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A new procedure for the study of the evolution of the solid phase in a moving solidifying ferromagnetic metal is proposed. The temperature distribution is controlled using eddy currents induced by a coil that covers partially the crucible surface and by cooling the rest of it, with an imposed crucible velocity. Analysis of the thermal field requires the solution of the time-periodic eddy-current problem coupled with the thermal diffusion problem. The nonlinearity of the B-H relation within the ferromagnetic material of the yoke and inside the solidified material cooled below the Curie point, as well as its dependence on temperature, are taken into consideration. Application of the polarization fixed point method allows the construction of an integral equation for eddy currents and always ensures the convergence of the iterative solution. At each time step, the heat diffusion equation is solved through a standard finite element technique, with the thermal conductivity and the specific heat capacity dependent on temperature.