Existence of a weak solution to a nonlinear induction hardening problem with Leblond–Devaux model for a steel workpiece

• Published in: Communications in nonlinear science & numerical simulation 2022-04, Vol.107, p.106156, Article 106156
• Main Authors: Le, Van Chien, Slodička, Marián, Van Bockstal, Karel
• Format: Article
• Language: English
• Subjects: Conduction heating; Conductive heat transfer; Eddy currents; Electromagnetic induction; Heat transfer; Induction hardening; Magnetic fields; Magnetic induction; Mathematical models; Monotone operators; Nonlinear equations; Nonlinear induction hardening; Nonlinear systems; Nonlinearity; Ohmic dissipation; Operators (mathematics); Phase transition in steel; Phase transitions; Potential operators; Resistance heating; Steel; Thermodynamics; Workpieces
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2021.106156

In the present paper, we investigate a mathematical model for an induction hardening process of a steel workpiece. We describe the electromagnetic process by the eddy current equations, which are coupled with the energy balance in the workpiece via the heat equation and the kinetic phase transition. The nonlinearities of the system are caused by a nonlinear relation between the magnetic induction and the magnetic field, together with a nonlinear heat conduction and Joule heating. We propose a time discretization for the variational system, and perform some a priori estimates for discrete solutions. By aid of Rothe’s method and the theory of monotone operators, we show the convergence of the proposed scheme and the existence of a weak solution to the variational problems. •A mathematical model for a nonlinear induction hardening process of a steel workpiece is investigated.•The electromagnetic process is described by Maxwell’s equations coupled with the heat equation and the Leblond–Devaux model.•A convergent time discretization scheme based on backward Euler’s method is proposed.•The existence of a weak solution to the variational problem is addressed.