Re: [femm] Diamagnetism

[David Meeker <dcm3c@xxxxxxxxxxxxx>]

Keith Gregory wrote:
> Diamagnetic and paramagnetic effects are difficult to measure 
> experimentally and are just as difficult to model. I would concur with 
> Dave, that a very fine mesh would be required and it may be that FEMM 
> doesn't work to sufficient significant figures to deal with the problem. 
> What precision do you use Dave?
>
> Keith.
All floating point values in femm are stored in double precision.  It wouldn't
be that hard to change things to use more precision, but double has seemed
to be adequate before.

In the conjugate gradient linear equation solver used for magnetostatic problems,
the equations that it solves have the form: A x = b
where A is a positive (perhaps semi)definite matrix and x and b are vectors,
with x the vector to be determined.  Let residual  r be defined as:  r= b
- A x.   Internally, the error metric used is: Error = Sqrt[(r.r)/(b.b)]
where the dot denotes inner product.  The solver keeps on going until the
error is reduced to less than the amount prescribed in the "solver precision"
box of the "problem definition"  The default is 10^(-8), which means that
the RMS error for any particular nodal value is one part in 100,000,000.
 The program will let you specify up to 10^(-16).

Nonlinear problems have slightly less precision.  The outer Newton iteration
must, of necessity, have less precision than the linear solver used inside
it (otherwise, things would never converge). The program's tolerance for
the nonlinear "outer loop" is:
Sqrt[ (x[k]-x[k-1]).(x[k]-x[k-1])/(x[k].x[k])] < 100*(solver precision)

For time harmonic problems, a similar solver criteria is used, except that
the residual that you use as a measure of convergence is really sort of a
pseudo-residual....we'll get into that some other time.

If you specify a high precision, the solver precision is probably not an
issue.  However, the discretization error from the mesh itself can be relatively
large--it many not be possible to mesh finely enough to get the level of
accuracy that would be required to resolve the effects of diamagnetic materials.

Dave.

--
David Meeker