Mathematical Physics - Volume II - Numerical Methods

2.1 Finite element application to solving of one-dimensional problems

39

By replacing the boundary conditions (2.51) - (2.53) into equation (2.50), and solving for C 0 , C 1 and C 2 , we obtain: C 0 = − ku i , (2.54) C 1 = − k h ( u i + 1 − u i − 1 ) , (2.55) C 2 = − ( 2 h ) 2 k ( u i + 1 − 2 u i + u i − 1 ) . (2.56)

By replacing (2.54) - (2.56) into (2.50) and introducing ξ = 2 z / h we obtain:

1 2 ξ ( 1 + ξ ) ·   u i − 1 u i u i + 1

  .

u = − 1

2 ξ ( 1 − ξ ) ( 1 − ξ )( 1 + ξ )

(2.57)

We can now introduce ψ 1 = − 1 square interpolation functions, Fig. 2.3.

2 ξ ( 1 − ξ ) , ψ 2 = ( 1 − ξ )( 1 + ξ ) , and ψ 3 = 1

2 ξ ( 1 + ξ ) , i.e. the

ψ (ξ) 1

ψ (ξ) 2

ψ (ξ) 3

1

1

ξ

1

2

3

Figure 2.3: Quadratic interpolation function.

It is possible to introduce higher order interpolation functions in the same way, i.e. in the general case, Lagrange interpolation can be applied to an interval of [ − 1 , 1 ] , wherein we adopted a non-dimensional coordinate system ξ = 2 z / h in the case that all elements have the same length h . In this general case, the non-dimensional coordinate ξ is introduced using the following expression:

2 z − ( z k + 1 + z 1 ) z k + 1 − z 1 .

ξ =

(2.58)

In accordance with designations shown in Fig. 2.4.

. . .

k k +1

1 2 3 4

ξ

ξ =-1

ξ =+1

0

Figure 2.4: Non-dimensional coordinates ξ .

In the general case, the i -th Lagrange interpolation function of the k -th order is:

( ξ − ξ 1 )( ξ − ξ 2 ) . . . ( ξ − ξ i − 1 )( ξ − ξ i + 1 ) . . . ( ξ − ξ k + 1 ) ( ξ i − ξ 1 )( ξ i − ξ 2 ) . . . ( ξ i − ξ i − 1 )( ξ i − ξ i + 1 ) . . . ( ξ i − ξ k + 1 ) .

ψ i ( ξ ) =

(2.59)

Apparently, the following holds:

ψ i ( ξ j ) =

1 for i = j 0 for i̸ = j

(2.60)

,