Page 36 - Using ANSYS for Finite Element Analysis A Tutorial for Engineers

IntroductIon to FInIte element AnAlysIs • 23

d 2 () − 12 − 2 −
x = = 1 2 1 2 = 11 = 11 .
2 a − ( 10) − 10
0 3 1

d 3 () − 1 3 2 −
x = = 1 2 − 4 1 = 14 = 14 .
3 a − ( 10) − 10
0 4 3

1.2.3.2 inversion

[]{} c
ax = {}
−1
−1
a [] []{} = [] {}
a
a x
c
I []{} = [] {}
−1
a
c
x
−1
a
∴{} = [] {} c
x
Example:
2
−  1 3 −  x 1   
2
   x   
1
 2 −4 2   2  =   using inversion,

 

  0 4 1  x   
3
  3
x 1   12. 11 02  2  .
 2
41 
.
.

  

∴  x x    01 02   =  . 
11 
.
 = 02.
.
1

2
 x     − 
 3
.
 3   −  08. −04 02. .    14 
1.2.3.3 gaussian elimination
General system of n equations with n unknowns:
a  a  x   c 
1
 11 n 1   1   

    =  
 


nn 
 a  n1 a  x n   c n 