where β = 1, 2, 3 is another dummy-index notation. Since
e
Ne N j
(b b
)
1
=
+ cic j
i
(30)
()
i
j
2
xα xα
2∆e
and
⌠ ⌠ N e dxdy = ∆
e
(31)
β
⌡⌡
3
e
we have
~e + ~e + ~e
⌠ ⌠ e N e N ej
a
a2 a3
(
)
~
a Nβ
dxdy = 1
bi b j + cic j
e
i
(32)
x
β
12 ∆
e
α xα
⌡⌡
e
The second term of Equation 29 is
~e
bβ ∫ N β N e N ej dxdy
∫
e
i
e
()
()
~
~
~e
2
2
N ej dxdy + b e ∫ N e N ej dxdy + b k ∫ N e N ej N e dxdy
j∫ i
= b ie ∫ N e
∫ i
∫ i
(33)
k
e
e
e
∆e ~ e
(
)
~ ~e
=
2 bi + 2b e + b k
j
60
for i ≠ j , and
17
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