On the analysis of axis-symmetric eddy-currents

This work is primarily concerned with the numerical simulation of linear time-harmonic electromagnetic field problems and to a lesser extent with some applications of these simulations to non-destructive testing. Chapter 1 provides a broad overview of the work and also pauses briefly to discuss the original inspiration of the project; eddy-current non-destructive testing. Chapter 2 looks in some depth at the underlying mathematical structure of the problem. With the aid of Tonti diagrams we systematically develop equations for the potential functions. The equations for the potentials as they originally stand do not have unique solutions. In order that there be a single unique solution to the problem gauge constraints must be enforced. One particular approach to enforcing gauge constraints, that of augmenting the operator, is treated in some depth and generality. In Chapter 3 having determined the equations governing the flow of eddy-currents and having determined that, given suitable constraints, they admit of only one solution the next problem to advance itself is how to determine this solution. Two broad classes of solution techniques are discussed: the calculus of variations and the method of weighted residuals. Following Tonti and Magri we show that every linear partial differential equation can, for a suitable choice of bi-linear form (see Appendix A) be regarded as the Euler-Lagrange condition of a functional. It is shown that none of the commonly applied variational principles are extremal principles for the general case of complex partial differential equations. With no obvious benefit accruing from a variational formulation of the eddy-current problem our attention was turned to the related technique of extended operators. It is shown that extended operators are especially useful in enforcing inhomogeneous boundary and interface conditions. Chapter 4 addresses itself to the problem of trying to interpret the flow of eddy-currents in terms of the few simple problems for which analytical solutions are known (Hammond's (1964) and Zaman's ( 1980) problems being the best known examples). The problems treated in Chapter 4 provide a basis for interpreting the results generated by the finite element program FINEEL developed in Chapter 5. Finally, in Chapter 5, same of the finer points of the finite element method are discussed: matrix assembly, solution techniques etc. and some typical results generated by the program are discussed in the light of previous results. Chapter 6 concludes the work by making a few suggestions as to which areas would profitably repay further research.