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

Induction heating is a frequently used technology in both fundamental and applied research. It is heavily exploited in the industry for processing materials by heat treatments. In addition, it is viewed as a promising tool in medicine, particularly as a part of therapeutic strategies for treating cancer diseases. Thus, in order to optimize (i.e., enhance and tune) the performance of the induction heating process, several aspects must be considered, including the design of the magnetic coils, features of the magnetic fields applied, coupling of magnetic and thermal fields, and the material’s characteristics. To tackle this complex problem, numerical mathematical models are often used. The results of which can help in understanding the role of the various parameters on the performance of the induction heating. Here, we present an alternative mathematical approach to solve the induction heating problem using Fredholm integral equations of the second kind with a singular kernel. To reduce the computation time, the Nyström method has been adopted. As the kernel function shows a singularity, a singularity subtraction has been involved in the developed mathematical procedure. Furthermore, the error features of the Nyström method with the singularity subtraction have been described, and convergence conditions of the proposed computational algorithm have been thoroughly identified. Although special conditions for the kernel function and the integration rule are needed, the method shows lower computing times, competing well with those of traditional finite-element based routines. The applicability of the developed methodology is demonstrated for the simulation of induction heating the body of a metal object.

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A Formulation for a Nonlinear Axisymmetric Magneto-Heat Coupling Problem with an Unknown Nonlocal Boundary Condition

Wang

Zhang

Kang

2023

Computational Methods in Applied Mathematics

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This paper investigates a nonlinear axisymmetric magneto-heat coupling problem described by the quasi-static Maxwell’s equations and a heat equation. The coupling between them is provided through the temperature-dependent electric conductivity. The behavior of the material is defined by an anhysteretic 𝑯-𝑩 curve. The magnetic flux across a meridian section of the medium gives rise to the magnetic field equation with the unknown nonlocal boundary condition. We present a variational formulation for this coupling problem and prove its solvability in terms of the Rothe method. The nonlinearity is handled by the theory of monotone operators. We also suggest a discrete decoupled scheme to solve this problem by employing the finite element method and show some numerical results in the final section.

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

Cited by 3 publications

References 26 publications

A numerical scheme for solving an induction heating problem with moving non-magnetic conductor

A numerical scheme for solving an induction heating problem with moving non-magnetic conductor

Solving magnetic induction heating problem with multidimensional Fredholm integral equation methods: Alternative approach for optimization and evaluation of the process performance

A Formulation for a Nonlinear Axisymmetric Magneto-Heat Coupling Problem with an Unknown Nonlocal Boundary Condition

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