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We present and discuss two variational formulations in 3-D for the magnetostatic problem. The first is based on a mixed formulation where the state variables are magnetic field H and magnetic vector potential A. In the second, we eliminate unknown A, and we calculate only H by a penalty method. For each method, a computer code has been implemented and we compare these methods with an International Problem for which experimental measures are available. Index terms - Magnetostatics, penalty methods, mixed finite elements, edge-facet elements.

A variational formulation is presented to solve Maxwell's equations with the electric and magnetic fields as variables in waves interacting with a conducting domain. This method couples a finite element method and a boundary integral method. The discretization uses Nedelec's edge elements for both fields. We finally present numerical results. Index Terms - Electromagnetic wave, finite element, integral method.

We here present the modelling of eddy currents and surface electric charges created by an electromagnetic field in a conductor. We used an electrical field formulation in which a finite element method inside the conductor is coupled with an integral method for the outside. Induced current is then directly given by Ohm's law. Multiply-connected domains are considered. The electric charges which appear at the surface of the conductor are calculated and capacitive effects are evaluated.

The separate modellings of the surface current distribution in a MRI antenna (with nothing around), and of the volume eddy current appearing in a patient placed in a given magnetic field, have already been described. When the patient is put in the magnetic field of an antenna, there appear mutual reactions. We here present an iterative method which allows us to compute in this last case the current density in both antenna and body, even when they are very close to each other.

We present a Boundary Integral Method (BIM) for the computation of the current distribution in an antenna with negligible skin depth, the unknown quantities being the overall current intensity and the scalar magnetic potential on the boundary. After computing these quantities, the current density distribution can be deduced and the resistance as well as the inductance of the antenna evaluated. The results are compared with experimental values.

We present a penalty h-formulation of magnetostatics in [R.sup.3] without any boundary conditions. It is implemented by coupling a Finite Element Method (FEM and a Boundary Integral Method (BIM). The discretization for the FEM uses Nedelec's edge elements which allows to describe the jump of the normal component of the magnetic field on the boundary. We describe the iterative method which takes into account the nonlinearity of the magnetic materials and we explain how the penalty parameter has to be chosen. We finally present numerical results concerning an International Problem for which experimental measures are available.

The perturbation of a static magnetic field by a massive ferromagnetic material is often described with the help of magnetic charges, located on its boundary if the permeability micron is constant. This description is no more accurate when some parts have a small thickness. We present a specific modelling of this last configuration. Each thin shell is replaced by a single layer of magnetic charges. The corresponding variational formulation, in which the magnetic charges are the unknowns, is established. Numerical tests confirm its validity.