Shape functions and numerical integration - Biomechanics: Concepts and Computation - page 303

17.4.1 Lagrangian elements

The shape functions of an element of order ( n − 1) in one dimension are chosen as

N a = l n − 1 . (17.29)

a

The quadratic shape function associated with node 1 of the element depicted in
Fig. 17.6 (with ξ 1 =− 1, ξ 2 = 0, ξ 3 = 1) satisfies

( ξ − ξ 2 )( ξ − ξ 3 ) 1

2 ξ ( ξ − 1) .

N 1 ( ξ ) = l 1 ( ξ ) =

ξ 1 − ξ 3 ) =

(17.30)

( ξ 1 − ξ 2 )(

Likewise it follows that

2 ,

N 2 ( ξ ) =− ( ξ + 1) ( ξ − 1) = 1 − ξ (17.31)

and

1
2 ξ

N 3 ( ξ ) = ( ξ + 1) . (17.32)

In two dimensions, the shape functions for the 9-node element as visualized
in Fig. 17.7 are formed by multiplication of two Lagrangian polynomials.
Leading to:

1
4 ξ ( ξ − 1) η ( η − 1)

N 1 ( ξ , η ) =

1
2 ( ξ + 1) ( ξ − 1) η ( η − 1)

N 2 ( ξ , η ) =−

ξ

1 2 3

Figure 17.6
A one-dimensional quadratic element,

− 1 ≤ ξ ≤ 1.

η

7 6 5

8 9 4

ξ

1 2 3

Figure 17.7
A two-dimensional quadratic element, −1 ≤ ξ ≤ 1, −1 ≤ η ≤ 1.