Research Topic : Coupled Finite Element and Integral Representation Methods in Electromagnetism

This work was achieved in collaboration with G. Caloz, IRMAR, University of Rennes 1 (UMR CNRS 6625). We are interested in determining the magnetic field created by an electromagnetic device formed of a metallic core and an inductor (such as an electromagnet or an actuator). This work is part of a project aimed to study optimal configurations of such an electromagnetic device where the shape of the metallic core or the shape of the inductor are optimised in order to satisfy a given criterion (e.g. obtain a uniform high intensity magnetic field as required for certain applications in Nuclear Magnetic Resonance such as the one described in Ref. [a] at the origin of this work).

The approach we have adopted is to write the magnetostatic problem, set in a three-dimensional unbounded domain, for the reduced scalar magnetic potential as main unknown. The problem is solved by using a formulation based on a coupling between an integral representation formula and finite element method. The particularity of this formulation lies in the fact that the boundary intoduced to bounded the computational domain can be taken arbitrary closed to, but distinct, from the physical border of the electromagnetic device which avoids the presence of singular integrals to be computed. Once calculated the magnetic potential, the magnetic field can be obtained at any point from an integral representation formula without need for numerical derivation and therefore without loss of accuracy with respect to the calculation of potential. The study of this method is the subject of publications [1] and [4]. The numerical implementation of the method in Fortran is done using the Melina Library developed at IRMAR, University of Rennes 1  by D. Martin.

We have also investigated 2 well-known numerical issues arising when using the reduced scalar magnetic potential for numerical computation purposes in electromagnetics. The integral representation formula for calculating the magnetic field from the magnetic potential is composed of two terms (a "single layer" integral term and a "double layer" integral term). For large values of the magnetic permeability, the two integral terms are nearly equal and of opposite sign so that it is impossible to accurately compute the magnetic field by summing the 2 terms directly. A similar phenomenon occurs when computing the total magnetic field which is the sum of the source field and the reaction field generated by the ferromagnetic core. We have studied this two numerical problems using an asymptotic expansion of the magnetic potential. In each of the two cases we have proposed a robust numerical method for the calculation of the sum of the 2 integrals that at the end appears to be similar to some approaches used in domain decomposition methods. This work is the subject of publications [2] and [3].

[a] C. J. Lewa and J. D. de Certaines. Selected-states magnetic-resonance spectroscopy: A potential method for huge improvement in sensitivity, Europhys. Lett. 35, p.713 (1996).


Publications related to this research topic

1- S. Balac and G. Caloz. Coupling of finite element and integral representation in magnetostatics. Technical Report (2006)
2- S. Balac and G. Caloz. The reduced scalar potential in regions with permeable materials : reasons for loss of accuracy and cancellation. International Journal of Numerical Modelling (20(4) : 163-180 (2007))
3- S. Balac and G. Caloz. Cancellation errors in an integral for calculating magnetic field from reduced scalar potential. IEEE Transactions on Magnetics, 38(4):1997?2002, 2003.
4- S. Balac and G. Caloz. Magnetostatic field computations based on the coupling of finite element and integral representation methods. IEEE Transactions on Magnetics, 38(2):393?396, 2002.