1
Direct coupling between magnetodynamic and thermal analysis allowed by a multi-harmonic decomposition of magnetic vector potential Romain Pascal1,2 , Jean-Michel Bergheau1 and Philippe Conraux2 Abstract— This paper deals with the numerical simulation of coupled magnetodynamic and thermal problems for induction heating applications. The electromagnetic problem is solved by using the so called Harmonic Balance Finite Element Method (HBFEM). This method enables to calculate accurately eddy currents and consequently the power losses due to Joule effect. In a perfectly simultaneous way, the power is introduced in a thermal model to compute temperature distributions. The coupled finite element formulation is detailed and the HBFEM results are compared with the one obtained with a transient magnetodynamic simulation. Keywords— simulation, induction heating, finite element, magnetic vector potential, magneto-thermal coupling, harmonics, Fourier series, non-linearity, eddy currents.
the same finite element. We detail the formulation of the problem and compare the results obtained using the proposed approach with a more classical one based on transient magnetodynamic simulations. II. Magnetodynamic analysis By now, numerous magnetodynamic formulations have been proposed to solve eddy current problems. Here, we use the magnetic vector potential A(x, t) [4] in a 2D or axisymmetrical formulation. From Maxwell’s equations, neglecting displacement currents, the magnetodynamic equation describing a time non-linear problem can be written as:
I. Introduction
T
HE numerical simulation of induction heating processes has to face to several difficulties. A first difficulty comes from the high non-linearities of the problem. Indeed, here we have to deal with magnetic non-linearities (magnetization law), thermal non-linearities (latent heat, radiation,...) and the strong coupling between both phenomena (temperature dependency of the electromagnetic properties, power losses due to Joule effect). Therefore accurate results ask for a fully coupled magnetodynamic and thermal simulation. The second difficulty arises from the time constants of both phenomena which differ considerably. One way to achieve a coupled simulation is to use a staggered approach. At each thermal time step, the temperature distribution is calculated using the power losses due to Joule effect previously computed. Then, the new temperatures are used to calculate new power losses and so on up to convergence. It is evident that such an approach leads to unreasonable computation times when complex 3D industrial components are involved. Therefore, new methods have to be investigated. An alternative approach consists of determining an equivalent reluctivity such as the magnetodynamic fields can be looked for under a complex form, see ref [1]. Unfortunately, the method gives rather coarse results as far as only the first term of the Fourier series is in fact considered. The method we propose hereafter rests upon the Harmonic Balance Finite Element Method proposed in [2] [3]. The magnetodynamic and thermal fields are coupled within 1 Laboratoire de Tribologie et Dynamique des Sytmes LTDS, UMR5513 CNRS/ECL/ENISE E-mail:
[email protected],
[email protected] 2 Systus International E-mail:
[email protected]
∂A(x, t) + rot (ν . rot A(x, t)) − J0 (x, t) = 0 ∂t
σ
(1)
In two dimensional case A(x, t) = A(x, t)ez and in axisymmetrical case A(x, t) = A(x, t)eθ . The non-linear constitutive relations B = ν (kBk, θ) H and J = σ (θ) E must also be taken into account in order to totally define the eddy current problem. To improve the method presented in the introduction, one can consider several terms in the Fourier series of the electromagnetic variables. The gain in accuracy, keeping obviously a good respect of the non-linearities, is then counterbalanced by an increase of the size of the problem. In fact, we have to consider more unknowns and the matrix of the system is then expanded. That way, we assume that all electromagnetic variables, for instance magnetic vector potential A, flux density B and current density J, are approximated by a Fourier series where ω is the fundamental angular frequency. And especially, the non-linear time-periodic solution A(x, t) of equation 1 is decomposed in a sum of m harmonic components: A(x, t) =
m X
Akc (x) cos(kωt) + Aks (x) sin(kωt)
(2)
k=1,3
Then, time variation of equation 1 is eliminated by solving the following equations, with l varying from 1 to m: Z 1 T ∂A σ + rot (ν . rotA) − J0 . cos(lωt) dt = 0 (3) T 0 ∂t 1 T
Z 0
T
∂A σ + rot (ν . rotA) − J0 . sin(lωt) dt = 0 (4) ∂t
2
Each equation 3 or 4 corresponds to a system of m equations with the 2m unknowns corresponding to the harmonics of magnetic vector potential: Akc (x) and Aks (x) with k varying from 1 to m. Then, it is treated using the classical finite element method and solved to determine the spatial distribution of harmonic unknowns in the whole space. All details for discretisations and the complete finite element formulation will be provided in a futur extension of the present paper. III. Thermal analysis The evolution of temperature distribution as a function of time inside a material is governed by the following equation: dH ρ − div(λ. grad θ) − Q = 0 (5) dt where: ρ, H, λ and θ are respectively mass density, enthalpy, thermal conductivity and temperature. Q is an internal heat source. Due to the presence of eddy currents in material, a volumic distribution of currents is calculated from the electromagnetic analysis. By using Ohm’s law, the power density dissipated is then deduced and injected through Q in heat conduction equation 5. Thermal problem is solved with finite element method. For more details see ref [5]. IV. Direct coupling between magnetodynamic and heat transfer analysis Today, two coupling methods called ”weak” and ”strong” already exist. In paper [6], the second one is achieved. At each thermal time step, a few loops between several calculations of power density and temperature lead to the convergence of an accurate temperature at each node of the mesh. This process is numerically very slow because it needs a transient magnetodynamic simulation at each coupling iteration. Therefore, this method is certainly accurate but unfortunately long, especially in 3D. In previous schema, the Joule power density is given by a time-averaged value over one electromagnetic period. Here, this calculation can be performed directly: Z 1 T J(x, t). E(x, t) dt (6) Q(x) = T 0 Both magnetodynamic and thermal problems are solved together within the same finite element. The global unknown vector Ui contains now 1 component for temperature and 2m harmonic components for magnetic vector potential. This is illustrated with figure 1. Final spatial solution is finally built with finite element formulation. V. Conclusion This paper presents a new approach for fully coupled magneto-thermal problems. Final implementation in the computer code SYSMAGNATM of the SYSWORLDTM family has been achieved. The full paper will present comparisons between the proposed method and ”weak” or ”strong” coupling methods based on transient magnetodynamic simulations.
Instant t + ∆t ther
Instant t Thermal time scale
θ
θ A 1,c A 1,s U = t
A A
1,c 1,s
Resolution with a
A k,c A k,s
Ut + = ∆t ther non-linear iterative schema
A m,c A m,s
A k,c A k,s A m,c A m,s
Q ( U t + ∆t
Q ( Ut )
ther
)
Fig. 1. Direct electromagnetic and thermal coupling
References [1] E. Vassent, G. Meunier, and J.C. Sabonnadiere, “Simulation of induction machine operation using complex magnetodynamic finite elements,” IEEE Transactions On Magnetics, vol. 25, no. 4, pp. 3064–3066, July 1989. [2] S. Yamada, K. Bessho, and J. Lu, “Harmonic balance finite element method applied to nonlinear ac magnetic analysis,” IEEE Transactions On Magnetics, vol. 25, no. 4, pp. 2971–2973, July 1989. [3] R. Albanese, E. Coccorese, R. Martone, G. Miano, and G. Rubinacci, “Periodic solutions of nonlinear eddy current problems in three-dimensional geometries,” IEEE Transactions On Magnetics, vol. 28, no. 2, pp. 1118–1121, March 1992. [4] O. Biro and K. Preis, “On the use of the magnetic vector potential in the finite element analysis of three-dimensional eddy currents,” IEEE Transactions On Magnetic, vol. 25, no. 4, pp. 3145–3159, July 1989. [5] J-M. Bergheau, G. Mangialenti, and F. Boitout, “Contribution of numerical simulation to the analysis of heat treatment and surface hardening processes,” in Heat Treating. October 1998, pp. 681–690, ASM International. [6] J-M. Bergheau and Ph. Conraux, “Fem-bem coupling for the modelling of induction heating processes including moving parts,” in Journal of Shangai Jiaotong University, June 2000, vol. E-5, pp. 91–99.
