Analysis and numerical simulation of an induction–conduction model arising in steel heat treating

The goal of steel heat treating is to create a hard enough part over certain critical surfaces or volumes of the workpiece and at the same time keeping its ductility properties all over the rest of the workpiece.We consider a mathematical model for the description of the heating–cooling industrial process of a steel workpiece. This model consists of a nonlinear coupled partial differential system of equations involving the electric potential, the magnetic vector potential, the temperature, together with a system of ordinary differential equations for the steel phase fractions. Due to the different time scales related to the electric potential and the magnetic vector potential versus the temperature, we introduce the harmonic regime, leading to a new system of nonlinear PDEs. Finally, we have carried out some 2D numerical simulations of this heating–cooling industrial process.