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

感应加热存在于各种与工业相关的工艺中。典型的工业感应加热线由一系列螺旋形线圈组成，由高频交流电提供能量，从而产生时变的磁场。一根细长的圆柱杆穿过线圈，以通过涡流的阻性损失来加热。当钢坯到达生产线末端时，它具有所需的温度，然后可以进行锻造/切割/淬火。在数学上，这个问题是由二维或三维区域(包括盘管、周围空气和钢坯)中的麦克斯韦方程和钢坯内部的热方程耦合而成的。后者给出了适当的边界和初始条件，以及取决于钢坯内部磁场的右侧。右侧可以有许多不同的定义，但所有可能的定义都会导致尖锐的热层(皮肤效果)。对于高频应用，这种集肤效应可以是毫米量级的，因为典型的坯料的长度是米量级的。因此，这个过程的物理维度是非常各向异性的(即，它们呈现出一个与其他物理维度大小不同的物理维度)。由于以下问题，这类问题的数值近似非常具有挑战性：i)区域的几何形状：由于钢坯是又长又薄的一块(它的宽度大约是几厘米，而它的长度可以达到几米)，使用重现这一点的网格(这是各向异性网格)是必不可少的。Ii)。热边界层：层内非常强的梯度导致不稳定性，这些不稳定性通过虚假振荡表现出来，数值温度呈现在上方和下冲，这使得离散的解是非物理的。因此，基于上述挑战，我们提出了以下工作计划：i)正性保持方法，即尊重问题的物理边界的有限元方法的推导和分析。Ii)。适应性战略。推导出适当的改进/取消定义策略，旨在处理层的尖端随时间移动的事实。三)。在AFRC的背景下执行该计划。

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.