Re: Re: Etesla6 math questions

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Jim Lux by way of Terry Fritz <teslalist-at-qwest-dot-net>" <jimlux-at-earthlink-dot-net>

Actually, it's pretty darn simple...

In fancy terms, what you're trying to do is iteratively solve an equation
which says that the second partial derivative is equal to zero...


However, in simple terms, what it means is that things have to vary
smoothly, and it turns out that for some very useful cases (electric fields,
heat flow, etc.) you can set up a big matrix of little cells, guess at the
initial values, and then iterate each cell as follows:

Calculate the average of all the neighbor cells (say you've got a 2 d matrix
cell(i,j)... avg= 1/4 *(cell(i,j+1)+cell(i,j-1)+cell(i+1,j)+cell(i-1,j))
Replace the value of cell(i) by alpha*cell(i)+ (1-alpha)*average of
neighbors.

Repeat for all the cells
Set the cells at the boundaries (the tesla coil, in this case) back to their
original values (if you changed them)
Repeat the process, a bunch of times..

(The heat flow analogy is this:  The heat flowing into and out of each cell
has to sum to zero)

The bigger alpha is, the slower it changes, but the better(higher quality)
the eventual solition is.  The smaller the cells are, the better the
solution is (it's calculus, ideally you want infinitesimally small chunks)

In a 3 d problem, there are potentially many, many, many cells, which can
take substantial computer time. If there are symmetries in the problem, you
can take advantage of them.. In the Etesla case, it's symmetrical around the
vertical axis, so you only need to calculate a single 2-D "slice" from the
center out, allowing for the fact that the "neighbor cell" on the inside is
smaller than the "neighbor cell" on the outside, when calculating averages..

Where the field is changing fast, you want smaller cells, so clever programs
use "variable grid density" to compute more detail where it's changing,  and
less where it's not.

In an old FORTRAN IV textbook by McCracken (70's), there's a fine example of
computing temperature distributions around a square pipe using this FEM
technique, with a number of nice "line printer plots" of the output.

One can also predict weather by this means using a somewhat more complicated
function (Navier Stokes) as the basis, rather than the simpler second
partial derivative=0.




----- Original Message -----
From: "Tesla list" <tesla-at-pupman-dot-com>
To: <tesla-at-pupman-dot-com>
Sent: Friday, January 31, 2003 7:01 PM
Subject: re: Re: Etesla6 math questions


 > Original poster: "Peter Lawrence by way of Terry Fritz
<teslalist-at-qwest-dot-net>" <Peter.Lawrence-at-Sun.COM>
 >
 > Terry,
 >        here is the part I did not understand the first time around and
still
 > don't understand:
 >
 >
 >  > The charge density certainly is non-uniform.  But that is controlled by
the
 >  > shape and E-Tesla does the "E-field relaxation thing" to find the
E-field
 >  > (voltages) around the coil.
 >  >
 >  > When we do the relaxation matrix, the charge density on the parts works
 >  > out!!  Sharp edges get high fields do to high densities.  Smooth edges
get
 >  > low fields do to low densities.  The "relaxation" does this field
density
 >  > adjustment for us.
 >
 >
 > where can I read about the "relaxation thing", this seems to be the crux
of
 > the matter. It seems to be a well known numerical analysis of fields
 > technique, but I've never taken more that an introductory numerical
analysis
 > class and have not heard of this, and it might be a bit more involved than
 > you could describe in email...
 >
 >
 > thanks,
 > Pete Lawrence.
 >
 >