Re: [femm] (Femm 3.3a works!) coenergy vs. stress tensor calculations of magnetic force

[David Meeker <dmeeker@xxxxxxxx>]

James Rabchuk wrote:

> (I do have a question for Dave, here. I shouldn't be lazy, and just 
> look at the code, but I'll ask the source. In the implementation of 
> the weighted stress tensor, are you doing a volume integral over the 
> entire air region of the model? That seems to be what's implied from 
> your description and the reference you posted. Also, you mentioned 
> that you weight elements in more tightly meshed regions more than you 
> do elsewhere. I wasn't quite sure how that would work. If that applies 
> over a single contour, then don't you need to keep the integration 
> over the entire contour at the same weight, in order to get the proper 
> "sum" of momentum flux across the boundary?) 

Yes, the "weighted stress tensor" does integrate over the entire air 
region in the model. The program "weights" each element by solving an 
extra PDE where the value of potential in that PDE is 1 over the 
selected region and 0 over non-air regions and boundaries. The 
direction of the gradient of the resulting solution gives the "normal" 
direction needed in computing the stress tensor, and the amplitude of 
the gradient is sort of the weighting given to that element. The mesh 
density is used to assign "permeabilities" in the extra PDE that is 
solved, i.e. regions with a dense mesh have a high "permeability", so 
that more of the contours flow through that region rather than parallel, 
less highly meshed regions.

A good way of understanding what the integral is doing is to consider 
the paper by Arkkio:

Determination of forces and linearized parameters of radial active 
magnetic bearings by finite element technique
Autila, M.; Lantto, E.; Arkkio, A.;
Magnetics, IEEE Transactions on , Volume: 34 Issue: 3 , May 1998
Page(s): 684 -694

In this paper, the air gap is an annular region. They essentially 
average all of the concentric paths for the stress tensor that you could 
draw in the gap. The "weighted stress tensor" is just a generalization 
of this idea to regions that have a more complicated geometry--in 
Arkkio's case, one can write down a valid weighting function by 
inspection (i.e. weighting function w = (ro-r)/(ro-ri) where ri is the 
rotor's radius and ro is the radius of the pole face). This isn't 
generally possible with more complex geometries, so FEMM has to solve an 
extra PDE to compute a valid weighting function.

Dave.

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