Re: [femm] Force calculation on an axisymmetric model

[David Meeker <dmeeker@xxxxxxxxxxxxxxxxx>]

Agnaldo Souza Pereira wrote:

> I have also found some difference in calculating force
> in axisymetric model: when calculating repulsion forces
> between two cylindrical blocks, one, a superconductor in
> mixed state, and the other, a NdFeB magnet, I've found
> different results using FEMM and student's Quickfield.
> I don't know how to explain this difference.
>
> Agnaldo.

I think that Quickfield correctly implements the integral for Maxwell's
stress tensor, but only a rather limited number of node points are allowed
in the student version (between 250 and 500, depending on which version you
are using). A lot of elements are generally required to get "good" force
calculations. Lots of elements are required because you are integrating
basically B^2 over a contour. Now, flux density B is an order less accurate
than vector potential A, which is the quantity that's actually obtained in
the finite element solution--B is basically the derivative (to be exact, the
curl) of A. In computing force, squaring B exacerbates the problem by
doubling the error again. The only way to decrease the error is to add more
elements.

If anyone is interested, I can post (in the e-group "files" section) the
files that model an axisymmetric problem two cylindrical magnets acting upon
one another. If the magnets are relatively far away from one another, there
is a closed-form expression for force that one can derive by idealizing the
magnets as simple dipoles. This analytical result can be used as a sanity
check for numerical results. To test things out, I modeled 2 cylindrical
magnets that were 0.5" long and 0.5" in diameter, with the faces of the
magnets separated by 3". I assumed that the magnets had unit relative
permeability and a Hc of 10^6 A/m. The results for force were:

Force from analytical expression derived from a dipole approximation:
0.02486 N
Force from femm 3.0 using a mesh of about 6000 nodes: 0.02469 N
Force from Amperes using a 1000 element mesh: 0.025067 N
Force from Quickfield 3.4a using a 499 node mesh: 0.03883 N

Now, both the femm and amperes solutions are within 1% of the analytical
solution, whereas the student quickfield solution comes in about 56% high.
Again, I'd guess that the difference arises solely from coarser the mesh
density in the quickfield solution.

At any rate a good way to get an idea of how many digits you can believe in
either a femm or quickfield force computation is to integrate the stress on
different contours around the part that you are interested in. In an exact
solution, the results should be identical. However, the results from a
finite element solution, which is only approximate, will be slightly
different. The differences between contours give you an idea of the order
of the error in the calculations.

Dave.


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