A. Bossavit

A rationale for 'edge-elements' in 3-D fields computations

A justification for the use of finite elements with edge-attached degrees of freedom (edge-elements) is sought in some mathematical structures that underlie eddy-current theory. An introduction to so-called mixed methods in magnetostatics is given, and two dual methods for eddy-current computations, both making use of edge elements, are shown to derive from these considerations. >

Edge-elements for scattering problems

It is shown that edge elements, as vectorial finite-element approximations fields like e and h, can be used to solve scattering problems. This can be done in essentially two ways, according to which of the two fields d or b one most wishes to get in divergence-free form. Apparent drawback of edge elements (due to the increased number of degrees of freedom in comparison with nodal elements) are more than compensated by economies due to greater sparsity. Moreover, they could help get rid of unwanted spurious modes when resonance frequencies of waveguides of complicated shape are computed. >

A mixed fem-biem method to solve 3-D eddy-current problems

A method is proposed to solve eddy-current problems in three dimensions. It is based on a special blend of FEM and BIEM techniques, and can be applied to non-linear situations. It has been implemented in the linear harmonic case and used in particular for studies on non-destructive testing. We first expose the mathematical foundation, next the dicretization technique. The current possibilities of the code are described and illustrated by computer graphic displays.

On the numerical analysis of eddy-current problems

Abstract Starting from Maxwells equations, we derive a general variational formulation suitable for eddy-current problems in non-magnetic, non-moving conductors disposed in an arrangement of closed multiply-connected circuits. The related problem of open circuits is considered and the validity of the notion of impedance is discussed. A finite element approach is described and illustrated by two examples, one of them from a real problem in electrical engineering.

Forces in magnetostatics and their computation

The use of unconventional (edge‐based) finite elements is proposed for the computation of a static magnetic field in a system of deformable conductors and ferromagnetic bodies (with a linear b‐h law). Thanks to a virtual‐work approach, the associated forces can be computed at the element level.

A theoretical approach to the question of biological effects of low frequency fields

A human body in the vicinity of an overhead cable is subject to two kinds of electrical effects, inductive and capacitive, which together are responsible for tiny currents. Recent concerns about possible biological effects of such currents have made their accurate computation necessary. The author address the mathematical modeling that has to precede such computations. By an asymptotic analysis of the full set of Maxwells equations (displacement currents not neglected), which takes into account the smallness of the electrical permittivity and the large penetration depth, the author arrives at a standard conduction problem, to which both kinds of effect contribute source terms: inductive effects give a right-hand side, and capacitive effects give a nonhomogeneous boundary condition. Numerical results are displayed. >

3D eddy-current distribution in a tokamak first wall during a plasma disruption using “TRIFOU”

In fusion reactor studies there is a lack of knowledge concerning the electromagnetic-type of phenomena generated by a plasma disruption event (rapid quenching of the plasma current). The induced eddy current distribution in space and time in the passive conducting structural components surrounding the plasma ring needs to be accurately investigated. TRIFOU is a full 3D eddy-current computer program based on a mixed FEM and BIEM technique, using the magnetic field, h, as a state variable. It has already been used in various areas of interest including static or rotating machines, non-destructive testing, induction heating, and research devices such as tokamaks. It can take into account various geometries and a wide range of physical situations (time dependency, physical properties, etc.). The present application is related to the eddy-current situation arising from a strong electromagnetic transient generated in the NET (Next European Torus) first wall segment. With respect to previous numerical simulations, the general 3D approach for the current density shows different eddy current circulations in the front/side shells and in the stiff back plate. The results obtained by TRIFOU are illustrated by means of advanced computer graphic displays and an animation movie.

Electromagnetic transients induced in the tokamak first wall by a plasma disruption using 'TRIFOU'

A complete 3-D electromagnetic analysis of the inboard first wall of the ITER fusion device has been performed using the 3-D general-purpose eddy-current code TRIFOU, based on a mixed FEM (finite element method) and BIEM (boundary integral equation method) technique and using the magnetic field as a state variable. The main objectives were to estimate the circulation of induced currents and to calculate the distribution of body forces from the coupling of eddy-currents and magnetic fields. The use of TRIFOU has demonstrated that eddy currents and Laplace body forces in complex 3-D situations can be predicted accurately. >

Eddy-current Problems

This chapter solves a non-stationary model, and the starting point is Maxwells system with Ohms law. The chapter introduces the main simplification, often described as the “low-frequency approximation,” which consists in neglecting the term of “Maxwell displacement currents,” as well as the method—implemented under the code name Trifou. The key idea of the Trifou method is to reduce the computational domain to the conductor to not to discretize the air region around. This method provided the first solution of general applicability to the three-dimensional eddy-currents problem. The chapter discusses the concept of cuts for each current-loop, which is a kind of associated Seifert surface, formed of faces of the mesh.

Mixed elements and a two-phase free-boundary problem in magnetostatics

Magnetostatics consists in finding vector fields b and h such that div b = 0, rot h = j (j is given), and b = Г(h), where Г is the subgradient of a convex functional U (the magnetic coenergy). When U is the sum of a quadratic functional and of a support functional, a 2-phase free-boundary problem results, a “vector” Stefan-like problem so to speak, because the unknown is a vector-field, not a function like e.g., temperature. The special structure of the equations calls for special “mixed” finite elements. We show how they help keep at the discrete level some interesting “complementarily” properties of the problem which are present at the continuous level.