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Published online by Cambridge University Press: 06 July 2010
5A. Introduction
In this chapter we consider a formulation for computing eddy currents on thin conducting sheets. The problem is unique in that it can be formulated entirely by scalar functions—a magnetic scalar potential in the nonconducting region and a stream function which describes the eddy currents in the conducting sheets— once cuts for the magnetic scalar potential have been made in the nonconducting region. The goal of the present formulation is an approach via the finite elements to discretization of the equations which come about from the construction of the scalar potentials. Although a clear understanding of cuts for stream functions on orientable surfaces has been with us for over a century [Kle63] there are several open questions which are of interest to numerical analysts:
(1) Can one make cuts for stream functions on nonorientable surfaces?
(2) Can one systematically relate the discontinuities in the magnetic scalar potential to discontinuities in stream functions by a suitable choice of cuts?
(3) Given a set of cuts for the stream function, can one find a set of cuts for the magnetic scalar potential whose boundaries are the given cuts?
In preceding chapters we have alluded to the existence of cuts, though we have not yet dealt with the details of an algorithm for computing cuts. The algorithm for cuts will wait for Chapter 6, but it is possible to answer the questions above. Section 5B gives affirmative answers to the first two questions by using the existence of cuts for the magnetic scalar potential to show that cuts for stream functions can be chosen to be the boundaries of the cuts for magnetic scalar potentials.
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