6.2. SOLVING THE MAGNETOSTATICS PROBLEM
179
system is therefore singular.
difficulty. 0
We shall return to this apparent
6.2.3 "m-weak" properties
We should now proceed as in Section 4.1, and answer questions about
the quality of the approximation provided by a m • We are satisfied that
b m = rot a m is solenoidal and that n. b m = 0 on the S b boundary, but
what is left of the "weak irrotationality" of h m = ~1 b ? And so forth.
/ n
S h
0
,
//
b
sh
FIGURE 6.8. "m-weak" properties of the vector potential solution. Left: The
circulation of h m = ~'1 rot am is zero along the circuit y that joins barycenters
h • • •
around any inner edge e = {m, n}. Middle: If e belongs to S, the clrculahon is
null along the open path 7. Right: The "variationally correct" mmf is obtained
by taking the circulation of the computed h m along the "m*-line" o joining sh0
h , •
and S 1, or along any homologous m -hne o'.
It would be tedious to go through all this again, however, and there
is more fun in
guessing
the results, thanks to the analogies that Tonti's
diagrams so strongly suggest. So we can expect with confidence the
following statements to be true:
• For an~y DoF vector a, the
term (RtM2(~ -1) Ra)e
=
~D h. rot w e,
where h = ~ rot(~
e ~ E
ae We)' is the circulation of h along the smallest
closed m*-line (closed rood S, in the case of surface edges) that surrounds
edge e (cf. Fig. 6.8).
• The circulation of the computed field, h, vanishes along all
m*-lines which bound modulo S h (cf. Fig. 4.6, right).
• The circulation of h is equal to I along all m*-paths similar to
(or its homologue of) on Fig. 6.8.
Exercise
6.3. Prove all this.
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