Relating Design Variables to Properties

For the analysis model to vary as the design variables are changed, its properties and/or its shape must be expressed in terms of the design variables. You need to provide these relations as part of the design model specification. This section discusses analysis model property changes.

Relating Design Variables to Shape Changesdiscusses shape variations.

Definition of Analysis Model Properties

Analysis model properties are defined as those quantities that appear on Bulk Data property entries. Plate thicknesses, area moments of inertia, elastic spring stiffnesses, elastic modulii, coefficients of thermal expansion, grid point locations, and so on, are all properties that can be written as functions of design variables. See the DVCREL1, DVCREL2, DVGEOM, DVGRID, DVMREL1, DVMREL2, DVPREL1, and DVPREL2 bulk data entries in the NX Nastran Quick Reference Guide for details on the properties that you can define as design variables.

Type-1 Design Variable-to-Property Relations

You can relate design variables to analysis model properties in either of two ways: with linear relations or with NX Nastran’s equation input capability. A linear relation is described using a DVPREL1 (Design Variable-to-Property RELation, type-1) Bulk Data entry.

Linear design variable-to-property relations are of the form

Equation 2-8.

where p_jis the j-th property, expressed as a linear combination of the design variables.

Type-2 Design Variable-to-Property Relations

An equation, or type-2 relation, is defined using a DVPREL2 Bulk Data entry that references a DEQATN (Design EQuATioN) entry. A property expressed using these relations can be written as

Equation 2-9.

where the j-th property is a function of a collection of design variables and constants. The function is defined using a DEQATN Bulk Data entry.

The same design variables may appear in either type-1 or type-2 relations. The block diagram of Figure 2-2may be helpful in understanding the role of the DVPREL1 and DVPREL2 entries in defining property variations.

Figure 2-2. Design Variable-to-Property Relations

The DVPREL1 entry defines linear property variations and requires only design variable input.

All necessary constants are supplied directly on the entry. Thus,Figure 2-2shows a single input and a single output from the DVPREL1 block.

On the other hand, the DVPREL2 entry relies on an equation to describe the (generally) nonlinear property variations. This equation may have design variables (DESVAR) as well as table constants (DTABLE) as input. As shown inFigure 2-2, the result is again an analysis model property variation.

Reference by Property Entry

In design variable-to-property relations, analysis model properties are referenced by property entry, rather than on an element-by-element basis. This relation is shown schematically inFigure 2-3. The property IDs PID1 through PIDm are referenced on the DVPREL1 and DVPREL2 entries. Since each of these property groups may contain a large number of elements (EID1 through EIDn), you can control many elements by using only a small set of DVPREL-type entries. For example, to modify plate thicknesses, all that is required is to reference a PSHELL ID on a DVPREL entry. All elements in the property group will then change accordingly.

Figure 2-3. Design Model Reference by Property Entry

Identification of Property Entry Items

You must identify a selected property entry item either by its field position on the property Bulk Data entry or by its word position in the element property table (EPT) that is generated by NX Nastran. In general, it is far easier to refer to the field position on the Bulk Data entry. In some instances, (BEAM elements in particular) reference to the element property table must be made instead. None of this is really as difficult as it sounds and is explained in the section on special modeling considerations at the end of this section.

To illustrate both type-1 and type-2 design variable-to-property relations, consider the stiffened panel section ofFigure 2-4. The example is test problem library example number D200X7, presented inStiffened Plate. Three design variables are each related to the plate thickness, web thickness, and cross-sectional area of the web cap, respectively. By grouping all of the elements in the base plate into one property group, their thickness can all be controlled by a single design variable. This same grouping is also used for the elements of the web as well as for the cap section elements.

Figure 2-4. Stiffened Panel Test Problem The base plate thickness is linearly related to a design variable according to

Equation 2-10.

This linear relation can be defined with a DVPREL1 Bulk Data entry as follows:

PSHELL, 1, 1, 0.15, 1

$

$...Define the design variables:

$

$DESVAR,ID, LABEL, XINIT, XLB, XUB, DELXV DESVAR, 1, T-PLATE,0.15, 0.001, 10.0

$

$...Relate the design variables to analysis model properties

$ (linear relations, so use DVPREL1)

$

$DVPREL1,ID, TYPE, PID, FID, PMIN, PMAX, C0, , +

$+, DVID1, COEF1, DVID2, COEF2, ...

DVPREL1,1, PSHELL, 1, 4, 0.01, , , , +DP1

+DP1, 1, 1.0

The PSHELL 1 entry defines the analysis model property group (to which a number of elements belong). The DESVAR 1 entry defines design variable 1 (T-PLATE) with an initial value of 0.15 and lower and upper bounds of .001 and 10.0, respectively. Design variable 1 is, in turn, related to PSHELL property group 1 thickness via DVPREL1 number 1. The 4 in DVPREL1 number 1, field 5, states that it is field 4 on the property entry (the thickness) that is to be varied. The continuation line on the DVPREL1 entry gives the functional relationship defined byEq. 2-10.

Field 6 on the DVPREL1 entry defines the minimum allowable value of the property; here, a minimum thickness of 0.01. This value enforces a lower bound, even though greater than the design variable’s lower bound. SeeStiffened Platefor the remainder of the design model.

Frequently, you may want to specify the bar cross-sectional dimensions in the design model instead and use this data to compute the resultant cross-sectional properties, such as areas and moments of inertia. Since these properties are often nonlinear functions of the design variables, this approach requires the use of type-2 design variable to property relations. Probably the easiest way to introduce type-2 relations is with a simple example:

Figure 2-5. Cap Section Detail

Assume that the cap section of the stiffened plate ofFigure 2-4is a rectangular section bar as shown inFigure 2-5. Suppose the bar is characterized by the cross-sectional dimensions b and h and is oriented with respect to principal planes 1 and 2 as shown. A stress recovery location, such as point c, might be defined as well. The Bulk Data describing the analysis and design model relations could be written as follows:

$...ANALYSIS MODEL PROPERTIES:

PBAR, 120, 220, 0.12, 1.6E-3, 9.0E-4, , , , +

+, -.15, .2

$

$...BAR CROSS-SECTIONAL DIMENSIONS:

$ESVAR, ID, LABEL, XINIT, XLB, XUB, DELXV

DESVAR, 10, B, .3, .1, 1.0

DESVAR, 11, H, .4, .1, 1.0

$

$...BAR PROPERTY RELATIONS, A, I1, I2:

$DVPREL2,ID, TYPE, PID, FID, PMIN, PMAX, EQID, , +

$+, DESVAR,DVID1, DVID2, ..., , , , , +

$DVPREL1,ID, TYPE, PID, FID, PMIN, PMAX, C0, , +

$+, DVID1, COEF1, DVID2, COEF2, ...

DVPREL1,260, PBAR, 120, 12, -.5 , , , +

+, 10, -.5

DVPREL1,261, PBAR, 120, 13, .05, , , , +

+, 11, .5

PBAR 120 defines the bar properties corresponding to the initial cross-sectional dimensions.

Note that the initial values of the design variables listed on the DESVAR entries agree with the initial properties. If any discrepancies exist between the initial analysis model properties (given on the property entries) and those calculated based on the initial values of the design variables, the properties based on the design model have precedence. The design model data is then used to override the analysis model properties. In this example, the design and analysis model agree, and no overrides take place.

Since the bar area and moments of inertia are nonlinear in the design variables, they must be defined using equations. These relations are defined on the DEQATN entries 501, 502, and 503.

The input arguments are assigned via the DVPREL2 entries 250, 251, and 252. The DESVAR identifier on the first continuation of each of these entries indicates design variables 10 and 11 are to be used as input (b and h) for each of the referenced equations.

Note

Override of analysis model data by design model data provides a convenient method of restarting in design optimization. See the restart example inRestarts in Design Optimization.

Equation Arguments

The input arguments defined on the DEQATN entries are formal arguments, which are defined at run time by the actual arguments provided on the DVPREL2 entries. The rules for defining these equations follow directly from FORTRAN syntax requirements. However, all arguments are assumed to be real numbers. FORTRAN intrinsic functions may also be used. (Some of these functions, such as ABS and MAX, are not recommended since they may yield functions having

discontinuous first derivatives. The sensitivity analysis as well as the optimization results may be incorrect if any of these discontinuities are encountered.)

Note

The available intrinsic functions and their uses are listed with the DEQATN bulk data entry in the NX Nastran Quick Reference Guide.

Minimum Allowable Property Values

Field 6 on the DVPREL2 entries defines PMIN, which is the minimum allowable value for the particular property during design optimization. Generally, its purpose is to prevent properties from taking on values near zero. The default is 0.001, which is acceptable for many applications.

Here, I₂is already less than this default, and I₁may take on values less than the default as well (based on the minimum B and H values). Therefore, we override the default in both cases and specify a PMIN of 1.0E-5 on DVPREL2 251 and 252, based on engineering judgment.

A similar situation is seen in the DVPREL1 relations that in this case, relate the location of the stress recovery point c to the values of the design variables. The y-coordinate location is a negative quantity, but the default PMIN for stress recovery locations is -1.0E+35 (see the DVPREL1 and DVPREL2 Bulk Data entry descriptions). Just to be explicit, a negative PMIN has been provided on DVPREL1 260. DVPREL1 260 and 261 respectively, define the following linear relations:

Equation 2-11.

Stress Recovery Locations: Words of Warning

Note the error that may have resulted if the location of the stress recovery point had not been allowed to vary. Since the stress computations would not have included the direct effects of changes in the stress recovery point, the stresses seen by the optimizer would not be those that the engineer had intended. This would certainly lead to incorrect results. Of course, other stress recovery points would probably also need to be defined to ensure recovery of the maximum stresses.