Page 207 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
As before, the diffusion term can be integrated after applying Green’s lemma. The
diffusion matrix for the elements inside the domain is
∂[N] ∂[N] ∂[N] ∂[N]
T T 199
[K 1e ] = k + k d

∂x 1 ∂x 1 ∂x 2 ∂x 2
b b i b j b i b k c c i c j c i c k
 2   2 
k i k i
=  b j b i b 2 j b j b k  +  c j c i c 2 j c j c k  (7.115)
4A 2 4A 2
b k b i b k b j b c k c i c k c j c
k k
The stabilization matrix is
T
T
t ∂[N] ∂[N] ∂[N] ∂[N]
[K 2e ] = u 1 u 1 d
+ u 2 d
2
∂x 1 ∂x 1
∂x 1 ∂x 2
T
T
t ∂[N] ∂[N] ∂[N] ∂[N]
+ u 2 u 1 d
+ u 2 d
2
∂x 2 ∂x 1
∂x 2 ∂x 2
2
 
u 1 b + u 2 b i c i u 1 b i b j + u 2 b i c j u 1 b i b k + u 2 b i c k
u 1 t i 2
=  u 1 b j b i + u 2 b j c i u 1 b + u 2 b j c j u 1 b j b k + u 2 b j c k 
j
4A 2 2
u 1 b k b i + u 2 b k c i u 1 b k b j + u 2 b k c j u 1 b + u 2 b k c k
k
u 1 c i b i + u 2 c u 1 c i b j + u 2 c i c j u 1 c i b k + u 2 c i c k
 2 
u 2 t i 2
+  u 1 c j b i + u 2 c j c i u 1 c j b j + u 2 c j u 1 c j b k + u 2 c j c k  (7.116)
4A 2 3
u 1 c k b i + u 2 c k c i u 1 c k b j + u 2 c k c j u 1 c k b k + u 2 c k
The forcing vectors along the boundary edges are (assuming ij as the boundary edge)
 
N i

∂N i ∂N j ∂N k n
[f 1e ] = k  N j  {φ} d n 1
0 ∂x 1 ∂x 1 ∂x 1
 
N i

∂N i ∂N j ∂N k
+ k  N j  {φ} d n 2
∂x 2 ∂x 2 ∂x 2
0
 
b i φ i + b j φ j + b k φ k
= k b i φ i + b j φ j + b k φ k n 1


4A
0
 
c i φ i + c j φ j + c k φ k
+ k c i φ i + c j φ j + c k φ k n 2 (7.117)


4A
0
 
N i
t ∂N i ∂N j ∂N k n

[f 2e ] = u 1 u 1 N j  {φ}

2 0 ∂x 2 ∂x 2 ∂x 2
 
N i
t ∂N i ∂N j ∂N k n

+ u 1 u 2 N j  {φ} d n 1

2 0 ∂x 2 ∂x 2 ∂x 2