Physically compatible finite element methods for an electrically-induced heating problem

Induction heating is present in diverse industrially-relevant processes. 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. A long and thin 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.

Mathematically, this problem is modelled by the coupling of Maxwell's equations in the two or three-dimensional domain (including the coils, the surrounding air and the billet), and the heat equation inside the billet. The latter presents appropriate boundary and initial conditions, and a right-hand side that depends on the magnetic field inside the billet. The right hand side can take many different, but all possible definitions lead to a sharp thermal layer (the skin effect). This skin effect can be of the order of milimitres for high-frequency applications, for a typical billet has of the order of meters in length. So, the physical dimensions of this process are extremely anisotropic (i.e., they present one physical dimension of a much different order of magnitude from the others).

The numerical approximation of this sort of problem is extremely challenging, due to the following issues:
i). The geometry of the domain: since the billet is a long and thin piece (its width is of the order of a few centimetres, while its length can attain several meters), the use of meshes that reproduce this (this is, anisotropic meshes) is essential.

ii). The thermal boundary layer: the very strong gradients inside the layer lead to instabilities that manifest themselves by spurious oscillations, with numerical temperatures presenting over and undershoots, that make the discrete solution non-physical.

So, based on the challenges stated above, we propose the following work programme:
i) A positivity-preserving method, i.e., derivation and analysis of a finite element method that respects the physical bounds of the problem.

ii). The adaptive strategy. Derivation of an appropriate refinment/derefinment strategy aimed at dealing with the fact that the tip of the layer moves in time.

iii). Implementation of the scheme in the AFRC context.