Design sensitivity is the calculation of derivatives of constraint functions with respect to design variables. While a knowledge of these derivatives is useful in its own right, the derivatives are required in many efficient optimization methods. Constraint derivatives are also required in some reanalysis methods. It is shown where the sensitivity coefficients fit into the scheme of a basic organization of an optimization procedure. The analyzer is to be taken as MSC/NASTRAN. The terminator program monitors the termination criteria and ends the optimization procedure when the criteria are satisfied. This program can reside in several plances: in the optimizer itself, in a user written code, or as part of the MSC/EOS (Engineering Operating System) MSC/EOS currently under development. Since several excellent optimization codes exist and since they require such very specialized technical knowledge, the optimizer under the new MSC/EOS is considered to be selected and supplied by the user to meet his specific needs and preferences. The one exception to this is a fully stressed design (FSD) based on simple scaling. The gradients are currently supplied by various design sensitivity options now existing in MSC/NASTRAN's design sensitivity analysis (DSA)

Wallerstein, D. V.

English

NASA Technical Reports Server

Ns -l  4s
DESIGN ENHANCEMENT TOOLS
IN MSC/NASTRAN
D. V. Wallerstein
The MacNeal-Schwendler Corporation
Los Angeles, California
PRECEDLNG PAGE Bt.ANK NOX FJt,,,ME_
505
https://ntrs.nasa.gov/search.jsp?R=19870002315 2020-03-23T14:50:08+00:00Z
DESIGNSENSITIVITYIN OPTIMIZATION
Design sensitivity (ref. i) is the calculation of derivatives of constraint
functions with respect to design variables. While a knowledge of these derivatives
is useful in its own right, the derivatives are required in many efficient
optimization methods. Constraint derivatives are also required in somereanalysis
methods. Figure i shows where the sensitivity coefficients fit into the schemeof a
basic organization of an optimization procedure (ref. 2). In the context of this
paper the analyzer is to be taken as MSC/NASTRAN.The terminator program monitors
the termination criteria and ends the optimization procedure when the criteria are
satisfied. This program can reside in several places: in the optimzer itself, in
a user written code, or as part of the MSC/EOS(E_ngineering Operating System)
currently under development. Since several excellent optimization codes exist
and since they require such very specialized technical knowledge, the optimizer
under the new MSC/EOSis considered to be selected and supplied by the user to meet
his specific needs and preferences. The one exception to this will be a fully
stressed design (FSD) based on simple scaling. The gradients are currently supplied
by various design sensitivity options now exisiting in MSC/NASTRAN'sdesign
sensitivity analysis (DSA).
ANALYZER
DSA /
L
PROGRAM
BLOCK OF DATA
Figure i
506
DESIGNCONSIDERATIONS- USERISSUES
Figure 2 outlines several implementation issues that were considered for the
current MSC/DSAand are still valid for _uture enhancements. From a user stand-
point_ th= computations of the gradients should be a natural extension of the finite
element analysis (FEM). At the sametime, the user should not be constrained by the
FEMto the degree that he or she is forced to makeunwanted adaptations of design
requirements. To this end, MSC/DSAhas been designed to have generality in modeling
complex designs and to allow for design variables which are not explicitly defined
in terms of standard finite element properties such as area, second momentsof area,
and thickness. By the same token, MSC/DSAmust not and does not impose any size
restrictions on the analysis. On the other hand, through its ability to link many
finite element property cards and material cards into a single design variable,
MSC/DSAallows for computational efficiency by reducing the number of design
variables needed for analysis. The output is in two forms. A matrix is generated
with both the current value of the constraint and the values of the gradients output to
the data base and to a general FORTRANfile. To aid the user in interpreting the
results, this same information is output in table form with both numeric and user
supplied BCDidentifiers.
• EASE OF USE
• GENERALITY TO MODEL COMPLEX DESIGNS
• CAPACITY TO SOLVE LARGE PROBLEMS
• OUTPUT EASY TO COMPREHEND
Figure 2
507
DESIGNCONSIDERATIONS- ANALYSISISSUES
Figure 3 gives the three prime issues considered in selecting the approach used
for MSC/DSA. The general philosophy followed was that MSC/DSAbe considered a post-
processor to a standard MSC/NASTRANanalysis. This requirement was easily met
through use of data-baslng. Through the data base the LU decomposition of the
stiffness matrix is recovered so that only backward operations are used to recover
the solution vector needed to compute gradient terms. An important feature of the
implementation is the table driven nature of MSC/DSA. Final recovery of all
information needed to form gradients is handled by standard MSC/NASTRANstress,
force, displacement, or modal recovery modules. This means that the only new
features added to MSC/NASTRANare those which form the necessary correlation tables
between constraints and their derivatives and the needed code to form the right hand
side needed for solution vector recovery. The net result is ease and reliability in
system maintenance.
• RESTARTFROMPRIMARYANALYSIS
• ONLYBACKWARDOPERATIONSUSEDTO RECOVERSOLUTIONVECTOR
• DATAORGANIZATION, ASSEMBLY AND RECOVERY
Figure 3
5O8
DEFINITIONOFTERMS
Figure 4 lists the meaning of someterms used in the following discussion. It
is appropriate, however, to mention two things at this point. First, MSC/DSAdoes
not impose any constraint equations. It merely returns the value of the constraint
at the current point in design space. Second, the approach is the design space
approach and not the adjolnt method. The adjoint method is discussed only for
comparison purposes. The two methods arise from the technique used to determine the
value of the second term in the expression for _i"
{b} - Vector of Design Variables
{u} - Vector of Displacements
{P} - Vector of Loads
[K] - Structural Stiffness
{_i} - Vector of Adjoint Variables
• i(b, u) _ 0 - Constraint Equation
6_ i = _--_--_ {6b} + _-_--_ {6u} - Sensitivity Coefficient
B = b/b o - Normalized Design Variable
Aij - Sensitivity Coefficient for ith Constraint and jth Design Variable
Figure 4
509
SENSITIVITYMETHODS
As mentioned in the previous figure, the difference between the adjoint method -
sometimes called behavior space or state space and the design space method is how
the {_u} term is replaced by {_b} in the expression for _i" These two methods are
depicted in figure 5. Notice that both require LU decompositions and formation of
the [H] matrix. The effect of the right hand sides is discussed in the next figure.
{6u} _ {6b}
ADJOINT METHOD
5_iI
DESIGN SPACE METHOD
[K
Figure 5
510
THEORYCOMPARISONSOFTWOAPPROACHES
Figure 6 gives a direct comparison of the two methods. The adjolnt method uses
the {_i/_u} vector to generate the right hand side. Only active constraits are
used to generate this vector; hence, the number of unknownequations at any given
point in design space is equal to the numberof active constraints. In the design
space method the right hand side is represented by the matrix [H]. This matrix is a
true pertubation of the load vector for a change in each design variable and
represents the perturbed equilibrium state of the structure. There is a column for
each load vector perturbed by a design variable.
Approach Adjoint Method Design Space
Unknowns {_i} 5[._]
Set of equations to be solved
Number of unknown vectors
to be determined
Number of active
constraints
Number of design variables
times
Number of load conditions
Figure 6
511
IMPLEMENTATIONCOMPARISONF TWOMETHODS
Figure 7 comparesthe two methods from a different point of view. Generally,
discussion of which method to use stems from the number of right hand side
vectors. But such discussion rarely takes into consideration the efficiency of
modern equation solvers. When this efficiency is taken into consideration the
number of unknowns to solve for takes on less signlflcances than other consider-
ations. Two major advantages of the design space approach have already been
alluded to. First, the [H] matrix represents a perturbation of equilibrium and
its formulation fits neatly into an existing finite element code (also it must be
formed in the adjoint method) and forms a natural load vector. Second, the formula-
tion of the gradients fits in naturally with existing general purpose finite element
code data recovery operations. Finally, a major expense in either method is the
forming of the necessary correlation tables between constraints and design variables
and the forming of the correlation between design variables and individual element
properties.
Item
Fewactive constraints
Adjoint Method
Small advantage
Design Space
Many active constraints
Few design variables
Small advantage
Forming right hand side using
existing code structure
Major advantage
Forming gradients using existing
code structure
Major advantage
Forming necessary correlation
tables relating element variables
to design variables and response
variables to constraints
Major expense Major expense
Figure 7
512
THEDESIGNSPACEINCREMENTALPPROACH
Figure 8 shows the necessary incremental equations required for the design
space approach. Notice that perturbation to nodal equilibrium occurs from two
sources, a change in stiffness arising from design variable perturbation of the
element stiffness matrices and a change in the actual load vector arising from the
same source. Also note that for accuracy and conslstancy a total solution vector
not an incremental one is formed.
[K°]{AUB} = {APB} - [£KB]{U° }
{u}- {u°} ÷ {AUB}
Figure 8
513
SOMEMOREDEFINITIONS
Figure 9 contains some definitions unique to this paper.
are mostly self explanatory.
Q - MSC/NASTRANOUTPUTSUCHAS:
These definitions
• Displacement
• Stress
• Force
QB _ OUTPUT VALUE USING NEW PROPERTY ORIGINAL SOLUTION VECTOR
QU _ OUTPUT VALUE USING ORIGINAL PROPERTY NEW SOLUTION VECTOR
Q0 _ OUTPUT VALUE USING BASE LINE RUN
Figure 9
514
ACTUALGRADIENTCOMPUTATION
Figure I0 gives the actual gradient computation as used by the design space
approach. The top equation is used for "element" type constraints (meaning element
stress or force) with self terms. Self terms arise whena specific element is used
as a constraint and one or more of its physical properties (such as area, second
moment of area, etc,) is included in a design variable and the derivative of the
constraint with respect to that specific design variable is required. The bottom
equation is used for "element" type constraint without self terms and displacement
type constraints. The incremental change in design variable is represented by ABj
and QLimi are user supplied limit values used in constraint evaluations. The upper
signs are associated with maximumtype constraints and the lower signs are associated
with minimumtype constraints. The various values of Q comedirectly from stress,
force, or displacement output files standard to MSC/NASTRAN.
B. ujQi3 o oQi Qi Qi
or
uj o
Qi Qi
Aij = +-I QLimliABj $I QLimliABJ
Figure i0
515
CURRENTUSERINTERFACE
Figure II gives the current user interface scheme. MSC/DSAis currently
designed to run as a post-processor to the standard MSC/NASTRANstatic solution
sequence (SOL 61), modal solution sequence (SOL 63), and buckling solution sequence(SOL 65). The MSC/DSAsolution sequence numbers correspond to each of the basic
solution sequences. To aid in relating specific constraints to specific design
variables two new case control cards have been defined. Constraints are defined via
the DSC@NScards and design variables are defined via the DVAR-DVSETcards.
SOL51I
SOL53
SOL55
EXECUTIVE
SENSITY 1
SET2 I
CASE CONTROL
DSCONS
DVAR
DVSET
BULK DATA
Figure ii
516
RELATINGCONSTRAINTSO DESIGNVARIABLES
The user may be interested in the derivatives of every constraint with respect
to every design variable. If this is so, just set SENSITIVITY= ALL. On the other
hand, it may be more reasonable in analysis to relate specific constraints to
specific design variables as depicted in figure 12. In this figure, structure
located at A is considered far enough removed from structure located at B that
derivatives of constraints at A with respect to design variables at B (or vice
versa) would be meaningless. The SET, SET2,SENSITIVITYcombination allows the user
to define specific constraint design variable relationships. This combination shown
relates in section A constraints I through 4 to design variables 70, 80, and 90. In
section B the relationships are constraint 300 related to design variables 1 and
3; and constraints I00, 200, and 500 related to design variables I, 3, and 4.
A B
F---I Constraints
Design Variables
SET i = I THRU 4
SET 2 = 300
SET 5 = i00, 200, 500
SET 30 = 70, 80, 90
SET 31 = 4 Defines sets of DVAR cards
Then the following SET2 card may be defined:
SET2 = 18 (1,30), (2,33), ((5), (31,33))
and the following SENSITIVITY card is specified as:
SENSITIVITY - 18
Defines sets of DSCONS cards
Figure 12
517
CONSTRAINTDEFINITION
Figure 13 shows how constraints are defined with the DSCCNSbulk data card.
Each constraint must have a unique DSCIDfor internal and external identification.
The LABELis for user convenience in identification of output. TYPEis any one of
the following: DISP, FCRCE,STRESS,LAMDA,or FREQ. ID identifies the actual grid
element. CCMPidentifies the specific displacement component or stress or force
component. LIMIT and CPTdefine the equations used for the constraint. If LIMIT =
0., then plus or minus the constraint value is returned depending on the value of
CPT.
i 2 3 4 5 6 7 8 9 I0
ID COMP LIMITDSCONS
DSCONS
DSCID
21
LABEL
DOOR
TYPE
DISP 4 1 .06
OPT
MAX
IConstralnt Value_ °
(Upper Limit) _i = [_i_ T...... 1.0" Sign (Limit)
or
(Lower Limit) _i = 1.0" Sign (Limit)
(Constraint Value) °
Figure 13
518
DESIGNVARIABLEDEFINITION
Figure 14 shows the DVAR-DVSETcards needed for design variable definition.
Each BID must be unique and define a design variable. Again, LABEL is a user
convenience. DELTAB defines the incremental change in normalized design
variables. VID points to a DVSETcard or cards. The VID on the DVSETcard need not
be unique. TYPEdefines the property being modified and FIELD defines the specific
field on the property card being modified. PREFand ALPHAalong with DELTABdefine
the actual change in property value. PIDI etc. define the specific property cards
modified.
I 2 3 4 5 6 7 8 9 I0
DVAR BID LABEL DELTAB VID VID VID VID VID ABCI
DVAR I0 DOOR .01 2 3 6 60 i00
I DV DVSET
I DVAR => Bi i
l DVSET _ Property Card Field l
p = p
o o
I
+ _ Pref[Ii + AB)_-I] ,' I! MATcard i
cards
Figure 14
519
OUTPUT
Output in MSC/DSAis of two types• Printed output gives the current constraint
values and their derivatives along with all the user identifying labels and matrix
output for use in optimization programs• The user will find the printed output so
clean and self-explanatory that there is no need to discuss it. Figure 15 shows the
form of the matrix output• The matrix is design constrained DC wide and its rows
are grouped by design variables• The number of rows in each group is the total
number of loads• The very first block gives the current constraint values•
T
LD
I
T
LD
T
LD
DV 1
DVj
_' DC "-y
_i) ,.(I)• • " _i • " °
_(2) ,.(2)
i " " " _i " • "
_°_ . . . _.°, . . .
A(1) A(1)
ii " " " il " " "
A(LD) (LD)
ii " " " All " " "
A(1) (LD)
lj " " " Ail
A(LD) (LD)
I] " " " Aij
Figure 15
520
SENSITIVITYCOEFFICIENTS
The primary use of sensitivity coefficients is in optimization. However, it is
often useful to look at them in their own right as tools that begin to give the
analyst a "feel" of the structure. Figure 16 summarizes their meaning. For
example, a positive sensitivity coefficient means that for an increase in design
variable the value of the constraint will increase. This will move the design
closer to a MAX constraint or further from a MIN constraint. Similar statements
hold for a negative sensitivity coefficient.
A-c-'>+ B INCREASE
AB
A_ < 0 + B DECREASE
AB
Figure 16
521
EXTENSIONTO SUPERELEMENTANALYSIS
Currently MSC/DSAdoes not allow for substructuring or, as it is called in
MSC/NASTRAN,superelement analysis. The inclusion of MSC/DSAinto superelement analy-
sis requires the establishment of ground rules as listed in figure 17. To this end,
design variables are considered global in nature. This meansthe design variable must
have a unique definition across all superelements. This requirement is necessary be-
cause design variables linking can extend across superelement boundaries. DSC_NS
cards are local to superelements since they can define local quantities. DVSETcards
are local to superelements since they point to property cards which (under current
superelement development) are local to superelements. SENSITY cards are by super-
element case control. _BJF (structural weight) is accumulated for all superelements
and its derivatives computedfor all design variables.
• DVAR CARDS ARE GLOBAL
• DSCONS CARDS ARE LOCAL TO SUPERELEMENT
• DVSET CARDS ARE LOCAL TO SE
• SENSITY CARDS ARE BY SE CASE CONTROL
• OBJF CARD ABOVE SUBCASE LEVEL
Figure 17
522
USERDEFINEDCONSTRAINTRELATIONSHIPS
Currently, constraints are restricted to displacements, or specific stress or
force output. It would be advantageous to define response values such as those
depicted in figure 18. Here it is assumedthat the constraint is related to the
panel stress, either through the distortion energy relationship or through the
utilization relationship.
RESPONSEVALUE-- MAX(_, B2 ]
(TENSION)
B2:I- + °LI2 + 4
2 Y
(COMPRESSION)
aL = MIN(o, _, Cy)
Figure 18
523
EXTENSIONOFCONSTRAINTRELATIONSHIPS
Figure 19 demonstrates the proposed extension of the constraint card to include
user defined relationships. Here the constraint points to a user defined equation
rather than a specific stress component. Other than the first two fields, the EQN
card is free field. The equations are written using standard FORTANtype
nomenclature and include standard type functions such as M_D, MAX,MIN, S_RTetc.
The Fi expressions are key word type expressions defined via the FUN1card. The
FUN1card returns a value to Fi based on either the specified stress component
directly or a table look-up relating the stress value to some functional relation-
ship such as a current value of an allowable.
i 2 3 4 5 6 7 8 9 I0
DSCONS DSCID
DSCONS 17
LABEL TYPE
SX123 STRESS
ID COMP LIMIT OPT EQUID
MAX 34417
EQN EQID -- FREE FIELD ........
EQN 34 MAX (SQRT( FI** 2+F2" *2- FI*F 3), ABC
+BC (F4+ SQRT( F4** 2+4 .* F5** 2)))
VALUE = EQN (FI, F2, ... FII)
VALUE = MAX (/FI 2 + F22 - FI * F3
FUN1 NAME A COMP TID ALPHA BETA B
FUN1 F7 2 32
F i -- [A(y) a+ B]B
Y = Function (Value of COMP)
Response Value = EQN (FI, F2, ... FN)
Figure 19
524
MSC/E@S
Figure 20 is a schematic of the MSC/E@S(Engineering Operating System)
currently under development. The keys to this system are the MSCData Base and the
new (currently under code development) MSC/NDDLexecutive. The NDDL(N__astranData
Definltion L__anguage)provides the foundation for the development of a compl--ete
engineering data managementsystem to support the various MSCanalytical modules
shown on the various spokes of figure 20. Since the NDDLprovides a means of
specifying a logical data structure definition it provides for the unique
identification and addressabillty of the data and provides for the definition of the
interdependencies of the data. Data Base managing will include such items as
automatic storage and retrieval of data for ease of use; data recovery, integrity,
and security procedures to insure data validity; and all current GIN@(general
purpose input/output routines) calling sequences. Notice that the axle of the
figure represents an extension of GIN@to include direct user interface between a
user defined programming system (where for example the actual optimization programs
will lie) and the NNDL and hence the Data Base. Through this interface new entry
points into functional modules are provided so that the user's optimization routines
can for example initiate new MSC/NASTRANanalysisor MSC/DSAanalysis, or query the
Data Base. Although the user is expected to supply his own optimization, note that
one spoke includes a MSC/FSD(Fully Stressed Design) option. As envisioned, the
user will have, through the interface module and the NNDL,access to any spoke or
the rim or both.
Figure 20
525
I.
o
REFERENCES
Arora, S.J.; and Haug, E.J.: Methods of Design Sensitivity Analysis in Struc-
tural Optimization. AIAA J., vol. 17, no. 9, 1979, pp. 970-974.
Sobieszczanski-Sobieski, Jar.slaw; and Rogers, James L.: A Programming Sys-
tem for Research and Applications in Structural Optimization. Proceedings
International Symposium on Optimum Structural Design, llth ONR Naval Struc-
tural Mechanics Symposium, Oct. 19-22, 1981, University of Arizona, Tuscon
Arizona, pp. 11-9 to 11-21.
526


Design enhancement tools in MSC/NASTRAN

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