The preceding difficulties are effectively resolved with approximation concepts. In NX Nastran, there are essentially three categories into which these approximations fall. These categories and the control you have over each are first itemized below. The remainder of the section discusses each of these items in greater detail.
• Design variable linking:
Design variable linking allows you to keep the number of independent design variables to a minimum, leading to a well-formulated design model. You link design variables as part of the design modeling process. Any combination of three possible methods can be used:
grouping of the analysis model properties, effective use of the DVPREL1 and 2 relations, and explicit linking of the design variables via the DLINK entry. In addition to reducing the computational overhead (both during sensitivity analysis as well as optimization), design optimization results interpretation is usually simplified.
Note
Design variable linking not only reduces the cost of design sensitivity and optimization, it also makes results interpretation easier.
• Constraint regionalization and deletion:
This process temporarily removes non-critical, redundant constraints from consideration on the assumption that this reduced set still contains adequate information to efficiently guide the design. NX Nastran performs this task automatically, although you do have some high-level control of the process. You can use the DSCREEN, DRESP1, and DRESP2, and DRESP3 entries to provide this constraint screening control.
Note
Constraint deletion reduces the cost of sensitivity analysis by temporarily deleting design constraints and the associated responses. The sensitivity of these responses does not need to be computed for the current design cycle. Formal approximations allow the optimizer to make design changes without having to invoke a full finite element analysis each time.
• Formal approximations:
Formal approximations allow the optimizer to make design changes without having to invoke a full finite element analysis each time.
In the code, the objective function and all retained constraints are cast in terms of
high-quality approximations that are explicit in the design variables. The optimizer refers to these approximations when it requires function evaluations instead of the costly and implicit finite element analysis. These approximations are constructed automatically in NX Nastran according to one of three possible methods. (This choice is problem-dependent.) You can choose from direct linearization, mixed method, and convex linearization. The method is selected by setting APRCOD to 1, 2, or 3 using the DOPTPRM Bulk Data entry. The mixed method (APRCOD = 2) is the default.
Approximate Model
Figure 3-1illustrates the connection between the finite element analysis, the approximate model, and the optimizer. Note that the approximate model, constructed using the results of the finite element analysis, provides the optimizer with the design information it requires instead of directly from the finite element analysis. The approximate model in Figure 3-1 incorporates the effects of the reduced set of design variables, the screened set of constraints, and the approximated set of structural responses.
Figure 3-1. Approximation Concepts
Design Variable Linking
The optimizer improves the design by changing the values of the design variables. The complexity (and computational cost) of the overall process tends to increase as the number of design
variables is increased. One of your design modeling goals should be to describe the permissible design changes using as few design variables as necessary. (A problem consisting of 200 to 300 independent design variables is currently considered a large design task in NX Nastran.) On the next few pages is a discussion of the methods that you can use to achieve this goal.
Linking by Analysis Model Properties
One of the more efficient methods you can use to minimize the number of independent design variables is to consolidate the number of analysis model properties that are referenced in the design model.
The situation is shown inRelating Design Variables to Properties,Figure 2-3. A DVPRELi entry (i = 1 or 2) defines a design variable-to-property relation for a particular property entry identified by its property ID, or PID. A large number of elements, in turn, might reference this property entry. Since they all reference the same PID, all elements in this group will undergo equal variations as the design variables are changed.
By considering the design implications when defining property groups, you may be able to reduce their numbers. A smaller number of designed property groups might then require fewer independent design variables, since groups of elements have already been “linked” by their property groups memberships.
Linking with Linear Relations
If the number of analysis model property entries cannot be reduced, it may be worthwhile to consider explicit methods of reducing the number of independent design variables. One way you can do this is by using linear design variable-to-property relations.
The DVPREL1 Bulk Data entry is used to expresses an analysis model property as a linear function of design variables or
Equation 3-1.
In this equation, the i-th property piis a function of the set of design variables x1through xn. One feature ofEq. 3-1is that it can be used to express a large number of properties in terms of a much smaller set of design variables. This is shown in the following example.
Example
Assume an optimal plate thickness distribution is to be determined using a combination of constant, linear, and quadratic basis functions as shown in the figure. The design variables act as multipliers of these basis functions that are, in turn, used to specify the element plate thicknesses.
Figure 3-2. Regions of Validity for the Plate Thickness
Figure 3-3. Basis Functions
Suppose the plate thicknesses are to vary in the x-axis direction, but are to remain constant in the y-direction. Evaluating the basis functions for each of these ten thickness stations and writing these design variable-to-thickness relations in matrix form, we get
Equation 3-2.
Reduced Basis Formulations
The thicknesses of the ten element groups are controlled using only three design variables, X1, X2, and X3. Each of the matrix columns correspond to constant, linear, and quadratic basis functions, respectively. Each column is a vector, and the dimension of each of these vectors is greater than the number of such vectors. Relations of this sort are termed reduced basis formulations.
In general, a reduced basis formulation can be expressed as
Equation 3-3.
The most powerful form ofEq. 3-3occurs when the number of rows of [T] is much greater than the number of columns or M >> N. This concept is used extensively in connection with shape optimization; a large set of grid coordinate variations can be governed by a much smaller set of design variables (seeRelating Design Variables to Shape Changes).
Listing 3-1shows the Bulk Data entries that can be used to implement the design formulation ofEq. 3-2. Since the reduced basis formulation is just a set of linear equations, DVPREL1 Bulk Data entries can be used exclusively. Also, since each of the plate strips are of constant thickness, each element group shares a property Bulk Data entry for a total of ten such entries over 40 plate elements.
$PSHELL,PID, MID1, T, MID2, 12I/T3 MID3
PSHELL, 101, 100, 1.0, 100, , 100
$DESVAR,ID, LABEL, XINIT, XLB, XUB, DELXV DESVAR, 1, X1, 0.33, -1.0, +1.0
DESVAR, 2, X2, 0.33, -1.0, +1.0 DESVAR, 3, X3, 0.33, -1.0, +1.0
$
$DVPREL1,ID, TYPE, PID, FID, PMIN, PMAX, C0, , +
$+, DVID1, COEF1, DVID2, COEF2, ...
DVPREL1,201, PSHELL, 101, 4, 0.01, , , , +
DVPREL1,210, PSHELL, 110, 4, 0.01, , , , +
+, 1, 1.0, 2, 0.1, 3, 0.01
Listing 3-1.
The three design variables, or basis function multipliers, are defined on the three DESVAR entries. Each has an initial value of 0.33; therefore, the initial design is defined as that which has a one-third contribution from each of the basis functions. The lower and upper bounds on the design variables are –1.0 and +1.0, respectively, which allow negative as well as positive contributions from each of the basis functions. The thicknesses are never allowed to assume negative values due to the 0.01 value of PMIN, or minimum allowable property value, in field 6 of each of the DVPREL1 entries.
Since the design model overrides the analysis model in design optimization, the initial analysis model properties will be set equal to the values computed from the design model specification.
For this reason, no effort has been made to calculate the initial thickness distribution on the set of PSHELL entries. These thicknesses have arbitrarily been specified as 1.0 since they will just be overridden anyway.
Note
A table is output whenever a design model override of the analysis model occurs. A sample of this table is shown inDesign Optimization Output.
Each DVPREL1 entry defines a row in the reduced basis matrix. The continuation lines of each entry lists the design variables and coefficients in each of the linear relations. Note that each of these continuation lines is simply a row in the reduced basis matrix.
Limitations in Reduced Basis Vector Formulations
In the previous example, the design variable-to-property relations were defined in terms of a reduced basis formulation. Essentially, the optimizer’s task is to find the best combination of these basis vectors. This combination might not yield the true optimum, and in fact only does so if some combination of the basis vectors is able to uniquely define the optimum. Including other basis vectors or allowing each property to vary independently probably will yield a different optimum, perhaps even one having a lower structural weight.
Which final design is more useful? If an arbitrary variation of plate thicknesses is permissible (assuming it can be manufactured, of course), and it did not require an unreasonable number of design variables to obtain this level of generality, then an independent type of relation may be preferable. That is, each plate element thickness can be a function of just a single design variable. However, in the preceding example, the basis function approach was able to guarantee that the resulting plate section had a smoothly varying thickness (in a stepwise fashion, that is). This may be a more desirable design solution, even though the objective function might be reduced further if this condition were relaxed. In short, the design model formulation represents a statement of design expectations and an implicit acknowledgement of its limitations.
Linking with User-Defined Equations
If the number of analysis model properties cannot be changed or reduced and a reduced basis formulation with DVPREL1 type relations is not sufficient or preferred, use of the DLINK (Design variable LINKing) Bulk Data entry should be considered. Its purpose is to provide explicit linear relations among the design variables themselves.
The DLINK entries define relations of the type
Equation 3-4.
Where the items in braces are separately vectors, and the items in brackets (0 and C1) are matrices. Eq. 3-4indicates that design variable linking effectively partitions the vector of design variables into an independent set and a dependent set . Each DLINK entry (each row) in Eq. 3-4defines a single dependent variable as a linear combination of an independent set of design variables. The dependent design variable still needs to be defined using a DESVAR entry.
The DLINK entry just describes the explicit linking relations.
A dependent design variable can be used in the design model in exactly the same way as an independent design variable. It can be used in connection with either property or shape optimization, on DVPREL1 and DVPREL2 relations, and so on. There are no restrictions. The optimizer, however, need only consider those variables that are in the independent set. The corresponding dependent design variable changes are determined from the independent set using the linking defined byEq. 3-4.
Example
Quite often, the DLINK entry may be the only way to implement a given design variable linking scheme. For example, imagine that a rectangular cross-sectional bar consisting of ten elements is to be designed. This bar should be smoothly tapering (in a stepwise fashion, at least), and for purposes of this example, the shape functions ofFigure 3-3, are used.
Suppose the base dimension of this rectangular section is to be a constant 5 mm with the section height as the design variable. This choice yields a total of ten variables similar to the thicknesses of the previous example. However, the only way to describe the cross-sectional properties in terms of the cross-sectional dimensions is through the use of equations. The figure below shows a representative cross section and element orientation, and the equations for area, I1and I2. These equations have been evaluated for an initial h of 10 mm.
Since the design variable-to-property relations for I1and I2are not linear in the design variables, DVPREL1 relations cannot be used to express the reduced basis formulation as before. The only way this can be implemented is by explicitly linking together the design variables themselves together using DLINK entries. Three additional design variables are defined and used as basis function multipliers. These design variables form the independent set. The dependent design variables, linked to the independent set, define the element dimensions. The corresponding element properties are defined using these dependent variables, which are input to DEQATN entries via DVPREL2 entries.
The design model definition is given inListing 3-2. The analysis model contains ten independent PBAR entries since all ten bar sections in the model are to vary.
$PBAR, PID, MID, A, I1, I2
$DESVAR,ID, LABEL, XINIT, XLB, XUB, DELXV DESVAR, 1, X1, 1.0, -1.0, +1.0
DESVAR, 2, X2, 0.0, -1.0, +1.0 DESVAR, 3, X3, 0.0, -1.0, +1.0
$
$...DEPENDENT DESIGN VARIABLE SET:
$DESVAR,ID, LABEL, XINIT, XLB, XUB, DELXV DESVAR, 11, H1, .01, 1.E-4, 1.0 DESVAR, 20, H10, .01, 1.E-4, 1.0
$
$...DESIGN VARIABLE LINKING, Hj = Hj(X1,X2,X3) ; j=11,20:
$DLINK, ID, DDVID, CO, CMULT, IDV1, C1, IDV2, C2, +
$+, IDV3, C3, ...
$...TYPE-2 DESIGN VARIABLE TO PROPERTY RELATIONS:
$DVPREL2,ID, TYPE, PID, FID, PMIN, PMAX, EQID, , +
$+, DESVAR, DVID1, DVID2, ..., , , , , +
$+, DTABLE, CID1, CID2, ...
DVPREL2,41, PBAR, 101, 5, 1.0E-12,, 40, , + +, DESVAR, 11
DVPREL2,42, PBAR, 102, 5, 1.0E-12,, 40, , +
+, DESVAR, 12
DVPREL2,43, PBAR, 103, 5, 1.0E-12,, 40, , +
+, DESVAR, 13
DVPREL2,44, PBAR, 104, 5, 1.0E-12,, 40, , +
+, DESVAR, 14
DVPREL2,45, PBAR, 105, 5, 1.0E-12,, 40, , +
+, DESVAR, 15
DVPREL2,46, PBAR, 106, 5, 1.0E-12,, 40, , +
+, DESVAR, 16
DVPREL2,47, PBAR, 107, 5, 1.0E-12,, 40, , +
+, DESVAR, 17
DVPREL2,48, PBAR, 108, 5, 1.0E-12,, 40, , +
+, DESVAR, 18
DVPREL2,49, PBAR, 109, 5, 1.0E-12,, 40, , +
+, DESVAR, 19
DVPREL2,50, PBAR, 110, 5, 1.0E-12,, 40, , +
+, DESVAR, 20
Listing 3-2.
The basis function multipliers are x1, x2, and x3. Their initial values of 1.0, 0.0, and 0.0 correspond to a constant height beam section. Negative lower bounds allow a negative contribution from any of the basis functions.
The dependent variables are defined using DESVAR entries 11 through 20. Since the dependent variables define the bar element heights (i.e., nonnegative physical dimensions), their lower bounds are specified as small positive numbers to avoid negative bar cross-sectional heights.
Note
The optimizer never proposes a design that is outside the bounds placed on the design variables. (This often results in a more effective method of applying bounds than is PMIN or PMAX on the DVPRELi Bulk Data entries.)
The dependent variables are related to the independent set on DLINK entries 21 through 30.
Each DLINK entry defines a row of the reduced basis matrix similar to the DVPREL1 entries of the previous example. This dependent set of design variables is input to equations in DVPREL2 type relations.
DVPREL2 entries 41 through 50 specify the area moments of inertia about axis 1 for each of the bar element properties as indicated by the 5 in field 5. Note that they all reference the same DEQATN entry 40 containing the functional relation between h and I1. Each of the DVPREL2 continuations describe the equation input arguments, consisting of just a single design variable.
The 1.0E–12 value for PMIN in field 6 ensures that the property does not take on values less than this bound.
For brevity, the equations for areas and (and DEVPREL2) for areas and I2for each property group have not been listed since the method used to define these relations is identical to that shown here. Without relating the design variables to changes in bar areas, weight minimization, for example, will not be possible. Also, if stresses due to bending are in the design model, the dependence of the stress recovery locations on the design variables should also be defined.