Shape functions and numerical integration - Biomechanics: Concepts and Computation - page 304

1
4 ξ ( ξ + 1) η ( η − 1)

N 3 ( ξ , η ) =−

1
2 ξ ( ξ + 1) ( η + 1) ( η − 1)

N 4 ( ξ , η ) =−

1
4 ξ ( ξ + 1) ( η + 1) η

N 5 ( ξ , η ) =

1
2 (

N 6 ( ξ , η ) =− ξ + 1) ( ξ − 1) ( η + 1) η

1
4 ξ

N 7 ( ξ , η ) =− ( ξ − 1) ( η + 1) η

1
2 ξ

N 8 ( ξ , η ) =− ( ξ − 1) ( η + 1) ( η − 1)

N 9 ( ξ , η ) = ( ξ + 1) ( ξ − 1) ( η + 1) ( η − 1) . (17.33)

17.4.2 Serendipity elements

For serendipity elements no internal nodes are used. Consider a 'quadratic ele-
ment', as depicted in Fig. 17.8 (right). The shape functions of the corner nodes are
defined by

1
4 (1 − ξ )(1 − η )( − ξ − η − 1)

N 1 =

1
4 (1 + ξ )(1 − η )( + ξ − η − 1)

N 2 =

1
4 (1

N 3 =

+ ξ + η + ξ + η −

)(1 )( 1)

1
4 (1

N 4 = − ξ )(1 + η )( − ξ + η − 1) , (17.34)

while the shape functions for the mid-side nodes read

Lagrange Serendipity

7 6 5 4 7 3

8 9 4

8 6

1 2 3 1 5 2

Figure 17.8
Example of a Serendipity element compared to the 'equal order' Lagrangian element.