Approximation for the Primary Variable using Piecewise Quadratic Basis Functions

For the purpose of deriving the finite element equations, we approximate the primary variable u(x) as follows:

(8.9)

Here, are the piecewise quadratic basis functions and N n is the number of nodes of the domain. When the piecewise linear basis functions are used, the number of nodes of the domain ( N n ) is equal to the number of elements ( N ) plus one. For the case of piecewise quadratic basis functions,  N^_n = 2N +1. Thus, for 4-element mesh of Fig. 8.3, N_n is 9. Here, the unknown coefficients have the following interpretation. At node , we have

(8.10)

Using equation (8.1), we get

(8.11)

Thus, is the value of the primary variable at node ' i '. This is the characteristics of the Lagrangian interpolation functions. As stated in section 6.1, the values , j = 1, 2., are called the degrees of freedom (abbreviated as dof ).

In section 8.1, we stated some properties of the basis functions . Now we shall discuss some additional properties. For this purpose, let us differentiate equation 8.9. Then, we get

(8.12)

Further, assume that the primary variable u ( x ) is a constant function with the value unity. Thus

u ( x ) = 1,

for x (0, L ).

(8.13)

Then

for x (0, L ).

(8.14)

Note that, for such a function, the nodal value of u(x) is unity at all the nodes. Thus,

for j = 1,2 ., .

(8.15)

Substituting equations (8.13 - 8.15) in equations (8.9) and (8.12) we get

,	(8.16a)
	(8.16b)

When ' j ' is even, the basis function is completely defined over the element j /2. It is a bubble function as shown in Fig. 8.3. Note that, its derivative is not continuous at the end points and or at the nodes j -1 and j +1. When ' j ' is odd, the basis function is defined over the 2 elements ( j +1)/2 and [(( j +1)/2)-1]. Thus, it shares a common node 'j' of these 2 elements. The derivative of this function is discontinuous at the end nodes ( j -2) and ( j +2) as well as the common node 'j'. Thus, u is discontinuous at the points , and . The functions and are defined over only 1 element, namely 1 st and N th . Therefore, they are discontinuous only at the end points of these elements. Unlike the properties given by equations (8.16 - 8.17), the last property will be different when ' p ' is different.

For k^th element, the only non-zero basis functions are 2k -1, 2k and 2k +1 (see Fig. 8.3). Thus, the approximation (8.9) for the element ' k ' becomes

(8.17)

Note that the local notation for , and is , and respectively. For the convenience of the element wise evaluation of the stiffness matrix and force vector, we introduce a local notation for the degrees of freedom , and . In the local notation, we call them as , and . Then, the expression (8.17), in local notation becomes

(8.18)