Page 201 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

193
CONVECTION HEAT TRANSFER
where [C e ] is the elemental convection matrix, that is,

[C e ] = u 1 −11 (7.93)
2 −11
The values of the derivatives of the shape functions are substituted in order to derive
the above matrix. The diffusion term within the domain
is integrated as
 
∂N i
∂[N] ∂[N] n  ∂x 1  ∂N i ∂N j φ i
T n
k d
{φ} =   k d

∂x 1 ∂x 1
 ∂N j  ∂x 1 ∂x 1 φ j
∂x 1
 
∂N i ∂N i ∂N i ∂N j
n

 ∂x 1 ∂x 1 ∂x 1 ∂x 1  φ i
= k   d

 ∂N j ∂N i ∂N j ∂N j  φ j
∂x 1 ∂x 1 ∂x 1 ∂x 1
n

k 1 −1 φ i
=
l −1 1 φ j
n
= [K 1e ]{φ} (7.94)
where [K 1e ] is the elemental diffusion matrix, that is,
k 1 −1
[K 1e ] = (7.95)
l −1 1
The characteristic Galerkin term within the domain
is integrated as
 
∂N i
T
t ∂[N] ∂[N] t  ∂x 1  ∂N i ∂N j φ i n

n
u 2 1 {φ} d
= u 2 1   d
2
∂x 1 ∂x 1 2
 ∂N j  ∂x 1 ∂x 1 φ j
∂x 1
 
∂N i ∂N i ∂N i ∂N j
t  ∂x 1 ∂x 1 ∂x 1 ∂x 1  φ i
2
= u 1   d
2
 ∂N j ∂N i ∂N j ∂N j  φ j
∂x 1 ∂x 1 ∂x 1 ∂x 1
n

t 1 1 −1 φ i

2
= u 1 2 l −1 1 φ j
n
= [K 2e ]{φ} (7.96)
where [K 2e ] is a stabilization matrix,
t 1 1 −1
[K 2e ] = u 2 1 (7.97)
2 l −1 1