“…The direct problem solves a magnetic field analysis coupled to a thermal transient, for a given geometry of the inductor and given value of supply current. The magnetic problem is solved in time-harmonics conditions in the magnetic vector potential, A ̇, and scalar electric potential, V ̇, using a finite element model imposing the gauge of Coulomb,∇ ⋅ A ̇ = 0(Di Barba et al , 2008; Meunier, 2008; Morisue, 1993, 1990): where µ 0 and µ r are the vacuum and relative magnetic permeability, respectively ( µ r = constant in the core region, µ r = f(H,T) in tube region, depending on T , temperature, and H , magnetic field, µ r = 1 in other regions); σ is the conductivity of the material ( σ = f(T) in the tube) and ω = 2 πf with f frequency of the supplied current (Di Barba et al , 2018d, 2018e). The typical mesh for the magnetic problem discretizes the geometry in Figure 1 in 64,300second-order elements, and, typically, it is composed of 145,600 nodes.…”