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Lagrange shape functions of degree p are constructed by partitioning into p sub‐intervals. The length of the sub‐intervals is typically but the lengths may vary. The node points are , and . The ith shape function is unity in the ith node point and is zero in the other node points:

(1.50)

These shape functions have the following important properties:

(1.51)

For example, for the equally spaced node points are , , . The corresponding Lagrange shape functions are illustrated in Fig. 1.3.

Exercise 1.5 Sketch the Lagrange shape functions for .