Electromagnetic Wave Scattering by Perfectly Conducting Surfaces in .NET

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1.2 Electromagnetic Wave Scattering by Perfectly Conducting Surfaces
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dx'dy'GFS(PR) [(x - x')Fy(r') - (y - y')Fx(r')]
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dx'dy'(G(R) - GFS(PR))[(X - x')Fy(r') - (y - y')Fx(r')]
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+ [(y - y') af~~; y') _ (f(x, y) - f(x', y'))] Fy(r') }
The last terms on the right-hand side of the integral equations of (6.1.48)(6.1.49) are small since rd > h. In terms of matrix notation the SMFSIA procedure is as follows. The surface integral equations (6.1.41) and (6.1.42) are cast into a matrix equation by the method of moments. This gives
(6.1.50)
Then, the original matrix is decomposed into the sum of a strong matrix, a block Toeplitz flat-surface part, and a weak remainder as
(z(s)
+ Z(FS) + z(W)) x = b
(6.1.51)
In (6.1.51), Z(s) is a matrix corresponding to the integrals of (6.1.48) and (6.1.49) with PR < rd' The strong matrix is a sparse matrix. The flat surface matrix corresponds to the second term with GFS(PR) in (6.1.48) and (6.1.49) with PR ;:: rd. The flat surface matrix is a block Toeplitz matrix. The weak remainder matrix elements consist of the differences of the Green's function G(R) - G FS(PR) connecting the two points whose horizontal distance is greater than r d and corresponds to the last terms on the right hand
63-D WAVE SCATTERING FROM 2-D ROUGH SURFACES
sides of (6.1.48) and (6.1.49). The weak and the flat surface matrix elements are nonzero only for those points whose horizontal interaction distances are greater than rd. Next, the matrix equation is rearranged to take an iterative form. The calculation procedure is, for the first-order and higher order solutions
(Z (Z
=(8)
=(8)
=(FS)
-(n+1)
(6.1.52) (6.1.53)
=(FS)
(+1) n
-(n+1)
=b-Z
=(w) (n )
(6.1.54)
The flat surface impedance matrix Z must be on the left-hand side of (6.1.53) for the SMFSIA to work for 2-D surfaces. For each order of solution x(n), the matrix equations (6.1.52) or (6.1.53) is solved by the conjugate gradient method. Note that the product of Z with x can be computed by a 2-D fast Fourier transform (FFT) algorithm which makes conjugate gradient iteration more efficient. The iteration of and (6.1.54) is carried out until the error norm
Z(w) X
=(FS)
=(FS)
through (6.1.53)
liZ :K(n) - bll Ilbll
(6.1.55)
falls below a threshold. In this section, an error norm of 1% is used for all numerical simulations.
(B) SMFSIA/CAG
We can further improve the SMFSIA by using the flat surface as a canonical grid (CAG). This method is called SMFSIA/CAG. For the weak remainder matrix elements, Green's function is approximately equal to the Green's function of the horizontal distance between the two points. Green's function can be expanded in a Taylor's series about the horizontal distance between the two points.
G(R) _ G
= (ikR - 1) exp(ikR) _ (ikpR - 1) exp(ikpR)
4 ~ R3 4 ~PR 3
L am(PR) Zg2)m
(6.1.56)
where Zd = j(x, y) - j(x/, y/). The larger the rd the less number of terms we need in (6.1.56). In this section, we keep up to the sixth term in the Taylor
1.2 Electromagnetic Wave Scattering by Perfectly Conducting Surfaces
series. In the following, the first 3 coefficients are listed for reference. 2exp(ikpR) 3 kexp(ikPR) 3 exp (ikPR) al ( ) = - k PR - Z 2 + 3 (6.1.57)
4~PR 4~PR 4~PR
a2 ( R P)
= - z
k3exp(ikpR)
32~PR
+ 6k2exp(ikPR) + 15 k z 32~PR
exp (ikpR)
32~p~
ex _ 15__p- -,- (i- -,- kp;-R3---,--) -;C a ( )=k exp(ikpR) 3 PR 4PR 192~
(6.1.58)
10ik3exp (ikpR) _ 42k 2exp (ikPR)
192~
192~PR
_ 96ik eXP (ik pR ) 2 196~PR
+ 96exp(ikp3R)
196~PR
(6.1.59)
The important property of above coefficients is that they are translationally invariant. In terms of the matrix equation, the iterative procedure is then
(Z(s)
+ Z(FS)) x(n+l) = lj(n+l)
(6.1.60)
-(n+l)
=b- LJZm
'"'" =(w) (n)
(6.1.61)
where is the expanded form of the weak matrix. The updated right-hand side is calculated by the FFT. Like the SMFSIA, SMFSIA/CAG has an adjustable parameter rd (the neighborhood distance). Furthermore, in SMFSIA/CAG there is a second adjustable parameter, which is the number of Taylor series coefficients of (6.1.56). These two adjustable parameters of rd and the number of Taylor series terms are interdependent. They are chosen to optimize the CPU. The numerical simulation results are presented in terms of the normalized bistatic scattering coefficient as normalized by the incident power
Z;:)
rah(
cPs)