Page 94 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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Note that the outward normal at node 1 is 1. Also, note that the gradient terms of
Equation 3.233 become zero at node j as N i =0at j. Now weighting the equation using
N j ,wehave THE FINITE ELEMENT METHOD
 
2

θ i µ ζ e θ i  0 
1
−11 + 21 + −dθ (3.236)
ζ e θ j 6 θ j  
dζ
In this case, the gradient term disappears for node i as N j is zero at node i. The outward
normal value of point j is −1 (see Figure 3.28). The element characteristics are given by
 
dθ
 
2  
1 1 −1 µ ζ e 21 θ i dζ


+ + (3.237)
−1 1 12
ζ e 6 θ j  dθ 
+ 


dζ
For the given problem with ζ e = 0.2, which is a non-dimensional element length, l/L
2
(Figure 3.28), and µ = 3, the element characteristics for the first element are derived as
follows:  
dθ
 
 
5.2 −4.9 θ i dζ
 
+ (3.238)
−4.9 5.2 θ j  dθ 
+ 


dζ
In a similar fashion, we can write the element characteristics equation for all the other
four elements. On assembling over all the five elements, we obtain
 
  

5.2 −4.9 0.0 0.0 0.0 0.0


0.0  θ 1
 
   
 −4.910.4 −4.9 0.0 0.0   0.0



  
0.0   θ 2
     
0.0 −4.910.4 −4.9 0.0 0.0   0.0
  θ 3
  = 0.0 (3.239)

 0.0 0.0 −4.910.4 −4.9 0.0 θ 4  
     
0.0 0.0     
0.0
 
 0.0 −4.910.4 −4.9 θ 5   
   
0.0 0.0 0.0 0.0 −4.9 5.2 θ 6  
dθ 
 
dζ


where θ 1 ,θ 2 ,... ,θ 6 are the temperature values at all the six nodes. The assembly proce-
dure has already been discussed in the previous chapter. Further details on the assembly
procedure are given in Appendix C. Note that dθ/dζ at node 1 is zero because of the zero
flux boundary condition but we also have the boundary condition at ζ = 1, as θ = 1. The
resulting nodal simultaneous equations can be written as
5.2θ 1 − 4.9θ 2 = 0.0
−4.9θ 1 + 10.4θ 2 − 4.9θ 3 = 0.0
−4.9θ 2 + 10.4θ 3 − 4.9θ 4 = 0.0
−4.9θ 3 + 10.4θ 4 − 4.9θ 5 = 0.0
−4.9θ 4 + 10.4θ 5 − 4.9θ 6 = 0.0
θ 6 = 1.0 (3.240)