Isoparametric Elements

As mentioned above, to form a mesh over a general region the elements must be allowed to take more general shapes. This is done by using the parent elements and transforming them by some mapping. The essential idea underlying this centres on the mapping of the simple geometric shape in the local coordinate system into distorted shapes in the global Cartesian coordinate system. The mapping from local to global coordinates will take the form

where  is the number of points defining the geometry of the element.

The functions  will clearly satisfy the following relation

where  is the Kronecker delta. Hence there is a one-to-one correspondence between the nodes in the parent element and the distorted element in the global coordinate system.

Under special circumstances the same shape functions can also be used to specify the relation between the global  and local  coordinate systems. If this is so the element is of a type called isoparametric; the four-node quadrilateral is an example. The coordinate transformation is therefore

where the N are the shape functions for the four-noded quadrilateral element.

In the previous chapter it was shown that element properties involve not only N but also their derivatives with respect to the global coordinates (x,y) which appear in the matrices B and D. Further, products of these quantities need to be integrated over the element area or volume.

Derivatives are easily converted from one coordinate system to another by means of the chain rule of partial differentiation, best expressed in matrix form by

where J is the Jacobian matrix. The determinant of this matrix,  , must also be evaluated because it is used in the transformed integrals as follows:

Under certain circumstances, for example those shown in Figure 2.7, the Jacobian becomes indeterminate. When using quadrilateral elements, reflex interior angles should be avoided.

It is clear that these ideas are applicable to all elements whether one, two or three dimensional.