.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 Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359 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 Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359 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: Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359 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. Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359 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. - Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359 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. Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359 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.  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 ~  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.  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 Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359 :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 Downloaded by UNIVERSITY OF ADELAIDE on October 25, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1994-1359 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.