∫ i
N Pds = LP
(88)
P
1 P
3
∫
N iP ds =
L
(89)
4
P
()
()
1 P
2
2
∫ N i N j ds = ∫N i N j ds =
P
P
P
P
L
(90)
12
P
P
Now substituting the Equations 80 and 81 into Equation 78, we have
~ e
~
η
$
∑
ds + ∑
N iP ~
∫ a
∫ a ∇ η∇ N i dΩ -
∫
∫
∫
-
b ηN i dΩ = 0 (i = 1, ..., N)
e
$
$
(91)
e
n
Pi
ei
P
e
(I)
(II)
(III)
where ei refers to those elements around node i where Nie(x,y)=0, and Pi refers to the two
boundary segments on each side of node i when i is a boundary node.
For element e with three nodes i, j, and k, where η , given by Equation 82, is
$
substituted into the ith, jth and kth Equation of 91, and hence the second terms come out
(∫∫~∇ N dΩ )η + (∫∫~∇ N ∇ N dΩ )η + (∫ ~∇ N ∇ N dΩ )η
∫a
IIe =
2
e
e
e
e
e
a
a
$
$
$
i
i
i
i
j
j
i
k
k
e
e
e
= A ei ηi + A ej η j + A ek ηk
$
$
$
(92)
i
i
i
IIej = A eji ηi + A ejj η j + A ek ηk
$
$
$
(93)
j
IIe = A ei ηi + A e j η j + A ek ηk
$
$
$
(94)
k
k
k
k
Likewise, the third terms in Equation 91 become
31
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