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(5.1.23) (5.1.24)
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For the first three terms,
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5 FAST METHODS FOR ROUGH SURFACE SCATTERING
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If we retain three terms in the Taylor expansion (NT = 3),
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= '""'" (Z(w) - Z(w)(O)) ~ mn mn
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1.2 Formulation and Computational Procedure
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(5.1.30) We make use of FFT in the calculation of Ym in (5.1.30). For example, for the second term in (5.1.30), -2(j(x m )) L:~=1 al~~d) f(xn)u n , we calculate in d the following manner: (1) pre-multiply Un by f(x n ) to get f(xn)u n (2) calculate L....J
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In the BMIA/CAG method, the Z is decomposed into a sum represented by the Taylor series expansion. Thus
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=(w)
= L....J Zm
rn=O
'""'" =(w)
(5.1.31)
=(w)
The m = 0 term corresponds to that of a flat surface. The form of Z m is such that it consists of terms that are products of a diagonal matrix TTl a translationally invariant matrix Zd, and a diagonal matrix T s : (5.1.32) where T s is a function of the coordinates x' of the scattering source, while T r is a function of the coordinates x of the field.
(B) Iteration Based on Updating the Right-Hand Side
Let X(O) and X(n) represent the zeroth-order solution and the nth-order solution, respectively. They obey the equations
=c =(S)-(n+l) -(n) Z X =c
=(s)-(O)
(5.1.33) (5.1.34)
where
d n ) represents the updated right-hand side with
-(0)
= C - L....J Zm X
-(n)
=C- Z
=(w)-(n)
'""'" =(w)-(n)
(5.1.35)
5 FAST METHODS FOR ROUGH SURFACE SCATTERING
Note that for this case of iteration, only Z(s) is kept in the left-hand side of (5.1.34). A residual R(n) can be defined as follows such that its norm provides the stopping criterion for the iterative procedure:
R(n)
-z X(n) + C = _(Z(s) + Z(Wlyx(n) + C
IIR(n)II/IIClI
(5.1.36)
x 100%. From
where the normalized L-2 norm is defined as (5.1.33)-(5.1.36), it follows that
-(0) R =
=(s)-(O) =(w)-(O) X - Z X
+ C = -z
=(w)-(O) " =(w)-(O) X = - L..t Zm X
(5.1.37)
-(n+l)
= -Z X
=(S)-(n+l)
=(W)-(n+l)
+ -C = -(n+l) C
-(n)
(5.1.38)
Thus the residual vector can be computed readily from the updated righthand sides. In the numerical results illustrated in this section, the stopping criterion of the iterative solution is set at 0.1%.
Computational Complexity
For the TE case the matrix is symmetric. The bandwidth b is usually much smaller than the order of the matrix N. To take full advantage of the banded matrix Z(s), a direct banded matrix solver is used to solve (5.1.33) and (5.1.34). The LU decomposition requires O(b 2 N/2) operations, while the backsubstitution only requires O(2bN) operations. The
=(w)_ Z(w) X
product is
computed by the FFT. Therefore, we can evaluate Z X in r N (log N) + sN operations (where r accounts for the number of FFTs and s accounts for the number of pre- and post-multiplications before the FFT). The computational complexity up to the nth-order solutions are O(nb 2 N)+O(nrN log N +nsN).
(C) Solution Based on Complete Impedance Matrix and Conjugate Gradient Method (CGM)
Another iteration approach is to keep the entire impedance matrix on the left-hand side. Then we apply a conjugate gradient method (CGM) to the matrix equation with the matrix decomposition.
( =(S) Z
+ L..t Zm
~ =(W))
(5.1.39)
1.3 Weak Matrix and Unknown Column Vector
For the CGM version, an initial guess of XeD)
0 is chosen. Let N c be
=(s) =(w)
the number of CGM iterations. By decomposing into Z and Zm and the use of FFT in conjunction with CGM, the approach requires O(Nc(bN + rNlog N + sN)).
Memory Requirements
The memory requirement of the strong matrix Z(s) is O(bN). The coefficients am(xd) in the Taylor expansions are translationally invariant. The storage requirement for Zm , m = 0, 1,2, ... , M, is O((M +1)N). The total memory requirement for the algorithm is O(bN + (M + 1)N). In the simulations, the bandwidth b is an adjustable parameter. In the updated right-hand-side approach there is a minimum bandwidth bmin for which the iteration process works. It requires many more iteration steps to converge at b = bmin than at a larger bandwidth. Therefore, in the simulations, b is chosen to be greater than the bmin so as to reduce the number of iteration steps. For the CGM iterative approach discussed above, the bandwidth can be smaller than the one used in the updated right-hand-side approach. This is because the bandwidth in this case depends on the accuracy of the Taylor series expansion. As a result, this approach requires less computer memory, and therefore it is ideal for very large surface lengths. However, it usually takes more iteration steps to converge. In Section 1.5, only the 2500 wavelength surface examples are performed by applying the CGM iterative approach.
=(w)