Re: [femm] discontinuities in stress tensor mask in thin air

[David Meeker <dmeeker@xxxxxxxx>]

This is sort of an interesting problem, because it is possible to write 
down a simple analytical solution for the force between two magnets if 
they are idealized as dipoles (a good approximation if the magnets are 
fairly far away from one another). This result can then be used to 
sanity-check the results from femm. The results from femm shouldn't be 
exactly the same as the analyitcal solution (because the magnets aren't 
really point dipoles), but the analytical result should be pretty close 
to the finite element one.

I've attached a zip file containing a two sample problems looking at the 
on-axis interaction between two magnets in free space. One is 2D 
planar, and the other is axisymmetric. There is also a Mathcad worksheet 
(saved as Version 7) that has the analytical solution for the force, as 
well as a the force results for various mesh densities. There is also a 
pdf printout of the Mathcad worksheet.

A couple of things to note: The interaction forces are really quite 
weak, which makes it harder to get good force results. The force 
between dipoles has a (1/distance)^4 dependency for axisymmetric 
problems, and a (1/distance^3) dependency for 2D planar problems. 
Second, in this case, there is an obviously "good" contour for a stress 
tensor surface integral--one that pretty much follows the line that 
bisects the two magnets. 

Anyhow, in both cases, things converge to solutions that are within a 
couple percent of the of the dipole approximation with increasing mesh 
density. Somewhat amusingly, the stress tensor surface integral has a 
faster rate of convergence in this case. I think that what is going on 
is that I'm using the "best possible" contour for the stress tensor 
surface integral, whereas the volume integral gets somewhat polluted by 
less accurate field values near the surface of the permanent magnets 
(where the field changes its strength and direction relatively 
suddenly). I should probably head-scratch a little bit more about how 
the program computes the weighting function used for the volume stress 
tensor integration (i.e. use one that is explicitly derived from a 
posteriori error estimates or something so that the surfaces of the PMs 
are avoided, perhaps like in the McFee paper).

Dave.
--
David Meeker
email: dmeeker@xxxxxxxx
www: http://femm.berlios.de/dmeeker

Attachment: zip00048.zip
Description: Zip archive