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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?

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  • $\begingroup$ The first boundary describes the influence of the local magnet. $\endgroup$ – Ulrich Neumann Dec 1 '17 at 17:38
  • $\begingroup$ 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. $\endgroup$ – Ulrich Neumann Dec 1 '17 at 17:41
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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]

\[Phi][t, x, y] == If[(v t + x - b/2)^2 + y^2 <=r^2, 1 - Exp[-t/T], 0] 

Since you have two equations defined on the interior, and only one unknown, the system is overdetermined, which is why Mathematica is giving you this error message.

(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 $\dot{\Phi} \neq 0$.)

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  • $\begingroup$ That's it. Thank you $\endgroup$ – Ulrich Neumann Dec 2 '17 at 11:14

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