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This paper deals with the acceleration of the hybrid finite-element/boundary-element analysis of three-dimensional (3-D) eddy-current problems by means of the fast multipole method. An adaptive truncation scheme for the multipole expansion of the 3-D Laplace Green function is proposed. As an application example, the TEAM workshop problem 28 is considered. The computational cost without and with fast multipole acceleration is discussed. Index Terms--Boundary-element method, fast multipole method, finite-element method, hybrid techniques, Laplace function.

In this paper, different numerical techniques belonging to the harmonic and the time formalism have been confronted through the calculation of the performances of a three-dimensional microwave filter. Among those techniques, the Newmark and Lanczos algorithms appear to be particularly well adapted for the numerical design of such applications. Practical aspects of the implementation and the use of these techniques are pointed out by the authors. Index terms - Bandstop filters, cavity resonator filters, eigenvalues/eigenfunctions, finite element methods, frequency domain analysis, modeling, time domain analysis.

A method for defining electric global quantities, i.e. currents and voltages, in finite element magnetodynamic h-formulations is presented. It makes use of edge finite elements, which enables to directly define currents in a strong sense as well as to exploit the discrete weak formulation to define voltages in a natural weak way, without the need of any supplementary integral formula. It is applied to massive and stranded inductors for defining voltage sources and for the coupling with circuit equations. Index terms - Finite element method, coupled problems, magnetodynamics, inductors, electric circuit.

A software environment for the treatment of discrete problems is presented and used for teaching the finite element method in electromagnetism. It makes use of a user-friendly data structure which gives students a great freedom in testing in an evolutive way various numerical techniques on practical problems, thanks to a total control on all the components of the modeling process. By this way, the transition from theory to practice is made quite directly and easily. Index terms - Education, Computer aided software engineering, structured programming, software tools, finite element method, modeling, electromagnetic fields.

Starting from the well known analytical formula for the eddy current losses in electrical steel laminations, saturation and edge effects are studied by means of 1D and 2D finite element models of a single lamination. A novel method for directly including the laminated core energy dissipation in a time stepped 2D model of a complete (rotating) machine is proposed. By way of example the method is applied to a tooth model with enforced flux waveforms. Index terms - magnetic fields, eddy currents, finite element methods, magnetic losses, skin effect

Transformation methods are a very powerful tool in finite element modelling. In many cases, an adequate mapping transforms the problem into an easier one or allows to take advantage of the symmetries. This paper demonstrates that any mapping can be handled automatically provided the classical vector analysis approach is given up for the benefit of a differential geometry approach. As a first example, it is shown that axisymmetrical problems need no more a particular treatment provided the mapping of the cylindrical coordinates on the cartesian ones is considered as it is. Furthermore, a novel axisymmetrical formulation is proposed which relies on one further transformation and improves considerably the quality of the interpolated field. Transformation methods are also of great help to model the infinite space by means of finite elements. Many authors have presented such transformations which are often instances of the same general shell transformation that is presented here. Index terms - Transforms, differential geometry, TEAM Workshop problem 11, object oriented programming, software design/development, Interpolation.

The aim of the proposed method is the projection of a field belonging to a given function space (continuous or not) onto a discrete one spanned by finite element basis functions. This technique is useful for imposing inhomogeneous boundary conditions or volumic source fields, for calculating a dual field given the primal one or for mesh to mesh interpolation. Index terms - Finite element methods, interpolation, polynomial approximation, modeling.

This paper presents a general deterministic optimization procedure to compute the spatial repartition of material property - dissipative and nondissipative part - which minimizes physical or numerical reflection in physical and Maxwellian absorbers. The reflection performances of particular absorbers are compared from many points of view and deeply analyzed. The obtained results can usefully guide the designer in the choice of a particular absorber for the treatment of practical or numerical applications. Index terms - Absorbing media, electromagnetic reflection, finite element method, modeling, optimization methods.

This paper presents the analysis of a busbar for a three-phase system. The best connection of the different phases is proposed to derive the optimal dissipation of the heat flux inside the busbar. The implementation of a magnetothermal coupled model which can take into account radiation exchange and the electric circuit parameters is necessary. A comparison of numerical results with experimental results is made on an industrial busbar. Index terms - Application, coupling, eddy currents, electromagnetic heating, finite element methods, thermal devices.

The aim of this paper is to present an experimental convergence study of several high order curl-conforming finite elements. Degrees of freedom are first mathematically defined in order to interpolate directly a set of analytical functions. The convergence is then analyzed on these functions before being compared with two- and three-dimensional finite element computations. Index terms - Finite element methods, interpolation, convergence of numerical methods, error analysis, modeling.

This paper presents the analysis of a transverse flux induction furnace. The originality of the analysis presented in this paper is to use successively three different formulations in order to solve the magneto-thermal coupled problem. The Whitney elements (node based, edge based, facet based and volume based elements) are shown to be a convenient mean to perform naturally that coupling and to fullfil convergence conditions for the iterative solver. Index terms - Coupling, eddy currents, electromagnetic heating, finite element methods, induction heating, thermal devices.

A method for defining global quantities related to fluxes and circulations is proposed in the frame of the finite element method. The definition is in perfect accordance with the discretized weak formulations of the problems. It therefore enables a natural coupling between local and global quantities in various formulations, while keeping a symmetrical matrix for the system, and then is open to the coupling of physical problems. Applications are given for electrostatics, magnetostatics and magnetodynamics. Index terms - Finite element method, coupled problems, electrostatics, magnetostatics, magnetodynamics, electric circuit.

A general computer aided description environment for the treatment of discrete problems, based on a concise structure for both development and application levels, is described and applied to the finite element method. Its characteristics reveal its great ability to welcome in an evolutive way a wide range of physical models and numerical methods, with any coupling of them. It is therefore adapted to various activities, such as research, collaboration, education, training and industrial studies. Index terms - Computer aided software engineering, software design/development, structured programming, software tools, finite element method, modeling, electromagnetic fields.

A new approach for error estimation in finite element computations is presented. This method may be applied to a wide range of modeling (magnetostatics, magnetodynamics, scattering problems). The main idea is to reconstruct, from the finite element solution of a primal formulation, a field belonging to the space of the dual finite element formulation. The error is then computed through the non fulfillment of the constitutive relation by the couple: primal solution-dual solution. 2-D problems with analytical solutions are treated to prove the efficiency of the method. Then the error estimator is used to adapt the mesh for a 3-D magnetostatic problem. Index terms - Error analysis, finite element method, adaptative meshing, modeling, magnetostatics, magnetodynamics.

Mixed formulations of Maxwell's equations are presented and the nature of the matrices appearing in these formulations is discussed. The point is made to their indefinite character. A method for solving these unusual matrices is then developed and applied to a test problem.

A general method to compute source fields in magnetostatics or magnetodynamics is presented for inductors of any shape. That source field is not the physical one because the zero divergence condition is not satisfied. However, the freedom so obtained is exploited to minimize its support as well as to reduce the CPU time. The use of edge finite elements enables its rigorous construction. A test problem illustrates the method.

Several characteristics of magnetostatic and magnetodynamic mixed formulations are presented and discussed. The use of nodal, edge, facet and volume mixed elements enables their natural and rigorous discretization. Their advantages compared with conventional formulations are pointed out.

The performances of a finite element (F.E.) programme are closely related to the richness of its F.E. library. An Object-Oriented organisation for F.E. programmes is proposed in this paper which intends to fit closely the mathematical structure of any kind of finite element. This philosophy, leads to a concise and efficient F.E. programme. Moreover it allows the user to develop himself easily his own formulations.

The paper presents a method to control the quality of finite element solutions using the error in constitutive relation. An a posteriori error estimator is built up. Its construction is general and gives quantitative results about the accuracy of the solution. Both problems of control of quality and mesh optimisation are also discussed. Several examples are presented.

A sequence of finite element spaces built on tetrahedra, hexahedra and prisms, is presented, and a gauge condition for vector potentials is shown to be well defined in these spaces. Results are presented for a modified vector potential formulation for 3D eddy current problems.

This paper describes the computation of waveforms of electrical current and voltage in inductances, taking into account the hysteresis phenomenon and electric circuit equations. The non-linear transient magnetic field is computed with the finite element method. The classical non-linear case is compared to th hysteretic one: it is shown that hysteresis has a crucial importance on the waveform by introducing asymmetry and damping, and by modifying the natural frequency of the oscillations.

This paper presents a numerical modelling of an electromechanical relay connected with an electric excitation circuit. This transient modelling not onl takes into account the classical electromagnetic equations of the device but also the movement and circuit equations. The use of the finite element-boundary element coupling method facilitates the computation of the movement while the actual coupling with circuit equations is necessary for an accurate and reliabl representation of transient phenomena.

A generalization of the Whitney complex is proposed, which is associated no mor with simplices, i.e. tetrahedra in three dimensions, but with collections of three kinds of geometric elements: tetrahedra, hexahedra and prisms. Nodal, edge, facet and volume finite elements, i.e. mixed elements, associated with collections of those geometric elements, are defined. Base functions for approximation spaces relative to these finite elements are defined and their properties are established. A geometric interpretation of these functions is given.

This paper presents a method which enables coupling between equations of electric circuits consisting of lumped elements RLC and a magnetic field model. The coupling between the finite element and the boundary element methods is used to compute the magnetic field produced by conductors excited by an electric circuit. The conductors involved in this computation may be connected according to any circuit topology and mixed with lumped elements.

Coupling between edge finite elements, nodal finite elements and boundary elements for the calculation of 3D eddy currents is presented. Its aim is to allow an adequate treatment of different formulations using scalar and vector variables. Three kinds of edge elements - tetrahedron, hexahedron and prism - are investigated and compared.

This paper gives a solution for the problem of unacceptable inaccuracies sometimes observed in the fields computed with the classical axisymmetrical model (i.e. first order finite elements with auxiliary potential V = A/r). Two methods are proposed here to improve the accuracy of the results: Isoparametrical second order elements and first order elements with a suitable coordinate transformation. Both are also considered for nonlinear problems.

For linear systems excited with sinusoidal constraints, an exact dynamical modelling can be performed with a phasor (complex number) approach. For nonlinear systems (exhibiting saturation, hysteresis and higher harmonics), a time stepping model is very time consuming. A method to model such nonlinear systems with an approximated phasor approach is presented. Determination of equivalent material characteristics is discussed and applications are given.

Nonorthogonal coordinate systems were used in computational electromagnetism in which differential forms represented specific electromagnetic quantities. This coordinate system elicited the final element modeling of thin or infinite domains. The incorporation of vector calculus in the transformation method resulted in systematic generalization.

Compares several methods for the computation of the magnetic field in thin plates. Challenges in the computation of the magnetic field in thin steel plates; Methodology of the study; Transmission conditions for thin magnetic plates.

Describes a model of the magnetostriction by a finite element scheme for magnetostatic cases. Mechanical problem; Hypothesis and finite element; Computation of the magnetoelastic energy; Results and conclusion.

Presents a study that focused on the use of a finite element-boundary element coupling method to compute the two-dimensional magnetic field in multiconductor systems with magnetic materials. Features of finite element formulation for conductors; Way in which the total current in a conductor is expressed; Aspect important in eddy current computation.