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6.1994-1359

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.ISON OF VARIOUS MOVE LIMIT STRATEGIES IN
STRUCTURAL OPTIMIZATION
*'
Daniel G. Hyams
Mississippi State University
Jackson, Mississippi
Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359
Georges M. Fade1 "2
Clemson University
Clemson, South Carolina
Abstract
Many recent improvements have been made in the
field of optimization, among which is the introduction of the
theory of approximations to improve problem convergence
and reduce the computational burden of repeated full analyses. This theory states that an approximationcan be made in
the neighborhood of the current design point, and the optimizer can make repeated calls to this approximationin lieu of
the analysis program (usually finite elements). However,
when such approximationsare introduced, a strategy must be
formulated which determines the n-dimensional domain
around the current design point where the approximation is
still applicable. The bounds of this domain are called move
limits and typically, they are uniformly applied to all the design variables according to the experience of the user.
The objectives of this research are to compare two
recently proposed strategies for the calculation of move limits, and propose either new or hybrid methods that allow the
design problem to converge to the correct answer in the least
number of iterations.
This paper treats the move limit as a function of two
parameters: the curvature of the objectivelconstraintfunctions at the current design point, and the effect of a particular
design variable on the constraints. The curvature of the function is indicated by the magnitude of the exponent obtained
by theTwo-Point Exponential approximation1. The effect of
the design variable on the constraint can be determined by effectiveness coefficients2.
The Two-Point Exponential approximation uses
previous design histories to find a better fitting approximation. This approximationuses an exponent which is obtained
by matching slopes at the previous design point. This expo*lMechanical Engineering student.
Research Experience for
Undergraduatesparticipant at Clemson University
Assistant Professor, Mechanical Engineering Department,
Member AlAA and ASME.
'2
Copyright 01994 by the American Institute of Aeronautics and
Astronautics, Inc. All rights reserved
nent provides a measure of curvature3for eachconstraint and
with respect to each design variable. Next, depending on the
magnitude of the exponents specific to a single design variable, move limits can be assigned based on some absolute
minimum and maximum allowable limits that the user prescribes.
Using the Kreisselmeier-Steinhauser 6 . S . ) function2, an overall effectiveness coefficient which relates the
effect of a particular design variable on all of the constraints
simultaneously can be derived. Then, based on the variation
between these effectiveness coefficients, move limits for
each variable can be computed. These move limits are also
based on some absolute minimumand maximum that the user
prescribes.
Both strategies are implemented and compared.
Speed of convergence is tabulated for various minimum and
maximum allowable move limits. In particular, the variations of the individual move-limits computed at each iteration for both methods are compared to identify differences
between the two strategies and devise new methods.
Five variations and combinations of the two strategies are tested on a structural design problem. The comparison of these move limit strategies shows that the method
based on the curvature of active constraints provides the best
balance between accuracy and speed of convergence.
Introduction
In many modern engineeringdesign problems, optimizationmust be used. Typically, theengineer mustfirstestimate the initial design parameters, analyze this design
through some analysis process (finite elements in the case of
structural design), and then check to see if the design is optimal; i.e. no improvement in the objective function can be
1
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made without violating one or more constraint functions. If
the optimum design isnot achieved, then the analysis is connected to an optimizer. This optimizer attempts to minimize
the objective function while satisfying the constraints typically by using gradient information. During the process, the
design parameters are changed, and the optimizer makes repeated calls to the analysis program to evaluate subsequent
designs. This iterative process demands an extremely large
amount of computer time for real-world problems, which in
turn raises the cost of the design.
requires a minimal amount of effort.
Many other approximations were developed over
the last two decades. Some are more appropriate to certain
types of problems. others try to over-constrain the problem to
ensure a feasible solution. Notable approximations are the
reciprocal appr~ximation~>*~,
the modified approximation7,
the Conservative Convex or Hybrid approximations.
By introducing any of these approximations,however, the problem mentioned earlier arises: how can we determine the distance from the current design point which
limits the applicability of the approximation? Until recent research in the subject, engineers have simply chosen these
move limits, usually constant throughout the optimization
and uniform over the design variables, based on their experience with the particular problem they are dealing with. This
paper summarizespast research in variable move limit strategies and proposes several new combinationsor modifiiations
of these strategies.
A major improvement to this classical design approach is to use the approximation theory. This theory states
that one can approximate the given problem, optimize this
approximation, and then check for global convergence.
However, since the approximations are only valid in a reduced space, some mechanism to avoid misleading the optimizer is needed. A flowchart of this process is shown in
Figure 1.
Many approximation schemes can be used, but the
most common is the simple Taylor series expansion of the
functions around the design point. This is given by the following equation:
By choosing appropriate move limits, the speed of
problem convergence can be improved drastically. Also, current research in multi-disciplinary optimization demands an
effkient method for assigning move limits -- by setting these
correctly, one problem may be optimized from the viewpoint
of different engineering fields. In effect, the m w e limit
would define an "allowable change" in a particular design
variable for multi-disciplinary optimization.
Because the first derivatives at the design point are
already known either directly from the finite element analysis, or from some m e r e d a t i o n scheme, this approximation
Initialize Variables
I
Convergence?
\
I
Approximation
Yes
I'
L
Optimizer
Figure 1. Design Optimization Process
A
'
Previous research
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Recently, several attempts have been made to automatically determine these move limits. Bloebaum2used expert system rules and "effectivenesscoefficients"togenerate
her move limits. This method is based on her argumentthat if
a design variable does not have a significant effect on the
constraints, then it should be allowed more leeway in its
move. However, if changing a designvariableresults in a significant change in the constraints, then its move limit should
be restricted. Bloebaum also uses heuristics to control the
move limits. This part of her procedure is not investigated in
this paper.
Thomas and Vanderplaats9 used heuristic rules to
determine move limits. If the maximum constraint violation
has increased during the last design iteration, then all move
limits are decreased by 500/0. Also, if a design variable hits
the same upper or lower limit on two consecutive design
cycles, then its individual move limit is perhaps too restrictive; hence, it is increased by 33%.
Fade1 and Cimtalay3 proposed to use the exponents
from the two-point exponential approximation to determine
the move limit. Since the exponent introduces a degree of
curvature into the approximation, one can use the value of
this exponent to indicate the nonlinearity of the functions and
therefore the magnitude of the individual move limit.
In this paper, we propose to use the information provided to us by the two-point exponential approximation and
the effect of the design variable on the constraints to assign
the best move limit to a design variable.
Derivation of Two-Point Exponential Approximation
The two-point exponential approximation attempts
toclosely model the objective and constraintfunctions by using an exponentp and substituting XP for X in the Taylor series:
resulting in the equation:
with the exponent evaluated according to
where the subscript 1 refers to the previous design point, and
the subscript 0 refers to the current design point from which
the approximation is carried out. Note the similarity in form
to a second derivative.
Using this method, an exponent p is computed for
each function and with respect to each design variable, forming the following matrix:
p l l p12 p13 ... p l m
p21 p22 p23 ... p2m
p3 1 p32 p33 ... p3m
[::
::
::
1:
1::
pnlpn2 ... . . . pnm
1
where m is the number of design variables and n is the number
of functions (constraints + objective). This matrix of expe
nents is the set of exponents used in the approximatingequation unless the magnitude of the exponents is larger than l , or
less than - 1. In such a case, the limiting cases of +1 or - 1 are
substituted for the computed exponent in the approximation.
These limits were found necessary to ensure convergence1.
Derivation of Move Limit Stratepies
In the optimization process, move limits must be
imposed on the design variables to prevent the design from
being driven too deeply into the infeasible region, where the
optimizer cannot recover. In this paper, the move limit is
treated as a function of two parameters: the curvature of the
functions in relation to the design variable in question, and
the effect of the design variable on the constraints.
The exponents in Fadel's two-point exponential
approximationprovide a very convenient measure of curvature of the functions. The effects of a design variable on the
constraints can be measured by effectiveness coefficients as
defined by Bloebaum2. Initially, both methods are
compared, then hybrid methods are generated to try take advantage of both methods and ultimately derive a robust and
efficient methodology.
Method 13:
If the computed exponent is greater than the range
allowed in the two-point exponential approximation(- 1 to
+ I ) , then the approximation introduces an error by forcing
this exponent to be restricted. Iffp(x)is the value of the function obtained with the computed exponent p, and fl(x) is the
value of the function computed with the maximum allowable
exponent (+I), then the two approximations,when looking at
only one design
- variable, are
x
=
x
+
[
- 11"
P3
ax
Now, we can assume a relative change of A% (themove limit)
in the design variable
X = xo + Ax0
and estimate the error in the approximationby subtracting the
two functions:
which can be rearranged to give:
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All of the terms on the left hand side of the equation are
known; therefore, we can group them as one constant.
where eij is the effectiveness coefficient, gj is the constraint
function,F is the objective function, and4 is the design variable. To determine the overall effectivenessof a design variable on all of the constraints simultaneously,it is necessary to
form a cumulative constraint. This is accomplished by using
the Kreisselmeier-Skinhauser functionlo:
The same procedure can be repeated for negative
exponents with a similar result. In this case, the exponent is
forced to -1; therefore, the function fl(x) is given by:
and the error term can be put in the form:
( -l
-
af
+ A ) P - l
XOdx
P
This function has theproperty that for a large choice
of Q (a user defined parameter) only the most critical
constraints participate in the cumulative constraint. Taking
the derivative of this function analytically, we can write
A -Wi
A+I
By selecting an acceptable error term W, we can use
an exponential function to fit the resulting curve of A versus
p. The plots of A versusp for various W's is shown in Figures
2 and 3 for positive and negative values of the exponent respectively. In this paper, W is selected as .2 and the resulting
curve-fit is:
from which we can write the effectiveness coefficients in
terms of the cumulative constraint:
for all positive exponents greater than +1 and:
for all negative exponents less than -1.
Now, we can use these coefficients to determine the
move limits. The standard deviation of all of the effectiveness coefficients must be determined by the equation:
Using these exponential functions to describe this
relationship, we can construct a complete function of A for
every p, as shown in Figure 4.
The maximum and minimum move limits are user
selected so that the engineer may still have reasonable control
over the problem.
The design variables witheffectivenesscoefficients
falling within one standard deviation are assigned a move
limit based on a linear distribution between the bounds,
based on the equation
Method 2;
This move limit strategy is based on the effect of
each particular design variable on the constraints. This is
evaluated through effectiveness coefficients as proposed by
~ l o e b a u m ~Effectiveness
.
coefficients attempt to quantify
the impact of a particular design variable on the design and
are expressed as
where&, a n d h a are user prescribed maximum and minimum move limits, and e' is the lower bound of the effectiveness space defined as:
Effectivenesscoefficients falling outside of these bounds are
assigned a maximum move limit if they are above the upper
bound, or a minimum move limit if they fall below the lower
bound.
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Com~arisonof the two methodst
One standard problem is used in this paper to
compare the two methods. This is the 25 bar transmission
tower with stress and displacement constraint^'^.
The finite element anal sis package STAP1lis used
in conjunction with CONMINIY,an optimizer based on the
usable- feasible directions method. In all test cases, the criteria for convergence is .001.
The problem consists of 7 designvariables, 50 stress
constraints, and 12 displacement constraints. The design
data for this problem are E=104 ksi, e=0.1 1b/in3, minimum
cross sectional area .O1 in2, stress limit 40 ksi, and displacement limit .35 in. The loading conditions are shownbelow in
Table 1, and the 25-bar tower is shown in Figure 5.
Table 1. Loading conditions for 25-bar test problem
this particular case, both methods use the same number of iterations. Both methods converge to the same result and both
are slightly more expensive in terms of number of iterations
than the reciprocal approximation. Figure 10shows the same
graphfor move limits between 80% and 20%. In this case, the
reciprocal approximation requires 21 iterations to converge,
the effectivenessmethod 2 1 iterationsand the 2 point approximation 17 iterations.
These tests show the sensitivity of the methods to
the user input maximum and minimum move limits. Both
methods depend on a variety of parameters such as the thickness of the constraint, the way the methods handle special
cases like the cases where the derivatives are 0, or one variable at the design point is 0, or the ratio of two variables at
successive design points is 1.
One interesting observation when comparing the
two methods is that the two point approximationmethod determines the move limit from either the largest or smallestexponent computed by the method for each design variable.
The method does not distinguish between active or non-active constraints, and uses all the functions to find the extremas. the effectiveness method is based on a completely
different methodology. The effectiveness coefficients consider all the constraints and their different degree of "activity" in the process. These effectivenesscoefficients are then
averaged and deviationsfrom these averages are used to determine move limits.
These differences prompted us to investigatehybrid
methods that combine both methodologies. The following
alternatives are presented.
Method 3:
This method is essentially a hybrid of Method 1and
Method 2. The curvature of the function is taken into consideration, but the effectiveness coefficients are applied if the
specified curvature conditions are not satisfied.
The results comparing the number of iterations and
the accuracy of the results are displayed inFigures 6.7 and 8.
Figures 6 and 7 show the progression of the value of the objective (volume) for two different move limit bounds
(80%-10%. and 50%-10%) for the reciprocal approximation, the effectiveness based algorithm and the two point algorithm. Figure 8. shows the progression of variable 4 and
the computed move limits for the three cases and m w e limits
bounds of 50% and 10%. In this figure, the three graphs are
offset by 5 in the x direction to be able to discern the differences between the methods. We note that this particularvariable often hits the lower move limit in all three cases. Figure
9 displays theprogression of the move limits computed by the
twomethods. The figure shows that in somecases, onemethod might either oscillate between maximum and minimum
allowable moves whereas the other might modify the moves
as the optimization progresses and slowly tighten the allowable range of the move limits to converge on the solution. In
Method 3 is based on the premise that a low curvature function should be allowed the maximum move limit.
The approach used in this paper is to allow the maximum
move limit if the maximum exponent for a design variable
falls into the range of - 1 to +l. If this condition is false, then
the move limit is applied based on effectivenesscoefficients.
Method 4:
Method4 imposes the same condition as above, but
also considers high curvature of the functions. If the maximum exponent is outside of a preset value then the minimum
move limit is imposed. If the maximum exponent is within - 1
to 1, the maximum move limit is allowed. The design variables with intermediate exponents have move limits decided
by their respective effectiveness coefficients.
To determine the "high exponent" range, we can
again look at the W error terms derived in Method 1. Setting
W=.2, we can numerically plot two curves of A versus p.
Now, we can choose an acceptablecutoff point by examining
where these curves become asymptotic horizontally, and
choose an acceptable derivative to describe this condition. In
this paper, a derivative of 1.0 was chosen (move limit only
changes by one percent if exponent changes by 1) and the derivatives were taken by finite difference along each curve to
find the location of this derivative value. Choosing W=.2, the
negative maximum exponent was -15.5 and the maximum
positive exponent was 17.0.
Table 2. Summary of Move L i t Strategies
limit.
-
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Method 5:
-
5
A matrix of move limits A(i j) are computed by the
mesa function to correspond to each p(i j). Then the
average of the move limit matrix is taken across each
row.
6
One p,,,,p,i,
is picked for each row of matrix p(i,j),
but only exponents from the active constraints are allowed. The mesa function then translates these exponents to move limits.
7
A matrix of move
A(i j) are computed by the
mesa functionfor each p(i,j). Then, the average is taken along each row of A(i,j) of only the move limib
generated from active constraints.
Method 6:
Method 7;
Method 7 modifies Method 1 by only considering
exponents whose respective constraints are active (i.e. within
a certain tolerance of zero). A matrix of move limits are computed by the mesa function for each of these exponents, and
the average is taken for each design variable.
Summary of Method=
The seven variable move limit strategies presented
in this paper are summarized in Table 2 below:
Results:
The seven methods were tested on the case defined
in the preceding section. Evaluation of these methods is
based on the number of iterations required for convergence
and accuracy of the result.
Each of the methods was run with varying maximum and minimum move limit ranges. The average and
standard deviation of the number of design cycles for each
method provide an indication of how well each method performs. These runs are compared to traditional optimization
methods.
-
I f ~ a x , ~ m i n f ~ r e a c h r ~ isintherange-1..1,
~~fp(ij)
maxlmum move limit is allowed. I f h a x .Pminis Out
of range -155.17. then minimum move limit is
forced. Otherwise, effectiveness coefficients determine the move limit.
Method 5 also uses the mesa function presented in
Method 1; however, instead of picking the maximum exponent, a matrix of move limits A(i j ) are computed for every
p(i,j) and then averaged along each row for each design variable.
Method 6 modifies Method 1 by only considering
exponents whose respective constraints are active (i.e. within
a certain tolerance of zero). The maximum exponent is then
picked for each design variable from only this set, and the
mesa function is used to translate this exponent into a move
limit.
- --
If p,,, p,i, for each row of p(ij) is in the range - 1.. 1,
maximum move limit allowed. Otherwise, effectiveness coefficients determine the move limit.
From these results, the following conclusions were
drawn:
1. For the particular case of the 25 bar transmission
tower with stress and displacement constraints, method 3
matches exactly the effectiveness method (2). This means
that the effectivenesscoefficient is the controlling parameter
in the move limit calculation, and the exponents that fall in
the -1 to +1 range do not play an active role.
2. Method 4 which uses the exponents when they
fall between -1 and +1 or are lower than -15.5 or higher than
17 and uses the effectivenesscoefficients otherwise, typically does not converge to the correct result. The method is affected significantly by the user supplied limits.
3. Method 6 whichuses the mesa function and only
considers activeconstraintsprovides the best performance in
terms of number of iterations, and insensitivity to user supplied limits.
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Table 3. Comparison of number of iterations for various methods and ranges of move limits.
Note:
* Only the largest limit is applicable to the linear, reciprocal and 2 point exponential approximations
** Converges to a slightly higher optimum.
Conclusion:
Two move limit strategies are compared and hybrid
methods are derived to achieve the best balance between reduction of the number of iterations and insensitivity to user
supplied move limits. The mesa based method converges
consistently to the correct result independently of the user
supplied limits. However, the number of iterations can vary
significantly depending on these limits. This method relies
on the most critical exponent across all the constraints and
objective of the problem. The effectiveness method which
uses statisticalmeans and the degree of constraint violationor
activity converges in a similar number of iterationsto method
1. This method shows less dependency on the user supplied
limits, but does occasionallyconverge to a suboptimalresult..
The remaining methods show that the use of active
constraints in the calculation of move limits is imperative.
The combination of active or violated move limits and mesa
function consistentlv converges to the correct result and uses
the s&e number ofiterations independentlyof the user supplied limits.
Note that these methods presuppose that the user
does not know which approximation would yield the best result for the vroblem at hand. The number of iterationsneeded
to converge to the correct result may be slightly higher than
that of the most appropriate approximation method, but can
be significantly smaller than the number of iterations needed
with an inappropriate method such as the linear approximation in the case presented.
References
[I] Fadel, G.M., Riley, M., Barthelemy, J.F., 'Two Point
Exponential Approximation Method for Strucmal
Optimization." Structural Optimization, Vo1.2.. pp.
117-124,1990.
[21 Bloebaum, C.L., "Variable Move Limit Strategy for
Efficient Optimization." 32nd Structures, Structural
Dynamics. and Materials Conference, Baltimore, MD
April 1991.
[31 Fade1,G.M.. Cimtalay, S., "Automatic Evaluation of
Move-Limits in Structural Optimization" Submitted to
Structural Optimization, 1992.
[41 Storaasli, 0.0..Sobieszczanski-Sobieski, J. "On the
Accuracy of the Taylor Approximation for Structure
Resizing" AIAA J. 12, pp. 23 1-233.1974.
[51 Noor, A.K., Lowder, H.E., "Structural Analysis via a
Mixed Method". Comp & Struct. 5, pp. 9- 12,1975.
[6] Austin, F., "A Rapid Optimization Procedure for Structures Subjected to Multiple Constraints." Proceedings of
the AIAAIASME 18th Structures, Structural Dynamics,
and ~
~ conference,
~
&pp. 71-79.
~
l
~
[7] Haft.%R- T-9 Shore. C-P-,Approximation method for
combined thermdstructural design. NASA TP- 1428,1979
181 Starnes, J.H. Jr., Haftka, R.T.. Preliminary design of
composite wings for buckling, stress and displacement
constraints. J. Aircraft 16, pp564-570,1979.
[91 Thomas, H.L., Vanderplaats, G.N., and Shyy, Y-K. "A
Study of Move Limit Adjustment Strategies in the
Approximation Concepts Approach to Structural
Synthesis." Fourth ATAA/LTSAF/NASA/OAI Symposium
on Multidisciplinary Analysis and Optimization,
Cleveland, OH, pp. 507-512,1992.
[lo] Kreisselmeier, G., Steinhauser, R., "Systematic
Control Design by Optimizing a Vector Performance
Index," Proceedings of the IEAC Symposium on Computer
Aided Design of Control Systems, Zurich, Switzerland,
1979.
Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359
[Ill Bathe, K., Wilson, E.L., Numerical Methods in Finite
Elements, Prentice Hall, 1976.
[I21 Vanderplaats, G.N., "CONMIN, a Fortran Program for
Constrained Function Minimization. User's Manual."
NASA Technical Memorandum, TM-X-62,282,1973.
[13] Haug, E.J., and Arora, J.S., Applied Optimal Desiw Mechanical and Structural Svstems. Wiley Interscience
example4.6. 1979.
Exponent P
Figure 3. Plot of A vs negative p for Various &or Terms
MAXIMUM MOVE LIMIT
t--------
MINIMUM MOVE LIMIT
-1.0
1
2
i
J
5
6
7
R
9
1
0
Exponent P
Figure 2. Plot of A vs positive p for Various Error Terms
Figure 4. Mesa Function
(
1
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:o
[tention number
Fiooure 7. Progression of objective volume
SO
iW l 3
- 10 move limits
I'
:O
i9
Iteration number
Fi,m 6. Progression of objective volume
Fiome 8. Progression of design variable 4
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Move Limit on DV4 30%
I
10
- 20%
20
Iteration Humber
Figure 9. Progression of move limits 50%-1Wo move.
Figure 10. Progression of move limits 80%-20%move.
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