A tutorial on the method of moments
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A Tutorial on the Method of Moments
Ercument Arvas1 and Levent Sevgi2
1
Department of Electrical Engineering and Computer Science Syracuse University Syracuse, NY, USA
T
1. Introduction
he Method of moments (MoM) is a general procedure for solving linear equations. Many problems that cannot be solved exactly can be solved approximately by this method.
2
Electronics and Communications Engineering Department Doğuş University Zeamet Sokak 21, Acıbadem – Kadıköy, 34722 Istanbul, Turkey
Abstract
Dedicated to the 87th birthday of Roger F. Harrington The Method of Moments (MoM) is a numerical technique used to approximately solve linear operator equations, such as differential equations or integral equations. The unknown function is approximated by a finite series of known expansion functions with unknown expansion coefficients. The approximate function is substituted into the original operator equation, and the resulting approximate equation is tested so that the weighted residual is zero. This results in a number of simultaneous algebraic equations for the unknown coefficients. These equations are then solved using matrix calculus. MoM has been used to solve a vast number of electromagnetic problems during the last five decades. In addition to the basic theory of MoM, some simple examples are given. To demonstrate the concept of minimizing weighted error, the Fourier series is also reviewed. Keywords: Numerical electromagnetics; Method of Moments; MoM; linear operator equations; integral equations; expansion functions; basis functions; Fourier series; test functions; numerical electromagnetics code; NEC
P /2 P /2
with P = π , then only A0 and A1 in Equation (1) are
1 1 e(x = ) cos2 பைடு நூலகம் − + cos 2 x , 2 2
(7)
is then identically zero. That is, the series exactly equals the function f ( x ) . On the other hand, for the periodic rectangular function shown in Figure 2 with P = 2π , we have f ( x) 4 sin 3 x sin 5 x + + .. . sin x + 3 5
IEEE Antennas and Propagation Magazine, Vol. 54, No. 3, June 2012
and
wn ( x ) = sin ( 2nπ x P )
in
Equation (6) yield Equations (3) and (4), respectively. Summarizing, the Fourier series is an expansion of a known periodic function, f ( x ) , in terms of sine and cosine functions. The expansion coefficients are determined so that the weighted error is zero. Depending on the function f ( x ) , the Fourier series, S ( x ) , converges to in some sense. For example, if f ( x ) = cos 2 x non-zero, and the series S ( x= ) 1 + cos ( 2 x ) 2 . The error, Figure 1. The normalized capacitance of the parallel-plate capacitor as a function of the plate distance, simulated with the MoM (the plates were identical and square, with width = length = 2a). Here, P is the period of the function, the expansion (or basis) functions are sine and cosine functions, and the expansion coefficients are given by A0 = 2 ∫ f ( x ) dx , P −P /2
(2)
π
(8)
2 2nπ x An = dx , ∫ f ( x ) cos P −P P /2 Bn = 2 2nπ x dx . ∫ f ( x ) sin P −P P /2
P /2
(3)
Let S N be the series with N terms so that S1 = 4sin x π , S= S1 + 4sin 3 x 3π , S= 2 3 S 2 + 4sin 5 x 5π . Figure 2 shows that as N increases, the Fourier-series representation better resembles the function f ( x ) ; however, it never equals f ( x ) . Non-periodic functions may also be approximated by Fourier series inside a finite region by assuming the finite region to be the period of that function. In this case, it should
For example, consider the simple problem of the parallelplate capacitor. The approximate analytical formula for the capacitance is C0 = ε A d , where A is the area of each plate and d is the distance between them. This formula neglects the fringing fields, and is inaccurate except for very small d. In a later section, we use the MoM to compute a moreaccurate capacitance (including the fringing effects) for arbitrary d. Figure 1 shows the computed capacitance, C, normalized to C0 as given above. The figure shows the limitation of C0 even for quite small values of d. The MoM owes its name to the process of taking moments by multiplying with appropriate weighing functions and integrating. It has been applied to a broad range of electromagnetic (EM) problems since the publication of the book by Harrington [1]. A comprehensive bibliography is too vast 260
to be given here. A selected short list includes [2-9]. A quick Internet search yields a very long list of publications on the MoM. To reinforce the concept of minimizing error or residual, we first visit the familiar Fourier series in Section 2. We then present the basics of the MoM in Section 3. Section 4 includes details of some simple examples of application of the MoM. Section 5 summarizes some of more advanced work using the MoM. Finally, Section 6 gives the conclusions.
2. Approximating a Periodic Function by Other Functions: Fourier Series (FS) Representation
A known periodic function, f ( x) , can be represented by a Fourier series (FS): f ( x) ≈ A0 ∞ 2π nx 2π nx + ∑ An cos + Bn sin . (1) 2 n =1 P P
Ercument Arvas1 and Levent Sevgi2
1
Department of Electrical Engineering and Computer Science Syracuse University Syracuse, NY, USA
T
1. Introduction
he Method of moments (MoM) is a general procedure for solving linear equations. Many problems that cannot be solved exactly can be solved approximately by this method.
2
Electronics and Communications Engineering Department Doğuş University Zeamet Sokak 21, Acıbadem – Kadıköy, 34722 Istanbul, Turkey
Abstract
Dedicated to the 87th birthday of Roger F. Harrington The Method of Moments (MoM) is a numerical technique used to approximately solve linear operator equations, such as differential equations or integral equations. The unknown function is approximated by a finite series of known expansion functions with unknown expansion coefficients. The approximate function is substituted into the original operator equation, and the resulting approximate equation is tested so that the weighted residual is zero. This results in a number of simultaneous algebraic equations for the unknown coefficients. These equations are then solved using matrix calculus. MoM has been used to solve a vast number of electromagnetic problems during the last five decades. In addition to the basic theory of MoM, some simple examples are given. To demonstrate the concept of minimizing weighted error, the Fourier series is also reviewed. Keywords: Numerical electromagnetics; Method of Moments; MoM; linear operator equations; integral equations; expansion functions; basis functions; Fourier series; test functions; numerical electromagnetics code; NEC
P /2 P /2
with P = π , then only A0 and A1 in Equation (1) are
1 1 e(x = ) cos2 பைடு நூலகம் − + cos 2 x , 2 2
(7)
is then identically zero. That is, the series exactly equals the function f ( x ) . On the other hand, for the periodic rectangular function shown in Figure 2 with P = 2π , we have f ( x) 4 sin 3 x sin 5 x + + .. . sin x + 3 5
IEEE Antennas and Propagation Magazine, Vol. 54, No. 3, June 2012
and
wn ( x ) = sin ( 2nπ x P )
in
Equation (6) yield Equations (3) and (4), respectively. Summarizing, the Fourier series is an expansion of a known periodic function, f ( x ) , in terms of sine and cosine functions. The expansion coefficients are determined so that the weighted error is zero. Depending on the function f ( x ) , the Fourier series, S ( x ) , converges to in some sense. For example, if f ( x ) = cos 2 x non-zero, and the series S ( x= ) 1 + cos ( 2 x ) 2 . The error, Figure 1. The normalized capacitance of the parallel-plate capacitor as a function of the plate distance, simulated with the MoM (the plates were identical and square, with width = length = 2a). Here, P is the period of the function, the expansion (or basis) functions are sine and cosine functions, and the expansion coefficients are given by A0 = 2 ∫ f ( x ) dx , P −P /2
(2)
π
(8)
2 2nπ x An = dx , ∫ f ( x ) cos P −P P /2 Bn = 2 2nπ x dx . ∫ f ( x ) sin P −P P /2
P /2
(3)
Let S N be the series with N terms so that S1 = 4sin x π , S= S1 + 4sin 3 x 3π , S= 2 3 S 2 + 4sin 5 x 5π . Figure 2 shows that as N increases, the Fourier-series representation better resembles the function f ( x ) ; however, it never equals f ( x ) . Non-periodic functions may also be approximated by Fourier series inside a finite region by assuming the finite region to be the period of that function. In this case, it should
For example, consider the simple problem of the parallelplate capacitor. The approximate analytical formula for the capacitance is C0 = ε A d , where A is the area of each plate and d is the distance between them. This formula neglects the fringing fields, and is inaccurate except for very small d. In a later section, we use the MoM to compute a moreaccurate capacitance (including the fringing effects) for arbitrary d. Figure 1 shows the computed capacitance, C, normalized to C0 as given above. The figure shows the limitation of C0 even for quite small values of d. The MoM owes its name to the process of taking moments by multiplying with appropriate weighing functions and integrating. It has been applied to a broad range of electromagnetic (EM) problems since the publication of the book by Harrington [1]. A comprehensive bibliography is too vast 260
to be given here. A selected short list includes [2-9]. A quick Internet search yields a very long list of publications on the MoM. To reinforce the concept of minimizing error or residual, we first visit the familiar Fourier series in Section 2. We then present the basics of the MoM in Section 3. Section 4 includes details of some simple examples of application of the MoM. Section 5 summarizes some of more advanced work using the MoM. Finally, Section 6 gives the conclusions.
2. Approximating a Periodic Function by Other Functions: Fourier Series (FS) Representation
A known periodic function, f ( x) , can be represented by a Fourier series (FS): f ( x) ≈ A0 ∞ 2π nx 2π nx + ∑ An cos + Bn sin . (1) 2 n =1 P P