Re: visualizing fields near charged objects

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Fri, 9 Feb 2001, John S. Denker wrote

> At 08:12 AM 2/9/01 -0500, Ludwik Kowalski wrote:
>
> > 1) Is it appropriate to perform simple averaging for usual
> > cylindrical cells (dr=const)?
>
> No.

and went on to give the essential theory.  To put it into practice
here's what I did (and, I assume, what others are doing as well.)

The key difference is the first derivative term in the cylindrical
Laplacian which involves r itself.  This means that we need a
radial coordinate which can be simply generated as an index along
the side of the "universe."

 27     ^ (+ r direction)
 26     |
 25     c
 24   d a b -> (+z direction
 23     e
 22
 21 <-(radial index)

The derivatives at the position of cell a are then numerically
implemented as follows:

Laplacian (V) = (c+e-2a) + (c-e)/(2*r_a) + (b+d-2a)

which corresponds respectively to the terms in John D's

> ... the Laplacian is ...
>  (d/dr)^2 + (1/r) (d/dr) + (d/dz)^2.

where r_a is the radial index for cell a.  Setting the Laplacian =
0 and solving for a we get

  a = (b+c+d+e)/4 + (c-e)/(2*r_a)

The only difference from the usual Cartesian implementation is the
last term which becomes progressively less importan (as one should
expect) as r gets bigger.

Once the calculation is finished, finding the Laplacian for those
cells whose value was specified as a part of the boundary
condition gives the charge density (times some appropriate
constant) in those cells.

John Mallinckrodt      mailto:ajm@csupomona.edu
Cal Poly Pomona          http://www.csupomona.edu/~ajm