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The integration over dy and dx can be found in Gradshteyn and Ryzhik [1965]. Note that in (2.1.62), the result only depends on the ratios of the lengths of the three sides of the rectangular parallelepiped. Similar expressions can be derived for the other four faces.
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Method of Moments
Method of Moments
The method of moments (MoM) is a numerical technique that has been used extensively in the solution of electromagnetic boundary value problems. Many excellent texts have been written on this subject [Harrington, 1968]. The technique is used extensively in this book in Monte Carlo simulations. A characteristic of this technique is that it leads to a full matrix equation which can be solved by matrix inversion. In later chapters, we will describe techniques that can speed up the numerical solution of these matrix equations. With the use of Green's function, integral equations can be derived. Consider a one dimensional integral equation of the form
dx'G(x,x')f(x') = c(x)
(2.2.1 )
where G(x, x') is the Green's function, f(x) is the unknown for the domain a S x S b, and c(x) is known for a S x S b. To solve (2.2.1), two sets of functions are used in the MoM: basis functions and weighting functions. (1) Basis functions. A set of N basis functions in the domain of a S x S b is chosen. Let the basis functions be !I, 12, ... , f N. The unknown function f (x) is expanded in terms of a linear combination of these basis functions.
f(x) =
L bnfn(x)
(2.2.2)
The linear combination of fn(x) should well represent the unknown f(x) in the domain. Substitute (2.2.2) into (2.2.1), we have
L bn
dx'G(x, x')fn(x') = c(x)
(2.2.3)
The unknown coefficients b1 , b2, ... , bN are to be determined. (2) Next a set of N weighting functions (testing functions) WI (x), W2 (x), ... , WN(X) is chosen. Multiply (2.2.3) by wm(x) and integrate over the domain
dxwm(x)
dx'G(x,x')fn(x') =
dxwm(x)c(x)
(2.2.4)
2 INTEGRAL EQUATION FORMULATIONS AND NUMERICAL METHODS
This gives the matrix equation
LGmnbn =
(2.2.5)
m= 1,2, ... ,N, where
Gmn =
dxwm(x)c(x) = (wm,c)
(2.2.6) (2.2.7)
dxwm(x)
dx'G(x,x')fn(x') = (wm,Gfn)
where the inner product notation is used.
(1, g) =
Computational Considerations
dx f(x) g(x)
(2.2.8)
Generally (2.2.5) is a full matrix equation. We note the following (1) Matrix solution: To solve a full matrix equation of order N by full matrix inversion (e.g., Gaussian elimination) requires O(N 3 ) number of operations. This increases rapidly with N. (2) Matrix filling: To calculate G mn , m, n = 1,2, ... , N can be computationally intensive because there are N 2 values of G mn . Also G mn can require the evaluation of a double integral as given in (2.2.7). The matrix filling can be more computationally intensive than matrix solution because G(x, x') can be of a complicated form. Also since there are N 2 elements of G mn , this can impose a large memory requirement. (3) The study of fn, n = 1,2, ... , N is also an important subject as the choice of fn must well represent the correct solution. Often they have to satisfy differentiation and continuity properties.
Basis Functions
Basis functions can use full domain functions such as sines, cosines, special functions, polynomials, modal solutions, etc. A set that is useful for practical problem is the subsectional basis function. This means that each fn is only nonzero over a subsection of the domain of f. A common choice is the pulse function (Fig. 2.2.1a)
fn(x) =
if an "5:.. x "5:. bn otherwIse
(2.2.9)
Method of Moments
Figure 2.2.1 Common choices of basis functions: (a) pulse functions; (b) triangle functions.
where the interval a S x S b have been divided into N intervals with endpoints an and bn , n = 1,2, ... , N. Another choice is the triangle basis functions (Fig. 2.2.1b). In Fig. 2.2.1b we show fn(x) and fn+I(X). Note that fn(x) and fn+I(X) overlap.