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# On the analysis of axis-symmetric eddy-currents

thesis

posted on 2017-01-05, 03:07 authored by Bryan, Geoffrey RaymondThis 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.