[Getdp] [GetDP] Calculation of an Integral quantity

[Matt Koch]

Hello Olivier,

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

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. Thus, I would have current  
running in the z-direction (around the coil), and the resulting  
magnetic field would then be in the x-direction and y-direction, and  
at x=0, it would be in the y-direction only (the coil axis). When I  
run your problems (both torus and torus3D), I get the magnetic field  
in the z-direction, though. So, I must be completely misunderstanding  
what the problem is about.

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. 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

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?  
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?

3) Function Spaces

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"

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

Thus, as far as I can tell, neither quantity a2 nor quantity B2 has a  
function space that covers "Coil". Yet, in Quantity/Name, you  
integrate over "Coil" (using the "In Coil" statement, I assume). What  
am I missing?

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

Thanks,

Matt Koch


----- Message from castany at quatramaran.ens.fr ---------
     Date: Sat, 5 May 2007 17:59:16 +0200
     From: Olivier Castany <castany at quatramaran.ens.fr>
Reply-To: Olivier Castany <castany at quatramaran.ens.fr>
  Subject: Re: [Getdp] [GetDP] Calculation of an Integral quantity
       To: getdp at geuz.org


>> thank you kindly for your reply. I took some time this morning to
>> review your "Torus3D" example, trying to see if I could figure out
>> what X vs. XS is. I noticed a couple of things that are not clear to me:
>
> Hello,
>
> first, a warning : the file torus3D.pro calculates two different
> and independent things :
>
> 1) FEM Galerkin method exactly the same as in torus.pro (did
> you look at it ?)
>
> 2) Integral quantites
>
> I could have tried to make only the 2nd calculation, but after reading
> Bernhard Kubicek's comments in Tower3DBiot, I thought it would produce
> error messages if there were no "Equation" term, so I wrote 1) and 2)
> in the same file. I have no time now to try and see if it is possible to
> calculate only part 2).
>
> (a warning about Tower3DBiot : I think Bernhard made a mistake and
> should use ZS instead of Z ; in his example, however, this does not make
> a huge difference ; I've mailed him)
>
>> 1) In FunctionSpace, you define function spaces on "D_tot" and
>> "CoilSection", in Formulation, you define an Equation in these, but
>> you also use that FunctionSpace
>                ^^^^
> No : there are 3 function spaces : potentiel_vecteur_jauge,
> potentiel_vecteur, champ_mag
>
>> to define two Quantities that work on "Coil", namely "a2" and "B2".
>                                 ^^^^
> I don't understand that word in this context.
>
>> How can a Quantity be defined on a
>> region ("Coil") when it has no function space there?
>
> "a2" belongs to the function space "potentiel_vecteur" => a2 lives on
> D_tot (with other words : a2 is defined on D_tot)
>
> The term "In Coil" is the integration domain of the source point :
>
> a2( (X,Y,Z) \in D_tot ) = \int_{(XS,YS,ZS) \ in Coil} mu0 *
> Laplace[]{3D} * J_s_source[]
>
> (Laplace[]{3D} is a function of (X,XS,Y,YS,Z,ZS) and J_s_source[] is a
> function of (XS,YS,ZS,NormalSource))
>
>> 2) I did note your use of X vs XS, but I still do not understand. To
>> begin with, you seem to use J_s only, defined on X, and not
>> J_s_source, defined on XS. So what is the point of working with XS?
>
> I do use "J_s_source" ! See the definition of the integral quantities.
> Example :
>
>       { Name a2 ; Type Integral ; NameOfSpace potentiel_vecteur ;
>         [ mu0 *  Laplace[]{3D} * J_s_source[] ];
>         In Coil ; Jacobian JSur ; Integration I ; }
>
>> Does XS only become meaningful if I use certain functions?
>
> I think it is only meaningful in an integral quantity expression.
>
>> Can I define where I want my XS to be? If so, how?
>
> See my explaination of the term "In Coil" above.
>
>> 3) The expression "Q(X,...) = \int_{XS,...} q(X,XS,...)" you mention
>> below is pretty much what I am after, but I did not see where or how
>> you implemented this in "Torus3D"?
>
> In the definition of the integral quantities. See above.
>
> Regards,
>
> O.C.
>
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----- End message from castany at quatramaran.ens.fr -----