Objectives:
• To understand the procedure of discretization and interpolation.
• To learn the algorithm of Gauss quadrature.
1. 1D element shape functions
Consider a 4-node cubic element in 1D. The length of the element is 3 with the first node located at xi = -2. The remaining nodes are equally spaced.
1) Construct the Lagrange shape functions for this element.
2) Interpolate the function ue(x) (e.g. the displacement along the axis of a uniform bar) in the element when the nodal values are given as
![728_1.png](https://secure.tutorsglobe.com/CMSImages/728_1.png)
with the same unit as the coordinates of the nodes. (Writing out the expression for u°(x) would suffice).
3) Evaluate the derivatives of the shape functions, i.e. the Be matrix, and use it to find the expression for the derivative of u°(x) (i.e. the axial strain e(x) = due /dx).
4) Plot ue(x) and a°(x) obtained in (2) and (3).
5) What are the corresponding displacement field and strain field when the nodal displacements are given by
![235_2.png](https://secure.tutorsglobe.com/CMSImages/235_2.png)
Why is this result expected?
2. Gauss quadrature
1) Write a short Matlab (or whatever language you prefer) program to perform 2 point Gauss integration of a function f over a given interval (a, b). Inputs: a, b, and the functional values of f at the two Gauss points x1 and x2, f 1) and f (x2). (The program should calculate the actual coordinates of the two Gauss points). Output: the (approximate) value of the integral ∫fdx. (If you are able, write a program with adjustable order of integration.)
2) Check your program with the following integrals by comparing to the exact values:
0∫4 (x3+1)dx, -1∫2 (x4 + 3x2)dx, -1∫1 cos2Πx dx