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 $f$ with respect to $u$ by

$$\nabla_u f \;\; \mbox{and} \;\; \nabla_{uu} f$$

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

$$W^T \nabla_u f \;\; \mbox{and} \;\; W^T \nabla_{uu} f W,$$

where $W$ is defined by (2.11).

We thus need only supply derivatives with respect to $u$. Formally, we must define the function value $f$, possibly the gradient vector $\nabla_u f$ (i.e., the vector whose $i$-th component is the first partial derivative with respect to the $i$-th internal variable) and, possibly, the Hessian matrix $\nabla_{uu} f$ (i.e., the matrix whose $i,j$-th entry is the second partial derivative with respect to the $i$-th and $j$-th internal variables), all evaluated at $u$. We now describe how to set up the data for a given problem.