130
5 An Electromagnetic Model for Anisotropic Media: Green’s Dyad for Plane-. . .
Q
(zz)
mM =
δxδyδz
6
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
1
σ
z
klm
if M = m − 1
2
σ
z
klm
+
2
σ
z
klm+1
if M = m
1
σ
z
klm+1
if M = m + 1
(5.29)
The first entry in each of these matrices is the lower diagonal, the second the main
diagonal, and the third the upper diagonal.
The Q’s contain the reciprocals of the anomalous conductivities, which may
vanish when a flaw cell contains only host material. This condition forces the
corresponding unknowns to vanish, but we are still left with an indeterminate ‘zero
over zero’ expression to evaluate. In order to overcome this problem we form the
reciprocals of the diagonal entries of the Q’s of (5.29),
ν
(xx)
klm =
6
δxδyδz
σ x
klm σ x
k+1,lm
2(σ x
klm + σ x
k+1,lm )
ν
(yy)
klm =
6
δxδyδz
σ
y
klm σ
y
k,l+1,m
2(σ
y
klm + σ
y
k,l+1,m )
ν
(zz)
klm =
6
δxδyδz
σ
z
klm σ
z
kl,m+1
2(σ
z
klm + σ
z
kl,m+1 )
,
(5.30)
and multiply both sides of (5.18) by them. The resulting Q matrices are also tridiagonal, with the following non-zero entries:
Q
(xx)
kK =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
σ x
k+1,lm
2(σ x
klm + σ x
k+1,lm )
if K = k − 1
1
i fK = k
σ x
klm
2(σ x
klm + σ x
k+1,lm )
if K = k + 1
Q
(yy)
lL
=
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
σ
y
k,l+1,m
2(σ
y
klm + σ
y
k,l+1,m )
if L = l − 1
1
i fL = l
σ
y
klm
2(σ
y
klm + σ
y
k,l+1,m )
if L = l + 1