Page 206 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
198
for the scalar variable φ into the above equation, we obtain
n+1
n
∂[N]
∂[N]

The above equation is valid globally. On substituting the global spatial approximation
{φ}
−{φ}
T
n
n
[N] [N] d
=−u 1 [N] T {φ} d
− u 2 [N] T {φ} d

t
∂x 1
∂x 2
∂ ∂[N]
n
+ [N] T k {φ} d

∂x 1 ∂x 1
∂ ∂[N] n

T
+ [N] k {φ} d

∂x 2 ∂x 2
t ∂ ∂[N] n ∂[N] n
+ u 1 u 1 {φ} + u 2 {φ} d
2
∂x 1 ∂x 1 ∂x 2
t ∂ ∂[N] n ∂[N] n
+ u 2 u 1 {φ} + u 2 {φ } d
2
∂x 2 ∂x 1 ∂x 2
(7.109)
The above equation is valid only if all the element contributions in a finite element
domain are assembled. The elemental matrices are derived by applying the following for-
mula for integration over linear triangular elements:
a!b!c!2A

a b c
N N N d
= (7.110)
i j k

(a + b + c + 2)!
and for the line integral

a b c a!b!c!
N N N d = (7.111)
k
i
i
(a + b + c + 1)!
where A is the area of a triangular element and is the length of a boundary edge. Applying
the above formulae, we obtain the element characteristic equations as follows:
The mass matrix is
 
211
A

T
[M e ] = [N] [N]d
=  121  (7.112)

12
112
The convection matrix is
∂[N] ∂[N]
[C e ] = [N] T u 1 + u 2 d

∂x 1 ∂x 2
   
b i b j b k c i c j c k
u 1 u 2
=  b i b j b k  +  c i c j c k  (7.113)
6 6
b i b j b k c i c j c k
where
b i = y j − y k ; c i = x k − x j
b j = y k − y i ; c j = x i − x k
b k = y i − y j ; c k = x j − x i (7.114)