Pilar Salgado

A Finite Element Method with Lagrange Multipliers for Low-Frequency Harmonic Maxwell Equations

The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics. This system of partial differential equations is a model for the so-called eddy currents problem. After writing this problem in terms of the magnetic field, it is discretized by Nedelec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained if the curl-free condition is imposed on the elements in the dielectric domain. Then, the curl-free condition is imposed, at a discrete level, by introducing a piecewise linear multivalued potential. The resulting problem is shown to be a discrete version of other continuous formulation in which the magnetic field in the dielectric part of the domain has been replaced by a magnetic potential. Moreover, this approach leads to an important saving in computational effort. Problems related to the topology are also considered in that the possibility of having a nonsimply connected dielectric domain is taken into account. Implementation issues are discussed, including an amenable procedure to impose the boundary conditions by means of a Lagrange multiplier. Finally, the method is applied to solve a three-dimensional model problem: a cylindrical electrode surrounded by dielectric.

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.

A numerical method for transient simulation of metallurgical compound electrodes

The objective of this work is to introduce and numerically solve a mathematical model for unsteady thermoelectrical behavior of an electric arc electrode. The electrode is composed of several materials and one of them suffers a change of state during operation. Thermoelectric parameters depend on the temperature in each material. The volumetric heat source is the Joule effect. The mathematical model leads to a coupled non-linear system of partial differential equations. The axisymmetry of the domain allows us to simplify the problem when it is written in cylindrical coordinates. We use an implicit method for time discretization and a standard finite element method for space discretization. Numerical results corresponding to the simulation of a real industrial situation are given.

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.

Numerical simulation of some problems related to aluminium casting

Abstract In this paper we present several models describing some thermal, mechanical, electromagnetic and hydrodynamic phenomena arising from electromagnetic aluminium casting process. A numerical solution of these models is proposed and numerical results are given for industrial castings.

A Finite Element Method for the Magnetostatic Problem in Terms of Scalar Potentials

The aim of this paper is to analyze a numerical method for solving the magnetostatic problem in a three-dimensional bounded domain containing prescribed currents and magnetic materials. The method discretizes a well-known formulation of this problem based on two scalar potentials: the total potential, defined in magnetic materials, and the reduced potential, defined in dielectric media and in nonmagnetic conductors carrying currents. The topology of the domain of each material is not assumed to be trivial. The resulting variational problem is proved to be well posed and is discretized by means of standard piecewise linear finite elements. Transmission conditions are imposed by means of a piecewise linear Lagrange multiplier on the surface separating the domains of both potentials. Error estimates for the numerical method are proved and the results of some numerical tests are reported to assess the performance of the method.

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 treatment of realistic boundary conditions for the eddy current problem in an electrode via Lagrange multipliers

This paper deals with the finite element solution of the eddy current problem in a bounded conducting domain, crossed by an electric current and subject to boundary conditions appropriate from a physical point of view. Two different cases are considered depending on the boundary data: input current density flux or input current intensities. The analysis of the former is an intermediate step for the latter, which is more realistic in actual applications. Weak formulations in terms of the magnetic field are studied, the boundary conditions being imposed by means of appropriate Lagrange multipliers. The resulting mixed formulations are analyzed depending on the regularity of the boundary data. Finite element methods are introduced in each case and error estimates are proved. Finally, some numerical results to assess the effectiveness of the methods are reported.

Numerical analysis of electric field formulations of the eddy current model

This paper deals with finite element methods for the numerical solution of the eddy current problem in a bounded conducting domain crossed by an electric current, subjected to boundary conditions involving only data easily available in applications. Two different cases are considered depending on the boundary data: input current intensities or differences of potential. Weak formulations in terms of the electric field are given in both cases. In the first one, the input current intensities are imposed by means of integrals on curves lying on the boundary of the domain and joining current entrances and exit. In the second one, the electric potentials are imposed by means of Lagrange multipliers, which are proved to represent the input current intensities. Optimal error estimates are proved in both cases and implementation issues are discussed. Finally, numerical tests confirming the theoretical results are reported.

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