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Theoretical background

Curved, Isoparametric Elements

To ensure that a small number of elements can represent relatively complex forms, simple rectangles and triangles no longer suffice. Such elements can be distorted, however, into a more arbitrary shape i.e. the basic elements can be mapped into distorted forms as shown in the figure.

Once such a co-ordinate relationship is known, shape functions can be specified in local co-ordinates and by suitable transformations, the element properties established.

We can write the transformation in the form:

Note: The number of terms in the polynomial [R] is equal to the number of nodes in the element as shown in the table:

So, for the 8 noded quadrilateral shown perviously:

where each row of [C] corresponds to the insertion of particular nodal values of (x, h) into the row matrix [R] i.e. [C] is [R] evaluated at the nodal co-ordinates.

Hence we have found the constants a and therefore, any position x, y can be transformed to the corresponding x, h values,

- which is called the geometry shape function