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4.1.1 The Values and Derivatives Required

It is assumed that a nonlinear element type
is specified in terms of internal variables
**u**, whose names
are those given on the `ELEMENT TYPE`
data cards
in an SDIF file (if the element has no useful internal
variables, the internal and elemental
variables are the same and the
internal variables will have been named after the elementals), see
Section 3.2.15. An optimization procedure is likely to require
the values of the element functions
and possibly their first and second, derivatives.
These derivatives need only be given with respect
to the internal variables.
For if we denote the gradient
and Hessian
matrix of an element function
with respect to by

respectively, the gradient and Hessian
matrices with respect to the elemental variables
are

where is defined by (2.11).
We thus need only supply derivatives
with respect to . Formally, we must define the function value ,
possibly the gradient
vector (i.e., the vector whose -th
component is the first partial derivative with respect to the -th
internal variable) and, possibly, the Hessian
matrix
(i.e., the matrix whose -th entry is the second partial
derivative with respect to the -th and -th internal variables),
all evaluated at . We now describe how to set up the data for a
given problem.

** Next:** 4.2 Indicator Cards
** Up:** 4.1 Introduction to the
** Previous:** 4.1 Introduction to the