8.3 BOUNDED DOMAINS: TREES, H-~ 239
allows one to dodge the difficulties they might otherwise raise. Eddy-
current codes that were implemented, years ago, with cut-submodules
based on some of the above-mentioned premature methods, on which
current research is trying to improve, still work superbly in their respective
domains of validity [BT, RL].
j Jr--
/"
.......... x#
......
~-~ ~-/---/-~
ii' i
/
/
72
FIGURE
8.8. (Look first at Fig. 8.9 for the way Z, here in three components, is
constructed.) If some parts of the conductor (here made of four distinct connected
components) are knotted or linked, it may happen that the complement of C and
Z is
not
simply connected, although the cuts Z do play their role, which is to
forbid multivalued potentials in the outside region. (Apply Amp6re to circuits
71
and
72")
The adaptation of the previous ideas to evolutionary regimes is
straightforward, along the lines of 8.2.5.
8.4 SUMMING UP
What is special with eddy-current problems, and explains the almost
unclassifiable variety of methods to deal with them, is the difference in
the nature of the equations in the conductor and in the air. From the
mathematical point of view, we have "parabolic equations" in conductors,
"elliptic equations" in the air. For sure, passing in complex representation
makes them elliptic all over, but different operators apply in the two
main categories of regions: the "curl-curl" operator in the conductors,
the "div-grad" operator in the air. We saw in Chapter 6 how intricate
the relations between these two basic differential operators could be,
marked by deep symmetries as well as striking differences. In the eddy-
current problem, they coexist and must be compatible in some way at the
common boundary. No wonder there is such a variety of methods for
eddy currents! A few general ideas emerge, however:
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