Abstract
We derive and analyze a mathematical model for induction hardening. We assume a nonlinear relation between the magnetic field and the magnetic induction field. For the electromagnetic part, we use the vector–scalar potential formulation.The coupling between the electromagnetic and the thermal part is provided through the temperature-dependent electric conductivity and the joule heating term, the most crucial element, considering the mathematical analysis of the model. It acts as a source of heat in the thermal part and leads to the increase in temperature. Therefore, in order to be able to control it, we apply a truncation function.Using Rothe’s method, we prove the existence of a global solution to the whole system. The nonlinearity in the electromagnetic part is handled by the theory of monotone operators. To supplement our theoretical results we provide a numerical simulation using real physical constants.
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