Re: discontinuities in stress tensor mask in thin air

["mikeshonle" <mike@xxxxxxxxxxxxx>]

Hi, Dave, thanks for your detailed analysis and response. I have 
been doing some more experimenting, and have come up with the 
following:
First, I tried to replicate your results for the axisymmetric case, 
and I got exactly the same number using the line integral method but 
doing the WST method, I got .65 instead of .59.

Also, I went back and used the line integral in my test cases, and 
got much more consistant (vs. changing mesh spacing) results (within 
about 2% instead of 30% !). I also re-ran my moving magnet 
experiment and it was completely monotonic and almost completely 
monotonic in the derivative as well.

Let me know if you would like my project file in case you would have 
a chance to take a look and see if I'm doing anything wrong. As a 
test, I've been varying the mesh spacing in the region just outside 
the region that has the magnets, and that changes the computed force 
by a lot (as well as showing the "diffraction" of the force tensor 
lines).

Also, B.T.W, would you happen to know what the density of NIB 37 is?

Thanks so much for you help, and let me know if you need any more 
information.

Thanks,
-Mike



--- In femm@xxxxxxxxxxxxxxx, David Meeker <dmeeker@xxxx> wrote:
> 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@xxxx
> www: http://femm.berlios.de/dmeeker