+ 2Nlog(N)mFFT] + NSMCG [72Nlog(N)mFFTJ
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(6.3.39) where NCGM and NSMCG are the number of iterations in CGM matrix solver and the number of right-hand side updates, respectively, n is the number of sample points per >.2, and mFFT is the total number of FFT's and inverse FFT's. Note that in the above equation, the total number of iterations required for convergence depends on rd. As an example, consider a simulation with the following number of iterations NCGM = 110 and NSMCG = 4, with sampling of n = 64, r d = 2.5>', N = 16384, and mFFT = 120. The term containing the r d dominates with 1.55 x 1011 computational steps. Therefore, SMCG is effective for a moderate rms height so that the number of Taylor series expansion terms can be manageable and the neighborhood distance r d can be much smaller than the surface length. In this section, an error norm of 0.01 is used for all numerical simulations.
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Integral Form of Matrix Equations
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For completeness, we list the detailed integral form of the six matrix equations in the following
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6 3-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
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+ ikl771br;;)(PR) [;~]m F~n)(f') [Of~~Y) Of~~;Y') +
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(n+l) (_) r
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{ dx'dy' I~n+l)(f')G2(R) x') - (z - z')] ax'
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. [of (x, y) (y _ y') oy
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+ I~n+l)(r')G2(R)
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d 'd '{ Ok F(n+l)(-,)of(x,y) of(x',y') x y 2 1771g2 x r a a ' [of (x, y) of (x', y') oy oy'
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+ 2ok 1771g2 F(n+l)(-') y r +
JpR>Td
dX'dY'{I~n+l)(f')a62)(PR)
+ of (x', y') (x _
ax' x') - (z - z')]
. [Of (x, y) (y _ y') oy
+ I~n+l)(f')a62)(PR)
E2 0
(- Of~~Y) (x - x') [Of (x, y) (z - z') oy
+ Of~~;Y') (x + (y _
y')]}
x'))
+ I~n+l)(f') El a(2)(PR)
d'd ,{ok b(2)( )F(n+l)(_,)of(x,y)of(x',y') x y 2 1771 0 PR X r a y ax' [of(X,y) of(x',y') oy oy'
+ 2ok1771 b(2)(pR )F(n+l)(-') 0 y r
(6.3.40)
m=l JpR>Td
dX'dY'{I~n)(f')ar;;)(PR) [Zd]m
. [Of(X',y')(y_y')_ Of(X,y)(y_y')]
+ I(n) (f')ar;;) (PR) [ZdJm
. [-(Z - Z')
+ of (x', y') (y _
+ of(x, y) (x ax
x')]
3.1 Integral Equation and SMCG Method
+ I~n)(f')~a~)(PR) [Zd] m E2 PR
a f(x, y) (z _ z') ax PR
+ (x _ x')] }
ax ax'
JpR>Td
dX'dY'{iklTJlb~)(PR)[!! ]mFJn)(f')[-l- af(x,y)af(X',y')]
. +iF~n)(r')klTJlb~)(PR) [Zd]m [_ af(x,y) af(X"y')]} PR ax ay'
/n+l)(_)
~R<Td
dX'dY'{Iin+l) (f')G 2(R)
. [af(x"y') ( _ ,)_af(x,y)( _ ')] ax' y y ax y y + I~n+l)(f')G2(R) [-(Z - z')
+ af~~; y') (y - y') + af~~ y) (x - x')] + (x - x')]}
+ I~n+l)(f') :~ G2(R) [- af~~ y) (z - z') +
JAA<~
dX'dY'{iklTJlg2FJn+l)(f')
[-1- af(x,y) af(x"y')] ax ax'
[af~'~Y') (y _ y') _ af~x,y) (y _ y')]
+ iF~n+l)(f')klTJlg2 [_ af~~ y) +
JpR>Td
af~~; y')]}
dX'dY'{Iin+l)(f')a62)(PR)
+ I~n+l)(f')a62)(PR) [-(Z - z')
+ af~~; y') (y - y') + af~~ y) (x - x')] + (x - x')]}
2 + I~n+l)(f') :~ a6 )(PR) [- af~~ y) (z - z') +
JpR>Td
dX'dY'{iklTJlb62)(PR)FJn+l)(f')
[-1- af(x,y) af(x"y')] ax ax'
(6.3.41)
+ iF~n+l) (f')k 1TJlb62)(PR) [_ a f~~ y) af~~; y')] }
_L r
JpR>Td
dX'dY'{af!;)(PR) [Zd]m Ix (f') [_ af(x, y) of(x'; y') (y _ y') PR ax ax
+ af(x,y) [af(X',y')( x - x ') - (Z - Z')] - (Y - Y')] ay ax'
63-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
+a~)(PR) [;:]m Iy(r,)[af~~Y) [(Z-Z')- af~~;Y')(y_y')]
af~~Y) af~~;Y') (x _ x') + (x -
X')]
+ a~) (PR) [Zd] mIn (r') [af(x, y) (x _ x') PR ax
+ a f(x, y) (y ay
y') -
(z - z')]}
_ r t
m=l JpR>Td
dX'dY'{ik1T/1b~)(PR)
[Zd]m Fx(r') [af(x,y) _ a!(x';y')] PR ax ax _
+ik1T/1b~)(PR) [;:]m Fy(r') [af~~Y)
= In(r) +f d 'd
af~~;Y')]} +I~nc(r)
'{G (R)Ix(-') [_ af(x, y) af(x', y') (y - y') ax ax'
+ af(x, y) [af(x" y') (x _ x') - (z - z')] _ (y _ y')] ay ax' + G 1(R)I (r') [af(X, y) [(z _ z') _ af(x', y') (y _ y')] ax ay' y +
af~~Y) af~~;Y') (x _ x') + (x -
x')] ay y') - (z - Z')]}
+ G 1(R)In (r') [af(x,y) (x - x') ax +
+ af(x,y) (y ax'
Jd x'dy,{ ok T/1g1
F (-') [af(X,y) _ af(x"y')]
+ 2ok 1T/1g1 FY (-') [af(X, y) _ af(x', y')] } r ay ay'
AA>~
(l)(p dx 'd y '{ a o R )1x (-') [_ af(x,y) af(x',y') (y - Y') r ,:) ,:),
uX uX
+ af(x, y) [af(x" y') (x _ x') - (z - z')] _ (y _ y')] ay ax'
(1)( )1 + ao PRy (_') [af(X,y) r ax
[( _ ,)_af(x',y')( _ ')] Z Z ay' y y
af~ y) af~~; y') (x
_ x')
+ (x -
x')] ay y') _ (z - z')] }
+ a(l) (PR)In(r') [a f(x, y) (x _ x') o ax
+ a f(x, y) (y -
3.1 Integral Equation and SMCG Method
dX'dY'{iklTJlb~l)(PR)Fx(r')
D (-')
PR>Td
[a!(x,y) _ ax
a!(x'~Y')]
(6.3.42)
'k + z 1 TIl b(l)( PR) r 0
[a!(x,y) a!(x',y')]} ay ay'
63-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
JpR>Td
dX'dY'{a 61 )(PR)FJn+l) (r')
. [Of (x, y) (y _ y') + of (x', y') (x _ x') - (z - z')] oy ax' + a(l) (PR)F(n+l) (r') [- of (x , y) (x _ x') + of (x', y') (x - x')] o y oy oy'
+ a61)(PR)F~n+l) (r')
[a f~~ y) (z -
+ (y - y')] }
(6.3.43)
