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

Alexandre Kampouris wrote:

> > 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?

Yes, u1and u2 are complex variables.

> 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

yeah, you are qiute right.

> 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,

I am now computing the E field on a rectangular patch (self term).

> 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.)

I am sorry to tell you that I am not familiar with the programme of
NEC2, I use it directly. So I have no idea on the capabilities of INTX
in such a short time.

> 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.

Yes, I will try to computing the integration by series expansions.

> 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

Thank you ,Alexandre, you are so kind and so warmhearted.--

Mailing Address:
              Mr. Xinjun Zhang
Dept. of Microwave Telecommunication Engineering
              Xidian University
              Xi'an City,710071
                   China
Email: zxj_at_mwp.xidian.edu.cn
Received on Wed Oct 14 1998 - 18:11:50 EDT