A nonlinear element function is assumed to be a function of the problem variables , a subset of the overall variables . Suppose that has components. Then one can consider the nonlinear element function to be of the structural form , where we assign . The elemental variables for the element function are the variables and, while we need to associate the particular values with , it is the elemental variables which are important in defining the character of the nonlinear element functions.
As an example, the first nonlinear element function
for a particular problem might be
The example may be used to illustrate a further point. Although is a function of three variables, the function itself is really only composed of two independent parts; the product of with , or, if we write and , the product of with . The variables and are known as internal variables for the element function. They are obtained as linear combinations of the elemental variables. The important feature as far as an optimization procedure is concerned is that each nonlinear function involves as few internal variables as possible, as this allows for compact storage and more efficient derivative approximation.
It frequently happens, however, that a function does not have useful
internal variables.
For instance, another element function
might have structural form
In general, there will be a linear transformation
from the elemental variables to the internal ones. For example in
(2.6), we have