# How to calculate eddy current with NDSolve

I want to calculate the eddy currents inside a plane plate b,h, which moves with constant velocity v nearby a fixed magnet(radius r, magnetic flux normal to the plate)

My attempt

werte = {b = 1 , h = 1 , r = 0.25, v = 1, T = 0.01};
tsim = b/2/v;
NDSolveValue[{Laplacian[\[Phi][t, x, y], {x, y}] ==
Derivative[1 , 0, 0 ][\[Phi]][t, x, y]
,
\[Phi][0, x, y] == 0 (* initial condition *)
,
DirichletCondition[0, True]  (* if necessary? *)
,
(* magnetic flux normal to the magnet *)
\[Phi][t, x, y] ==If[(v t + x - b/2)^2 + y^2 <=r^2, 1 - Exp[-t/T], 0]}, \[Phi]
,
{x,-b/2,b/2}, {y, -h/2, h/2}, {t, 0, tsim},
Method -> {"MethodOfLines", "TemporalVariable" -> t, "SpatialDiscretization" -> {"FiniteElement"}}
]


results in this error message

NDSolveValue::overdet: There are fewer dependent variables, {\[Phi][t,x,y]}, than equations, so the system is overdetermined.


Is there any experinece solving such physical problems?

• The first boundary describes the influence of the local magnet. – Ulrich Neumann Dec 1 '17 at 17:38
• The second boundary is the initial condition, which is needed for the method of lines. The third boundary sets phi==0 on the boundary of the rectangular region. – Ulrich Neumann Dec 1 '17 at 17:41

You have two equations that are defined for all values of t, x, and y, rather than just on the boundaries:
Laplacian[\[Phi][t, x, y], {x, y}] == Derivative[1 , 0, 0 ][\[Phi]][t, x, y]

(Note also that the function $\Phi$ defined in the second equation does not satisfy the PDE given in the first equation, since inside the moving circular region $\nabla^2 \Phi = 0$ but $\dot{\Phi} \neq 0$.)