Dolores Gómez

Transient numerical simulation of a thermoelectrical problem in cylindrical induction heating furnaces

Abstract This paper concerns the mathematical modelling and numerical solution of thermoelectrical phenomena taking place in an axisymmetric induction heating furnace. We formulate the problem in a two-dimensional domain and propose a finite element method and an iterative algorithm for its numerical solution. We also provide a family of one-dimensional analytical solutions which are used to test the two-dimensional code and to predict the behaviour of the furnace under special conditions. Some numerical results for an industrial furnace used in silicon purification are shown.

Eddy-Current Losses in Laminated Cores and the Computation of an Equivalent Conductivity

This paper deals with the computation of eddy-current losses in laminated cores. In the first part we recall some approximate formulas existing in the literature and implemented in many electromagnetic computer packages. Then we assess their accuracy by making comparisons with the exact loss values obtained by numerical solution of the quasi-static Maxwell equations. In the second part we introduce a definition of an equivalent electric conductivity allowing us to replace the laminated core with a homogeneous isotropic or anisotropic medium. We compare the values of this equivalent conductivity with those obtained from some approximate formulas.

Mathematical models and numerical simulation in electromagnetism

1 The harmonic oscillator.- 2 The Series RLC Circuit.- 3 Linear electrical circuits.- 4 Maxwells equations in free space.- 5 Some solutions of Maxwells equations in free space.- 6 Maxwells equations in material regions.- 7 Electrostatics.- 8 Direct current.- 9 Magnetostatics.- 10 The eddy currents model.- 11 An introduction to nonlinear magnetics. Hysteresis.- 12 Electrostatics with MaxFEM.- 13 Direct current with MaxFEM.- 14 Magnetostatics with MaxFEM.- 15 Eddy currents with MaxFEM.- 16 RLC circuits with MaxFEM.- A Elements of graph theory.- B Vector Calculus.- C Function spaces for electromagnetism.- D Harmonic regime: average values.- E Linear nodal and edge finite elements.- F Maxwells equations in Lagrangian coordinates.

Numerical Mathematics and Advanced Applications

C. BOULBE, T.Z. BOULMEZAOUD, and T. AMARI 1 Laboratoire de mathématiques appliquées, Université de Pau et des Pays de l’Adour, IPRA Av de l’Université, 64000 PAU, FRANCE. c.boulbe@etud.univ-pau.fr 2 Laboratoire de mathématiques appliquées, Université de Versailles Saint Quentin, 45 av des Etats Unis, 75035 Versailles, FRANCE. boulmeza@math.uvsq.fr 3 Centre de Physique Théorique, Ecole Polytechnique, F91128 Palaiseau Cedex,FRANCE. amari@cpht.polytechnique.fr

Numerical analysis of a transient non-linear axisymmetric eddy current model

This paper deals with the numerical solution of an axisymmetric transient eddy current problem in a conductive non-linear magnetic media. This means that the relation between the magnetic field and the magnetic induction (i.e., the so-called B - H curve) is non-linear. We analyze a weak formulation of the resulting problem in the axisymmetric case, with the source term given by means of a non-homogeneous Dirichlet boundary condition. For its numerical approximation, we propose a fully discrete scheme based on a finite element method combined with a backward Euler time discretization. We establish its well-posedness and derive error estimates in appropriate norms for the proposed scheme. In particular, we obtain an L 2 rate of convergence of order O ( h + Δ t ) without assuming any additional regularity of the solution. Moreover, under appropriate smoothness assumptions, we also prove an L 2 -like rate of convergence of order O ( h 2 + Δ t ) . Finally, some numerical results, which confirm the theoretically predicted behavior of the method, are reported.

Magnetostatics with MaxFEM

In this chapter we solve several magnetostatics problems by using MaxFEM. For some of the problems we will provide the analytical and the numerical solution. The examples include linear and nonlinear magnetic materials.

Eddy currents with MaxFEM

In this chapter we solve several examples governed by the time-harmonic eddy currents model by using MaxFEM. For some of the problems we will provide the analytical and the numerical solution. We exploit that some problems can be approximated by 2D or axisymmetric models but some others will require a genuine 3D model.

A thermo-electrical problem with a nonlocal radiation boundary condition

A coupled problem arising in induction heating furnaces is studied. The thermal problem, which involves a change of phase, has a nonlocal radiation boundary condition. Convective heat transfer in the liquid is also included which makes necessary to compute the liquid motion. For the space discretization, we propose finite element methods which are combined with characteristics methods in the thermal and flow models to handle the convective terms. In the electromagnetic model they are coupled with boundary element methods (BEM/FEM). An iterative algorithm is introduced for the whole coupled model and numerical results for an industrial induction furnace are presented.

Numerical Simulation of Induction Furnaces for Silicon Purification

This paper deals with mathematical modelling and numerical simulation of induction heating furnaces for axisymmetric geometries. The mathematical model presented consists in a coupled thermo-magneto-hydrodynamic problem with phase change. We propose a finite element method and an iterative algorithm to solve the equations. Some numerical results for an industrial furnace used for silicon purification are shown.

Numerical Simulation of Magnetization and Demagnetization Processes

The purpose of this paper is to describe a finite-element method for the numerical simulation of magnetization and demagnetization processes in ferromagnetic pieces with hysteresis. In particular, the proposed methodology will be applied to compute the remanent flux density after a magnetic particle inspection test on specimens with cylindrical symmetry consisting of three stages: longitudinal magnetization, circular magnetization, and demagnetization.