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Shape Functions and Local Coordinate Systems

The most useful, simple, general shapes adopted for plane elements are the triangle and the quadrilateral. Elements of the latter shape are considered in this text. Figure 2.5 shows a rectangular element and a more general quadrilateral element.

Figure 2.5: Types of quadrilateral element
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The shape functions for the rectangle are given in Chapter 1, namely $N = (1 - x/a)(1 - y/b)$ and so on. Constructing similar shape functions in the global coordinates $(x,y)$ for the general quadrilateral results in very complex algebraic expressions.

Instead it is better to work in a local coordinate system as shown in Figure 2.6. The general point $P(\xi ,\eta )$ within the quadrilateral is located at the intersection of two lines which cut opposite sides of the quadrilateral in equal proportions. For reasons associated with subsequent numerical integrations it proves to be convenient to normalise the coordinates so that the line joining node 1 to node 2 has the normalised coordinate $\xi=-1$, the line joining node 3 to node 4 is $\xi =+1$. Similarly the line joining node 1 to node 4 is $\eta =-1$ and the line joining node 2 to node 3 is $\eta=+1$. In this system the intersection of the bisectors of opposite sides of the quadrilateral is the point $(0,0)$, while the corners $1,2,3,4$ are $(-1,-1), (-1,1), (1,1), (1,-1)$ respectively.

Figure 2.6: Local coordinate system for quadrilateral
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When this system is adopted, the shape functions for the general quadrilateral with corner nodes take the simple form:

$\displaystyle N_1$ $\textstyle =$ $\displaystyle \frac{1}{4}(1-\xi)(1-\eta)$  
$\displaystyle N_2$ $\textstyle =$ $\displaystyle \frac{1}{4}(1-\xi)(1+\eta)$  
$\displaystyle N_3$ $\textstyle =$ $\displaystyle \frac{1}{4}(1+\xi)(1+\eta)$ (2.1)
$\displaystyle N_4$ $\textstyle =$ $\displaystyle \frac{1}{4}(1+\xi)(1-\eta)$  

and these can be used to describe the variation of unknowns such as displacement or fluid potential in an element as before.

These local coordinates can clearly be extended to three dimensions for brick elements. The local coordinates for the triangular and tetrahedral elements are rather more complex. Details of these can be found in most of the standard texts on finite element analysis.

The use of these natural coordinate systems makes the derivation of shape functions much easier and leads to the definition of a set of standard or parent elements. The geometric shape of these elements is chosen to be convenient.


next up previous
Next: Isoparametric Elements Up: Element Matrix Contruction Previous: Element Matrix Contruction
Chris Greenough (c.greenough@rl.ac.uk): September 2001