Mathematical Physics - Volume II - Numerical Methods

3.4 Finite element interpolation

65

where ( ξ i , η i ) are node coordinates. By solving this system of linear algebraic equation in terms of a 1 , a 2 and a 3 , we obtain: a 1 = v 1 ,

a 2 = v 2 − v 1 , a 3 = v 3 − v 1 ,

(3.74)

By replacing (3.74) into (3.72), the following is obtained: v e h ( ξ , η ) = v 1 ψ e 1 ( ξ , η )+ v 2 ψ e

2 ( ξ , η )+ v 3 ψ e

3 ( ξ , η ) ,

(3.75)

where ψ e

i ( ξ , η ) are element shape functions: ψ e

1 ( ξ , η ) = 1 − ξ − η , ψ e 2 ( ξ , η ) = ξ , ψ e 3 ( ξ , η ) = η .

(3.76)

v , ( ) ξ η e h

v

v 1

v 2

1

e

2

v 3

3

ψ 3 ,

( ) ξ η e

1

1

2

e

3

1

ψ 2 ,

( ) ξ η e

1

e

2

3

ψ 1 e

, ( ) ξ η

1

y

1

e

2

x

ξ

3

η

Figure 3.11: Interpolation functions of a triangular finite element.