[Getdp] [GetDP] Calculation of an Integral quantity

[Olivier Castany]

> thanks for your quick reply, again. I have been trying to follow your  
> explanations, but I must confess, I am more confused then ever.

Sorry !

> 1) To begin with, it is my (probably false) understanding that the  
> y-axis is the symmetry axis and that the coil cross section is rotated  
> around that axis to give the coil body. 

Yes.

> Thus, I would have current  
> running in the z-direction (around the coil), and the resulting  

No. Run torus3D.pro. One of the output is J_s : this is the surface 
current on the coil. 

> 2) X vs. XS
> 
> It seems you are using a Green's function in your integral. Per GetDP  
> manual, these depend on X and XS. You also use your home-made function  
> J_s_source and then form the integral over a product of the two. This  
> now makes me believe that XS is the coordinate of integration, while X  
> is a general coordinate that remains.

Yes

> Would it therefore be fair to  
> say that the use of Type Integral in Quantity/Name with "In Domain"  
> presupposes the following:
> 
> F(X) = int(Domain) f(X,XS) dXS, where XS is in Domain

Yes. In my former message, I was writing it that way :

a2( (X,Y,Z) \in D_tot ) = \int_{(XS,YS,ZS) \in Coil} mu0 *
Laplace[]{3D} * J_s_source[] 

> If that were so, it would indicate to me that I would not have to use  
> Green's functions, but could integrate over my own functions only? 

Yes. You can write :

[ mu0 * 1/(4*Pi*Sqrt[($X-$XS)^2+($Y-$YS)^2+($Z-$ZS)^2]) * J_s_source[] ];

instead of : [ mu0 *  Laplace[]{3D} * J_s_source[] ];

The result is the same but it takes more time.

> Could you possibly confirm this? It is a lot of work to attempt this  
> in the types of problems I am interested in, so I need to be  
> reasonably sure I got the idea right. Furthermore, could I use a  
> quantity in this integral that is defined in another Name entry and  
> Function Space, but on the same domain?

I don't really understand the question. I would tend to answer yes.

Remark : I think these integral quantities are only calculated in the 
postprocessing. You can not apply a differential operator to them in 
the postprocessing. 

> 3) Function Spaces

In the following, I think you're mistaking the "domain of definition of 
a function f" and the "function space" to which f belongs.

Remember : 
- a function space is a set of functions (with a vector or affine space 
structure)
- the support of a function is its domain of definition

> As far as I can tell, the following is true about the function spaces  
> in your problem:
> 
> "D_tot" is Region 66 & 67
> "CoilSection" is Region 63
> "Coil" is Region 79
> 
> a) function space potentiel_vecteur_jauge supports "D_tot & CoilSection"
> b) function space potentiel_vecteur supports "D_tot "
> c) function space champ_mag supports "D_tot"

I think you can't say it that way : a function space "supports" nothing 
or no one.

I would say : "the support (i.e. domain of definition) of the functions 
which belong to potentiel_vecteur is D_tot"

[More precisely, in GetDP, the "Support" is not really the domain of 
definition, but rather the set of simplices for which the equation is 
built. Example : Support Region[{D_tot,CoilSection}] -> in terms of 
geometry, CoilSection is inculded in D_tot, however, D_tot are surfaces 
and CoilSection are lines, so they must be both mentionned]

[some people on the list know that far better than me... if I'm wrong, 
please, correct me]

> a) quantity a1 is defined on function space potentiel_vecteur_jauge
> b) quantity a2 is defined on function space potentiel_vecteur
> c) quantity B2 is defined on function space champ_mag

I would not say "on". a2 is an element of the function space 
potentiel_vecteur" (and as a consequence, its support is D_tot).

> Thus, as far as I can tell, neither quantity a2 nor quantity B2 has a  
> function space that covers "Coil".

They cannot "have a function space" (they belong to it).
a2 and B2 have almost nothing to do with "Coil". They are definied on 
D_tot. 

The values of a2 and B2 are calculated throught an integration where the 
integration point RS scans the surface "Coil".

> Yet, in Quantity/Name, you  
> integrate over "Coil" (using the "In Coil" statement, I assume). What  
> am I missing?

I think you're mistaking "domain of definition of a function" and 
"function space".

> Sorry for being so dense, but as I get older, I feel my thinking  
> capacity rapidly drain away ...

Tut, tut, tut ... I won't believe you !