236
CHAPTER 8 Eddy-current Problems
forces, those along the selected edges. Now, we'll have two different
kinds of DoF: Besides the mmf's along edges inside C, there are others,
associated with the nodes in the air and on the conductor's surface, which
can be interpreted as nodal values of the magnetic scalar potential, as we
shall see. Again, let us treat the contractible case first.
Let E c, as above, be the subset of edges
inside
C, that is, entirely
contained, apart from the extremities, in the interior of C. The set
N-Nc is composed of the nodes which are neither in int(C), nor in 3D.
Let E c be the number of edges in E C and No the number of nodes in
N- Nc. Last, call U (isomorphic to C Ec ÷ No) the space of vectors v =
{tI, F } = {ti e : e ~ E c, F n : n ~ N- Nc}, where the degrees of freedom tie
and F n are now unconstrained complex numbers. Let at last IK °~ be the
space of vector fields of the form
(32)
H ~e ~ E C n~
N-N C n
-- tie We nt- ~ F
grad
W n.
Proposition 8.5. IK °
is isomorphic to U.
m
Proof.
This amounts to saying that degrees of freedom are independent,
that is to say, H
~-
0 in (32) implies all ti e and
F n are
zero. We know
this is the case of the tie'S, by restriction to C (cf. Remark 5.2). As for the
Fn'S ,
the equality 0
= ~nFn
grad w n
~
grad(~ n
F n Wn) implies ~n F n
Wn
equal to some constant in the only connected component of D - C, a
constant which is the value of this potential on 3D, that is, 0. Again we
know (cf. Exer. 3.8) that all Fn'S must vanish in this case. 0
Proposition 8.6.
If C is contractible,
IK°m - IH°m
~
{H
E Wlm
" rot H
=
0 out
of C}.
Proof.
After (32), one has rot H
-- ~e ~ E
C He rot W e, and supp(rot w e) is
contained in the closure of C, so rot H
= 0 out
of C. Conversely, if
H
E
Wlm and if rot H
--
0 in D -C, which is simply connected, there
exists a linear combination • of the w e for n ~ N- Nc, such that H
--
grad
• in D- C, hence (32).
Now let
H g E W 1
be an approximation of the source field. We
m m
have IH g =
H g
n u IH 0
again, and we can "suffix everything with m"
m m m !
hence the desired Galerkin approximation for problem (23), the same,
formally, as (24):
find H
E
IHgm such that
(33)
fDi(OpH .H ' + ~C G-1 rot H. rotH' = 0 VH' E IH°m"
This is, in an obvious way, a linear system with respect to the unknowns
H e and F r¢ the form of which is similar to (31).
8.3 BOUNDED DOMAINS: TREES, H-cb
237
To build
Hgm , tWO
techniques are possible. The first one consists of
first computing
H g
by the Biot and Savart formula, then evaluate the
circulations I-Ie g of
H g
along all edges inside D. (For edges on the
boundary, one
sets He g "- 0,
which
does introduce some error, but compatible
with the desired accuracy, 7 if the mesh was well designed.) One then
sets Hgm = ~
e~ E Hge
We"
However, this does not warrant rot
H g --
0 where
Jg =
0, as it should
m
be, for the Biot and Savart integral is computed with some error, and the
sum of circulations of
H g
along the three edges of a face where no
current flows may come out nonzero, and this numerical error can be
important when the edges in question happen to be close to inductor
parts. This is a serious setback.
Hence the idea of again using the spanning tree method, which
automatically enforces these relations. But contrary to the previous section,
it's not necessary to deal with all the outside mesh to this effect. One will
treat only a submesh, as small as possible, covering the support of Jg. Of
the set E g of edges of this submesh, one extracts a spanning tree E t, and
one attributes a DoF to each co-edge the same way as in (28). One finally
sets Hgm
~- Z e~ E g - E t He g W e.
8.3.4 Cuts
Difficulties with the non-contractible case are about the same as in
Subsection 8.3.2. Holes are no problem: Just pick one node n inside
each non-conductive cavity within C and set ~n = 0 for this node. But
the "loop problem" arises again, for if D- C is not simply connected,
IK°m is strictly included in IH°m" Missing are the fields H that, although
curl-free in D -C, are only local gradients, or if one prefers, gradients of
multivalued potentials ~.
Hence the concept of "cuts", that is, for each current-loop, a kind of
associated Seifert surface (cf. Note 6 and Exer. 8.3), formed of faces of the
mesh. (Figure 8.7 will be more efficient than any definition to suggest
what a cut is, but still, recall the formal definition of 4.1.2: a surface in
D, closed mod C, that doesn't bound mod C.) One then doubles the
nodal DoF for each node of this surface (Fig. 8.7, right): to n÷ is assigned
theDoF ~* =~n, andt° n the nodal value ~* =~n +J'where J is
n+ - n_
the loop-intensity. Let N * be the system of nodes thus obtained, and
~,* w n, where the w~+ are supported on one side of Z, as
z Zn~N* n
7If this is not the case, one may always resort to solving the magnetostatics problem,
rot H g -- Jg and div H g = 0 in D, with the same boundary conditions.
238 CHAPTER 8 Eddy-current Problems
suggested on Fig. 8.7: Then • is multivalued in D- C, and the fields H
• * grad w do fill out IH ° It all goes again as if
= ~e~ EC I'Ie We q- ~n~ N * n n m"
there was one extra DoF (the unknown intensity J) for each current loop.
f
V
0 0
_
FIGURE 8.7. A cutting surface X, and the doubling of nodes in X. The loop
intensity J is also the circulation of the magnetic field along a circuit crossing
(along with the normal v) and is therefore equal to the jump of ~. Bottom right:
support of the nodal function Wn+.
The big difficulty with this method is the construction of cuts. Several
algorithms have been proposed [Br, HS, LR, VB], all more or less flawed
because of a faulty definition of cuts. All these early works assumed,
explicitly or not, that cuts must constitute a system of orientable (i.e.,
two-sided) surfaces, having their boundaries on 3C ("closed modulo C
but non-bounding", in the parlance of Chapter 4)mwhich is all right up
to now--but also,
such that the complement in D of their set union with
C,
that is, what remains of air after cuts have been removed,
be simply connected.
And this makes the definition defective, as Fig. 8.8 suffices to show: The
complement of the set union of C and of the Seifert surface X is
not
simply connected, but the three components of X do qualify as cuts
notwithstanding, for the magnetic potential • is effectively single-valued
outside C u X. See the controversy in the IEE Journal [B&] triggered by
the publication of [VB] for a discussion of this point. What cuts should
do is make every curl-free field equal to a gradient in the outer region
minus cuts. Credit is due to Kotiuga and co-workers [Ko] for the first
correct definition of a cut, a constructive algorithm, and an implementation
[GK].
It cannot be assessed, as the time this is written, whether this method
is preferable to the "belted tree" approach. This is the matter of ongoing
research [K&]. Let us, however, acknowledge that the problem of "knotted
loops" is really marginal. Even common loops are infrequent in everyday
work, because good modelling, taking symmetries into account, often
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