4.4.3 The INDIVIDUALS Data Cards

The INDIVIDUALS indicator card is used to announce the definition of function and derivative values and the transformation between elemental and internal variables for the types of nonlinear element functions required. The syntax for data cards following the indicator card is given in Figure 4.5.

Figure 4.5: Possible data cards for INDIVIDUALS

The one- or two-character string in field 1 specifies the type of data contained on the card. Possible values for the first character of the string are:

T

This card announces that a new element type is to be considered. The string etype-name in field 2 gives the name of the element type; the name may be up to ten characters long and must have been defined in the ELEMENT TYPE section of the SDIF file (see Section 3.2.15).

R

This card announces that information concerning the transformation between the elemental and internal variables for the element type is to be given. Such information is appropriate only for element types which have been defined with internal variables in the ELEMENT TYPE section of the SDIF file (see Section 3.2.15). The transformation is specified by the matrix $W$ of Section 2.2; only nonzero coefficients of $W$ need be specified here.

The string inv-name in field 2 contains the name of an internal variable (i.e., row of $W$). The name must be a valid Fortran name, see Section 3.1.2, and have been defined on an IV data line in the ELEMENT TYPE section of the SDIF file. The strings iv-nam in fields 3 and (optionally) 5 then give the names of elemental variables (i.e., columns of $W$). The names must be valid Fortran names and have been defined on EV data lines in the ELEMENT TYPE section of the SDIF file. The strings in fields 4 and (optionally) 6 contain the numerical values of the coefficients of $W$ corresponding to the row given in field 2 and the columns given in fields 3 and 5 respectively. These numerical values may each be up to 12 characters long. The entries of $W$ may be defined in any order.

As an example, the transformation (2.9) could be entered with three R data cards. On the first, field 2 would hold the name given to the internal variable $u_1$; field 3 would hold the name given to the elemental variable $v_1$ and field 4 would contain 1.0. Similarly field 5 would hold the name given to the elemental variable $v_2$ and field 6 would also contain 1.0. On the second, field 2 would also hold the name given to the internal variable $u_ 1$; field 3 would now hold the name given to the elemental variable $v_3$ and field 4 would contain -2.0. On the third card, field 2 would hold the name given to the internal variable $u_2$; field 3 would hold the name given to the elemental variable $v_1$ and field 4 would contain 1.0. Field 5 would now hold the name given to the elemental variable $v_3$ and field 6 would contain -1.0.

A

This card announces that an auxiliary parameter, specific to the current element type, is to be assigned a value. The string p-name in field 2 gives the name of the auxiliary parameter that is to be defined; this name must be a valid Fortran name, see Section 3.1.2, and have been previously defined in the TEMPORARIES section. The string in field 7 is an arithmetic expression. The assignment

$$\mbox{auxiliary variable named in field~2} \leftarrow \mbox{field~7}$$

is made, where again $\leftarrow$ means ``is given the value''; any variable mentioned in the arithmetic expression must either be reserved (see Section 4.4.1), or have been defined in the TEMPORARIES section. If in this latter case, the variable is integer or real, it must have been allocated a value itself either on a previous GLOBALS data card or on a previous A, E or I card for the current element type in the ELEMENTS section.

I

This card announces that an auxiliary parameter, specific to the current element type, is to be assigned a value whenever a second logical auxiliary parameter has the value .TRUE. The string, p-name, in field 3 gives the name of the auxiliary parameter that is to be defined; this name must be a valid Fortran name, see Section 3.1.2, and have been previously defined in the TEMPORARIES section. The string in field 7 is an arithmetic expression. The assignment

$$\mbox{auxiliary variable named in field~3} \leftarrow \mbox{field~7}$$

will be made if and only if the logical auxiliary parameter, l-name, specified in field 2 has the value .TRUE.; the logical parameter must have been previously defined in the TEMPORARIES section and allocated a value in the GLOBALS or INDIVIDUALS section. The arithmetic expression must obey the rules set out in the A section above.

E

This card announces that an auxiliary parameter, specific to the current element type, is to be assigned a value whenever a second logical auxiliary parameter has the value .FALSE. The string, p-name, in field 3 gives the name of the auxiliary parameter that is to be defined; this name must be a valid Fortran name, see Section 3.1.2, and have been previously defined in the TEMPORARIES section. The string in field 7 is an arithmetic expression. The assignment

$$\mbox{auxiliary variable named in field~3} \leftarrow \mbox{field~7}$$

will be made if and only if the logical auxiliary parameter, l-name, specified in field 2 has the value .FALSE.; the logical parameter must have been previously defined in the TEMPORARIES section and allocated a value in the GLOBALS or INDIVIDUALS section. The arithmetic expression must obey the rules set out in the A section above.

F

This card specifies the value of the nonlinear element. The string in field 7 is an arithmetic expression; the assignment

$$\mbox{nonlinear element function} \leftarrow \mbox{field~7}$$

is made; any variable mentioned in the expression must obey the rules set out in the A section above.

G

This card specifies the value of a component of the gradient of the nonlinear element. The string, iv-nam, in field 2 contains the name of an internal variable. The component of the gradient specified on the card will be taken with respect to this variable. The string must be a valid Fortran name, see Section 3.1.2, and have been defined on an IV data line, for a nonlinear element defined with internal variables, or an EV data line, for an element without explicit internal variables, in the ELEMENT TYPE section of the SDIF file. The string in field 7 is an arithmetic expression; the assignment

$$\mbox{derivative of element w.r.t. variable in field~2} \leftarrow \mbox{field~7}$$

is made; any variable mentioned in the arithmetic expression must obey the rules set out in the A section above. G cards are optional. However, once the user starts to form the gradient for an element type, any component not explicitly specified will be assumed to have the value zero.

H

This card specifies the value of a component of the Hessian matrix of the nonlinear element. The strings iv-nam in fields 2 and 3 contain the names of internal variables. The component of the Hessian specified on the card will be taken with respect to these variables. Either string must be a valid Fortran name, see Section 3.1.2, and have been defined on an IV data line, for a nonlinear element defined with internal variables, or an EV data line, for an element without explicit internal variables, in the ELEMENT TYPE section of the SDIF file. The string in field 7 is an arithmetic expression; the assignment

$$\mbox{second derivative of element w.r.t. variables in fields~2 and 3} \leftarrow \mbox{field~7}$$

is made; any variable mentioned in the arithmetic expression must obey the rules set out in the A section above. H cards are optional. However, once the user starts to specify the Hessian matrix for an element type, any component not specified will be assumed to have the value zero. The matrix is assumed to be symmetric and so the user needs only supply values for one of

$$\frac{\partial^2f}{\partial u_i \partial u_j} \;\; \mbox{or}... ...rac{\partial^2f}{\partial u_j \partial u_i} \;\;\;\;(i \neq j)$$

it does not matter which. Observe that defaulting Hessian components to zero gives a very simple way of inputing sparse matrices; however, as we stressed in the introduction, we do not generally recommend this method of specifying invariant subspaces.

The data started on an A, I, E, F, G and H card may be continued on a card whose first field contains an A+, I+, E+, F+, G+ or H+ respectively. Such cards contain an arithmetic expression in field 7 and no further data; the arithmetic expression must obey the rules set out in the A section above. At most nineteen continuations of a single assignment are allowed.

The data for a single element type must occur on consecutive cards and in the order given in Figure 4.5, excepting that A, I and E cards may be intermixed. A new element type is deemed to have started whenever a T card is encountered. The F card is compulsory for all element types; elements with useful transformations from elemental to internal variables must also have R cards. The data for a particular card type is considered to have been completed whenever another card type is encountered.