Induction heating is present in diverse industrially-relevant processes such as heating for forging operation, quenching, hardening, and brazing. A typical industrial induction heating line consists of a series of spiral-shaped coils energised by high-frequency alternating current that generates a time-varying magnetic field, where a cylindrical bar travels through the coils in order to be heated by resistive losses of the eddy currents. When the billet reaches the end of the line it has the desired temperature and can then be forged/cut/quenched.

The complete mathematical model of this problem consists of Maxwell’s equations in the 3-dimensional domain (including the coils, the surrounding air and the billet) coupled with the heat equation inside the billet. On previous work (see [KMcK25], a PhD thesis jointly supervised by the two supervisors of this project) the two-dimensional problem was studied both theoretically, and numerically, and a novel finite element method was proposed, analysed, and tested for this problem. In particular, the method showed very good numerical results when compared to experimental results, even outperforming the code used at the time at AFRC.

The work described above was restricted to the computation of the heat within the billet, without the computation of the magnetic field on the air surrounding it, and, especially, not including the movement of the billet. So, in this project the goal is to continue modelling this phenomenon and getting the numerical schemes closer to the realistic setting. More precisely, this PhD project has the following main objectives:

1. Propose, analyse and test numerically a strategy to generate adaptive meshes.
   The main goal is to propose a strategy of refinement/coarsening of the computational mesh. The main motivation for this lies in the fact that, as time advances, a very thin thermal boundary layer is created (as a consequence of the “skin effect”, a thin layer of very high-intensity magnetic field). The tip of this layer moves in time, and the thickness of the layer also varies, from very thin when the heat just gets to the tip, to covering the whole width of the domain. This small features cannot be resolved by a uniform mesh, since this would imply a prohibitive computational cost. In fact, the thermal layer can be of the order of millimetres, in a billet of the order of1 meter in size, and thus a uniform computational mesh resolving these features would contain hundreds of millions of elements, which makes it completely untrack table. So, to reduce the computational complexity, and even to make this problem possible in “regular” computers, adaptive strategies are a must. The strategies will be based on previous work by one of the supervisors (e.g., [ABR17]) to deal with the space components of the problem, and, to deal with the possible multiple time scales, we will take inspiration in [CGS20] as a guide to refine/coarsen in time.

2. The extension of the finite element method to the more realistic case where the billet moves.
   This case is particularly challenging as it involves changing the physical setup in time. This can be tackled in two main ways. One alternative is to re-mesh the domain at each time step, which would make the computational strategy too expensive to be of practical interest. Another avenue, which is the one we will take in this project, consists on using fictitious domains strategies, where the moving domain evolves over a background fixed finite element mesh. The main challenge in this situation, other than the mathematical analysis of the proposed method (which is far from immediate) is the interpolation of results between meshes at different time steps. To address this point we will use the computational capabilities of FenicsX, which is the computational package used by K. MacKenzie in her PhD thesis, and in which her numerical results were obtained.
3. Implementation of the resulting schemes in the realistic AFRC context.
   The main goal will be to continue the process that started in [KMcK25]. That is, to test the resulting methods for realistic, three-dimensional geometries. The results obtained will be matched to experiments ran at the AFRC, and qualitative, as well as quantitative, comparisons will be carried out.

References

[ABR17] A. Allendes, G.R. Barrenechea, and R. Rankin. Fully computable error estimation of a nonlinear, positivity-preserving discretization of the convection-diffusion-reaction equation, SIAM J. Sci. Comput., 39 (5), A1903 A1927, (2017).

[CGS20] A. Cangiani, E.H. Georgoulis, M. Sabawi. A posteriori error analysis for implicit-explicit hp-discontinuous Galerkin timestepping methods for semilinear parabolic problems. Journal of Scientific Computing 82 (2), (2020).

[KMcK25] Katherine MacKenzie: “Finite Element Methods for 2D Induction Heating Problems”. PhD thesis, University of Strathclyde.