Mathematical Physics - Volume II - Numerical Methods

Chapter 3. Comparison of finite element method and finite difference method

54

obtained in the following form:

    W 1 3 l 3 l

   

W 2

W 3 − 6 − 3 l 6 + 6

W 4

W 5

W 6

6 3 l

3 l l 2

0 0 0 0

2 l 2

2 EI l 3

[ K ] =

− 6 − 3 l

− 3 l + 3 l − 6 3 l

l 2

− 3 l + 3 l 2 l 2 + 2 l 2 − 3 l

l 2

0 0 0 0

− 6 3 l

− 3 l l 2

6

− 3 l

− 3 l 2 l 2

where

( 1 )

W 1 = W W 2 = W W 3 = W W 4 = W W 5 = W W 6 = W

1

( 1 )

2

( 1 )

( 2 )

= W = W

3

1

( 1 )

( 2 )

4

2

( 2 )

3

( 2 ) 4 , wherein we can remove rows and columns related to W 1 , W 2 , W 5 i W 6 , since they equal zero due to boundary conditions. Thus we obtain

l 3

12 0 0 4 l 2 =

l 3

l 2

2 EI

16 EI

12 0 0

(3.32)

[ K ] =

Global mass matrix is obtained in the same way:

ρ Al 420

l 2

156 0 0

[ M ] =

(3.33)

If we solve the eigenvalue problem:

[ K ] W = λ [ M ] W

(3.34)

where W = frequencies: W 3 W 4 is the eigenvalue vector, and λ is the eigenvalue. We obtain two natural (3.35) By comparing coefficients in the above expressions obtained by analytical and numerical methods (finite difference and finite element methods, Table 3.1) with exact results, it can be concluded that the finite element method is more efficient in the case of simple modeling. ω 1 = 22 , 7 L 2 r EI m , ω 2 = 82 , 0 L 2 r EI m .

Table 3.1: Result comparison.

Exact Analytical FDM FEM 21,46 22,48 15,59 22,7 61,6 124,1 29,85 82,0