(
)(
)(
)
~ e2
~ e e
~ e e
∫ b N i dΩ ηi -
∫
∫ b N i N j dΩ η j -
∫
∫ b N i N k dΩ ηk
∫
IIIe = -
$
$
$
i
e
e
e
= Bei ηi + Bej η j + Bek ηk
$
$
$
(95)
i
i
i
IIIej = Beji ηi + Bejj η j + Bek ηk
$
$
$
(96)
j
IIIe = Be i ηi + Be j η j + Bek ηk
$
$
$
(97)
k
k
k
k
~
~
Assuming that the coefficients a and b also vary linearly on element e, i.e.
~ = ~ Ne + ~ Ne + ~k Ne
(99)
a
ai i a j j a
k
~
~
~
~
b = b i N e + b j N ej + b k N e
(100)
i
k
then using the relations (84), (85) and (86), we have
~
= ∫ ~ ∇ N e∇ N edΩ
∫a I J
e
A
IJ
e
(b I b J + c Ic J )∫e~ i N ei + ~ j N ej + ~ k N ek dΩ
1
∫a
=
a
a
4∆
e2
~ ~ ~
1
a i + a j + a k (b I b J + cIc J )
=
(I, J = i, j, k)
(101)
12∆e
~ e ~ e ~
e e
∫e
∫
b i N i + b j N j + b k N k N I N J dΩ
BeJ = -
e
I
∆e ~ ~ ~
~
b i + b j + b k + 2 b I for I = J
30
=-
(I, J = i, j, k)
(102)
∆ ~ ~ ~ ~ ~
e
60 b i + b j + b k + b I + b J for I ≠ J
32
Previous Page
