Integral Equation and SMCG Method in .NET

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3.1 Integral Equation and SMCG Method
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dX'dY'{_ik1b~1)(PR)I~n+1)(r')
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(6.3.44)
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dx'dY'{-ik 1T1 1E2b~)(PR)
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af(x,y)af(x';y')(y_y') ax ax
+ af~~ y) [af~:; y') (x - x') - (z - z')] - (y - y')] + a~)(PR)
[;i]
F~n)(r') [af~~ y) (z -
+ af~~; y') af~; y) (x - x')
af~~; y') af~~ y) (y _ y') + (x _ x')] + a~)(PR)
[af~~Y) (x _ x') + af~;Y) (y +
y') - (z -
[;iJ
F~n)(r')
- _ Fn(r) 2
Z')]}
----'------'-:--='---'-
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af(x', y')] ax'
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dX'dY'{G2F~n+1)(r') [_ af(x, y) af(x'; y') (y _ y')
ax ax ax'
(z - z')] _ (y _ y')]
+ af(x, y) [af(x" y') (x _ x') -
6 3-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
+ G2(R)F~n+l)(r') [af 1x , y) (z X
+ af(x', y') af1x, y) (X a~
- af(x',y') af(x,y) (y-y')+( x-x ')] ay' ax
+ G2(R)F(n+l)(r')
[af(X, y) (x _ x') ax
+ af(x, y) (y ay
y') -
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[af(x" y') (x _ x') - (z - z,)] _ (y _ y')] ax'
(2)(p + ao R )F(n+l) (-') [af(X, y) (Z y r ax
+ af(x', y') af(x, y) (
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_ af(x',y')af(x,y)( _ ')+( _ ')] + (2)( )F(n+l) (-') ay' ax y y x x ao PR n r .[ af x 1 , y) (x - x')
+ af(x, y) (y ay
y') - (z -
Z')] }
(6.3.45)
3.2 Numerical Results of Bistatic Scattering Coefficient The numerical simulation results are presented in terms of the bistatic scattering coefficient as normalized by the incident power. For an incident wave with a horizontal polarization (TE) we have lah(()s, cPs) and p~nc is given by (6.1.62) and (6.1.63) respectively. In medium 1, the co-polarized and crosspolarized scattered components of ~ for a = v (vertical polarization) and h (horizontal polarization) expressed in terms of the surface field components, Fx , Fy , Ix, I y, are respectively
1" =
dx'dy' exp( -ik/3') [{ Ix(x', y') cos()s cos cPs
+ Iy(x', y') cos()s sin cPs
' ,)af(x',y') . () - I x (X ,y ax' sm s
I (' ,)af(x',y') . ()} y x ,y ay' sm s
(6.3.46)
- T]{ Fx (x', y') sin cPs - F y(x', y') cos cPs}]
3.2 Numerical Results of Bistatic Scattering Coefficient
~ = ~:
dx'dy' exp( -ikjJ') [{Ix (x', y') sin <Ps - I y (x', y') cos <Ps} Fx (' ,y') 8f(x', y') sm () s . X 8x' (6.3.47 )
-I-. -I-. + T] {Fx (X " ) cos ()s cos cps + Fy cos (). CPs ,y s sm
_F(X'y,)8 f (x',y')s'n()}] y, 8y' 1 s
where {3' = x' sin ()s cos <Ps+Y' sin ()s sin <Ps+ f(x', y') cos ()s. The expressions for the transmitted waves are similar. The normalization is that the integration of reflected and transmitted power over solid angles gives unity for a lossless second medium. The rough surface has a 2-D Gaussian power spectrum given by
ly = lx 47Th
exp (_ K;l; _ K;l;) 4 4
We describe various accuracy tests using a 2-D dielectric rough surface. Then, we compute the solution of an electromagnetic wave scattering problem with up to 98,304 surface unknowns and up to 300 realizations. We first compare against matrix inversion (MI) for a small problem. In Figs. 6.3.1 and 6.3.2 the normalized bistatic scattering coefficient obtained by the MI is compared to the solutions obtained by the SMCG for co- and cross-polarized components, respectively, using a single surface realization. Surface lengths in the x and y directions are Lx = L y = 4.0'\ which gives an area of L 2 = 16,\2. The rms height is h = 0.2'\, with a correlation length of lx = ly = 1.0,\ and a relative permittivity of (3.0 + iO.2). The surface is sampled at 16 points per ,\2 to give 1536 surface unknowns. The neighborhood distance r d = 2.0'\. The incident angles for all illustrations in this paper are 0 0 ()i = 10 and <Pi = 0 . It can be observed that both polarized components of MI and SMCG curves almost completely overlap each other in Fig. 6.3.1.