7.6 Three-dimensional modeling and simulations Ë 247level. A prominent advantage of theTmethod lies in the fact that the state variableT
is only defined in the conducting region, avoiding the mesh of the domain outside the
conducting regions. Hence, this considerably reduces the number of degrees of free-
dom when the FEM technique is adopted to numerically discretize the mathematical
equations.
From a macroscopic point of view, the Maxwell’s equations are still valid to
describe the electromagnetic behavior of the HTSC. According to Eq. (7.1) in the quasi-
static approximation, the current densityJis a divergence-free vector, which allows
the introduction of a current vector potentialTexpressed as
J=∇×T, (7.64)
to which the Coulomb gauge is applied to guarantee the uniqueness of the solution,
i.e.∇ ⋅T=0.
Applying the Helmholtz’s theorem to vectorTyields
C(P)T(P)=^1
4 휋
X
V(∇耠⋅T(P耠))∇耠^1
R(P,P耠)
dV耠−^1
4 휋X
S(n耠⋅T(P耠))∇耠^1
R(P,P耠)dS耠+^1
4 휋
X
V(∇耠×T(P耠))×∇耠^1
R(P,P耠)
dV耠−
1
4 휋X
S(n耠×T(P耠))×∇耠1
R(P,P耠)dS耠, (7.65)
whereR(P,P耠) is the distance between the source pointP耠and the field pointP, the
superscript耠refers to the quantity at the source point,n耠is a unit vector out of the
surfaceS耠, and the coefficientC(P) takes the following values [30]:
C(P)=
. (^66)
(^66)
F
1 , P∈V耠(excludingS耠),
1 / 2 , P∈S耠,
0 , elsewhere.
(7.66)
According to the physical fact that the normal component ofJmust be zero on the
surface of the HTSC, i.e.Jn= 0 ,Tmust fulfil the boundary condition that [30]
n耠×T= 0. (7.67)Therefore, only the normal componentTnexists on all surfaces of the HTSC.