' 0

? kN

•?»*£?$'
ON THE CONVERGENCE OF AN ITERATIVE FORMULATION

OF THE ELECTROMAGNETIC SCATTERING

FROM AN INFINITE GRATING OF THIN NIRES

by

JERRY C. BRAND

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

NORTH CAROLINA STATE UNIVERSITY

*

Raleigh, North Carolina

May 1985

This research was supported by the National Aeronautics and Space

Administration through grant NSG 1588.
ABSTRACT

Contraction theory is applied to an iterative formulation

of electromagnetic scattering from periodic structures and a

computational method for insuring convergence is developed. A

short history of spectral (or k-space) formulation is presented

with an emphasis on application to periodic surfaces. The

mathematical background for formulating an iterative equation

is covered using straightforward single variable examples

including an extension to vector spaces. To insure a

convergent solution of the iterative equation, a process called

the contraction corrector method is developed. Convergence

properties of previously presented iterative solutions to

one-dimensional problems are examined utilizing contraction

theory and the general conditions for achieving a convergent

solution are explored. The contraction corrector method is

then applied to several scattering problems including an

infinit e grating of thin wires with the solution data compared

to previous works. Problems associated with extending the

contraction corrector method to two-dimensional iterative

formulations are outlined including the benefits of applying

this process to difficult practical problems such as knitted

mesh surfaces.
Ill

ACKNOWLEDGEMENTS

The author wishes to thank all the people who have

helped him in the completion of his graduate work. In particular, he expresses deepest appreciation to his major advisor, Professor J. Frank Kauffman, who was first a friend

and then a mentor in many situations. Special thanks are

extended to the people in the RF Subsystems Section for the

support they offered, and especially to Linda Snyder and

Susan Sanders for their help in typing.

The author also wishes to express his thanks to his

wife who gave so much support in the completion of this

dissertation and for the final typing of the dissertation.
IV

TABLE OF CONTENTS

Page

LIST OF TABLES v

LIST OF FIGURES vi

I. INTRODUCTION 1

II. REVIEW OF THE FUNDAMENTAL FORMULATION 4

III. MATHEMATICAL FORMULATION OF ITERATIVE

EQUATIONS, CONTRACTIONS AND FIXEDPOINT THEORY 12

IV. THE PERIODIC STRUCTURE: ' A CANONICAL

CASE AND THE REGIONS OF SOLVABILITY 23

V. THE PROBLEM OF AN INFINITE GRATING OF

THIN WIRES 48

VI. EXTENSION TO TWO-DIMENSIONAL PROBLEMS 61

VII. CONCLUSIONS 74

REFERENCES . 76

APPENDIX A 79

APPENDIX B .' 86

APPENDIX C 90
LIST OF TABLES

Page

Table 1. Example of a convergent iterative

equation 16

Table 2. Example of a non-convergent iterative

equation with convergence achieved by

applying the contraction corrector

method 19

Table 3. Example of a complex valued iterative

equation and the use of a complex contraction corrector o 21

Table 4. Example of a two-variable contraction

corrector method 67
LIST OF FIGURES

VI

Page

Figure 1. a) The planar periodic surface indicating electric current density

J over the conducting portion

and K over the aperture portion

mn

b) Unit cell definition ... o ... . g

Figure 2. Example of convergent iterative equation g(x) 15

Figure 3. Free-standing conducting strips with

Floquet cell width a, aperture size

b and conductivity a 32

Figure 4. Convergence of the aperture E field

for a=1.4X, b=0.6X, 6=0* case using

equation (40) 35

Figure 5. Convergence of the aperture E field

for a=1.4X, b=0.6X, 0=0° case using

equation (40) with the correction of

equation (53) applied/ from Tsao and

Mittra IB] 36

Figure 6. Convergence of the aperture E field

for a=1.4X, b=0.6A» 9=0° case using

equation (42). This is the contraction corrector method 38

Figure 7. Contraction factor for a cell width of

1.4X, incidence angle 0=0° with H-wave

39

polarization and <*>=0°
vii

Page

Figure 8. Contraction factor for a strip width of

0.6x, incidence angle 0=0° with H-wave

polarization and $=0° .......... 40

Figure 9. Contraction factor for a strip width of

0.1X, incidence angle of 0=0° with

H-wave polarization and $=0° ......' • 42

Figure 10. Convergence survey using contraction

criteria for 0°<0<90°, $=0°, a=0.9X,

and b=0.4X. .... ........... 43

Figure 11. a) Non-convergent results of the variational corrected iterative equation

for 0=70° ........ ..... . 45

b) Non-convergent results for the basic

iterative equation with 0=70° ... . ^g

Figure 12. Convergent results obtained from the

contraction corrector method of equation

(30) with "5=0°, 0=70°, a=1.4X and

b=0.6A .............. ... .

Figure 13. Planar grid of thin conducting wires. . . 49

Figure 14. Convergence properties of the variational correction scheme using both E^ ' and

the Dirac delta function as the test

function with 0=$=0°, a=0.25X and

b=0.247X ................. 50

Figure 15. Definition of reflection coefficients as

described in [22l ............ 53
viii

Page

Figure 16. Comparison of results obtained from Wait(2j

and equation (30) for various cell widths

with <J>=0° , 0<G< 90°, and o= « S 55

Figure 17. Reflection coefficient as defined in

Figure 15 for $=90°, 0°<0<90°, a=0.247X

and a=°°S

Figure 18. Several cases of finite conductivity

with 0=0°, 0°<G<90°, a=0.25X and

b=0.247X ................. 58

Figure 19. Illustration of converged accuracy for

various number of terms with a=0.25A,

b=0.247X, 0=$=0° and a= « S ........ 60

Figure 20. Actual mesh surface, 69

Figure 21. Floquet cells outlined on the actual

mesh surface 71

Figure 22. Single mesh path of the actual mesh

surface 72

Figure 23= Grid representation of the segmented

single mesh path 73
I. INTRODUCTION

The electromagnetic scattering properties of periodic

structures have been of interest to engineers and scientists for many years and various methods for solving the

scattering problem have been presented [l] - [8] . These

methods include solutions generated from static approximations, averaged boundry conditions, the method of moments,

physical optics, the geometrical theory of diffraction, and

a combination of these techniques. However, certain frequency regions or certain geometrical configurations (or

both) cause grave difficulties .which cannot be overcome by

the methods mentioned above .

. Tsao and Mittra P] presented a novel iterative

technique based on earlier spectral approaches which

can be traced to original work performed by Bojarski [10] .

This technique, called the spectral-iteration approach, extended the scattering solution capability to regions previously untouched by other methods. In particular, Tsao

and Mittra [9] examined the scattering from periodic structures and their applications to frequency selective surfaces. The present author's area of interest is the problem of determining the reflection propeties of mesh surfaces [ill . Mesh surfaces are used in many applications,

but the most current application is for conducting reflectors on space-born antennas. The mesh surface is a complex
2

periodic structure whose reflecting properties are not

readily analyzed by the methods mentioned above. For instance, an attempt to solve the mesh problem by the method

of moments would require a special set of basis functions

as well as an enormous amount of computer memory. Analysis

of the mesh surface by the spectral-iteration technique

would be of great value since the method is essentially independent of geometry, i.e., does not require explicit

knowledge of appropriate basis functions and does not require extreme amounts of computer storage. However, the

basic iterative scheme suffers from convergence problems

that are associated with most iterative formulations [12] .

This work treats basic iterative techniques, presents a

background on the convergence problems associated with

iterative techniques, details the formulation of a corrective scheme to ih:sure convergence of the iterative technique, and applies a version of the corrective scheme to

the specialized problem of a parallel wire grating.

Current techniques for solving complex scattering problems are limited by the difficulties outlined above. The

technique described herein provides a basis for the further

development of solutions to complex problems. Additionally,

the solution methodology provides a useful concept which

may be applied to problems outside the realm of electromagnetic scattering.
3

Portions of the background material presented below

follow directly from the referenced works and credits are

listed for these. The notation used herein is self concained with attempts to follow the references as closely as

possible to maintain a common base, but allowing for differences to insure clarity of the equations. This is included

for consistancy, completeness, and continuity. The reader

is advised to consult the references for greater detail.
II. REVIEW OF THE FUNDAMENTAL FORMULATION

Bojarski [10] considered that field quantity Y(x) and

a source density w(x) were governed by the differential

equation

L y(x)=-L w(x) (1)

•f W

where the form of the operators for an n-dimensional

problem are / i \

%-E

 ailn) 5 P-1.2,— n (2)

i=o

 v P /

and Lw= £ bi(lx^) 5 P-1.2.— n (3)

subject to the constitutive equation

w(x) = q(x)

The generalized integral representation of equation (1)

is given by

w(x')dx' + v. (x) (5)

where g(x, x1) is the appropriate Green function satisfying

equation (1) with w(x) replaced by <s(x), the Dirac delta

function and Hfinc being the externally applied field.
5

Equation (5) is the basic equation which is to be solved by

application of a transform technique.

The Fourier transform of a function f(x) is given as

f(k) = / f(x)ejk*

x L dx (6)

and the transform pairs formed are noted as f(k) •*•»• f(x).

Thus, by taking the Fourier transform of equations (1), (2),

and (3), the k-space formulation of the problem becomes

L (kmk)=L (k)w(k) (7)

T "

£ a.(jk)1 (8)

i=o

 x

m

k)-Z Vjk)

where j is the usual imaginary counter-clockwise rotation

operator. The integral equation (5) then becomes

(10)

still subject to the conditions of equation (4). This

approach yields two algebraic equations which may usually

be solved with much less effort than equation (5). With
6

this thought in mind, we now examine the application of

this procedure to the problem of electromagnetic scattering

from a periodic structure.

The electric field E generated from an equivalent magnetic source R can be represented by

E(x,y,z)= — VxF(x,y,z) (11)

where F is the associated electric vector potential of the

source and e is the permittivity of the medium [13] . The

relationship between F and K is established with the use of

the position vector r and the free space Green function

A /\

. ' . -j (k-r) =

G = - - I (12)

4ir|fl

by

 -F(r)=/G(r,fI)'K(rl =/G(r,f c )dr' (13)

I

A

where k and r are the respective unit vectors of the problem.

From this, the magnetic field intensity H can be derived

from Maxwell's equations and is given by

S(x.y.z)— J«e F(x,y,z)+ vy^x.y.z) (14)

where p is the permeability of the medium. For the xy

planar set of magnetic currents shown in Figure 1, F =0 is
obtained from equation (13) since z=0 and G is then a

function of x and y only. Allowing the medium to be that

of free space, i.e. e=e o andy= y , and since F =o, equation (14) is expanded in Cartesian coordinates x and y to

yield for z=0

H (x,y)= -r^-

s

 3 J am

a.

3F

2

3.

3F,

a,

9F,

(15)

where k = 01^:1 e is the propagation constant. The quantity given by equation (15) is a general representation for

the scattered magnetic field intensity generated from a

planar source of magnetic current in the xy plane.

Consider the planar periodic surface shown in Figure 1

to be the source distribution for che magnetic field of

equation (15). Upon substituting equation (1.3) into equation (15) and taking the Fourier transform of equation (15)

we obtain

mn

2 2

o mn

amn'mn

k

 2

K -

(16)

B y)^'
ORIGINAL PAGE-IS

OF POOR QUALfTY 8

Figure 1. a.) The planar periodic surface indicating

electric current density J over the

mn

aperture portion,

h.) Unit cell definition.
for the transformed magnetic field where

- k sinG cos*

a o

(17)

a

_k sinesin$

o

(18)

are the Floquet modes [14] and

'k CL 8 ^

 mn' mn'

(19)

is the Fourier transform of the dyadic Green function [9] .

The discrete nature of equation (16) results from taking

the Fourier transform of a periodic function [15] . The

final step in completing the process of obtaining a useable

form of the magnetic field intensity in the aperture is

accomplished by taking the inverse Fourier transform of

equation (16), noting K=E Xz, using the equivalence theorem

3.

and applying the appropriate boundary conditions on

H

s(x,y) at z=0 [9] . This leads to

, 2 2

H

tine

(x,y) = -2

J

mn

mn

a

mn

(20)
10

where ^a(a ^mn) ^"s tne transf°rmed electric field in

the aperture and ^tinc^

x»y^ is the incident tangential magnetic field. The inverse of equation (20) is obtained by

extending the operations over the entire unit cell. The extended operator is necessary to insure the operations indicated by equation (20) are in the domain of the solution.

That is, the boundary conditions of the problem are satisfied. This extension is accomplished by including the electric current density over the conducting portion of the

unit cell. The electric current density is included by introducing the truncation operator and the complement truncation operator defined .as

T(f(f)) = 0 for r on the conductor

(21)

T(f(f)) = f(f) for r in the aperture

and

T (f(f)) = 0 for r in the aperture

_ (22)

T (f(r)) = f(r) for r on the conductor

This allows equation (20) to be extended over the unit

cell as
am

amn8 mn

ko2 "amn

11

2

(23)

with J(x,y) reprsenting the electric current density on the

conductor. Equation (23) is a form which can readily lend

itself to solution by iterative techniques.
12

III. MATHEMATICAL FORMULATION OF ITERATIVE EQUATIONS

CONTRACTIONS AND FIXED-POINT THEORY

The use of iterative equations to solve mathematical

problems has been around for many centuries dating back to

B.C. 600 when this technique was used to solve problems

such as determining the square root of three. Beginning

with this early exploration of iterative techniques came

the plights of correct problem formulation and convergence

of the solution. Later methods, such as Newton's method of

solution, alleviated such problems but often forced other

constraints on the problem. One such constraint is that

the initial solution estimate needs to be in a region

reasonably close to the desired solution. Tsao and Mittra

[9] used this spirit of a priori knowledge in the form of

a variational correction to remain near the solution point

in their problem formulation. However, as will be demonstrated later, the cases presented in their work did not require such a correction and, indeed, they even noted where

the iteration method failed even using such a correction.

The idea of using the iterative scheme to solve general scattering problems leads one to seek a way of determining the criteria for convergence and a way of generating a

convergent iterative formulation. Fixed-point theory and

contractor theory [16, 17] are applied to formulate a con-
13

vergent iterative solution to the scattered fields derived

in Chapter II.

The general idea of fixed-point theory revolves around

finding the solution to an equation such as

f(x) =y=x 2 -6x+5 = (x-5) (x-1) = 0 (24)

This is accomplished by forming an iteration function as

xn+1 = g<V (25)

and finding the intersection y=g(x ) and y=x which yields

the fixed-point and solution to equation (24). This is

accomplished by starting with an initial guess x , substituting into equation (25) to generate x,, and continuing

this process until x +1=x +e where e is the allowed error.

At this time x=x , is the approximate solution to equation

(25). The conditions for a convergent solution are in

general:

a.) on a closed interval I, g(x) maps I into itself

b.) g(x) is continuous on I

and c.) g(x) is differentiable on I and

de(x)

< 1

Thus the following theorem may be stated

Theorem 1. Let g(x) satisfy conditions a.), b.), and

c.). Then g(x) has exactly one fixed

point x* in I and starting with any XQ in

I, the sequence x , x, , ... , x generated

by equation (25) converges to x*.
14

The process outlined by the above theorem is illustrated in

Figure 2. The fixed-point x* is often said to be a point

of attraction for g(x).

 x

n+i is also said to contract to

x*.

To illustrate the use of this theory, one root of equation (24) will be determined. Let

(26)

and find

d g(x) = -* (27)

then, the interval I is defined as xtl and IcC-3,3]. Note

that the known root x=l is in this interval and

-T— g(x) =1/3. We then expect that the iteration function

will converge to the fixed-point x*=l which is the correct

solution. The iterative process is contained in Table 1.

The question then arises, what happens when the sequence

will not converge?

To generate an iterative equation that will convege we

must meet the conditions listed in Theorem 1 above. As an

example, let us examine the roots of

f(x) = x2 - x - 6 = (x+2) (x-3) (28)
1

5

:\

a N

s01s8

S• «l

S

SSs

X! bOCo

ctf O

"

0

•*->c0) bO<»cooo<DQ, X! CM0)3bO•H
16

f(x) = x2 - 6x +5 = 0

x

2 + 5 g(x) **

X

0

xl

x2

X3

X4 -

X5

*^£

6

g(x)

2.0

1.5

1.208

1.077

1.026

1.009

1.003

xn-l

x*=x 1.000

n

Table 1. Example of a convergent iterative equation
17

which can be seen to be -2 and 3. By forming the iterative

equation

= xn+l=xn

(29)

and noting

(30)

we see that the iterative equation will not converge. However, by properly forming a new iterative equation G(x)

that has G(x) x=-2

<1 or G(x) x=

, the roots

may be obtained. Although manipulation of equation (29)

could yield a suitable G(x) [12] we seek a G(x) that can

be formed by a method known as relaxation. The relaxation

process generally uses a portion of the n iteration along

t~Vi

with a portion of the iterated n (i.e. n-t-1) iteration to

"relax" the process. The form of the relaxed iterative

equation is

G(x (l-R)g(xn) (31)

where R is called the relaxation constant. The condition

=0

x=x

(32)

n

may be used to determine an optimum R such that convergence
18

of G(x) is assured. Performing the operation of equation

(32) on equation (31) yields

4- G(x)

dx

= R+U-R) 4- S<x)

dx

x=x

= 0 (33)

x=x

n n

and solving for the linear correction R

where g'(x)= -7— g(xj. This process of finding the optimum

n

R is called the contraction corrector (a name to be explained later in this section) and is used to iteratively

solve for the root x=3. This is illustrated in Table 2.

Equivalence between the contraction corrector and Newton's

method can be shown and is illustrated in Appendix C.

Note that g(x ) would not converge by itself and thus

 G(x)

is necessary for a correct solution to be obtained. As a

further note on the iterative process, the conditions that

g(x) maps I into itself is not trivial. For instance, note

that

f(x) = x2 + 2 = 0 (35)

will not yield a solution from an interative process unless

x is allowed to be complex and g(x) or G(x) maps into the

complex plane. The iterative equation G(x) may even require a complex relaxation or complex contraction correction constant and precautions governing complex algebra
19

-

 xn+l = xn - 6

x_.,= G(x ) =

n = Rx

n=

(1

-

R)

2 xn

n

0

1

2

3

4

X

n

1.0

7.0

4.24

3.23

3.01

R

n

2.0

1.07

1.13

1.18

1.2

n

1

2

3

4

5

6

2x

g(xn)

-5.0

43.0

11.98

4.43

3.06

^

3.0001

3.0006

3 . 0036

3.0216

3.130

3.798

G(xn)

7.0

4.24

3.23

3.01

3.0

g(xn)

3.0006

3.0036

3.0216

3.130

3.798

8.424

Table 2. Example of a non-convergent iterative equation

with convergence achieved by applying the contraction correction method.
20

and operations must be observed. In particular, the analytic nature of f(x), g(x), and G(x) in the interval I

should meet the requirements of Theorem 1. An illustration

of the nature of these problems is given in Table 3. In

two cases, a real x never maps x into the complex plane

and, as such, never yields a solution. Even applying the

contraction corrector method with a real x does not give a

solution. Only the one case using a complex x and the contraction corrector method converges. This follows from the

conditions of Theorem 1. Since problems have been observed

in the simple problems above, a proper choice of action is

to pursue the corrective method in a more general sense.

Stakgold Cl8l has formed a very sound collection of

theorems and definitions on the idea of metric spaces and

their transformations, and the following material is condensed from his book with the advice that the reader consult his work for proofs and greater detail.

Definition 1. A transformation L of a metric space

x into itself is Lipschitz continuous

if there exists a p, independent of u

and v, such that

d(Lu,Lv)_< pd(u,v) for all u,v£x

where d(C,£) is a proper metric in x. When the definition

above holds for some fixed \p( <1 , the operation of L is

called a contraction on X.
f(X) - X - X + 2

21

g'(Xn) 2X_

g'(XQ)

G<Xn> - 6 X + (1 - B ) gCX i

ff*(X ) — 1 n n n n n

n

0

1

2

-

X

n

1.00

3.00

11.00

-

g(xn>

3.00

11.00

123.00

diverges

n

0

1

2

3

xn

4.24 ••• J4.24

2.18 + J2.34

1.13 + jl.45

0.64 •«• jl.25

n

0

1

2

3

4

5

6

Xn

2.00

0.67

-4.67

-1.91

-0.34

1.11

—

g(xn>

1.88 + J35.95

1.81 •»• J10.24

1.20 * J3.29

0.85 •*• jl.59

g(xn)

6.00

2.44

23.78

5.66

2.12

3.24

—

g'cxn>

4.00

1.33

-9.33

-3.83

-0.69

2.23

—

Bn

1.06 - jO.07

1.09 - jO.13

1.13 - J0.28

1.05 - JO. 40

6n

1.33

4.00

0.90

0.79

0.41

1.81

—

G(Xn)

0.67

-4.67

-1.91

-0.34

1.11

-0.62

-

G(XD)

2.18 •»• J2.34

1.13 + jl.45

0.64 t- jl.25

O.bO * jl.31

X* - 0.5 -f jl.32

Table 3. Example of a complex valued iterative equation

and the use of a complex contraction corrector.
22

Theorem 2. Let the operator L be a contraction

on a complete metric space X. Then,

u=Lu has one and only one solution

in X which may be obtained from any

initial starting point u in X.

Thus, u -*• u* (the fixed-point) as

n-*-» and u* exists and is unique in

X.

Note that u may have multiple components, i.e., u=u(z,,Z2,..

. ,z ) where z. are complex coordinates. P as given by definition 1 is the description of the measure of a general derivative of an operator. Thus, the process used in the single variable examples earlier may be extended to higher dimensional spaces with more general operators. This theorem

allows that the relaxed corrective iterative process outlined earlier is indeed a contraction, hence the name contraction corrector. This nomenclature is used to differentiate this scheme from other corrective schemes applied to

iterative processes. The use of this definition and theorem as applicable to the spectral-iteration approach of equation (23) is detailed in the following section.
23

IV. THE PERIODIC STRUCTURE; A

CANONICAL CASE AND THE REGIONS OF SOLVABILITY

The total electric field for the structure shown in

Figure 1 may be expressed in terms of the incident and

scattered fields as [9 ]

^

E,

Z

mn

amn mn o - mn

~°mn mn

v mn

ran

x

Z

mn I

amn mn

B -a B

mn mn

2

 1

mn Is,

K

1

mn J

(36)

°mn' mn

mn

-k -z z

mn

e

for z<0

a

where

x

mn

~ mn '°r ko

2

 » °mn2 + Bmn2

1

 tor k.1 < a* + * z

(37)
24

= =

and E and E represent the transformed electric field

a a r

and are respectively the reflection and transmission coefficients of the Floquet modes. Deriving H from equation

(36) and by enforcing the boundary conditions on the tani gential magnetic fields across the cell at z=0, equation .

(23) is generated again. This exercise is presented to

complete the field description and to illustrate the relationship between the aperture fields and the reflection and

transmission coefficients as the coefficients will be of

interest later in the section. Equation (23) is the beginning point to cast the problem in an iterative form.

The summation in equation (23) represents a discrete

Fourier series (DPS) for an infinite duration (i.e., periodic) sequence [19] . This representation allows for direct

transformation between the (x,y,z) domain and the (k ,k >k )

x y z

domain. Since the functions are represented by sequences

with complex exponentials having a periodicity of 2ir/m and

2ir/n, one period of the aperture distributions, that is one

cell of the structure, can be used to completely specify the

transform. The use of one period to represent a periodic

function in this manner is known as the discrete Fourier

transform (DFT). It is important to note that the transformation mentioned above is exact and this infers that no

aliasing will occur when the DFT is performed. Aliasing

can occur when a non-periodic function is truncated and
25

and transformed with a DFT. Since information about the

original function is lost in the truncation, performing the

inverse transform can never return the original function.

However, when using exactly one period of the periodic function, the original function can be reconstructed when the

DFT and inverse DFT are performed. The most common form of

DFT algorithm is the fast Fourier transform (FFT) in which

special properties of the DFT are exploited to decrease the

computation time of the transform. Throughout this section,

the DFT and its inverse are represented by F and F respectively. Implicit functional dependence is also used to

present a "cleaner" form of "the equations.

With the above thoughts in mind, we may solve equation

(23) in terms of E~ as:

,-1

G"1 • F V

J)

 -

 Htinc (38)

and, for clairity write equation (23) as:

T (J) = T

c c

ti

tine +

,-1 8 T(Et) (39)

Note that the subscript indicating aperture field has been

removed from equations (38) and (39) since the truncation

operator is included and the tangential field E over the

entire cell is used. Solving for E in equations (38) with
26

T (J) substituted from equation (39) yields

«• (n+1)

E

t

p

= F

-1

*.F [T(lt<»>)] "

Htinc

(40)

This is the form of iterative equation given by Tsao and

Mittra [9] and is the basic equation to which a scheme

called the contraction corrector will be applied. The con

traction factor must be less than 1.0 to have a convergent

solution [18] and the contraction corrector scheme will

allow this condition to" be met.

The iterative equation for one-dimension (E or E ^

x y}

will converge if the conditions for a contraction given by

Stakgold [l&] are met. The proper metric for the space X

and the operators in equation (40) is:

d(u,v) = (41)

where u =( ^ > g2, ... , £n> and v =(?1» ?2"" ' ^ are the

vectors in X and Ct z, represent the components of each vector. The u and v used in the metric d(u,v) may be any u,v

in X since the contraction must hold for all u,v in X. One

This choice

would allow the contraction process to be observed and

logical choice for u and v is E and E
27

would give the relative error between each iteration. This

also allows the regions of convergence to be determined.

That is, for decreasing error, the solution is converging

and for increasing error diverging. Since we are interested in the contraction to a fixed-point, a wiser choice for

u and v is E and ^E~ + e) where e is some small complex

number. Thus, at each component of E~ we essentially

have the ability to determine the general derivative of the

operator at each element. The contraction corrector is a

vector R with one element for each component of E

The form of R is exactly that of equation (31) for a onevariable problem extended to an N component one-dimensional

problem. This powerful concept allows the regions of convergence to be determined and allows the formulation of a

convergent solution as

R

(n) •' E(n) + (? - R(n)) • L E(n) (42)

with L being the operator defined by equation (40). Uniqueness of the solution is guaranteed since the problem was

formulated from Maxwell's equations via a vector magnetic

potential and the appropriate boundary conditions were

applied. Note that this representation is for a one-dimensional scattering problem. Additional conditions are necessary for the general solution of two-dimensional problems

[12] . These conditions and other considerations for two-
28

dimensional problems are discussed in Section VI.

The contraction corrector R = <Ri » RO'"' ' Rn must

be determined in a general operator sense to behave as R in

equation (34). This implies that closer examination of Definition 1 is warranted. The determination of an operator

derivative must be formulated in terms of a contr'action.

The derivative must be expressed in an analytical form that

allows the contraction to be determined numerically for

each R. of R. Thus, equation (42) will guarantee a convergent solution when R is properly determined. The contraction is related to the .continuity of the operator (transformation) L through Definition 1 as:

d(Lu, Lv) < P d(u,v) . (43)

with d(C,C ) given by equation (41). The determination of

the contraction of L at u = u and v =(u + Au)is a representation that is thought of in operational calculus as the derivative of the operator L for vanishing Au. Equation (43)

is then written as:

d[Lu, L(u + Au)] <_ p d[u, u + Au] (44)

If the vector u has m components and equation (44) is expanded

using equation (41), the result is:
29

,. +Au)-Lu, + L(u_+Au) - Lu0 1 i\ \ 4 A

< P

Ku1+Au)-u1

•J

•(45)

Eliminating terms in the denominator of equation (45) yields

+ . L(u +Au)-Lu_

f

< P (46)

mlAu l

Now, since m>l, a stronger condition is written

I

2 + |L(u2+Au)-Lu2|2

 +.-.+I

P (47)

Consider the following: given

12 ; I 12 ^ Ai [!• "*

 a2 *—* am (48)
30

Is the inequality true? Squaring both sides yields:

m

(49)

m

a. m

and the inequality is seen to be valid for non-zero a..

Equation (47) is then written as:

L(u,+Au)-Lu, Au)-Lu L(um+Am)-Lu

Au

< p (50)

and then may obviously be further reduced for each component

t0:

 L(ui+Au)-Lu1

1 Pi (51)

Au

noting that -f^P-

 < P»

From equation (51) it is

be defined and the final form of equation (51) becomes:

From equation (51) it is seen that a complex-valued p may

c

i

L(u.+Au )-Lu.

Au

= P

Ci

(52)

This form is the measure of the derivative of the

operator L at each component and may be used to solve for

the contraction corrector R. at each component u- e u.
31

Even though a powerful tool has been developed to

allow the computation of a convergent one-dimensional solution to electromagnetic scattering, the background problems

surrounding the contraction corrector scheme should be examined. This.background material is of interest since the

breakdown of prior methods and the reasons for non-convergent solutions gives insight into what particulars of the

scattering problem cause the difficulties. In particular,

known geometrical problem areas are: conductor width, cell

size, aperture size, and the angle of incidence and polarization of the impinging plane wave. The understanding of

the difficulties arising for certain values of those parameters can provide vital insight when attempting to extend

the solution to two-dimensinal configurations, especially

when mesh structures are involved. The reference point for

this background investigation is the work of Tsao and Mittra

[9] . A complete iterative flow is detailed in Appendix B.

Consider the case of the free-standing strips shown in

Figure 3. This planar configuration of thin, perfect conductors was the prime example used in the work of Tsao and

Mittra [9] . The case of a cell width of 1.4X , aperture

width of 0.6x , and incidence angle 0=0* was of particular

interest. Although the iteration equation (40), would perform adequately, Tsao and Mittra decided to apply a correction step to the iterative process. The idea of being

"near" the solution in numerical methods (i.e., Newton's
32

Conducting

Strips

>

H-nc

finc

$5r•^ifi$, , v'.-y-»fjf f*Kl:-^J ivjE-S-M

4

X

Figure 3.

Free-standing conducting strips with Floquet

cell width a, aperture size b and conductivity

o.
33

Method) to either insure or speed convergence is commom

place. The correction step was chosen to be a form of the

Method of Moments [28] .

The amount of amplitude error of IT may be calculated from processing E with equation (40) and using an

amplitude correction C determined via a variational formula

calculated with Galerkin's method. This correction is

given by [9] :

C =

tan, tan

tan

(53)

where:

(54)

ds

aperture

with "*" denoting conjugation and ds the outward surface

normal. The value of C yields a weighted average error

based on the first moment's variation of the operator using

E

 n as the test function. C will be equal to one when E

represents the averaged approximate solution to the problem.

This statement implies that oscillations of the solution

could average out and yield a solution that is incorrect.
34

Numerical difficulties could then arise if a problem generated such a solution from the iterative process. Great

care is then warranted with this type of numerical method

as is true with any numerical scheme. Additionally, as previously noted for iterative processes, just being near the

correct solution in amplitude does not guarantee a convergent process.

The geometry given in Figure 3 for a=1.4X, b=0.6X, 0=0°

and H-wave (E.. parallel to strip edges) incidence is a 11 no

problem that converges by equation (40) without the aid of

the variational correction of equation (53). What then is

the useful purpose of the correction C? The most useful

purpose of C is to speed convergence since an iterative

equation which is a contraction will converge from any initial estimate and will diverge if not a contraction. Another useful (but applied with caution) purpose is the indication of accuracy of the iterated solution. The problem

above is taken from Tsao and Mittra [9] . The convergence

of the problem is shown in Figure 4. Figure 5 illustrates

the convergence of E of the problem using equation (40)

corrected by C of equation (53). The iterations of both

Figure 4 and Figure 5 are quite similar; however, the E n

shown in Figure 5 are converging more rapidly then the E

in Figure 4 by one iteration. This is a very small difference

in convergence rates and indicates that the methods behave

similarly.
3

5

bec•H-P 0

,

O -H

3 t-(

•a -P

c wOO0)0)a

BX

CD O

CO•OIIftIIaSO«H•H<H-pJH0)d

CO•H-Po*0)

O -HCD

oC <

D

0) (0

h O <D> o c o O II o o

•<*a)bC•H

fe
36
37

Equation (42) is also used to solve the problem above.

The contraction corrector R is obtained using equation (52)

along with equation (34). The application of this contraction corrector yields results for E shown in Figure 6.

Note that the first iteration is the same as the first

iteration in Figure 4. However, the second and third iterations shown in Figure 6 are improved over those in Figure 5.

Even though this convergence improvement is by one iteration, it is pointed out that the contraction corrector

guarantees convergence (except for possible numerical computational problems). The case for using the contraction corrector over the other methods is not well presented in the

preceeding example. The basic iterative equation and the

variational corrected iterative equation break down and diverge for various cases. The following material is presented to establish bounds for the regions of solvability for

the preceeding problem and to demonstate the usefullness of

contraction theory in this matter.

The basic iterative formulation is found lacking for

various cases. Conductor size, aperture size, cell size

and incidence angle are the parameters that are varied and

may cause convergence difficulties, The first case to be

examined is that of constant Floquet cell width and varying

aperture size. In paticular, let a = 1.4X and 0.005A <_ b

_< 1.395X for perfectly conducting strips, incident angle 9=0

and H-wave polarization. The contraction factor for this
3

8

r< C CO O • -H O -P II O f^ QJ • -P

.-< C

^fl O • o II 0-pCO •HCO o<H730•H<HIW0fn0 O D, -H cd -POJ 0 3 X! 0« • •P 0 T3O •H bD Si O G -P •H 0 0 CO g(HO

IM

O

fi0 0 M M -

P

JH OS O

0 O 0 > O F-i

C OM

O II O

u o o CO0Vl bO•H

CT3LJ 3
3

9

I

8

sa

0)ococ-PTJa)oo<Ho•PocS «HCoo

rt

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oII•3Cat CO«5

1S1

•H^

rt rHOD

,

0)

rf oIIobCCai
POO

R QUALrTY

40

»

oc0•o•Hoco•o«Ho•M•a-P01oJ

oOIICo03

tsl

•HOQ

.

c

_ *»

£ 3 j "

i

,^£

•

0O0£c0•H•p0?H^Jco0000)

Oj

1K-P•H?0oII0Q)

rl

hfl

c0*

<*>

M
41

case is graphed in Figure 7. Recall that the solution will

diverge for a contraction factor greater than one. These

non-convergent regions are where the contraction corrector

must be applied to insure covnergence. Next, the cell size

is varied as 0.9x <. a <_ 1.4x for constant conductor size,

i.e., (a-b) = constant, and 0= 0° with H-wave polarization.

Figure 8 shows the contraction factor for a conductor size0.8X . This last example contained a conducting strip that

was a large portion of a wavelength. The case of a conductor size 0.1X and cell width 0.2 A _< a _^ 0.9 A for H-wave polarization and 0= 0° is presented in Figure 9. This example

represents a small strip and the contraction factor associated with its geometry. The last case to be examined is cocerned with the angle of incidence of the impringing plane

wave. A cell width a = 0.9X , conductor width b = 0.4 A

and aperture width 0.5X is chosen for the convergence survey illustrated in Figure 10. Note that the contraction

factor bumps above 1.0 around 0=8 ° and diverges from 1.0

for 0>60°. This indicates that the basic iterative scheme

will have problems with convergence around 0=8 ° and will

s^

not converge for 0>60°. The above examples offer several

cases that point out problem areas in constructing a convergent iterative solution.

The contraction corrector may be applied to construct

a convergent solution when the basic iterative scheme
0)oc<p •o•Hoc•H<HOAP•O•PM

erf J*Oo•Po

erf COo

erf COO-3bC

•e-C

erf CO•HP

erf N

erf Oa

erf1OIIO(UtHt£C

crf

8N

8N
4

3

88 IHCDc <

*

O • •H O •p II O X) cdfn T3 •P C C rt OO .< bO 0) C • •H O CO II 9 01A >> O 0) O > II M ©

.

CO

O

0) o O C5 C V CU CD W V ^ o <u o C (-. o o O «H 0) hO•H

fa

in

IA
44

breaks down. Consider the case of Figure 3 with a = 1.4 A,

b = 0.6X and 0= 70° . At this incidence angle, the iteration equation (40) and variational correction of Tsao and

Mittra [9] break down and do not converge. Figure 11 illustrates that after six and seve'n iterations respectively,

neither method converges. However, Figure 12 reveals that

the contraction corrector method converges after six iterations and has a small error over the conductor. Herein

lies the reason the contraction corrector method is preferred. Even when the basic method and variational correction schemes fail to converge, the contraction corrector method assures that solution can be reached and does

so with reasonable ease.

The foregoing examples have demonstrated the ability

of the contraction criteria to be applied to one-dimensional

scattering problems to determine the regions of solvability. Additionally, the contraction corrector scheme has

been utilized to achieve convergence when the basic iterative schemes failed. The examples presented give indications that the contraction corrector can be successfully

applied to planar wire surfaces.
0) •PoOJooaco• H•P

rtdOP ot> "o o OT fi P O

rl VICo

CO0) -pcCooIc

trf 0)0>•HPd!H0)

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OF POOR QUALITY

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> 1 I

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4->00^fc4Oo

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48

V. THE PROBLEM OF AN INFINITE GRATING

OF THIN WIRES

The structure shown in Figure 13 has been studied over the

years by various techniques to determine the reflection coefficient from the grid [1 , 2, 20 ]. The reflection coefficient may be obtained from the transform of the aperture E

field in the form of the Floquet modes. For the case of

cell widths less than ^/2, this coefficient is obtained

from the n = m = 0 element of the E~ variable for the

• mn

case of plane wave incidence as [9]

~(n) -(n) -..

F = E- = E+ - E . (55)

The first atempt to solve the problem with a X/'4 cell and

A/600 radius wire was performed with the variational correction iterative process using E n as a test function in the

variational scheme. This method fails to converge for various sampling rates and various numbers of Floquet modes,

and a more suitable test function is sought [21] . The

Dirac delta function is chosen as the test function and

gives acceptable results for small angles of incidence.

The delta function samples the response and allows point

matching to be utilized to improve the variational correction factor. The convergence properties achieved with these

two test functions are shown in Figure 14. Since the iteration method should be convergent for all incidence angles,
C-COJL
5

0

CO

co3•H•a0)

COen

co

0) .c

co

x^ tp^

-u

T3 O C CSi

cti C

. IW

•PCO0)

.c .c o

•P -M 0) II

o A A

«H JQ -P

O T3

bfl CO C

co c a at

0) -H

•H CO C *<

•M 3 o m <D D -M • a E o o

oven

!-> J3 3 c3

O, O «M

to «

0) do

O C -*-> O C O (H || a) -H a> <i Ml -P TJ II ti O O> Q) (U O C )H ^ -P O O -H -H O O Q ^ 0)h300

oo
51

the development of a different form of correction is a

logical path to follow.

The iterative process of equation (40) is one derived

for E~ from itself. That is, the E was determined from

E~ with an intermediate step in the iterative process

used to determine the surface current density J . The

iterative process then has two distinct parts and it is

reasoned that two distinct correction factors are required.

By weighting previous iterations of both E ' and 7(n)'

 lt:

is found that convergence can be obtained for small cell

size and near normal angles of incidence. However, it is

also determined that the stability and speed of convergence

is critically dependent on the weights chosen. It is then

realized that a solution may only be obtained by an "accident" as mentioned before. The hit-and-miss approach required by the weighted iterations may allow the determination of a proper solution but the recursive time required

could be excessive. These facts then drive the iteration

equation toward a solution obtained by the contraction corrector method.

The cases chosen for examination are A/8, A/4, and A/2

cell widths with both TE and TM incidence. The TE case corresponds to incidence angles <J> = 0° and 0° _< 0_< 90° while

the TM case corresponds to incidence angles $= 90° and Ou

<_ 9_< 90° , with either E". or H. respectively co-

"""" *^~ •!• *• l^- i. Lit-
52

polarized to the wires. Since the determination of reflection from two-dimensional mesh reflector antenna surfaces

is the application toward which this work is directed, the

reflection coefficients must be carefully scrutinized to

ascertain the best definition of reflection. Wait [2] defines a suitable normal reflection coefficient and a comparison to his results will be made. The specular reflection is of interest in antenna work and the construction

of the geometry required is given in Figure 15 [22] . When

E. is entirely co-polarized with the grid wires, the two

me °

reflection coefficients agree in definition. With TM polarization (vertical) incident, a pseudo-Brewster angle is expected at certain incidence angles, Variations of magnitude are also expected to occur with changing wire conductance. Before examining, various cases and comparing the results to those of previous authors, the wires must be converted into equivalent strips to yield a structure equivalent to that used by the previous authors LI, 2, 20, 27-1 .

Harrington [29 ] relates that equivalence between thin

metallic rectangular strips and small radius wires is obtained if

b = 4w (56)

where b = radius of the wires and w = width of the strips.

Thus, using equation (56), we are able to use the strip ana-
ref

t

V

X

O 0

tran

'tran

0 o o

H

tran

** >

m

- e

\

\

\

.--^*"0trS

f

i

(

-*• y

Figure 15. Definition of reflection coefficients

as described in [22] .
54

lysis to determine the reflection from a grid of cylindrical

wires. A further characterization of the wires is required

when the wires are of finite conductivity. Schelkunoff [23]

gives the internal impedance of a thin cylindrical wire as

*0(Yb)

(

where nm= j^/( o^ + jo,^)"* and Y = jiy^ + jcoej **

with ^ ,£_, and a being the electrical properties of the

m m m

wire and K , K, being modified Bessel functions. These

principles may now be applied to the cases detailed above.

The cases of cell widths of A/8, A/4, and A/2 are now

presented with a wire radius of A/600. Good agreement is

obtained between the results of the contraction correction

method and the resuLts of Wait Czl for incidence angles of

both *= 0°and $= 90°. The case of *= Ou is illustrated in

Figure 16. Recalling the other definition of reflection co

efficient given in Figure 15, the reflection coefficients

for the case of $= 90° are shown in Figure 17. The expected pseudo-Brewster angle is observed as a reflection minimum near the angle of 67*. The internal impedance of the

wire becomes important in the problem when the conductivity

of the wires becomes finite. Instead of the tangential

electric fields cancelling over the conducting portion of

the cell, they must combine to support the current at the
5

5

LD

CD Lf

)

(\J

o>

CDoos>r-CDun

U)

01ooo

ro

CO

•M JC

•H +J at -O *«w 0) C

O

o

8

CO

•O0)

•«->oCOW0) <HOoCO

O-H

CD

<5)

CD IID•oeOO) V

)

~ v| o C O o•H * •M o

rt O 3 II o1

 e.

0)

CJ csJ 0)p3bO

uoi.
5

6

UJ

1

0 -a

Ji C

3 sS

C CM

0) ,0

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I*-H h 0) O I

I

jr «H D

0) IH3be

u>
57

surface as

E + F = T 7

tine tscat o int (58)

where E = scattered electric field and I = current

U S C 31 O

along the conductor. The results of a finite conductivity

are shown in Figure 18 where equation (58) has been included

in the iteration equation. The results resemble that given

by Jordan and Balmain [30] for a lossy ground and follow

the idea that the grid represents a shunt impedance. Overall, good agreement is obtained with previously computed and

measured results.

As a final note, some concern has been expressed over

the non-convergence of the iterative technique and over the

fact that increasing the number of Floquet modes not not increase the accuracy of the solution [24] . The first concern has been addressed in the preceeding sections while the

second will be addressed now. When an iterative equation is

in a non-convergent region, the solution may only be arrived upon by an accident of computation unless special methods

of computation have been included.

The consequence of adding additional Floquet modes is

adding additional points that are in the non-convergent region. The result of this consequence can range from producing a large variation in the iterated solution to helping
58

•H rtf

> N

•H •

•P O O II

3 J3

T3

C T3

O C

O Oj

•p

•H

C

•H

<H

«H

O

(0

0)

rt

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II

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V|

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0) o

> O

0) II

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•

QO

0)

Pi

3

bo
59

the iterated solution. The control over the results is nil

with a non-convergent iterative scheme. However, when the

contraction corrector method is applied, the expected increase in accuracy with increasing Floquet modes is obtained. Of course, a limit on accuracy still exists due to the

finite register length of the computing machine used. The

case of a X/4 cell width, X/600 radius wire with infinite

conductivity and $=0=0° is chosen to illustrate the above

comments. A plot of converged accuracy is shown in Figure

19. The usual leveling off accuracy is noted. This is due

to the higher order Floquet modes having less and less

effect on the solution, i.e., the farther away wires contribute less and less to the fundamental cell when a convergent process is used.
60

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61

VI. EXTENSION TO TWO-DIMENSIONAL PROBLEMS

The present analysis has been limited to fields having

only one spatially varying component. When a second spatially varying component must be added, the difficulty in

applying the contraction corrector scheme is increased.

Each field component must have its own iteration equation

as can be seen from equation (23). These equations are

coupled, however, and this leads to new constraints that

must be applied to the problem Cl2l . For iterative equations of the form

xn+l

(59)

in a space X, the sufficient but not necessary conditions

for convergence are

9f + 8f

13T -57

_ _ .

8x 3y
62

for all x,y in X where p< 1. As was observed in the onedimensional case, the conditions for convergence revolve

around the derivatives of the iterative equations. The determination of the partial derivative follows the form of

equation (52) and may be written for f(x ,y ) as n n

Ax' yn)

"

f(xn'y

n

)

where f indicates the partial derivative of f(x,y) is Jv

taken. Again, it is seen that contraction theory allows

the measure of the derivative to be calculated. The conditions to insure convergence must now be determined.

The conditions given in equation (60) are sufficient

and not necessary and as such are fairly strong conditions.

Thus if a stronger condition is applied, the stronger condi'

tion is also sufficient. Such conditions are

•

 fy ' Sx ' 6y - 0.0 (62)

These conditions can not be met unless a new set of equations are formed using the contraction corrector method.

The method for the one-dimensional case required the combining of x and x , together to form a new iteration equation. The two-dimensional case requires the combining of
63

the x and x , along with f , f , g , and g to meet the

n n * x x y x y

requirements of equation (60). A form of equation that

conforms to the plan outlined above for x +, is

-

a >f (xn- V+6>V B S(xn-V (63)

The first term of equation (63) is that of the one-dimensional case. The second term of equation (63) is seen to

be equal to zero when y approaches the fixed-point y*.

Similarly, the iterative equation for y +, is:

V ^[3x

n-

 9 n

(64)

The complex constants a, 3 ,Y * and 3- are the contraction

corrector terms that must be utilized to force equations

(63) and (64) to meet the convergence requirements of equation (60).

The partial derivatives of equation (63) are set equal

to zero and are found to be :

(65)

yw * (1-

<"Wyn)+6-'iy*n,yn> -

From these equations, (3 may be solved in terms of a as
64

1 -a)fx(xn»yn:

g (x ,y ) x n n

which may be substituted into equation (65) to find:

a = Y " " (68)

If g (x ,y ) =0, a is found from equation (65) as:

x TI n

a = (69)

-

1

which is seen to be the same form as equation (34) in the

one-dimensional case. This is expected since F (x ,y ) is

x n n

not affected by g (x ,y ). 6 is found from equation (66) A il II

as:

( a -1) f

1-g y

for gx(xn,yn) = 0 (70)

The case g,,(x ,y ) = 1 is excluded for the problems of

y n ' n r
65

4

interest as this would indicate multiple solutions. Multiple solutions are excluded from the problems prsented herein since they are derived frotn Maxwell's equations and

appropriate boundary conditions and, as such, uniqueness is

guaranteed. The case that the denominator of equation (68)

equals to zero requires closer examination. This case

would yield an a •*» . This implies that x = 0 as seen from

equation (63) and this case is a trivial one for the problems of interest. Therefore, this special case is not considered. The a and $ determined above then allow the

iteration equation (63) to be solved for x , = F(x ,y ).

n+l n J n

With this corrected value of x +, , a similar set of contraction correctors may be calculated for y and y

solved for. The contraction coefficients y and 3 are calculated as:

Y +(1 - Y) gw(x^,1 ,y

3

 --

and

~

 g

x

(xn+l'yn)

(72)
66

When f (x

n+i»yn)

 = 0» arguments similar to those previous

hold and the following correctors are obtained as:

(73)

(74)

The Y and 9 are then used in equation (64) to find

G(x ,,y ). The iterative process is then repeated until a

suitable accuracy has been achieved. An example of this

two-dimensional contraction corrector scheme is laid out in

Table 4.

Even though the foregoing text and results of Table 4

*

have indicated that the contraction corrector method may be

successfully applied to analytical two variable problems,

it remains to be seen that the theory is directly extendable to numerical methods. Current results of the application of equations (63) and (64) with the correctors of equations (67) through (74) as applied to the one-dimensional

problems of Sections IV and V agree with the previous data.

However, the application of the contraction corrector

scheme to a two-dimensional canonical problem has yielded

poor results. The problem appears to be the numerical evaluation of the partial derivatives. Without the accurate de-
67

f(x,y)=x=2x+2y g(x,y)=y=x2+2y

F(x,y)=x=ax+(l-a)f(x,y)+By-$g(x,y)

G(x,y)=y=yy+(l-Y)g(x,y)+Sx-3f(x,y)

n

0

1

2

3

x

n

1

4

14

y

n

1

3

22

rHv«

^n'V

4

14

72

«<VV

3

22

240

n XQ yn a B y 9 F(xu ,yn ) G(xn ,yn )

0 1 1 0.67 0.67 0. 5

1 0.5 1 - - 0 1 - -0.25

2 0.5 -0.25 -

x =0.5 y =-0.25

Table 4. Example of a two-variable contraction

corrector method.
68

termination of f , f , g , and g , it is doubtful the con- x y x y

traction corrector method will be successful in arriving at

a convergent solution. Another problem that is not readily

apparent is the stringent conditions placed on the partial

derivatives. Invoking such strong conditions as given in

equation (60) may cause computational problems and eased or

adjusted conditions may be necessary uo reach a viable solution method. These problems are being investigated and

useful results are expected in the near term [25] .

The usefulness of the two-dimensinal contraction corrector can not be underestimated. For instance, when a

plane wave strikes a one-dimensional grating at an incidence angle that induces

 :current along two axes, a crosspolarized scattered field will be generated. The twodimensional nature of the problem must be addressed to obtain the valuable cross-polarized information. The solution for reflection of waves from actual mesh surfaces is

also of interest and involves the use of the method detailed in this dissertation.

An illustration of an actual mesh surface is shown in

Figure 20. The material is molybedenum wire coated with

gold and woven in a double-knit fashion. Reflector surfaces of space-deployable antennas have surfaces of this

nature and their reflection characteristics must be known

to accurately predict their electrical performance. Figure
69

Figure 20. Actual mesh surface.
70

21 illustrates the periodic nature of the mesh surface. It

can be seen from the outlined boxes that Floquet cells may

be formed. This periodic nature may be capitalized on and

used to formulate the mesh surface into a suitable problem

for the contraction corrector scheme.

Figure 22 is the discrete representation of one single

mesh path. The discretation of the mesh path is accomplished with twenty-eight, approximately equal length,

straight segments. With these segments defined, it is possible to overlay a grid onto the segments to achieve a representation of the mesh surface. The equally spaced points

on the grid represent the sampled points required for the

Fourier Transform of the solution method. This typical

grid representation of the mesh surface will allow the determination of the reflection characteristics of the surface. Other surface weaves and materials may be placed on

similar grids and their reflection characteristics calculated. Figure 23 illustrates the application of the grid

to the segmented single mesh path.
71

Figure 21. Floquet cells outlined on the

actual mesh surface.
72

Figure 22. Single mesh path of the actual mesh surface
73

Figure 23. Grid representation of the segmented

single mesh path.
74

VII. CONCLUSIONS

A new method called the contraction corrector method

has been developed to insure the convergence of the onedimensional spectral-iteration approach when solving electromagnetic scattering problems. The method was presented

beginning with basic examples of the iterative method and

progressing to detailed iterative operator theory. Computed data generated from the scheme was critiqued and compared to work previously presented and was found to agree well,

The ability to predict the regions of solvability with the

contraction corrector scheme was demonstrated. The method

was then utilized to solve for the reflection coefficients

from an infinite grating of thin wires and the results were

compared to previously published data. The ideas of generalization to a two-dimensional problem were discussed. A

solution technique for the two variable contraction corrector method was developed and numerical difficulties in

applying the method were discussed. A sketch of an actual

mesh surface was presented and the application of the spectral-iteration approach was detailed. A general reference

list and an over-view of both the k-space problem formulation and the spectral-iteration approach were also included.

The material presented in this dissertation gives a

solid foundation for future research, and indicates that a
75

suitable method for analyzing scattering from arbitrary

mesh surfaces can be developed. The recommendations for

future research should be directed toward understanding the

the shortcomings of the two-dimensional problem as detailed

in Chapter VI. With the proper constraints placed on the

two-dimensional problem and the correct evaluation of the

partial derivatives, a suitable method of solution is deemed

obtainable. Another alternative is the reformulation of

the problem with another method of attack. Many numerical

methods have yet to be applied to the problem and one may

will be suited for the two-dimensional scattering problem.
76

REFERENCES

[1] G.G. Macfarlane, "Surface impedance of an infinite

parallel wire grid at oblique angles of incidence,"

IEE, No. 93, Part III A, pp. 1523-1527, 1946.

[2] J.R. Wait, "Reflection at arbitrary incidence from a

parallel wire grid, "App. Sci. Res., Section B, vol.

4, pp. 393-400, 1954.

[3] R.E. Collins, Field Theory of Guided Waves, McGrawHill, New York, 1960.

[4 ] R.B. Kiebuntz and A. Ishimaru, "Scattering by a periodically apertured conducting screen," IRE Trans. Ant.

and Prop., vol. AP-9, pp. 506-514, Nov. 1961.

[5] N. Amitay and V. Galindo, "The analysis of circular

waveguide phased arrays," BSTJ, pp. 1903-1932,

Nov. 1968.

[6 ] C.C. Chen, "Transmission through a conducting screen

perforated periodically with apertures," IEEE MTT,

vol. MTT-18, pp. 627-632, Sept. 1970.

[7 ] S.W. Lee, "Scattering by dielectric-loaded screen,"

IEEE Trans. Ant. and Prop., vol. AP-19, pp. 656-665,

Sept. 1971.

[8 ] J.P. Montgomery, "Scattering by an infinite periodic

array of thin conductors on a dielectric sheet," IEEE

Trans. Ant. and Prop., vol. AP-23, No. 1, pp. 70-75,

January 1975.

[9 ] C.M. Tsao and R. Mittra, "A spectral-iteration

approach for analyzing scattering from frequency

selective surfaces," IEEE Trans. Ant. and Prop., vol.

AP-30, No. 2, pp. 303-308, January 1982.

[10 ] N.N. Bojarski, "K-space formulation of the electromagnetic scattering problem," Tech. Rep. AFAL-TR

71-75, March 1971.

[11 ] J.C. Brand and J.F. Kauffman, "Analytic considerations

for calculating the complex reflection characteristics

of conducting mesh antenna surfaces," Int. Sci. Radio

Union, Albuquerque, NM, May 1982.

[12 ] S.D. Conte and C. DeBoor, Elementary Numerical

Analysis, McGraw-Hill, New York, 1972,.pp.-44 - 50.
77

[13 ] R.F. Harrington, Time-Harmonic Electromagnetic Fields.

McGraw-Hill, New York, 1961, pp. 27 - 347

 y« - 100.

[14 ] V. Galindo and C.P. Wu, "Numerical solutions for an

infinite phased array of rectangular waveguides with

thick walls," IEEE Trans. Ant. and Prop., vol. AP-14,

No. 2, pp. 149-158, March 1966.

[15] A.B. Carlson, Communication Theory, McGraw-Hill, New

York, 1975, pp. 56 - 58.

[16 ] W.M. Patterson, Iterative Methods for the Solution of

a Linear Operator Equation in Hilbert Space-A survey,

Lect.Notes in Math, 394,Springer-Verlag, New York,

1974.

[17 ] M. Altraan, Contractors and Contractor Directions

Theory and Application, Marcel Dekkar, New York, 1977.

[18] I. Stakgold, Green's Functions and Boundary Value

Problems. John Wiley, New York, 1979, PP. *** -zo9.

[19] A.V. Oppenhiem and R.W. Schaffer, Digital Signal

Processing, Prentice-Hall, Englewood Cliffs, NJ,

1975, Chapter 3.

[20] M.I. Astrakan, "Averaged boundary conditions on the

surface of a lattice with rectangular cells," Radio

and Elec. Phys., No. 8, pp. 1239-1241, Aug. 1964.

[21 ] J.C. Brand and J.F. Kauffman, "The application of

the spectral-iteration approach to conducting mesh

reflector surfaces," Int. Sci. Radio Union, Houston,

TX, 1983.

[22] E.G. Jordan and K.G. Balmain, Electromagnetic Waves

and Radiating Systems, Prentice-Hall, Englewood

Cliffs, NJ, 1968> pp. 144 - 150.

[23] S.A. Schelkunoff, Electromagnetic Waves, Van Nostrand,

New York, 1943, pp. 264.

[24] J.P. Mongtomery, "Analysis of frequency selective

surfaces using the spectral-iteration approach,"

Int. Sci. Radio Union, Houston, TX, 1983.

[25] C. Christodolou, J.C. Brand, and J.F. Kauffman, to

be published.

[26] R. Mittra, W. Ko, Y. Rahmat-Samii, "Transform approach

to electromagnetic scattering," Proc. IEEE, vol. 67,

No. 1, pp. 1486-1503, Nov. 1979.
[27] M. Kontorovich, V. Petrun'Kin, N. Yesepkina, and

 78

M. Astrakhan, "The coefficient of reflection of

a plane electromagnetic wave from a plane wire mesh,"

Radio Eng., vol. 7, pp. 222-231, 1962.

[28 ] R.F. Harrington, Field Computation by Moment Methods,

Reprinted by Roger F. Harrington, Cazenovia, New York,

1968, pp. 127 - 131.

[29] R.F. Harrington, Time-Harmonic Electromagnetic Fields,

McGraw-Hill, New York, 1961,pp. 223 - 225.

[30] E.G. Jordan and K.G. Balmain. Electromagnetic Waves

and Radiating Systems. Prentice-Hall, Englewood

Cliffs, NJ, 1968, PP. 629 - 635.
79

APPENDIX A

The FORTRAN source code that follows will allow the

user to solve for the electric field in the aperture of

an infinite grating of thin wires as well as the reflection

coefficient from the grating. The program is written in

FORTRAN 4 and should run on most FORTRAN compilers with no

modifications except the unit numbers on the read and write

statements. The plot routine must be user supplied and the

program modified accordingly. The input variables are specified in comments internal to the program. Comments are

also included to help the user follow the flow of the program. Some of the variables are self explanitory and the

others are listed below. A suitable DEL, the derivative

increment, has been found to be (0.01,0.01) for most cases.

E(I) Electric field in the aperture

JC(I) Current density on the strip

G(I) Transformed Green function

EUT(I) Dummy E field variable

CREF Complex reflection coefficient

HI Incident H field

ED(I) Incremented E field in the aperture

ALF(I) Contraction corrector

CK? Constants used in the iterative equation

K Propagation constant

EOUT(I) Output variable used for the external plot

Z Internal impedance of the strip
80

Lines

170-400

410-520

530-610

620-640

650-720

740-920

930-990

1020-1110

1120-1130

1140-1150

1160-1200

1210-1230

1260-1290

1300-1370

1380-1440

1450-1510

1520-1580

1590-1610

1630-1670

1680-1740

1750-1780

1790-1840

1850

Program Flow

Input data

Initialize parameters

Calculate incident E and H fields

Calculate thin wire impedance

Set up aperture/strip sample space

Calculate Floquet propagation constants and

Green function transform

Initial aperture E estimate

3.

Calculate constants in iterative equation

Store untransformed E

a

Take FFT of l"a and (E& + DEL)

Perform G JE

£t

Take inverse FFT of G E

a

Perform complement truncation T (F[G E ])

C 3.

Perform FFT on T (F[G E ])

C £L

Perform G"1 (F[T (F[G E ])])

C £L

(F[T (F[G El)])

C £L

Calculate reflection coefficient

Perform inverse FFT on G~

Plot set up

Calculate contraction corrector

Apply contraction corrector if desired

Calculate finite conductivity effects

Perform FFT on first iterated E

81

Test for five iterations, next iteration

go to 1160

1880-1980 Output external plot and stop execution,
PAGE S3

Of POOR QUALFTY gl

C

C****** FINAL.S   DERIVATIV E CORREC T ** ****

C Author Jerry C* Brznri

DAT E August 12, I^^S

All rights reserved

Dirension all arrays

COMPLEX EC51?),JC(M2>«G(51i:)fEUT(5i;),DE L

COMPLEX PI,CPEF,HI

COMPLEX ED(512),ALF(51?),F9If £(512),U1(512) ,E[)UT(5^2)

COMPLEX J»CK1(512) ,CK2(5U),CK(512),7

DIMENSION ?OUT(512tIDf? )

RE« L K,K2

A = Floquet cel l dimension

F = Strip size

WRITE(6,*) * INPUT FLOQLET CELL SIZE , STRIP SIZE'

URITP(6,*) ' NORMA11TE D IN WAVELENGTHS *

R£AD(5,*)A,B

FREQ = Freauency in Hertz

WRITE«5,*) * INPUT F=>EQUENC y IN H£PT?'

MA> = FFT si2e = Number of sair.cLes per cell

IW = lOySC^AX ) ; i.e. ^AX=?**1 U

IP = 0 NO clct; IP = 1 »»lot;

NOTE..,,. External plot routine required

IAC = C No correction;

IA!>=1 Contraction corrector scheme applied

WRITE(6,*) 'INPUT ^f LOfaZCN), PLOT OPTION, ADJUST OPTICN*

R£AD(5,*> ""AX, IW, IF,!A!i

SIP

 s Conductance cf strip

UAITE(6,«) * INPUT CONDUCTANC E OF ST»IP IN SIE-ANS*

READC5,*) S1G

TH = Theta angle of incidenc e

DEL = Comple» increment useo ir derivativ e routine

PH = Phi angle o* Incicence

WRITE(6,*> * INPUT TH£TA ANGLE, DERIVATIV E DELTA, PHI ANGLE '

48? Rt*CK 5,*)TH , DEL,PH

URITE(6,<»6) DEL

16 F09MATC-',' DEL = ',iE10.4)

2 9 0 FORMATC-',' ADJUSTMEN T USE D

C Initialize routine constants

TPlsPI*2.

C*2,997956?+8

ETAsSQRTfUU/FP)

ITER=C

J=(D.C,1 .3)

«>D=18C./D1

C Calculat e incident electric and magnetic fiel d components

C for electric field paralle l to hires, i.e. no cross-
C COlarirat ion includec oiatr^M*. ^

IFfPH.LT.45.) EINC»1. C J2 ^l PASEIS

IFfPH.GE.45.) EINC=CCSfTH-DS )

 W TOOR

 QUALITY

IFCPM.LT.45.) STH=C .

IFfPH.GT.45.) STH=T H

Hls1./ETA*fCOS(TH»CR)+J*SlN(TH*DR) )

IFfPH.LT.45.) w:=1./FTA«COS(TH*DP )

Z=SQRTfTPI*FREO*UU/2./SlG>*(1.?,1.0)/TPI/ 6

WRIT5C6.15) Z

15 FORKATf-*, ' Z=C,?=1C.?,-f-,El0.2f-)-)

C Calculate number of samples on strip and in aperture

NsIFlX(TAU/A*FLOAT(K*.X»

V AX1 = I»«X+1

WRITEC6f ? 3)

IF(NLGT.MAX) GOTO

K=TPI/ALAMP

SK=K*SIN<TH*PR>*COS<°H*l,P)

C Calculat e Green function transfor m

DO 4C ISI.^AX

U=TPI*(I-1)/A-SK

GOT O 60

50 U=TPI*fI-M«X-1)/A-SK

IFfU.GE.r2)GCT O 7C

G(I) = -J*5Q«?T(K2-U )

GOT O 44

?C G(I)=-SQRTfU-K2)

4A G(I) = G(D-SStf*SS*:/C(!)

4C COK'TINUE

C Initial F fiel C estimate

DO 32C 1=1,N

E(I)=(1.C,r.G)

32C EOCl)sF(I)*DEL

DO 22C I=N 1 ,»A X

ED(i)sCu.OtO.O)

C NOTE.... Iterative for« useo ir this program is x =

C Calculate "BM ccrtion ct iterativ e equation

DO 11C 1=1, MAX

11C CK1 (I ) = HI*J«W*LU

CALL FFTCCKltIW,«AX >

DO 120 I=N1,»AX

12C CK?(I)=HI*W*UU/J

DO IA'0 1=1 ,N

CAL L FFTCCI f 2tIW,*»AX )

DO 13C 1=1, MAX

13C CK(l)*(Cr1(I)-»CK?(]»/6(Z )
00 30C 1 = 1, P»X 83

3DC U1CI)=E(J)

CALL FFT(£D ,IU,MAX3

CAIL FFT<E,IU,P!AX)

Pecin "AX" portion of iterative equation

PerforiB (6*E)

80 00 10C 1 = 1, M«X

EOC1)«CONJG<EDCI>*CC I ))

10C E(I)sCCNJG<E(I)*G<I)>

Ferfori" inverse transfer* ol (G*E)

CALL FFT(E,IW,MAX )

CALL FFT (ED ,IW,M*X )

WRITE (3,199) ITER

19S FORMMCICXtHO)

Perfor m truncation operation T(6*t >

00 150 1=1 ,N

ED<I)=CQ.O,C.O )

15C E( I)=(C. 0,0.0)

DO 16C I=N1,PAX

ED CI)=CONJP (?D (I)) /M^ X

F.(I) = CCNJGCE<I))/M* X

Calculate current density on strip

16C JC(I) = F(!)*J/W/UU-H

Perform inverse transfer* en T(G*t)

CAL L FFT(£,I W ,^AX)

CALL FFT<ED,IW,MAX )

Perform TCG*E)/G and acd constant

 trO"

DO 17C 1 = 1, M«X

EUT(Z)*CCNJG(E<2))/KAX

17C CONTINUE

Calculate reflection cce^ficiert

PI=J*SIN(STH*DR)/CCS(STH*LP )

A53 FO«?MAT(6X,* «EF = * , Fl u. = , cGx , 'CRE F= ' , F1 0. 2 ,1 X , F1 C .

C Perfcrir irverse t r ansf c r^at ion to obtain first iterate d

C electric field

CALL FFTCEUT, IUf PA>)

CALL FFT<EPUT »I«,M*X)

DO 2CC J =1 ,M«X

EDUTCI)=CONJG(EDUT<1))

20C EUT(I) = CONJG<EUT(D)

C Flct option

IFClP.LT.Cn GOTO 150

CALL JBPLOT (?UT ,l« A ) ,TT t K , tCuT )

180 ITER=IT£P+1

C Cal l subroutine to calculat e derivative of operator and

C contraction factor

CALL DF3IV(EUT,EDUf ,"AX ,PR IN E , AL F , I AD , D£L ,CO^ )

WRITE (6t3iD) CON

34C FOfiMAT(/,1^X,* CON1R«CTIC N = *vFlu.^)

C Perfcr* ccrrecticn on electri c tield; NCT£... If l«D= u
JDRIGINAL'PAGFIS

C ro adjustment is usec as ALF(1)=0. 0 ^ POOR QUALITY 84

00 350 1=1,N

EUT(I)*ALF(X>*U1CX)*<1.-»LF(X))*EUTCI )

35C EDUTCI)*EUT(I>+DEL

310 CONTINUE

C Calculate field on strip due to finite conductivity

00 21C I«Nl t*AX

EDUT<I)=-JC(T)*2*B

C Perfcrir transform of E fielc tc begin next iteration

CALL FFT(EUT,IW,«IAX)

CAL L FFT(EDUT,IU,MAX )

00 220 I=1fP»X

ED(I)=EDUT(I)

2tC E(I)=EUT(I)

IFC1TEP.GT.5 ) GOT O 2<-C

GOT O 6C

24C CONTINUE

DO 777 I=1,M«X

PINDEX=FLO'T(I-1)/FLOAT(VAX-1)* A

C Perfor m plot ftrcticns 1f-r externa l Plot routine

WR!TE(?2,7?fi) PINDEX,EOUTCI,1,1),EOUT(I,2 f 1),£OUT(I,3 t 1 )

*,ECUT<1, A,1) fEOUT(],r,1 )

WRITE(23,77£) SINDEXt EOUTU,1,?i,LOUT I I,2,2),EOU*(1,3 , 2)

+ ,ECUT<If4f?),EOUT( I,',2)

77F FORMAT(6F8.2)

777 CONTINU E

2 6 C FOPKATC-'tlTX,* TI*IL Y EXIT* )

?9f WR!TE(6,?9 )

? 9 FORM«T(*-*,* ERRO R I N W )

999? STC P

ENS

SUPROUTIME J°PLOT(*,INT ,iTER t touT )

COMPLEX «(1)

DIMENSION *^P(512) ,EOUT(512,10,2),PHASE(512)

N=ITER*1

DO 8C J=1,INT

AMP(1 ) = C*BS(«(!))

*II»AG(A (I) J

IF(ARS(PHX).LT.I.E-2:) FHX=1.t-2n

PHASE(I)=57.295*AT#NZ(PHY,PHX)

EOUT(I,N,1)=AKP(I)

EOUT( I,N,2)=PH«SE(1)

80 CONTINUE

RETURN

END

SURROUTINE DERIVC E ,EP,M AX,PKlH5,ALF,IAD,OEL,CON)

C This subroutine calculates tht derivative and

C contraction factor ol the operator

COMPLEX E(r*AX) ,ED(rAy)«ALF rn«x) « PRlKF(r.AX) .DEL

COK=C.C

PRIM E (I)=(FD(D-E(l)?/ntL

COK=CCN+fC*BS(PRIMf(!)*D'L))*"2
85

13 IF(!*D.LT.1> ALF(I)«~.n'

CON= SORT (COM FLO AT CMAX > >/C«faS(DEL)

RETUR N

END

SUBROUTINE FFTCAfHfN )

C This is the FFT subroutine tailed from the main program

COMPLEX A(K)fUty,T

j=r

00 7 Is1,N«"1

IF(I.6E«J) GOTO 5

T=»(J)

A(J)*ACX >

A(I)=T

5 K=NV2

6 IF(K.GS.J) GOTO ?

GOTO e

7 J=J+K

DO 20 L=1t"

LE=2**L

LE1=LE/E

DO 2? J = 1,I.E1

DC 1C I«JtK,L£

7=A(IP.)*U

A(IP)=A(!)-T

10 «(I)=A(I)*T

ao u=u*w

END
86

APPENDIX B

The general form of the basic iterative process is

given as

(n+1) -.-I

The linear operations indicated in equation (Bl) may be

easier to understand if a flow of the operations is presented.

Figure B-l illustrates the iterative process used in the

computer program of Appendix A. The initial estimate of

the electric field in the aperture is transformed and the

transformed Green function is applied along with Floquet's

theorem. The inverse transform is applied to this result

to obtain the current density on the strip. The inverse

of the previous steps is then performed to obtain the

first iteration of the electric field in the aperture.

That is, the current density is truncated over the aperture

and transformed, the inverse transform Green function is

then applied along with Floquet's theorem to obtain the

iterated aperture electric field. This iterated electric

field may then be truncated and the process repeated until

a suitable solution and tolerable error is-obtained.

Figure B-2 illustrates where the basic iterative pro-
8

7

FINAL

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89

cess may be interupted to accomodate the contraction corrector scheme. This point is chosen since an iterated

solution has been generated. With this iteration and any

previous iterations a filtering process may be employed.

The text pointed out the usefulness of the contraction

corrector scheme since it guarantees a convergent solution.

Thus, the iterative process can be viewed as a feedback

process whereby the errors of processing can be eliminated

and a useful solution of the problem obtained.
90

APPENDIX C

The optimum contraction corrector for the one-dimensional problem is shown to be equivalent to Newton's

method. The proof is as follows:

f(xn) = 0 (Cl)

G(xn> - xn+l

g'

- R) g(xn)

g'

<C2)

(C3)

g' g'-l-g'

x +

g

1

n, g'-l

-g(xn)

n

 + g'-l

(C4)

(C5)

g'-l

(C6)

g'xn - g(xn)

g'-l

(C7)

x

n .

 xn -

(g'-l) g'-l

(C8)

"

 x

xn+l = xn *

n

g'-l

x -

n

where g1 = — g(x)

dx x=x

n

(C9)