Page 202 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
194
The boundary term from the diffusion operator is integrated by assuming that i is a
boundary node, as follows:
n
∂[N] n N i ∂N i ∂N j φ i

T
[N] k {φ} d = k d
∂x 1 0 ∂x 1 ∂x 1 φ j
 
∂N i ∂N j n

N i N i φ i
= k  ∂x 1 ∂x 1  d
0 0 φ j
1 1 n
 
− φ i
= k  l l 
00 φ j
={f 1e } (7.98)
where {f 1e } is the forcing vector due to the diffusion term, that is,
  n
 φ i φ j 
− +
{f 1e }= k l l (7.99)
0
 
The boundary integral from the characteristic Galerkin term is integrated, again by
assuming that i is a boundary node, as
n
t T ∂[N] n 2 t N i ∂N i ∂N j φ i

2
u [N] {φ} d = u d
1 1
2 ∂x 1 2 0 ∂x 1 ∂x 1 φ j
 
∂N i ∂N j n

t N i N i φ i
2
= u   d
1 ∂x 1 ∂x 1
2 0 0 φ j
 
1 1 n
t − φ i
2
= u  l l 
1
2 φ j
00
={f 2e } (7.100)
where {f 2e } is the forcing vector due to the stabilization term
  n
t  φ i + φ j 
2
−
{f 2e }= u 1 l l (7.101)
0
2  
The forcing vectors are formulated by assuming that the node i is a boundary node.
Because of the opposite signs of the outward normals at the interface between any two
elements within the domain, these forcing vector terms vanish for all nodes other than the
boundary nodes. The remaining terms will have a value only at the domain boundaries. Also,
the boundary terms due to the CG stabilizing operator (Equation 7.101) can be neglected
during the calculations without any loss in accuracy.