[Getdp] basic permant magnet question - please help
Olivier Castany
castany at quatramaran.ens.fr
Sun Mar 18 00:56:55 CET 2007
> Equation {
> Galerkin { [nu[] * Dof{ Curl a}, {Curl a}]; In Domain;
> Jacobian JVol; Integration I1; }
> Galerkin { [hc[], {Curl a}; In Domain_M; Jacobian
> JVol; Integration I1;}
> }
>
> How does {hc[], {Curl a}} come into the equation? I believe that hc is
> Coercitivity of the magnet?
The relation between H and B in a magnet is complicated. If the magnet
is used in a narrow enough (H,B) domain around a working point, the
relation can be assumed to be linear.
Examples of working points are : (H,B) = (-Hc,0), (0,Br) or whatever
fits the physical situation.
In the case of a working point (H,B) = (-Hc,0), the relation in the
magnet is linearized as : B = mu * (H + Hc)
(outside the volume of the magnet, mu is different and there is no Hc)
The Maxwell equation without free current is : rot(H) = 0
which implies : rot(B/mu - Hc) = 0
The divergence-free field B is written as : B = rot(A).
If A is a solution, it satisfies :
\int_Domain rot(rot(A)/mu - Hc) * A' = 0, for all A'
Boundary conditions must be taken into account. Suppose the border of
Domain is made of two parts S1 and S2 with the following boundary
conditions :
- values of A are imposed on S1
- H // n on S2 (n = normal to S2)
Let us call AA the set of all possible A' with vanishing value on S1.
>From the previous equation, we deduce :
\int_Domain (rot(A)/mu - Hc) * rot(A') = 0, for all A' in AA
(the relation "rot(H) * A' = H * rot(A') + div(H x A')" has been used)
The two terms are written in GetDP in the following way :
Galerkin { [ 1/mu[] * Dof{d A} , {d A} ] ; ... }
Galerkin { [ - Hc[] , {d A} ] ; ... }
--
O.C.