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

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.