Steepest Descent Fast Multipole Method in .NET

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Steepest Descent Fast Multipole Method
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The fast multipole method (FMM) was invented by Rohklin [Rohklin, 1990; Coifman et al. 1993]. Michielssen and Chew [1996] used the steepest descent method to express the product of impedance matrix and column vector. Jandhyala et al. [1998a,b] applied the steepest decent fast multipole method (SDFMM) to 3-D rough surface scattering problem. Recently, FMM has been applied extensively to large scale electromagnetic boundary value problems. SDFMM has also been combined with the PBTG method [Li, 2000]. In the following, we apply the method for one-dimensional rough surface for the Dirichlet problem. In Sections 3.1-3.4, the theoretical analysis is presented. In Section 3.5, the computational algorithm and the computational complexity of the approach are given.
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3.1 Steepest Descent Path for Green's Function
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Steepest Descent Path for Green's Function
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The Green's function is
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9 (x, z) =
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1 dkzeikxlxleikzz~
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Later on x ----7 x - x', Z ----7 Z - z'. The integration contour is on the real k z axis. Next we make transformation to complex angle.
k z = k cos 0: k x = ksino:
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= k cos( 0:' + iO:") = k~ + ik~
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(5.3.3) (5.3.4)
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Balancing real and imaginary parts
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= k cos 0:' cosh 0:" = - k sin 0:' sinh 0:"
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(5.3.5) (5.3.6)
Similarly for k x
+ ik~ = k sin 0:' cosh 0:" + ik cos 0:' sinh 0:"
We also have dk z = -k sin 0: do:. The original contour integral is now in the complex noted the Sommerfeld integration path (SIP)
plane and is de-
9 (x, z) =
do: eiklxlsina+ikzcosa
The Sommerfeld integration path extends from 0: = 0 + ioo to 0: = 0 along the imaginary axis. It then goes from 0: = 0 to 0: = 7r along the real 0: axis. Finally it goes from 0: = 7r to 0: = 7r - ioo (Fig. 5.3.1). Since Ixl ~ Izl, the dominant exponential term is
eiklxlsina = ei(ksina'coshal/+iklxlcosa'sinhal/)
= eiklxl-klxlcosa'sinhal/
The steepest descent path
is defined by Re(sin 0:)
so that sin 0:' cosh
Figure 5.3.1 Steepest descent path and Sommerfeld integration path (SIP).
The saddle point is at a = 7f /2. The steepest descent path r is a contour of constant phase for the dominant exponential term, the amplitude of which decreases rapidly away from the saddle point. In the vicinity of (a', a") = (~, 0), by letting a' = ~ + (3, we have sin( ~ + {3) cosh a" = 1 or cos {3 cosh a" = 1. For small {3 and a", the equation becomes
(1 - ~2) (1 + (0';)2) = 1, giving a" =
-{3. Thus the steepest
descent path r has slope = -1 in the vicinity of (a', a") = (~, 0) as shown in Fig. 5.3.1. Since the major contribution comes from the vicinity of a' = ~, the integration can be discretized as follows:
(x, z)
47f Jr . Q
da eiklxlsina+ikzcosa
= _2_ '""'" eiklxlsinaq+ikzcosaq.6.a q 47f ~
If we use the transformation
= 7f - a and then let
anew --.
a, we have
(x, z)
47f Jr . Q 4~
3.1 Steepest Descent Path for Green's Function
where Q is the number of angles. This means that the sign of the cosine term in the exponent in (5.3.12) can be switched. Let BW be the bandwidth on either side of ci = l We sample evenly on the a' axis with interval ~a'. Let the number of angles Q be an odd integer.
2(BW) -1
. (sample evenly on real ai-axIs)
(5.3.14) (5.3.15)
." a q = a Iq + ~aq
a~ = ~ cosh a"
+ (q - l)~a'
(5.3.16) (5.3.17)
Note that a~ a~ > 0 for a~ < ~ and a~ < 0 for > ~. The selection of BW must be large enough to ensure that integrand becomes small enough. Note that BW depends on Ixi. When Ixi is small the contribution is from larger range of a. The worst case and the largest BW corresponds to Ixi = Xmin, where Xmin is the minimum separation in x. It is required that
= -- = -sin a~ cos f3 is an odd function of f3,