Re: NEC-LIST: Quick method on Complex Exponential Integration

At 10:34 PM 10/12/98 +0800, Jin Mouping wrote:

>There is a Complex Exponential Integration written in language
>MATHEMATICA 3.0. It is as following,
>
> Integrate[Exp[-I k u]/u,{u,u1,u2}] (1)
>
>where u1,u2 are complex variables.

I don't have the paper you quote on hand. But are you sure that the
integration variable U and bounds are complex?

If I'm not speaking nonsense (I looked at this quite a while back, so
please forgive me), the way it is written, this integral has an
analytical solution, expressed in the form of Sine and Cosine
Integrals. (Si and Ci), for which there are polynomial approximations
available from NETLIB. Just decompose it in a sin(x)/x and cos(x)/x
term, using Euler's identities. But things get more complicated in the
geometry of NEC2, and AFAIR there's no analytical solution after the
change of variables U = sqrt(rho^2+(z-z')^2).

Your expression looks a lot like the one computing the E field of a
current element at a certain range, computed by the NEC2 routines INTX
using numerical integration, routine which is called in turn by EKSCX.

Look under the writeup for the routine INTX, pp 209-215, in the NEC2
"program description - CODE", available on the WEB at

http://www.dec.tis.net/~richesop/nec/
or
http://www.qsl.net/wb6tpu/

INTX computes the contribution of the constant term (A) of the
expansion of the current distribution, the contributions of the sine
(B) and cosine (C) basis functions being computable using closed form
expressions in EKSCX. (These are defined on p. 14 of part I.)

The question of the treatment of the singularity when U-->0 is also
covered on page 211. This occurs when the calculation of the effect of
the current distribution on its own segment is attempted.

Adaptive Romberg integration is used, which is briefly described, and
also discussed in good numerical analysis books.

Since the B and C terms are quickly computable, the use of an
iterative technique for the A term *might* take a large proportion of
the fill time of the interaction matrix. I want to test one of these
days the relative speeds of these calculations, and maybe create some
sort of series approximation to accelerate this part a bit. But then,
if there was a faster way to do this, the very bright people at LLNL
would have used it. :-) So why try to improve their excellent work?

Alexandre

>Reference
>
>J.H.Richmond and N.H.Geary "Mutual Impedance of Nonplanar -Skew
>Sinusoidal Dipoles", IEEE Trans. on AP. Vol.23 No.3 May 1975 pp.412
Received on Wed Oct 14 1998 - 10:22:29 EDT