[Getdp] numerical saturation effect - 2

[Christophe Geuzaine]

m.fenner at gmx.net wrote:

> Dear Christophe,
> 
> first of all thanks for your comments on the wiki examples. I am
> going to test your suggestions asap.
> 
> 
> But here is another problem: a while ago I asked about the saturation
> of the magnetic flux in the core of a coil. I investigated the
> problem a little further:
> 
> 1. The physics says that the magnetisation M is dependent on the
> shape of the body of the magnetisable material. M is then given by:
> 
> M = chi / (1 + N * chi ) * H_ext
> 
> where chi is the magn. susceptibility and N is called demagnetisation
> factor, with 0 < N < 1. Exact values exist for simple geometries like
> a sphere: N = 1/3.
> 
> With chi -> infinity obviously M saturates. For the sphere you get M
> = 3 * H_ext.
> 
> The total flux is then B =  mu_0 (H_ext + M) = mu_0 * 4 * H_ext.
> Compared to the vacuum case the flux gain is 4.
> 
> 2. When I solve the problem of a small sphere in a long coil for
> varying mu_r I find that the value of {d a} (which should be the
> magnetic flux) has a gain of ~3!
> 
> This suggests that {d a} represents only the magnetisation. But from
> the formulation with
> 
> nu [ Sphere ]  = 1. / (murSphere * mu0);
> 
> I would not expect that. Do you have an explanation? (see also the
> attached files)

Matthias - IIRC, the exact solution inside the sphere is

b = (3 * b_0) / (1 + 2/mu_r)

i.e., the gain tends to 3 for large values of mu_r.


> 
> 3. One experiment to measure chi employs a torus with a coil wound
> around it to get rid of the demagnetisation factor. I wanted to model
> this problem. Though my .geo and .pro files seem to be okay (The
> current is wound around a torus anyway) I get a
> 
> Warning   : Null right hand side in linear system .... Solver    :
> Generalized Minimum RESidual (GMRES) 1  0.0000000e+00  NaN
> 
> I do get a solution in the very same geometry when I define a current
> density with a z-component (i.e. phi-component in zylindrical
> coordinates) in the coil region. Do you have any idea what goes wrong
> here? (again files are attached)

You cannot specify a current density in the x-y plane with this 2D 
vector potential formulation: the basic assumption used for establishing 
this formulation is precisely that the current density only has a 
component in the z-direction.

Best,

Christophe

-- 
Christophe Geuzaine
Applied and Computational Mathematics, Caltech
geuzaine at acm.caltech.edu - http://geuz.org