Page 38 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

IntroductIon to FInIte element AnAlysIs • 25
Solve using back substitution:

c n−1
x = n
n
n−1
a nn
n
x = 1  a − ∑ a x r  
ir
i
a   1, n+1 ri=+1 
ii
Example:
 22 1 x 1   9
     

 21 0  x 2  =  4 


  11 1 x 3   
 
6
 
1. Eliminate the coefficient of x in every equation except the first one.
1
Select a as the pivot element.
11
• Select a = 2 as the pivot element.
11
• Add the multiple −a / a 11 = − 22 = −1 of the first row to the
/
21
second row.
1
• Add the multiple −a / a 11 = −= − 05 of the first row to the
.
31
third row. 2
2  2 1  x 1   9 
 
 0 − 1 − 1  x   

   2  =− 5 



 
.
.
 0  0 05 x 3   15 
2. Eliminate the coefficient of x in every equation below the second
2
′
one. Select a as the pivot element (already done in this example).
22
2  2 1  x 1   9 
 
 0 − 1 − 1  x   

  2  =− 5 


0   0 05 x 3   15 

 

.
.
3. Solve using back substitution:
c ( )
3
''
x = 3 = 2 = 3
3
a 33 ( )
''
1
2
( −+
53)
x = = 2
2 − 1