2.1 The Electromagnetic Model Equations
23
R
y
klm (x, y, z) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
y − lδy
δy
for kδx ≤ x ≤ (k + 1)δx,
lδy ≤ y ≤ (l + 1)δy, mδz ≤ z ≤ (m + 1)δz
0
o t h e r w i s e
(2.17)
S
y
klm (x, y, z) =
⎧
⎪ ⎪ ⎨
⎪ ⎪ ⎩
(l + 1)δy − y
δy
for kδx ≤ x ≤ (k + 1)δx,
lδy ≤ y ≤ (l + 1)δy, mδz ≤ z ≤ (m + 1)δz
0
o t h e r w i s e .
(2.18)
Rewriting (2.11) and (2.12) in terms of these new functions gives
E
x
klm =
KLM
J
x
KLM
R x
KLM (x, y, z)
σ KLM
+
S x
K+1LM (x, y, z)
σ K+1LM
R
x
klm (x, y, z) + S
x
k+1lm (x, y, z)
dxdydz
E
y
klm =
KLM
J
y
KLM
R
y
KLM (x, y, z)
σ KLM
+
S
y
KL+1M (x, y, z)
σ KL+1M
R
y
klm (x, y, z) + S
y
kl+1m (x, y, z)
dxdydz ,
(2.19)
where σ KLM is the (uniform) conductivity of flaw cell KLM.
Because of the compact support of the basis functions, we have
R
x
KLM (x, y, z)R
x
klm (x, y, z)dxdydz =
1
3
δxδyδzδ kK δ lL δ mM
(2.20)
R
x
KLM (x, y, z)S
x
klm (x, y, z)dxdydz =
1
6
δxδyδzδ kK δ lL δ mM
(2.21)
R
y
KLM (x, y, z)R
y
klm (x, y, z)dxdydz =
1
3
δxδyδzδ kK δ lL δ mM
(2.22)
R
y
KLM (x, y, z)S
y
klm (x, y, z)dxdydz =
1
6
δxδyδzδ kK δ lL δ mM .
(2.23)
Putting these results into (2.19) gives
E
x
klm =
1
3
J x
klm
σ klm
+
1
6
J x
k+1lm
σ k+1lm
+
1
6
J x
k−1lm
σ klm
+
1
3
J x
klm
σ k+1lm
δxδyδz
(2.24)