Page 200 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
192
integration using the following formula:

b
a
N N d
= a!b!l (7.88)
i
j

(a + b + 1)!
and therefore derive the element matrices for all the terms in Equation 7.87. The term on
the left-hand side for a single element is
 φ − φ 
 n+1 n 
{φ n+1 n  i i 


T }− {φ } N i t
[N] [N] d
= N i N j n+1 n d

t
N j  φ − φ 
 j
j 
 
t
 φ
 n+1 n 
 i − φ 
N N i N j  t 
2 i 
= i 2 n+1 n d

N j N i N j  φ − φ 
 j
j 
 
t
 φ
 n+1 n 
 i − φ 
i 
l 21  t 
= n+1 n
6 12  φ − φ 
 j
j 
 
t
{φ}
= [M e ] (7.89)
t
where [M e ] is the mass matrix. For a single element, the mass matrix is given as

l 21
[M e ] = (7.90)
6 12
The above mass matrix for a single element will have to be utilized in an assembly
procedure for a fluid domain containing many elements. In Equation 7.89
 φ − φ 
 n+1 n 
 i
{φ}  t i 

= n+1 (7.91)
n
t  φ − φ 
 j
j 
 
t
In a similar fashion, all other terms can be integrated; for example, the convection term
is given by
n
∂[N] N i ∂N i ∂N j φ i
n
u 1 [N] T {φ} d
= u 1 d

∂x 1
N j ∂x 1 ∂x 1 φ j
 
l ∂N i l ∂N j
n

  φ i
= u 1  2 ∂x 1 2 ∂x 1 
 l ∂N i l ∂N j 
φ j
2 ∂x 1 2 ∂x 1
n

u 1 −11 φ i
=
2 −11 φ j
n
= [C e ]{φ} (7.92)