Piecewise Quadratic Basis Functions

Figure 8.3 shows the piecewise quadratic basis functions for a 4-element mesh. Note that, now, there are 2 nodes per element or totally 9 nodes. Further, the figure also shows the element numbering system and the global notation for the nodes and coordinates.

Figure 8.3: Piece wise quadratic basis functions for 4 - element mesh

To generate these shape functions, we note that, like the piecewise linear basis functions of Fig. 4.3, these functions should satisfy the following requirements:

• Each function must be 1 at node 'i' and vanish at all other nodes. Thus,

(8.1)

• Each function must have a local (or compact) support. Thus, each must vanish outside a certain interval of the domain.
  
• The set of must be linearly independent. That is, the solution of the equation involving scalar coefficients

for x [0, L ]

(8.2)

must imply that

for all i .

(8.3)

But, like the functions of Fig.4.3, the variation of these functions is not linear. Instead,

• Each function has quadratic variation with 'x'.

To generate the mathematical expression for these functions, we proceed as follows. We consider a typical element and the three basis functions which are non-zero over this element. This is shown in Fig. 8.4

Figure 8.4: Typical element k, the non-zero basis functions and

the local notation for the nodes, coordinates and basis functions

Note that, the basis functions which are non-zero on the element are , and . For the convenience of the element wise evaluation of the stiffness matrix and the force vector, at the element level, we introduce a local notation for these basis functions. In the local notation, we call them as , and . Similarly we introduce a local notation for the nodal coordinates images , and of the element . In the local notation, we call them as , and . Further, for the nodes 2k-1, 2k and 2k+1 of the element , we introduce a local numbering system. Now, we number these nodes as 1, 2 and 3. The local notation for the basis functions, the coordinates and the nodes is shown in Fig. 8.4.

Since each basis function is a quadratic function of 'x', we can write it as

,

for i = 1,2,3.

(8.4)

The unknown constants , and can be determined from the following condition. Note that at node 'j', that is at , the value of the function is '1' if i = j and '0' if i ? j .Thus,

(8.5)

Evaluating equation (8.4) at and then substituting in equation (8.5), we get 3 equations for 3 unknowns , and . By solving these equations for , and and substituting these expressions in equation (8.4), we get the basis functions in terms of the coordinates . We do these for each 'i'. The expressions for the basis functions are:

(8.6)

The above equations can also be written as

for i =1,2,3.

(8.7)

At the element level, these basis functions are called as the Lagrangian shape or interpolation functions .