14.7 Isoparametric Elements and Numerical Integration

In Section 14.4 the concept of shape functions has been introduced. Within each element u _h has been written as

The shape functions are simple polynomial expressions in terms of the coordinate x. For instance for a linear interpolation the shape functions are linear polynomials, according to

where x ₁ and x ₂ denote the position of the nodes of the element. In this case the shape functions are linear functions of the global coordinate x. It is appropriate in the context of a generalization to more-dimensional problems to introduce a local coordinate ?1 ? ? ? 1 within each element such that ? = ?1 and ? = 1 correspond to the edges of the element. With respect to this local coordinate system, the shape functions may be written as

This is visualized in Fig. 14.5.

Figure 14.5: Shape functions with respect to the global x-coordinate.

Computation of components of the element coefficient matrix and the element load array requires the evaluation of integrals of the form

Figure 14.6: Shape functions with respect to the local ?-coordinate.

These integrals may be transformed into integrals using the local coordinate system, according to

This requires the computation of the derivative dx/ d ? . For this purpose the concept of isoparametric elements is introduced. The shape functions N _i introduced for the interpolation of the unknown function u _h