~e
bβ ∫ N β N e N ej dxdy
∫
e
i
e
()
()
( ) dxdy
~e
~e
~e
e 3
2
2
= bi ∫ N i dxdy + bk1 ∫ N e 1 N e
∫
∫ k i
dxdy + bk2 ∫ N e 2 N e
∫ k i
(34)
e
e
e
∆e ~ e ~ e ~ e
(
)
=
3bi + bk1 + bk 2
30
for i = j ≡ i , where k1 & k2 are the other two nodes of element e. Now, Equation 29 can
be written as
~ + ~ + ~
a a2 a3
(
)
= 1
bi b j + cic j -
e
K
1,i , j
12∆e
∆e ~ e
(
)
~ ~e
2 bi + 2 b e + b k
when i ≠ j
60
j
(35)
∆e
(
)
~ ~e ~e
3b ie + b k1 + b k 2
when i = j ≡ i
30
[]
e
After computing the element matrix K 1 for all the elements (e = 1, 2, 3, ... , E), where E
is the total number of elements, we can assemble them into a "global" matrix [K1].
Equation 28 becomes
{ } [ ]{ }
1
1
{η}[ 1 ] η}
$ K {$
∑Ω ηe
T
T
I1 =
ηe =
e
K1
(36)
$
$
2 e∈ 1x3
2
3x3
3x1
In (36)
{η} = {η1 , η2 , η3 ,
⋅⋅⋅ ⋅⋅, ηN }
T
⋅$
$
$$$
(37)
where N is the total number of nodes in domain Ω
18
Previous Page
