Introducing undergraduates to the moment method

____

IEEE TRANSACTIONS ON EDUCATION, VOL. 38, NO. 4, NOVEMBER 1995 385

Introducing Undergraduates to the Moment Method W. Perry Wheless, Jr., Senior Member, IEEE, and Larry T. Wurtz, Member, IEEE

Abstract-Four application problems are presented which have been found to facilitate introductory instruction on numerical methods for undergraduate electrical engineering students. The method of moments (MOM) is introduced at the level of first principles, preparing students for subsequent advanced topics in matrix methods and developments through Linear vector space theory. The problems are such that the associated computer programming assignments are straightforward, minimizing distraction from the mathematical concepts. Numerical solutions may be obtained, or verified, with a pocket calculator in some cases. Computer programs to obtain more sophisticated graphics output and to expedite solution of equations with large matrices can be progressively added as supplementary student exercises.

1. INTRODUCTION

HE METHOD of moments (MOM) [1]-[3] has become T firmly established as a fundamentally important numerical method in applied electromagnetics. Most major research universities now incorporate MOM into their curricula, with formal presentation of the moment method frequently beginning with integral and integro-differential equations. This paper discusses four variations of a simple boundary-value problem that have been found useful in the process of introducing undergraduate electrical engineering students to the moment method. Additional variations can be devised. Appli- cations that can be successfully concluded, such as described here, help make students comfortable with matrix methods and increase their appreciation of the linear vector space theory inherent in the method. With deeper insight into how linear operator equations are transformed into a system of linear algebraic equations, students can proceed faster through more advanced material [4]-[6].

11. MOTIVATION Application of the moment method to engineering problems

of practical interest generally is not obvious or direct; prior experience with a variety of specific problems is essential. While facility with MOM only results from practice on advanced problems, there are pedagogical advantages to beginning with examples that allow the mathematical concepts to be clear, free from the complications of physical concepts.

There is a tendency on the part of instructors to intro- duce physical problems straightaway. Because curricula and students vary, starting with simple physical problems may be preferable in some cases. Our experience, however, has been that too many beginning students perceive physical problems to have idiosyncrasies that obscure the numerical

Manuscript received September 13, 1993; revised March 1, 1994. The authors are with the Department of Electrical Engineering, University

IEEE Log Number 9415219. of Alabama, Tuscaloosa, AL 354874286 USA.

method. Such perceptions, whether justified or groundless, nevertheless affect student attitude and performance. The introductory approach reported here takes advantage of the fact that students at this level are more familiar with the relevant mathematics than with the electromagnetics. Problems such as those presented in this paper stimulate interest and provide valuable positive reinforcement to students through the exhilaration of success in a reasonable time frame.

The material in this paper, as indicated by the title, is introductory in nature. It is intended that instructors will not dwell at length on these problems, but will promptly proceed to appropriate, and progressively more challenging, physical problems.

111. MOM FUNDAMENTALS

The emphasis here is on the basic MOM requirement of reducing a functional equation to a matrix equation, through consideration of inhomogeneous equations of the form

where C is an operator, g is a known excitation (source), f is the response (field), and where we seek to determine f with C and g both known. The basic solution procedure follows that described in [3], with essential points summarized as follows.

Represent the unknown function f by the series expansion m

n=l

where the an are constant coefficients, to be determined, and the series of functions fn, in the domain of L, are called basis or expansion functions. For a finite number N as the upper limit of summation, the result generally is an approximation o f f . It is possible to obtain an exact solution even with finite N , however, in those cases where some combination of the basis functions allows an exact representation of the solution. Substituting (2) into (l), under linearity, gives

N

(3) n=l

A set of weighting, or testing, functions wm is defined by (arbitrary) choice, and the inner product, taken to be the integral over the interval of the product of functions in accord with [3], is formed with the w,

N

n=l (4)

0018-9359/95$04.00 0 1995 IEEE

386 IEEE TRANSACTIONS ON EDUCATION, VOL. 38, NO. 4, NOVEMBER 1995

so that the progression m = 1 , 2 , 3 . . . , N yields a set of equations which can be expressed in matrix form as

( 5 )

where L is an N x N matrix and lan), 19,) are N x 1 column vectors, explicitly

L N 1%) = 1974

N

N

which makes the coefficients required for a solution available from the operation

Ian) -L-l N 19,). (7)

IV. BASIC EXAMPLE PROBLEM The examples that follow treat variations of the boundary-

value problem: Given excitation g(x) , find response f(x) in the interval 0 5 x 5 1 satisfying

(8) -- d 2 f ( x ) = g(x ) withf(0) = f(1) = 0. dx2

Clearly, the linear operator is C = - $. Functions for g(x ) can be selected so that simple exact solutions are available for comparison to the numerical method results.

While aspects of this problem with the specific source g = 1 + 4x2 are in [3], it is worthwhile and informative to study other cases in detail, as suggested by [7]; the specific sources g = 2(3x - 1) and g = 4(1 - 3x2), with several different basis and testing functions, are considered here.

Note that to say that the basis functions are in the domain of L requires that the functions satisfy both boundary conditions. Also, application of Galerkin's method [8], [9] means that the basis functions f n and testing functions w, are chosen to be the same. The Galerkin method is illustrated in Examples 1 and 3.

A. Example I Obtain a MOM numerical solution of the basic problem as

stated in (8) , using Galerkin's method with the choices of

Solution: The solution sought is f, which is related to the f n - - x - Zn+l = ~ ( 1 - x,) and g = 62 - 2 = 2 ( 3 ~ - 1).

basis functions f n through the series expression for f N

(9) n=l

as discussed in connection with (2) above. The number of basis functions combined into the composite solution function f is N, and the a, are coefficients to be determined by the numerical solution procedure. To apply Galerkin's method, it is required that

w, wn(x) = f,(x) = x(1 - 2"). (10)

As indicated in the previous section, the elements C,, of the matrix L are determined by integration, for n = 1 , 2 , . . . , N and m = 1,2, ..., N

N

while elements in the column vector 19,) are determined by

The N coefficients required in (9) to construct the final solution can be obtained by carrying out the right-hand product indicated in (7).

The case of N = 1 produces the following results in straightforward fashion:

(13) Cl1 = 0.333, 91 0.167, ( ~ 1 = 0.50

and f = alz(1- x ' ) = ;(l - z).

For N = 2 L= [,.333 0.5001

0.500 0.800 -1.00

(14) Ian) =L-' Ism) =

N

n 1

f = c a n [ 4 - z")] = 4 1 + a 2 f 2

=-x( l -x)+2(1-x2)=x2(1-z) . n=l

It is already known that this simple boundary-value problem

(15)

so that the numerical solution for N = 2 is precisely equal to the exact solution in this case. For other problems, where a finite sum for f fails to give an exact solution (see Examples 3 and 4), we expect to obtain approximate solutions which converge to exact results as N increases. A representative MAT LAB^^ 1101 program for the computational aspects of this problem is the following:

clear; N = input('VuZue of N...') for m = 1: N ;

for n = 1 : N ;

end; end; gm = gm'; Linv = inv(L); alphan = Linv * gm; z = 0 : .U1 : I ; z = 2'; f = zeros(length(x), I ) ; exact = (x."2). * ( 1 . - x); for n s= 1 : N ; f =f+(a lphan(ns ) .* ( z .* ( l . - z .Ans ) ) ) ; end;

Plotting statements are not included in the program above; for MATLAB, plotting by means of keyboard command entry is both flexible and convenient. Plots for the solutions N = 1

has the exact solution

fezac t = x2(1 - .>

gm(m) = ( m * ( m + l ) ) / ( ( m + 2 ) * ( m + 3 ) ) ;

L(m, n ) = ( m * n ) / ( m + n + I);

WHELESS AND WURTZ: INTRODUCING UNDERGRADUATES TO THE MOMENT METHOD

~

387

0.14- N=2 and

0.12-

/ 0.1 - /

N=l / /

/

0.04

0.021 ,/ / I / /

" 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0

X

Fig. 1 Numerical and exact solutions for Example 1

1.9 1

and N = 2 = exact are given in Fig. 1. If one escalates to higher values for N , a1 = -1 and a2 = +1 in all cases, but the higher coefficient values turn out to be zero. For example, a3 = a4 = 0 for the specific case N = 4.

A student learning exercise is to explore the fact that raising N to a sufficiently large value causes the matrix to become nearly singular or badly scaled, and to compare solutions by Gauss elimination against the direct matrix inversion algo- rithm. The Gauss elimination procedure can be invoked in MATLAB by changing the two lines

Linu = inu(L) alphan = Linu * gm

in the program listing above to the single statement

alphan = L\gm

using the left division operator which is provided in MATLAB in addition to the customary right division operator. For this particular problem, N greater than approximately 11 begins to cause reliability warnings for the standard inversion procedure, while the corresponding number using Gauss elimination is about N = 83.

B. Example 2

the solution f for For different source g(x) = -12x2 + 4 = 4(1 - 3x2), find

= 4(1 - 3x2), f ( 0 ) = f(1) = 0 (16) d2 f -~ dx2

by the use of point-matching. The choice of basis functions f , = x(1 - 2") is continued from Example 1 .

Solution: While point-matching [ 1 11, [ 121 may appear to be a different method to students, it can be shown readily [13] that point-matching is encompassed by MOM. To arrive at a point-matching solution, N equispaced points on the interval 0 5 x 5 1 are selected

m N ' x, = - m = 1,2, . . . , N (17)

and, associated with the match points, the Dirac delta function

20, = S(x - 2,) (18)

is selected for the testingweighting function. Solution of this problem allows students to see that the use of S functions as the method of testing causes the LHS (using the numerical method series solution for f ) and RHS of the problem equation, (16) in this case, to be precisely equal at the specific spatial match points where the S functions reside.

Evaluation of the elements lmn and g, again requires two integrations, which are expedited by the sifting property of the S function

@(x - x,)]dx = (n)(n + l)(g)n-l and

1

0 (20) gm = (9 , w,) = J [4 - 12x2] [S(z - xm)]dx

2 = 4 - (12)(G) .

Case results for three values of N follow. For N = 1

(21)

Using (17), x1 = 1. The LHS of (16) is -$ = -8, and the RHS (namely 4 - 12~ ' ) is also 4 - 12 = -8 at x1 = 1.

C l 1 = 2.0, 91 = -8.0, CY^ = -4.0, and f = a1z(1 - x') = 4x(x - 1).

The case N = 2 results in

and the corresponding series solution is 2

n=l (23) f = an[x(l - P)] = 5x(l - x) - 3x(1 - x2)

= x(2 - 52 + 3x2).

Eq. (16) for the f of (23) has LHS=RHS=l for x1 = -8 for 5 2 = 1.

and

Finally, for N = 3

r2.00 2.00 1.331 r 2.667 1

1%) =L-' 19,) = N

which allows computation of the three-term expression for f

(25)

3

n=l f = c a n [ 4 - xnll = Qlfl + a2f2 + a s f 3

= x( l - 2x + x3). We note that - 2 = 4 - 12x2 now, so that (16) is satisfied

for all x, as well as at x1 = ;,x2 = 5, and 2 3 = 1 . Fig. 2 has plots of the three cases against the exact solution, which is f = x( 1 - 2x+z3), so that the numerical solution for N 2 3 is seen to provide exact results for this problem. Larger N values


1 _ . . - . - . __ N=2 I I

\

4L51 \ \ \

\ N=l \ \ \ \ \

/ /

/ /

/ / - ,

I \

\

> - , - I

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -1

X

0.151 I

-0.1' I 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

Fig. 2 Numerical and exact solutions for Example 2. Fig. 3. N = 1 , 2 , 3 , 4 and exact solutions for Example 3.

will again drive the matrix L toward becoming singular, with concem in this case arising when N is approximately 19 or

Gauss elimination is used.

C. Example 3

N = 2 produces 0.067 N L= r . 1 3 3 0.2331

larger for direct matrix inversion, or approximately 99 when 0.233 0.419 I g m ) = [ I o . ~ ~ z ] (31) 5.3338

Ian) =L-' 1 % ) = N

which gives an approximate solution function of 2 Carrying over g(z) = 4(1 - 3z2) from Example 2, next

f = a, [x2(1 - x")] = x 2 ( 2 - 5.3332 + 3 . 3 3 3 ~ ~ ) . (32) consider the Galerkin's method solution to

with fn = z2(1 - 2") now. The major point of this exercise is to show that while the exact solution cannot be achieved with any finite number N of the new basis functions, we still may anticipate that the approximate solution will converge to the exact solution as N is monotonically increased. Solution: For Galerkin's method, we must take

and proceed to calculate L and lgm) according to N

1

n (mn)(7m + 3mn + 7n + 15) 3(m + 3)(n + 3)(m + n + 3)

. [ X 2 ( 1 - zmlda: =

n=l For N = 3, the vectors and matrix tum out to have element values

[,.133 0.233 0.309 L= 0.233 0.419 0.567

(33) 10.309 0.567 0.778_1 1-0.2331

r 14.5831

Numerical results for N = 1,2 ,3 , and 4 are shown in Fig. 3, again using MATLAB and plotted against the exact solution, already known from Example 2. Another plot, showing the results for N = 10 in order to make the trend perfectly clear, appears in Fig. 4. The approximate solution indeed approaches the exact solution more closely as N increases but, because the exact solution cannot be expressed as a finite series combination of the basis functions alfl + a2 f2 + a s f 3 . . ., the numerical solution remains an approximation for finite N . Using Gauss elimination, the upper size limit for N is approximately 68, compared to about 10 using direct matrix inversion.

1

(29) D. Example 4 gm = (g,w,) = J4(1 - 3z2)x2(1 - xm)& 0

Solve the boundary-value problem from (8) with g ( x ) = 4( 1 - 3x2) using triangle functions as the expansion functions and pulse testing functions, as defined below. The triangle functions, fn = A(z - xn), are

(8m)(2m+l) - - - 15(m+3)(m+5).

Convergence of the solution as N is increased is readily demonstrated. Starting with the case N = 1

and f = a1z2(1 -E') . (34) ell = 0.133 91 = -0.067 (yl = -0.500

389 WHELESS AND WURTZ: INTRODUCING UNDERGRADUATES TO THE MOMENT METHOD

L = ( N + 1 ) 2 N

0.151

- 2 -1 0 ". 0 0 0 -1 2 -1 ... 0 0 0

0 -1 2 ... 0 0 0

; . . 0 0 0 ... 2 -1 0 0 0 0 ... -1 2 -1

- 0 0 0 . . ' 0 -1 2

. . . .

-0.1' 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

Fig. 4 Ar = 10 and exact solutions for Example 3.

A

Fio 5 Trianole ( A ) and nnlw (IT) functions __\..,____ - - " -

about the N equispaced points over the interval 0 5 z 5 1 m

xm = - m = 1 ,2 ,3 ,..., N N + l (35)

and the testing functions for this problem are defined in terms of pulses II(z - 2,) by w, = ( N + l)II(z - z,). That is

The basic triangle function A and pulse function II are illustrated in Fig. 5.

Solution: The solution f is once more represented by the power series

(37) n=l

The operator L remains - & and g(z) is given in the problem statement. Proceeding with the calculation of values for the elements of matrix L, taking note of the effect of the second derivative on A

N

cmn = (L fn , wm) = 1

. ( N + l)rI(z - z,)dz = ( N + 1) J rI(z - x,)

-[-S(z - + 2S(x - x,) - S(z - zn+l)]dz 0 (38)

f +2(N + 1)2 m = n

where S is the Dirac delta function and L has the general

. (39)

1

0 gm = (g,w,) = J4(1 - 3 z 2 ) ( N + l)II(x - xm)dx

",+& = 4(N + 1) J (1 - 322)dz.

1

(40) The upper and lower limits are modified from the original values of 0 and 1, respectively, as required in order to conform with the defined extent of the pulse function in z (see Fig. 5). In view of (33 , it can be shown that

x m - g o

12m2 + 1 ( N + 1)2 '

g m = 4 -

Convergence characteristics are illustrated by considering the sequence of results for N = 2,3, ... . For N = 2

The next case, N = 3, gives

(43) -2.813 N 0 -16

-0.053 N

Aid of a computer program is enlisted for numerical results when N 2 4. A representative MATLAB main program listing follows:

clear; N = input('Va1ue of N...') Nplus = N + 1 ; for m = 1: N ; gm(m) = 4 . - ( (12. * m"2) + l ) / (Nplus l"2) ; for n = 1: N ; zf m == n; L(m, n) = 2. * (Nplusl"2); elseif abs(m - n) == I ; L(m, n) = -(Np/usI"2); else L(m, n) = 0.;

end; end; end; gm = gm'; alphan = L\gm; x = 0:O.Ol:l; z = z'; f = zeros(length(z), 1 ) ;

for ns = 1: N ; tri = triangle(x, N , ns); f = f + (aZphan(ns) * t r i ) ; end;

exact = x. * (1 - 2. * x + z."3);


0.15

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X

Fig. 6 Numerical and exact solutions for Example 4

where the triangle basis functions are each calculated sepa- rately by the .m file containing

function T = tn’angle(a, N , nt ) m = n t / (N + 1 ) ; lx = length(x); T = zeros(lx, 1 ) ; xdzff = zeros(lx, 1 ) ;

for xta = 1 : lx; zf zdzff(xtz)<l./(N + 1 ) ;

else T (x t i ) = T(z tz ) + 0.; end; end;

xdiff = &(a - ~ m ) ;

T(xtZ) = T(xt%) + (1. - (abs(z(zti) - am) * ( N + I ) ) ) ) ;

Results for N = 2,4, and 10 are plotted in Fig. 6 against the exact solution. For N 2 20, the numerical solution becomes essentially indistinguishable from the exact.

exercises is not a significant distraction from the important concepts.

The examples presented here should provide direct, in- class assistance similar to that realized at The University of Alabama for other educators involved in teaching MOM in undergraduate fields courses.

REFERENCES

[ l ] L. V. Kantorovich and G. P. Akilov, Functional Analysis in Normed Spaces, translated by D. E. Brown. Oxford: Pergamon, 1964, pp. 586-587.

[2] R. F. Harrington, “Matrix methods for field problems,” Proc. IEEE, vol. 55, no. 2, pp. 136-149, Feb. 1967.

[3] -, Field Computation by Moment Methods. New York Macmil- Ian, 1968.

[4] L. L. Tsai and C. E. Smith, “Moment methods in electromagnetics for undergraduates,” IEEE Trans. Educ., vol. E-21, no. 1, pp. 14-22, Feb. 1978.

[5] E. H. Newman, “Simple examples of the method of moments in electromagnetics,” IEEE Trans. Educ., vol. E-31, no. 3, pp. 193-200, Aug. 1988.

[6] D. R. Wilton and C. M. Butler, “Effective methods for solving integral and integro-differential equations,” Electromagnetics, vol. 1, pp. 289-308, 1981.

[7] C. H. Ma, University of Mississippi Short Course on Applications of Moment Methods to Field Problems, May 1973.

[8] L. V. Kantorovich and V. I. Krylov, Approximate Methods of Higher Analysis, 4th ed., translated by C. D. Benster. New York Wiley, 1959, ch. N.

[9] D. S. Jones, “A critique of the variational method in scattering problems,” IRE Trans. Ant. Prop., vol. AP-4, no. 3, pp. 297-301, 1956.

[lo] MATLAB is a registered trademark of The Mathworks, Inc., 24 Prime Park Way, Natick, MA 01760.

[ l l ] H. Y. Yee and N. F. Auden, “Uniform waveguides with arbitrary cross-section considered by the point-matching method,” IEEE Trans. Microwave Theory Tech., vol. M’IT-13, no. 11, pp. 847-851, Nov. 1965.

[I21 R. H. T. Bates, “The point-matching method for interior and exterior two-dimensional boundary-value problems,” IEEE Trans. Microwave Theory Tech., vol. MTT-15, pp. 185-187, Mar. 1967.

[13] R. F. Harrington, “Origin and development of the method of moments for field computation,” IEEE Antennas Propagat. Mag., vol. 32, no. 3, pp. 31-36, June 1990.

V. CONCLUSION W. Perm Wheless. Jr. (M’85-SM88) received both the Bachelor’s deeree I - ~ -

in physics and the Master’s degree in environmental sciences and engineering from the University of North Carolina. He completed the M.S. degree in electrical engineering at Georgia Tech in 1974 and from 1974 to 1980 was a

This paper four drawn from a of problems that has been found useful in the early stages of instruction on the moment method in undergraduate electrical engineering courses at The University of Alabama. Other educators can devise additional variations.

Student reaction to these problems has hen assessed

term and a final “exit interview” summary meeting at the end of each course. Students have endorsed these exercises as effective and worthwhile, helping them approach subsequent problems at the level of [4] and [5] with confidence. It appears that exercises of the nature and scope presented here make students more comfortable with matrix methods

consultant specializing in RF and microwave systems. His doctoral research was in applied electromagnetics at the University of Mississippi, where he received the Ph,D, degree in 1986.

He was an Acting Assistant Professor at the University of Mississippi from 1984 to 1986. After one year as an Assistant Professor at New Mexico State University, he joined the faculty of The University of Alabama in nscaloosa as Associate Professor of Electrical Engineering. His principal research interests include antennas, microwave circuits, and high-frequency measurements, in

Dr. Wheless is a registered professional engineer in the states of Mississippi and North Carolina,

through two with students during the

to

and increase their interest in more advanced applications of Larry T. Wurtz (M’88) received the B.S. and M.S. degrees in computer science and the B.S. degree in mathematics from Troy State University, Troy, Undergraduates have remarked that are motivated AL. and the M.S. and Ph.D. &flees in electrical eneineerinp in 1985 and

to learn more about numerical methods and field theory by

associated Computer progam assignments are valuable novice projects, and students report that the programming for these

1988, respectively, from Auburn”University, Aubum, -AL. He is an Assistant Professor of electrical engineering at The University Of

Alabama, Tuscaloosa. His research interests are in the areas of analog and digital integrated circuit design, hybrid electronics, RF and microwave circuit design, and computer architecture.

early Success with the family of problems discussed here. ne

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Introducing undergraduates to the moment method