Approximation of Order p

Shape functions , i =1,2,., p+1 for this case are given by (equation 8.8):

(11.33)

If we use equation (11.7) as the expression of mapping function to represent these shape functions in terms of the natural coordinate , Then the corresponding procedure is quite cumbersome. Therefore, we follow a simpler procedure which involves the use of expression (11.1) for the mapping function. This procedure is described in the next paragraph.

Fig. 11.4 Nodal coordinates for approximation of order p
(a)   For physical element k     (b)   For master element

Figure 11.4 shows the nodal coordinates for the physical element k and the master element. We choose the nodes to be equidistant. There are p+1 nodes . Thus, the nodes divide the element length into p equal intervals. For the physical element k , the x -coordinates of the nodes can be expressed as 

(11.34)

where h_k is the element length. For the master element, the element length is 2. Further since the node 1 is at the left end, -coordinate of node 1 ( ) is 1. Therefore, the -coordinate of the nodes can be expressed as

(11.35)

Note that the relation between the physical coordinate   of node j and its natural coordinate can be obtained from equation (11.1) as follows:

(11.36)

Then, subtracting equation (11.36) from equation (11.1) we get

(11.37)

Similar to expression (11.36), we can write

(11.38)

Subtracting equation (11.36) from equation (11.38), we get

(11.39)

Substituting equations (11.37) and (11.39) into the expression (11.33) and canceling out the factor a ^p both from the numerator and denominator, we get the following expression for the shape functions:

(11.40)

Here,  = -1 and the remaining nodal coordinates are given by the expression (11.35).

Note that, we get expressions (11.14 - 11.15) for the linear shape functions when we substitute p = 1,  = -1 and = +1 in equation (11.40). Similarly, we get the expressions (11.30 -11.32) for the quadratic shape functions if we substitute p = 2, = -1, = 0, = + 1  in equation (11.40).