3-D WAVE SCATTERING FROM 2-D ROUGH SURFACES in .NET

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6 3-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
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[-(Z - z') + of (x', y') (y o~
+ of (x, y) (x fu
x')]
+ G 1 (R)Fn (r') [- of~: y) (z - z') - (x - x')]}
Fn(T) 0 - - -2
(6.3.20)
dx 'd'{ -~ok 1T/1 - 92 I x (-') [Of (x, y) - -----',------'E2 of(x' , y')] y r E1 aX ax'
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Of(x',y')]} !=l' uy
dx'dy' { G2(R)Fx (r') [- Of~: y) oy [Of (x', y') (x _ x') - (z ax' z')
Of~:; y') (y -
+ of(x, y)
Z')] _ (y _ y')]
+ G2(R)Fy(r') [of~: y) (z _
+ Of~~; y') Of~~ y) (x x')]
Of~~; y') Of~: y) (y _
y') oy
+ (x _
+ G2(R)Fn (r')
(6.3.21)
. [Of (x, y) (x _ x') ax
+ of (x, y) (y -
y') - (z - z')] }
3.1 Integral Equation and SMCG Method
In terms of a matrix notation, the SMCG procedure is as follows. First, the above 6 scalar surface integral equations are discretized into a matrix equation by the moment method. Then, we choose the neighborhood distance r d as the distance which defines the boundary between the weak and strong element of the impedance matrix Z (for example, rd = 2>.). Let
= J(x - x')2 + (y -
y')2
(6.3.22)
represent the horizontal separation between two points on the rough surface (x, y, f(x, y)) and (x', y', f(x', y')). The strong matrix is a sparse matrix. For the weak matrix elements, we expand the Green's function in a Taylor's series about the flat surface, f(x, y) = o. 1 R ) exp(ik 1,2 R ) = ~ (1,2) ( ) G 1,2 (R) = (1 - ik ,2 47fR3 L....t am PR
(6.3.23)
gl,2
exp(ik1,2R) 47fR
L....t
b(1,2) (
(6.3.24)
where Zd = f(x, y) - f(x', y'). The above coefficients a~,2) (PR) and b~,2) (PR) are translationally invariant in the horizontal directions. In the numerical results of this section, we keep the expansion terms at 6 (M = 5) in (6.3.23) and (6.3.24). In the following the first 4 coefficients are listed for reference (6.3.25) (6.3.26)
(6.3.27)
(6.3.28) (6.3.29) (6.3.30)
6 3-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
b(1 '2) (PR)
. exp(~k12PR)
3ik12 - - 2PR - - - ' ''321f 321f
k3 2 -i 1,2 PR { 19 21f
3} +-321fPR
(6.3.31)
b~1,2) (PR) = exp (ik 12 PR)
k2 1,2PR 321f
13ik12 ' 1921f
1921fPR
(6.3.32)
The impedance matrix is decomposed into the sum of a strong and a weak matrix.
Z = Z(8)
=(8)
+ Z(w)
=(w)
(6.3.33)
where Z represents near field strong interaction and Z represents nonnear field weak interaction. Next, the weak matrix elements are expanded in a Taylor's series about the horizontal distance between the two points
=Cw)
= L-. Zm
~ =(w)
(6.3.34)
The zeroth term in (6.3.34) is called the fiat surface contribution
=(FS)
= Zo
=(w)
(6.3.35)
The iterative matrix-solving procedure is, for the first-order and higher order solutions
(Z (Z
=(8)
=(8)
=(FS)
(6.3.36) (6.3.37) (6.3.38)
=(FS)
(+1) n
-(n+1) ,,=(w) (n)
-(n+1)
=b- L-.Zm X
Equations (6.3.36) and (6.3.37) are solved using the conjugate gradient method (CGM). The fiat surface matrix Z which represents the lowest order Taylor expansion term is on the left-hand side of the matrix equation. Without the fiat-surface matrix on the left-hand side, we have observed that the iteration does not converge for rough surfaces with moderate rms heights. Thus, the terms strong and weak refer to the magnitude of the matrix elements, instead of their total contributions to the iterative matrix equation. The product of Z with x can be computed using a 2-D FFT algorithm. Updating the right-hand side is also calculated using the FFT. An
=(FS) =(FS)
3.1 Integral Equation and SMCG Method
additional advantage of the SMCG is that only the Taylor expanded coefficients need to be stored. With the number of Taylor series coefficient fixed at M = 5, for a given rough surface the computational complexity will depend on the number of CGM iterations (6.3.37), SMCG iterations (6.3.38) and the neighborhood distance rd. The total number of operations (multiplications) is approximately NCGM [256rinN