWG-AP combination
as compared to a tenth of a wavelength RWG-only meshing. With the number of AP
elements minimized and their usage location chosen wisely, matrix element computation
is significantly reduced from an all AP scheme. Therefore, for such geometries, the
combination of basis and expansion functions is a more effective solution method than

either AP or RWG used by alone.

For the mini-arrow geometry, we find that due to the small surface area of its side
surfaces, the meshing is geometrically constricted. On such a geometry, the reduction of
unknowns is not significant and the introduction of any AP elements to an RWG-only
implementation actually increases the solution cost since matrix elements must be

recomputed at each incident angle.

In a related effort in conjunctions with UIUC [3], we found that the conesphere geometry
with C2 continuity across the sphere-cone interface can benefit from an AP only type

solution. On the conesphere, the number of elements in the mesh away from the

41

discontinuity is kept large due to the curvature of the object. We found that the
introduction of RWG elements near the discontinuity had minimal affect in reducing the
design cost since the number of RWG tested and RWG expanded matrix element entries
was small compared to the AP tested AP expanded entries, as shown in Figure 5.1. An
AP-only solution method may be better than RWG-only; however, this depends on the
memory and processor resources available (AP-only is more processor intensive while
RWG—only requires more memory). Regardless, for the conesphere geometry a mixed
implementation reduces overall costs some, but the geometry lends itself to an AP only

mesh.

 

Figure 5.1: Graded Mesh on Square Plate vs. Conesphere

Conclusion

For all the objects considered, the AP only and RWG-AP combination RCS values we
observed were accurate relative to the traditional tenth of a wavelength sampled, RWG
tested, RWG expanded method of moments numerical solution. On large, smooth
surfaces with low curvature the RWG-AP combined basis and testing function method is

a more effective solution that solely an RWG or AP implementation.

42

2. Challenges

The usual long run times inherent in most numerical methods always have an
impact on the number of different implementations one can study. Several simple
and electrically small objects were considered in this research to quickly gain
some insight into the fundamental characteristics of each methodology. From this
information, we can begin to estimate how each might work on more numerically
intensive problems. As computational resources continue to become more
powerful with time, we can broaden the scope of our studies.

Discretizing an object into a consistently “good” mesh is a particularly difficult
task. It is an art form in itself, but is essential to quality numerical results. Since
the scattering problem is sensitive to discontinuities, a mesh that accurately
describes a testing object is crucial to attempting measured comparisons. Both
high quality meshing programs (e.g. SDRC IDEAS, PRO-ENGINEER, etc.) and a
highly talented mesh generation engineer are required to obtain high quality
meshes.

The essential difficulty with the method of moments matrix solution is the
requirement for large amounts of computer RAM. As frequency increases, so

does the need for memory.

3. Future Work

Following this research it would be beneficial to study this method with more objects and

at higher frequencies to broaden the knowledge base and understanding of the practicality

of the combined basis function method. Specifically, extended studies with curved

43

surfaces using six-point second order triangles have been presented in [3] and further

research would be very beneficial.

From a more general perspective, the possibility for other basis functions with the method
of moments has the potential to dramatically improve the scattering problem. Further
application of the fast multipole method [3] can impact the solution cost for large

matrices.

It would be of particular interest to consider applying the AP method to radiation studies
since antenna problems involve only one right hand side of the integral equation. Thus,
the advantages of AP might be realized without the disadvantage of needing to compute

the matrix elements for multiple incidence angles.

44

APPENDICES

45

APPENDIX A: FUNDAMENTAL THEORY

There are three sections to this fundamental electromagnetics theory appendix. We begin
with the independent large-scale form of Maxwell's equations. the fundamental starting
point for electromagnetics theory, and almost all electrical physics for that matter. The
vector potential quantities are then developed and finally a brief discussion of the Green's
function as implemented in the above research. These are included in an attempt at
completeness. Since we very well could have started with the results this development
derives, these basic concepts are included as an appendix. Hopefully their inclusion will
enable those not familiar with the electromagnetics discipline to understand and critically

evaluate the fundamental challenges faced when we apply the theory to real situations.

I. Maxwell 's Equations

Definitions: (All the following quantities are functions of time and a spatial position
vector r.)

E = electrical field intensity (Volts)

B = magnetic flux density

H = magnetic field intensity

D = electric flux density

J = moving charge density (Amperes per square meter)

p = charge density (Coulomb's per cubic meter)

46

Constants:

l
80 = 3—6;* 10 Farads per meter ...permittivity of free space

11,, = 47: "‘10-7 Henry's per meter ...permeability of free space

Parameters of a medium:

8 = 8,80 describes the permittivity in relation to free space and
it = urn.) describes the permeability relative to free space while
6 describes the conductivity of a medium.

In general, S is any open surface bounded by a closed contour C. Therefore, our
fundamental independent Maxwell's equations include F araday's (A. l) and Ampere's

laws (A.2).

(fie E'dl = —:,'—l, s fi'B d5 ...Faraday's law (A- 1)

0 (It

[1“ 4% B061]: 8 Sh'E d5 + L fi°J d5 ...Ampere's law (A.2)

Conservation of electrical charge requires

J; fi-J ds = ‘67.le dt. (A.3)

Equations (A.l)-(A.3) are in large-scale form. Assuming that C and S are not functions

of time, we can pull the time derivative inside the surface integral. Applying Stoke‘s

Theorem, I VX Vol? (13 = @V (11 , to the left hand side of(A. l) and moving the right

hand side of (A.1) to the left we get JS[VX E + "(,L, B] ‘15 dS = 0 . Since this holds for

any surface, the bracketed quantity = 0 or equivalently,

47

VXEZ—iB (A.4)

(It
If we apply Stoke's Theorem to the left of (A2), move the right hand side to the left and
combine the integrals, we can use the same argument as above (equation is true for any

S) to get,

iVxB=J+eofiE (A.5)

If we apply the Divergence Theorem, j'V'V dV = i V°15 0'8 , to the left hand side of

the constraint equation, combine the integrals, and note that it holds for all volumes v, we
get,
— __6L
V°J — .1. 0 (A6)

Equations (A.4)-(A.6) are the point form of the independent Maxwell equations. If we
take the divergence of both sides of (A.4), use the vector identity V-Vx A : 0 , and

invoke causality, we observe

V-B = O (A.7)
Taking the divergence of both sides of (A5), and use the same vector identity to get
% p = 8,, 73’: V'E I) if [EOV'E — P] = 0 . Time integrate both sides and invoke

causality to get the point form of Gauss' law.

V.E = L (A8)

0
In general media (not necessarily free space), we define auxiliary equations to include

non-zero magnetization M and polarization P, also functions of time and space.

48

1 . . .
H = T B — M (Amperes per meter) ...magnetic field intensrty

J = J i + 0' E (Amperes per square meter) ...total free current

D = 80E + P (Coulombs per meter) ...eleetric induction, flux density
In simple (linear, isotropic) media, we then define P and M in terms polarization
susceptibility xc and magnetization susceptibility 1.... P = 8.,XCE and M = X...“ So by
substitution

D = 80E + P = 80E + eUxCE = eo( 1+xc)E = e(r)E = e._,e..(r)E and

B = 1|.)(H + M) = 110(1+Xm)H = “(OH = 11.11.(r)H
The above equations are valid even for inhomogeneous media. By substitution into (A4)

and (A.5),VXE = ‘:71,#(T)H and tVX#(T)H : J + (%D. Ifwe assume the

media is homogeneous (permeability and permittivity are not functions of space), E and u

can come out of the derivatives, leaving
Vsz—aQ—fiH, (A9)
VxH=J+%D. (Am)
For use in the frequency domain, it is useful to define phaser notation for the vector field
quantities. We can write E(l',t) = E0 (r) 008 (wt +(pE (r)) where a) = 27rf. Using Euler's

equation to expand the cosine function into exponentials and considering the real part, the

following is equivalent,

E(r,t) : E0 (r) Re {61(wl+¢lj(r))} : Re {E(r)e¢5(l‘)elwl } (Al I)

49

We now make the following key observations:

1. All the vector field parameters can be written using a similar argument as for the
electric field.

2. The operator Re{ } commutes with addition, subtraction. integration. and
differentiation.

3. Time integration transforms to multiplication by jw in the frequency domain.

4. Since every parameter contains an e’l‘”t term, we can suppress it for notational
purposes.

The resulting time harmonic Maxwell Equations and the continuity equation, with vector

itm

field quantities having spatial dependence and a suppressed e term are:
Vsz—jwuH (A12)
VxH=J+jwa=J‘+(o+jwe)E (A13)
_ l
V-E——p(r) (A. 14)
8
V-H = 0 (A15)
V-J = jwp(r) (A.l6)

Notes:

0 To convert to the time domain, unsuppress the e""l

term and take the real part.
0 The above form of the equations is under the relatively strong assumption of

linear, homogeneous, isotropic media.

0 lfthe medium is free space, 8 = 80, u = 110, o = 0. If source free, .1 i = 0.

50

2. Boundary Conditions

For a comprehensive explanation of boundary conditions. see [9]. page 13. A summary

is stated here.

At the interface of two media with differing electrical properties, Maxwell's equations
dictate the following:
o The tangential components of the electric field across an interface between two
media with no impressed magnetic current densities along the boundary of the

interface are continuous.
nx(E2—El)=0 (A17)
0 The tangential components of the magnetic field across an interface, along which

there exists a surface current density Js (A/m) are discontinuous by an amount

equal to the electric current density.
15X(H2—H1)=Js (A.l8)
In a medium with infinite conductivity, the tangential components of E and H = 0. Thus
on the surface of a perfect electrical conductor (PEC), the tangential component of the

total electric field equals zero and the tangential component of the total magnetic field is

equal to the surface current density.

3. Vector Potentials

By making a change of variables, we can represent the electric and magnetic fields in
terms of intermediate variables, the electric and magnetic vector potentials. Using the

Lorentz gauge condition, we can manipulate the equations into a standard differential

51

equation with solutions. In this section, we will define the vector potentials and derive

the scalar Helmholtz equation.

Metic Vector Potem
Since we know VopH = 0 and the vector identity Vo(Vx V) = 0 . define A such that
aH=V><A. (A.l9)
By substitution into (A.l2),
Vsz—jw(VxA)=>VX(E+ja)A)=O (A20)
Since we have the vector identity Vx(—VV) = 0 , define (I)c such that
E = -V(I>, —j£0A. (A21)

Substitute (A. 19) and (A21) into (A.l3),

 

Vx[VXA]=J'+(o+jme)(—V<Dc—ja)A). (A22)
11

By homogeneity, can factor —I- out of the left hand side and multiply (A.22) by u.
[1

Define the wave number k such that k2 = —jwp (0' + wa ). Then (A.22) becomes
VxVxA=J'—u(o+jwe)V<Dc+k3A. (A23)

Using the Lorentz gauge condition,
_ fa)
(I). - k2 V'A, (A24)

and the vector identity VXVXV = V(V-V)—V2V , we can write (A.23) as

52

V2A+k2A =—11,J'. (A25)

Electric Vector Potegtial: (source free space.)

Using similar steps as above, since V-E = O , define F such that

E=—VXF. (A%)

Then by substitution into (A.l3),
VxH=J+jw,u(—VxF)=>Vx(H+ja)eF)=O, (A27)

We can then define “-Vwm 3 H = -V(Dm — jwllF . Using the same vector identities

and substituting into (A. 12),

VxVxF = jwp(—V(I>m —jpr). (A28)

_ jwé‘
By choosing the Lorentz gauge condition again, (pm “ T V°F and the same vector
identity as above, we can write (A.28) as
V2F+k2F =0. (A29)

Recall that A, F, E, HA)”, , and s are all functions of position.

4. Scalar Green 's Function

Let us consider the scalar Helmholtz equation from section 2 of this appendix,

V2w(r)+k2w(r)=—s(r), (A30)

53

 

Recall from above that u! is the unknown wave
function, s (r) is a known volume source density, and

k is the wave number. For a single point source r'

 

observed at r, s(r) = 5 (r —r') , we define the

 

Green's function, g(r|r'), as the field at any r due to Figure A-li Pit-51110" Vectors

r'. Thus the Green's function for the scalar Helmholtz equation is
V2g(r | r')+k2g(r| r') = —6 (r— r').

(A3 1) can be solved using the integral transform technique.

First, take the Fourier transform.

—(Af +Af +Af)g(2t|r')+k3g(}.

 

Then solve for g .

e—jix' (3- 1).!"
~ A r' = =
g( I ) (if—k3) (A-k)().+k)

 

 

Then. take the inverse Fourier transform.

00000

g(r|r')= (sir i .l lgo‘fl'flr'mdil

—oo—oo—oo

 

I") = —c '1'"

(A31)

(A.32)

(A33)

(A34)

To analytically evaluate the above integral (which has poles at :k = A ), switch to polar

A-space, evaluate over a pole excluding contour in the upper and lower half planes

exploiting Cauchy's Integral Theorem resulting in

g (r l I") = 3%,;6_"/AR where R = lr—r'l.

54

(A35)

Since g(r|r') is the point source solution and J(r') includes the magnitude and distribution

throughout a source region, by superposition,
A=uJ,J(r'>g(r1r')dV' (A...)

is a solution to (A25). For a surface current density, (A36) reduces to
A =uJSJtr9g<r1r94t (A...)

and is valid for open and closed 5.

55

APPENDIX B: VECTOR IDENTITIES

V is a vector function; w is a scalar function.

V-(VXV)=O
Vx(—VV)=O

VxVxV—VV-VzVZV

V-( WV) = wV-V + V-Vw

VX(Vw)= wVXV—VXVW

56

(8.1)

(B2)

(8.3)

(8.4)

(3.5)

APPENDIX C: INNER ICITE VERTIC ES

The following is the calculation as input to the SkyMesh2TM file. Based on the known
outer vertices and the desired out region thickness, using geometry, this calculates the

inner triangle vertices.

# Kite Definition (x-z plane)
klx =
kly =
klz =
k2x =
k2y =
k2z =
k3x =
k3y =
R32 =
k4x = 5.05
k4y = -3.85
R42 2 O

OONOWUTOOO
(DO
UTU'l

#Calculation of inner kite based on d and outer kite

Ml : (k2y—kly) /(k2x-klx)

M2 = (k3y-k2y)/(k3x—k2x)

M3 = (k4y-k3y)/(k4x—k3x)

M4 = (kly-k4y)/(klx-k4x)

X1 = (k2x+le)/2+d*(k2y-kly)/sqrt((k2y - klylAZ + (k2x - klxl‘Z)
x2 = (k3x+k2x)/2+d*(k3y-k2y)/sqrt((k3y - k2yl‘2 + (k3x — k2x)‘2)
x3 = (k4x+k3x)/2+d*(k4y-k3y)/sqrt((k4y - k3yl‘2 + (k4x - k3x)‘2)
x4 = (k1x+k4x)/2+d*(kly-k4y)/sqrt((kly - k4y)*2 + (klx - kdxlA2)
y1 = (k2y+klyl/2+(k2x+k1X—2*xl)/(2*M1)

y2 = (k3y+k2y)/2+(k3x+k2x-2*x2)/(2*M2)

y3 = (k4y+k3y)/2+(k4x+k3x-2*X3)/(2*M3)

y4 = (k1y+k4y)/2+(klx+k4x-2*x4)/(2*M4)

21 = (M4*x4+y1-Ml*xl-y4)/(M4-Ml)

22 = (Ml*xl+y2-M2*x2—yl)/(Ml-M2)

23 = (M2*x2+y3—M3*X3-y2)/(M2-M3)

w2 = M2*z2—M2*x2+y2
W4 = —w2

# Outer Kite # Inner kite

l klx kly klz 5 21 0.0 0.0
2 k2x k2y R22 6 22 W2 0.0
3 k3x k3y R32 7 23 0.0 0.0
4 k4x k4y k4z 8 22 W4 0.0

57

APPENDIX D: ADDITIONAL KITE SEPARATION COMPARISONS

The following sets of images are in addition to the discussion in Chapter 4.3: RWG

Region thickness for the kite.

Figure D.1 shows the AP results comparing different thicknesses of the RWG region.
The AP-WL-P and AP-OTT-P plots show close matching for all three thicknesses. The
AP-OTT-T plot is comparable to the MIX-OTT-T plot discussed in Chapter 4.3. Figure
D.2 shows closely matched RCS curves for the MIX-OTT-P and MlX-WL-P cuts.
Figure D.3 again shows closely matched RC S curves for the RWG only case. Refer to

Chapter 4.3 for further discussion.

IQ- Psunm Plot: AP-M-p Cut PhiPot

 

T fi

 

 

 

 

. >- — Slraighl
2° - - — 0:1 4
0:15
.3).
. 1‘
40» , ‘
e i I .s

8‘ 8"
“‘§

__ 3

N75}; .

 

 

 

148 1m 133

K11. Pattern Plot; AP-OTT Cut Phi Pol

 

 

 

 

—— Straight
- - — [i=1
. D=1 5

 

 

 

 

 

 

 

 

 

 

 

 

      

 

 

momma) 3030.404001]
Theta (dogmas) Theta (degrees)

Figure DJ: 10 GHz Kite RCS, AP Basis Functions

58

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  
 
 

 

 

 

  

 

 

 

tha Pattern Plot: Mil-WL Cut Phi Pol Kite Pattern Plot: Muir-OTT Cut Phi Pol
‘15 v r 1 Y Y Y T I r Y v v 7
— Stral ht 0 ‘ ‘
‘20 9 ‘ — Straight
- . — 0:1 - - — D=1
-25 . a 0:1 5 40 .. . 0:1 5 ,
.m P
.35 . ‘ 4
l
E 40 ' 1 <
2 1
u '45 h 1 \'
60» , i ' ' “ .
l | ' on F."
.fi 1 . . | l “I -(
i " 'f U i l L |
m '4 ‘I ‘ f -4
56 . . t’. 4
.70 A A A A A A A A A A A A A A A A. A
0 23 49 80 m 1(1) 120 110 160 TM -33 ~83 40 ~23 D Z) 40. 50 m
Phi (dogma) ' Theta (dagraas)
Figure D.2: 10 GHz Kite RCS, Mixed Basis Functions
Kite Pattern Plot: RWG-WL Cut Phi Pot
‘15 r v v v V I. r r 1
—— Straight
'20 — 0:1 ‘
- 0:1 5
.25 . .
.39 . .
.35 l ,
1
E 40 " I i
9 | I
g "5 ‘ a <
. 0 l \.
‘50 ' i | i I I
.. I . .
455, ' . . g .
‘ i 1
m ' ‘0’ a 1 F
V a
455 - , " 4
.70 A . A . A A A 4‘ A .1
D 23 40 H) m 100 120 140 1G] 1m
Phi (dagraaa)
Kite Pattern Plot" RWG-OTT Cut Theta Pol Kite Pattern Plot: RWGOTT Cut Phi Pol
0 r 0
— Straight — Straight
. _ 0:1 — 0:1
-10 _ o D=1 5 D“ 5
20
30 ' s i
E ' A
3 ' ~ 5 l .3
.. .' 1 r x. ..
.‘0 I i“ a
a .j
.50 > a l, A
80 ~ 4
_70 L . A . A . A I . m A A A '1 A A A I A
an ~60 ~40 ~23 D m 40 an m 33 ~60 40 -.’.0 U 20 40 so a)
Theta (degrees) Theta (degrees)

Figure D.3: 10 GHz Kite RCS, RWG Basis Functions

59

 

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6O

[1]

[8]
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[121

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[141

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62

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