fuzzy move limit evaluation in structural optimization - CiteSeerX

FUZZY MOVE LIMIT EVALUATION IN STRUCTURAL OPTIMIZATION Pierre Grignon*1 Georges M. Fadel*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 approximation can be made in the neighborhood of the current design point, and the optimizer can make repeated calls to this approximation in lieu of the analysis program. However, since the approximation is local, the allowable changes in design variables have to be restricted. These restrictions are the move limits. We propose in this paper to increase or decrease the move limits using a fuzzy logic algorithm. The fuzzy rules take into account two parameters that are the exponent computed by the Two–Point Exponential method, an approximation method, and the effectiveness coefficient which compares the slope of constraints to that of the objective. In spite of the fact that the fuzzy rules are still dependent on the user, the algorithm provides a very easy interface to the user to set the move limits according to appropriate criteria and achieve a low and steady number of iterations irrespective of the maximum move–limits input by the user. Introduction The use of approximations in structural optimization improves the convergence and reduces the computational time spent in analysis and design. This is accomplished by calling on an approximation routine instead of running the actual analysis code. Many local approximations have recently been developed as variations of the traditional Linear Approximation or Taylor Series Approximation. These include the Reciprocal Approximation, Two–Point Exponential Approximation, Hybrid Approximation, Convex Approximation, Quadratic and Quadratic Reciprocal Approximations. This proliferation shows the concern of engineers to reduce computational time and improve the design capabilities. ––––––––––––––––––––––––– *1 Graduate student. Member ASME *2 Assistant Professor, Mechanical Engineering Department, Member AIAA and ASME.

Copyright E1994 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved

However, when such approximations are 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. These bounds, at first input by the user, could be adjusted after each step using different move–limits strategies. Additionally, the move limits could be selectively applied to each design variable. A good choice of move–limits would improve the accuracy and convergence of the problem. The move–limits could be treated as being dependent on the curvature of the objectives and constraints at a design point, and/or on the effect of a design variable on the constraints. Fadel3, Bloebaum2 , and Thomas9 have developed three main strategies to calculate move–limits. The Two–Point Exponential method1 is an approximation method that uses an exponent found by comparing slopes at the two last design points. This exponent is a measure of non linearity or curvature for each constraint and objective with respect to a design variable. Fadel uses the magnitude of the exponent to determine the move–limits based on a user prescribed maximum and minimum move– limit. The effect of a design variable on the design is measured by Bloebaum through an effectiveness coefficient. Bloebaum uses both effectiveness coefficients and expert system rules to generate move–limits. Thomas uses an algorithm which decreases the move–limits of all the design variables when the maximum constraint violation increases during the last design cycle. Individual move limits are also adjusted by increasing the bounds when the artificial bounds are repeatedly hit. The methodology developed in this paper results from the following observations:

S

S

1– The rapidity of convergence depends on the maximum and minimum move–limits given by the user. 2– Fadel uses a function based on the exponent computed by the Two–Point Exponential approximation technique to reduce the move–limit when the design becomes non–linear. But if the design is linear, the move–limit is not increased.

as the Adaptive strategy. These three methods are presented below. First, the two point exponential approximation is derived since the curvature based method depends on it. Derivation of Two–Point Exponential Approximation The two–point exponential approximation attempts to closely model the objective and constraint functions by using an exponent p and substituting X p for X in the Taylor series:

ȍ(x * x )ǒēxēf Ǔ n

S

3– Increasing or decreasing the move–limit is a strategy developed by Thomas9 but the change is applied discretely, not continuously.

These observations, in addition to the information gained from the analysis are used to define a methodology for move limits. Initially, the methods used by the three references cited earlier are described, then the information available is listed, and the methodology derived is detailed before applying it to a sample problem and drawing conclusions. Previous research Fadel 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. This method is referred to as the curvature based method in this paper. Recently, several attempts have been made to automatically determine move limits. Bloebaum2 used expert system rules and ”effectiveness coefficients” to generate move limits. This method is based on her argument that 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 design variable results in a significant change in the constraints, then its move limit should be restricted. Bloebaum also uses heuristics to control the move limits. This method is referred to as Effectiveness based method. 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 50%. 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%. Their method is referred to

f(X) + f(X 0) )

i

0i

i

i+1

resulting in the equation:

ȍƪǒxx Ǔ n

f(X) + f(X 0) )

i

i+1

0i

pi

ƫǒxp Ǔǒēxēf Ǔ 0i

*1

i

i

with the exponent evaluated according to

ǒ Ǔ

ln ēf ēx 1i pi + 1 ) ln x 1i

* *

ǒ Ǔ

ln ēf ēx 0i ln x 0i

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: p12 p13 ... p1m ȱp11 ȳ p21 p22 p23 ... p2m p32 p33 ... p3m ȧ ȧ ȧp31 ... ... ... ... ... ... ... ... ȧ ... ... Ȳpn1 pn2 ... ... pnm ȴ where m is the number of design variables and n is the number of functions (constraints + objective). This matrix of exponents is the set of exponents used in the approximating equation unless the magnitude of the exponents is larger than 1, 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. Curvature Based Method3: If the computed exponent is greater than the range allowed in the two–point exponential approximation(–1 to +1), then the approximation introduces an error by forcing this exponent to be restricted. If fp(x) is the value of the function obtained with the computed exponent p, and f1(x) is the value of the function computed with the maximum allowable

exponent (+1), then the two approximations, when looking at only one design variable, are p x 0 ēf x f p(x) + f(x 0) ) x 0 * 1 p ēx

ƪǒ

Ǔ

ƫ

evaluated through effectiveness coefficients as proposed by Bloebaum2. Effectiveness coefficients attempt to quantify the impact of a particular design variable on the design and are expressed as

f 1(x) + f(x 0) ) (x * x 0) ēf ēx

e ij

Now, assuming a relative change of A% (the move limit) in the design variable x + x 0 ) Ax 0 the error in the approximation is estimated by subtracting the two functions:

ƪ(1 ) A)p * 1 * Aƫ x ēxēf

dg j dx i + dF dx i

where eij is the effectiveness coefficient, gj is the constraint function, F is the objective function, and xi is the design variable. To determine the overall effectiveness of a design variable on all of the constraints simultaneously, it is necessary to form a cumulative constraint. This is accomplished by using the Kreisselmeier–Steinhauser function10:

p

Df(x) +

0

1 K.S. + C + r

ƪȍ ƫ m

e r@gj

j+1

which can be rearranged to give: (1 ) A) p * 1 Df(x) + * A + W1 p x 0 ēf ēx All of the terms on the left hand side of the equation are known; therefore, they can be grouped as one constant. 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 f1(x) is given by: x f 1(x) + f(x 0) ) (x * x 0) x0 ēf ēx

This function has the property that for a large choice of r (a user defined parameter) only the most critical constraints participate in the cumulative constraint. Taking the derivative of this function analytically, the following equation is obtained: *1

dC + dX i

ƪȍ ƫ ƪȍǒ m

j+1

x 0 ēf

ēx

+

Using these exponential functions to describe this relationship, a complete function of A for every p can be constructed as shown in Figure 1. In this method, the maximum and minimum move limits are user selected so that the engineer may still have reasonable control over the problem.

Now, these coefficients can be used to determine the move limits. The standard deviation of all of the effectiveness coefficients must be determined by the equation:

s(e) +

ƪ

N

ȍ

1 (e * e ) 2 N * 1 i+1 i

ƫ

1ń2

The design variables with effectiveness coefficients falling within one standard deviation are assigned a move limit based on a linear distribution between the bounds, based on the equation

Efficiency Based Method2 This move limit strategy is based on the effect of each particular design variable on the constraints. This is

Ǔƫ

dC dX i ei + dF dX i

(1 ) A) p * 1 * A + W2 p A)1

By selecting an acceptable error term W, an exponential function can be used to fit the resulting curve of A versus p.

j+1

dg j r@g e j dX i

from which the effectiveness coefficients in terms of the cumulative constraint can be derived:

and the error term can be put in the form: Df(x)

m

e r@gj

Ai +

(e i * e l) (A max * A min) ) A min 2s(e i)

where Amax and Amin are user prescribed maximum and minimum move limits, and e l is the lower bound of the effectiveness space defined as: _ e l + e * s(e) _ e u + e ) s(e) Effectiveness coefficients 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.

S

Function Values (objective and constraints)

S

Degree of violation of constraints

S

Gradients of objective and constraints with respect to each design variable

S

Effectiveness coefficients (Ratio of slopes of constraints over slopes of objective.

S

Kreisselmeier Steinhauser coefficient

S

Errors in the approximation (compared to real analysis after convergence of the approximate problem)

Adaptive Strategy method9 The method implemented by Thomas, Vanderplaats and Shyy9 considers what happens during the optimization process. The authors noticed that, depending on the type of approximation used (conservative or not) the move limits could over constrain the problem. Their work is based on the premise that the move limit for each design variable should be based on the accuracy of the approximation with respect to that design variable. They start by stating that a simple approach would be to reduce move limits as the optimization progresses towards the solution. This method can however stop the optimization far from the optimum. Therefore, they look at a global accuracy measure which is obtained at the end of every design cycle when approximation results can be compared to real analysis results. Based on these results, if the magnitude of the error is over some percentage, then the move limits are globally reduced by some other prescribed percentage. This global approach worked well with some problems, but also lead to premature convergence if the move limits were reduced too quickly. Therefore, an adjustment of individual move limits was added to the procedure in which if a particular design variable hits the move limit bounds for two consecutive design cycles, then the move limit associated with that design variable was increased by some percentage. Proposed Methodology The three methodologies presented use various indicators which are available to the designer. These indicators depend on the amount of information available from the analysis. The critical information available from an analysis at each design iteration is listed below. This information can either be directly read from the analysis results files, or can be easily computed. All the information is available when performing a structural optimization run with a gradient based method. S

Design Variables

If two successive design points are known, this additional information can be derived: S

Exponent or measure of curvature

S

Ratio of derivatives

S

Ratio of design variables

S

Additionally, if the slope of a constraint is 0, if the ratio of design variables is 0 or 1, additional information can be gathered.

S

Size of move from previous design point

S

Move direction (increase or decrease of variables)

What can be done with this information? Coordinating its use to decide on the optimal move limit is the objective of this work. And, since we need to combine rules, we propose to use fuzzy logic to accomplish this combination. What is Fuzzy logic Digital computers operate uniquely with discrete numbers. Continuous sets such as 9, the set of real numbers, can only be approximated. Moreover, computers use a two state logic (True (1) / False (0)) which make uncertainty or statements such as ”it is very cold” or ”it a steep slope” difficult to represent. Fuzzy logic, introduced by Zadeh, is a logic with an infinite number of truth states ranging from True (1) to False (0). It also has its own truth table, allowing the use of common logic operators such as ”and”, ”or”, and ”not”. With Fuzzy logic, a truth value ranging from 0 to 1, can be assigned to any logic set of rules. Fuzzy expert systems use this truth value to make

decisions when strict two valued algorithms fail. During the past few years, fuzzy controllers have spread in commercial and industrial applications. They are based on ”If.. then ..” rules that generate a fuzzy output based on a fuzzy input. It has been shown14, that these controllers can curve fit any continuous function, linking the output and the input. Figure 2. illustrates the fuzzy zone of a comfort zone for temperature. Basically, below 60°F, we can say it is cold, between 60 and 80°F, it may be cold to some, warm to others, and there is a progressive change, not an abrupt one. Above 100°F, it is definitely hot, but, from 80°F to 100°F, it is also gets progressively hotter.

The 25 bar transmission tower problem consists of 7 design variables, 50 stress constraints, and 12 displacement constraints. The design data for this problem are E=10 4 ksi, r=0.1 lb/in3, minimum cross sectional area .01 in2, stress limit 40 ksi, and displacement limit .35 in. The loading conditions are shown below in Table 1, and the 25–bar tower is shown in Figure 5. Table 1. Loading conditions for 25–bar test problem Node

x

y

z

1

1 2 3 4

0.5 0.5 1.0 0.0

0.0 0.0 10.0 10.0

0.0 0.0 –5.0 –5.0

2

2 4

0.0 0.0

–100.0 –100.0

–5.0 –5.0

Why use Fuzzy Logic In the problem at hand, discrete information is generated by the analysis and the algorithms implemented. However, the rules used to generate the size of the move limit are fuzzy, with different considerations given to the curvature of the functions and to the magnitude of the effectiveness coefficient. The simplicity of the Fuzzy logic algorithm implemented by Henry Hurdon16 and its adaptability, made it an attractive substitute to a standard rule based method. Implementation The code developed by Hurdon15 considers two inputs to the Fuzzy controller: The temperature and Humidity of a room, and the output generated is the speed of a fan based on these two inputs. The code was modified to accept the exponent computed in the two point exponential approximation, and the effectiveness coefficient, as well as the mean and standard deviation of the coefficients. These are combined using a fuzzy logic algorithm to result in a multiplier for the move limit originally set by the user. The table illustrated in Figure 3. displays the multiplier output as a function of the two parameters. In this table, the values LOW, MED and HIGH are defuzzified into multipliers as illustrated in Figure 4. The product of the exponent and the effectiveness returns a value that generates a linearly interpolated value between some minimum and maximum multipliers that the user can modify. The aim was to be able to increase the move–limit bounds in the cases where the exponent is between 1 and –1, and the effectiveness coefficient indicates a large move limit, and reduce the move limits when both methods indicate smaller bounds. Results One standard problem is used in this paper to compare the methods. This is the 25 bar transmission tower with stress and displacement constraints13. The finite element analysis package STAP11 is used in conjunction with CONMIN12, an optimizer based on the usable– feasible directions method. In all test cases, the criteria for convergence is .001.

The results comparing the number of iterations and the accuracy of the results are displayed in Figure 6. The figure shows that irrespective of the original maximum move limit imposed by the user, the number of iterations is relatively constant. This contrasts with earlier work that shows a strong correlation between the imposed upper and lower maximum and minimum move limits that the previous algorithm used. The fuzzy algorithm is very easy to set up and especially to modify and adapt. The rules for the two parameters that affect the move limits can be changed in one statement, and the defuzzification graph dictates how the final move limits are related to both parameters. The table shown in Figure 3. can also be very easily modified, especially the bounds of the domains dry, moist and wet, or cool, warm and hot and the corresponding speeds which are the multipliers of the maximum move limit imposed by the user. Conclusion: This paper summarizes the work to date on move limits for structural optimization. It lists the information available to the engineer to make informed decisions about move limits and approximations. A Fuzzy algorithm that allow easy combination of rules is presented and tested on a sample problem. Even though the results to date are not conclusive, the algorithm shows promise, but most of all, flexibility and ease. References [1] Fadel, G.M., Riley, M., Barthelemy, J.F., ”Two Point Exponential Approximation Method for Structural Optimization.” Structural Optimization, Vol.2., pp. 117–124, 1990. [2] Bloebaum, C.L., ”Variable Move Limit Strategy for Efficient Optimization.” 32nd Structures, Structural Dynamics, and Materials Conference, Baltimore, MD April 1991.

[4] Storaasli, O.O., Sobieszczanski–Sobieski, J. ”On the Accuracy of the Taylor Approximation for Structure Resizing” AIAA J. 12, pp. 231–233, 1974. [5] 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 AIAA/ASME 18th Structures, Structural Dynamics, and Materials Conference, pp. 71–79.

[15] Hurdon, H, ”Fuzzy.c” a code available on FTP sites, Mar, 1993.

MAXIMUM MOVE LIMIT

1 Relative Change

[3] Fadel,G.M., Cimtalay, S., ”Automatic Evaluation of Move–Limits in Structural Optimization” Submitted to Structural Optimization, 1992.

0.5

MINIMUM MOVE LIMIT

0

–5

[7] Haftka, R. T., Shore, C.P., Approximation method for combined thermal/structural design. NASA TP–1428, 1979

0 Exponent P

5

Figure 1. Mesa Function

[8] Starnes, J.H. Jr., Haftka, R.T., Preliminary design of composite wings for buckling, stress and displacement constraints. J. Aircraft 16, pp564–570, 1979.

COLD

[9] 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 AIAA/USAF/NASA/OAI Symposium on Multidisciplinary Analysis and Optimization, Cleveland, OH, pp. 507–512, 1992.

60 [10] 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.

[14] Kosko, B. and Saturo, I. ”Fuzzy Logic”, Scientific American Vol. 269, No 1.pp76, July 1993.

100

EFFECTIVENESS e+ s 2

LOW

MED

HIGH

Figure 3. Fuzzy Controller table

T

STRESS AND DISPLACEMENT CONSTRAINTS

MULTIPLIER

4

2

.5 0.05 LOW

MED

HIGH FUZZY OUTPUT

Figure 4. Defuzzification graph

Figure 5. Twenty–five bar Transmission Tower VOLUME 17000

15000 20% MAX 13000

11000

9000 50% MAX 7000

5000

90% MAX 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ITERATIONS Figure 6. Progression of objective volume