2
3 e e
(η )
= ∑ N i ηi
e2
$
$
i =1
( ) ( N N ) ( N N )
Ne 2
e
e
e
e
(26)
1
1
2
1
3
{ } ( ) ( N ) ( N N ){η }
T
e2
= ηe N e N1
e
e
e
e
$
$
2
2
2
3
( ) (N N ) (N )
e2
N 3 N1
e
e
e
e
3
2
3
( x 1 ≡ x,
x 2 ≡ y ), i.e. Equation 25
where α = 1, 2 is the dummy-index notation and
represents the sum for α = 1 and α = 2. Note that { η e}T = [ η 1e η 2e η 3e] from Equation
$ $ $
$
20.
(
)
~
We may also assume that the coefficients ~, b in Equation 19 vary linearly on
a
element e:
3
3
~e
~
∑ N i a i , b = ∑ N ei b ie
~e =
e ~e
a
(27)
i =1
i =1
For the first part of Equation 19, we may write
(
)
1
~
I1 = ⌠ ⌠ ~(∇ η) - b η2 dA
2
a $
$
⌡⌡ 2
Ω
(28)
{ } [ ]{η }
1
= ∑ ηe
T
e
e
K $
$
1
2 e∈W 1x 3
3x3
3x1
where
⌠ ⌠ e N e N ej
~e
= ~β N β i
dxdy - b β ∫ N β N e N ej dxdy
∫
e
ae
e
K 1,i , j
(29)
x
i
α xα
⌡⌡
e
e
16
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