8.3 BOUNDED DOMAINS: TREES, H-~ 231
Remark
8.4. When jg is sinusoidal, using this scheme is a viable alternative
to solving (18) directly. One should use an informed guess of the solution
as initial condition, and monitor the time average over a period,
(~)2
-10~ft_ 2x/c0t
exp(i~ s) k(s) ds, duly approximated by a sum of the kind
1
N- ~._ 1 N
exp(i0~(m + 1/2 -j)St)k
m+ 1/2-j,
where N~t = 2~/0), the
period.-This will converge, relatively rapidly (in no more than three or
four periods, in practice) towards the solution K of (18). Such a "time
domain" approach to the harmonic problem can thus be conceived as
another iterative scheme to solve (18). Fast Fourier Transform techniques
make the calculation of time averages quite efficient. 0
8.3 BOUNDED DOMAINS: TREES, H-~
Now we change tack. Let D be a simply connected domain containing
the region of interest (here the conductor C and its immediate
neighborhood, Fig. 8.6), the inductor I, magnetic parts where ~t ¢ ~t 0, if
such exist, and suppose either that D is big enough so that one can
assume a zero field beyond, or that n x h = 0 on 3D for physical
reasons (as in the case of a cavity with ferromagnetic walls, used as a
shield to confine the field inside). We thus forget about the far field and
concentrate on difficulties linked with the
degeneracy
of the eddy-current
equations in regions where c~ = 0.
D
FIGURE 8.6. Computational domain D, containing the region of interest, and
large enough for the boundary condition n x h = 0 on 3D to be acceptable. (In
practice, the size of the elements would be graded in such a case, the farther from
C the bigger, the same as with Fig. 7.2.)
232
CHAPTER 8 Eddy-current Problems
8.3.1 A constrained linear system
Let m be a simplicial mesh of D. Functional spaces, a bit different now,
are IH = {H
E InLrot(W)"
n x H = 0 on 3D} and
(21) IH g = {H ~ IH : rot H
= Jg
in D - C},
(22) IH ° - {H ~ IH : rot H
-- 0
in D - C}.
Now we know the paradigm well, and we can state the problem to solve
without further ado:
find
H E IH g
such that
(23)
~D i
co ~t H. H' + ~c rs-1 rot H. rot H' = 0 V H' ~ IH °.
As we intend to enforce null boundary conditions on the boundary
of D, let us remove from N, E, 9" the boundary simplices, as we did
earlier in 7.3.1, and for convenience, still call N, E, J-, T the simplicial
sets of this "peeled out" mesh. Apart from this modification, the notation
concerning the spaces W p and the incidence matrices is the same as
before. In particular, W 1 is the spanmwith
complex
coefficients, this
m
timemof Whitney edge elements w e, for all e in E. As this amounts to
have null circulations along the boundary edges, a field in W 1 can be
m
prolongated by 0 to all space, the result being tangentially continuous
and therefore an element
of
In2rot(E3 ). So we can identify Wlm with a
subspace of
In 2
(E3)
Let us set IH = W 1 denote by IH the isomorphic
rot " m m*
finite-dimensional space C z, composed of all vectors I~ = {E e : e ~ E },
and call E = #E the number of inner edges. For v and ~' both in IH,
we set
(U, U') = ~ e~ z Ue . U'e
------E ee E
(Re[ue] + i Im[ue])" (Re[u'e] + i Im[u'e]).
(Again, beware: This is not the Hermitian scalar product.) This way, an
integral of the form fD Ct U. U', where ¢z is a function on D (such as kt,
for instance), possibly complex-valued, is equal to (Ml(Ct)u, c') when u
U'
-- Ee~E UeW e
and
U'--Ee~E eWe .
We know from experience the eventual form of the discretized
problem: It will
be find H
E IH g
such that
m
(24)
YD i
co gt H. H' + Yc rs-1 rot H. rot H' = 0 V H' ~ IH°m,
where IH g
and IH ° are parallel subspaces of IH .
m m m
8.3 BOUNDED DOMAINS: TREES, H-cI) 233
But these subspaces must be constructed with some care. One cannot
simply set
IN g - W 1 ~ IU g--.. {H E W 1 •
rot
H = Jg
in D- C}, because
Jg
m m m
has no reason to be mesh-wise constant (which rot H is, if H e Wa),
although this happens frequently. Failing that, W 1 c~ IH g may very well
m
reduce to {0}. So we need to find a subspace of W 1 that closely
m
approximate W~m c~ IH g. For this, let -q-c a y be the subset of "conductive''
faces, i.e., those inside C, faces of its boundary being excluded. Let us set
IHgm ={HE W lrn"
~fn. r°t H =~f n'f~ V f ~ Y-c}"
This time, IH g
m
characterized by
is in IH = W1, and the isomorphic space
I~ g
is
(25)
IH g -- {H E IH: (R H)f-- Jfg V
f ~ Y-c},
where
Jfg is
the intensity ~f n. Jg through face f and R the edge-to-faces
incidence matrix. As in Chapter 6, we shall abridge all this as follows:
]l-I g ---
{H E
IH : LH-- Lg},
where L is a submatrix of R, the dimensions of
which are (F- Fc) x E (F inner faces in D, minus the F c conductive
faces) and L g a known vector. Denoting by IH ° the kernel of L in IH,
we may now reformulate (24) like this: find H
~ IH g
such that
(26)
ico (Ml([.t) H, H') + (M2(13 -1) RH ,R H') =0 k/ H'E IH 0.
This is, as in Chapter 6, a constrained linear system, which can be rewritten
as follows:
(27)
i
co M I(B) + R t M2((~ -1 )R L t
L 0
0
L g
where the dimension of the vector-valued Lagrange multiplier v
is F-F c.
Exercise 8.2. Find a physical interpretation for the yrs.
One can very well tackle the system (27) as it stands (cf. Appendix B,
Remark B.3). But for the same reasons as in Chapter 6, one may prefer to
use a representation of IH g and IH ° in terms of independent degrees of
freedom. There are two main ways to do that.
8.3.2 The tree method
We suppose C contractible for a while. Let Nc, Ec, Yc denote the sets
of nodes, edges, and faces inside C. Let us form a spanning tree E
W (cf.
234 CHAPTER 8 Eddy-current Problems
5.3.2) for the mesh of the closure of D - C. (Some edges of the interface
3C will belong to ET and form a spanning tree for this surface.) Recall
that for each edge of E- E c
-
E T, or co-edge, one can build a closed chain
over D - C by adding to it some edges of E T, with uniquely determined
coefficients. The idea is to select as independent DoFs the mmf's along
the E C edges inside C and the
E T
edges of the tree. Then, the mmf's
along the co-edges can be retrieved by using (25), as explained in the
next paragraph. This vector of independent DoFs E~vill be denoted u.
The corresponding vector space, isomorphic to C Ec ÷ , is denoted U.
For each edge e of E c • E W set
H e -- U e.
Any other edge e e E is a
co-edge, and is thus the closing edge of a circuit all other edges of which
come from E W by construction. Call C (e)
c E T
the set composed of
these other edges, and c the chain-coefficient assigned to edge ~ e c (e)
by the procedure of 5.3.2. The circuit they form bounds a two-sided 6
and hence orientable polyhedral surface Ze" formed of faces of J"- Fc.
Each of the two possible fields of normals on E e orients its boundary
3~et as
we saw in Chapter 5. Let n be the one for which e and
3~ e are
oriented the same way. Now, assign the value
(28)
II e =
IZe n. ~g + Y_, ~, c(e) C~ U~,
as DoF to co-edge e. This completes the mmf vector H, hence a field H
--
E e ~ E He
We' associated with u. We'll denote this correspondence by H
= flU, Jg). (Beware: u is a vector, H
is a
field.) Let now
(29)
IKm g= {H" H - Ze~ E HeWe} ----- {f(ut Jg)" U E U}
be the span of these fields in IH m, and IKm ° be the parallel subspace,
which is obtained by exactly the same construction, but with
Jg --
0.
Thus constructed, IKgm and IK °m coincide with IH gm and IH°m .
As a bonus, the above construction gives an approximation
H g E W 1
of the source-field Hg: the field corresponding to u = 0, that is
m m
Hgm--
f(0, Jg). Note that
IK gm
= Hgm +
IK°m,
and hence IH gm = Hgm +
IH°m
as
well, that is to say,
(30)
IHgm -- {Hgm + H" H E
IH°m}.
6This is
not
supposed to be obvious (but please read on, and return to the present note
at leisure). The circuit based on a co-edge can be a knot of arbitrary complexity, so it's not
so clear that it always bounds an orientable and non-self-intersecting surface. But this is
true, being a theorem in knot theory. Such a surface, called a
Seifert surface
(cf., e.g., [Ro]),
always exists [Se], however tangled the knot may be. See Fig. 8.9 in Exercise 8.3 at the end.
8.3 BOUNDED DOMAINS: TREES, H-O 235
Since IK g and IK ° coincide with II-I g and IH ° all we have to do
m m m /
now is throw into (24) the expressions H = f(i3, 3 g) and H'= flU', 0) in
order to obtain a linear system in terms of u, the form of which is
(31) (ico M + N) u-r g
with M and N symmetrical, non-negative definite, and M + N regular.
But this time M and N largely differ from Ml(~t) and RtM2((~-I)R
(only the blocks relative to the edges of E c coincide), and overall, their
conditioning greatly depends on the tree construction. (To each spanning
tree corresponds a particular basis for the space IH2.) Not all trees are
thus equivalent in this respect, and finding methods that generate good
spanning trees is a current research subject.
The matrix ico M + N is not Hermitian, and this raises specific
algorithmic problems. So here begins the numerical work (to say nothing
of the programming work, which is far from run-of-the-mill), but we shall
stop there, because the modelling work is donemat least in the case when
C is contractible.
So how can the technique be generalized to the non-contractible case?
If there are only "holes", i.e., if C is simply connected but with a
non-connected boundary, no difficulty: Just build a spanning tree for
each connected component of D- C. The problem is with "loops".
Suppose for definiteness there is a single loop in C, as in Fig. 8.6. Then,
by a deep but intuitively obvious result of topology ("Alexander's duality",
cf. [GH]), there is also one loop in D- C. There are now two kinds of
co-edges, depending on whether the circuits they close surround the
conductive loop or not. (Note that those which do surround the loop do
not bound a polyhedral surface of the kind discussed above, that is,
made of faces in y-- Y-c, and this is what characterizes them.) Next,
select one of these loop co-edges, and add it to the initial tree, thus obtaining
a "belted tree". Thanks to this added edge, the circuits of all remaining
co-edges do bound, as we noticed in 5.3.2. Obviously (by Amp6re), the
DoF of the belt fastener is the intensity in the current loop. There is one
additional DoF of this kind for each current loop. With this, the key
result
(IKgra
and IK°m coincide with IHgm and I~m) stays valid, and
everything else goes unchanged.
8.3.3 The .--F method
The H-~ method is "edge elements and nodal elements in association"
and stems from the second way to obtain a set of independent degrees of
freedom. With the previous method, the DoFs were all magnetomotive
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