Quadratic Approximation ( p = 2 )

Shape functions , i = 1,2,3 for this case are given by (equation 8.6):

	(11.16)
	(11.17)
	(11.18)

Since p = 2, the mapping function (equation 11.7) becomes:

(11.19)

Note that the nodes 1 and 3 are at the end points of the element. Let us choose node 2 at the midpoint. Then

(11.20)

One can obtain the shape functions in terms of x by substituting equations (11.19-11.20) into the expressions (11.16-11.18). But this procedure is cumbersome. Therefore, we follow an alternate procedure which is more convenient. This procedure involves the use of expression (11.1) for the mapping function. It is described in the next paragraph.

Let ,  and  be the natural coordinates of the nodes 1, 2 and 3 respectively. Note that node 1 is at the left end (  = -1) and the node 3 is at the right end (  = 1). Further, the node 2 is at the center (  = 0). Thus,

(11.21)

The relation between the physical coordinate  of the node i and its natural coordinate (_i ) can be obtained from equation (11.1) as follows:

,
,
,	(11.22)

Therefore, combining equations (11.1) and (11.22), we get

(11.23)

Further, from equation (11.22), we get

,	(11.24)
,	(11.25)
,	(11.26)

Substituting equations (11.23 -11.26) in expressions (11.16 -11.18) and canceling out the factor a² from the numerator and denominator, we get

,	(11.27)
,	(11.28)
,	(11.29)

Using the values of ,  and  (equation 11.21), the above expressions can be simplified to

,	(11.30)
,	(11.31)
,	(11.32)

These shape functions are shown in Fig 11.3. The quantity h_k  denotes the element length . The nodes are marked by the symbol x .

Fig. 11.3  Shape functions for quadratic approximation (p = 2)
(a) For physical element k        (b)  For master element