Natural Coordinates

To facilitate numerical integration, we transform each (rectangular) element to a master (square) element for which the expression for numerical integration is available. The master element is a square of size two. This element is shown in Fig. 24.2. The coordinates are called as natural coordinates.

Figure 24.2 Mapping of a Rectangular Element onto the Master Element

Since the straight boundaries of a rectangular element parallel to (x, y) axes get mapped onto the straight boundaries of the master element parallel to axes, the mapping can be expressed by the following relations :

	(24.2)

Note that, x is independent of and y is independent of . To determine the unknown coefficients , we use the conditions that the nodes 1 and 2 of the rectangular element map onto the vertices (-1, -1) and (1, -1) respectively. Then, we get

	(24.3)

By solving these equations, we obtain the following expressions for :

(24.4)

Similarly, the unknown coefficients b i are determined form the conditions that the nodes 1 and 4 of the rectangular element map onto the vertices (-1, -1) and (-1, 1) respectively. Then, we get the following expressions for :

(24.5)

After substituting the mapping function expression (eq. 24.2), the finite element approximation (eq. 24.1) becomes :

(24.6)

where

	(24.7)

Note that, this expression is also a bi-linear polynomial in and .