[American Institute of Aeronautics and Astronautics 10th AIAA/ISSMO Multidisciplinary Analysis and...

American Institute of Aeronautics and Astronautics1

Robust Optimal DesignWith Differential Evolution

Man Mohan Rai*NASA Ames Research Center, Moffett Field, CA-94035

ABSTRACT

Traditional aerodynamic shape optimization has focused onobtaining the best design given the requirements and flowconditions. However, the flow conditions are subject to changeduring operation. It is important to maintain near-optimalperformance levels at these off-design operating conditions.Additionally the accuracy to which the optimal shape ismanufactured depends on the available manufacturing technologyand other factors such as manufacturing cost. It is imperative thatthe performance of the optimal design is retained when t h ecomponent shape differs from the optimal shape because o fmanufacturing tolerances and normal wear and tear. Theserequirements naturally lead to the idea of robust optimal designwherein the concept of robustness to various perturbations is builtinto the design optimization procedure. The imposition of thisadditional requirement of robustness results in a multiple-objective optimization problem requiring appropriate solutionprocedures. Evolutionary algorithms have been used successfullyin design optimization. Here a new evolutionary method formultiple-objective optimization is presented. It draws upon ideasfrom several genetic algorithms and evolutionary methods; one o fthem being a relatively new member to the general class o fevolutionary methods called differential evolution. The capabilitiesof the evolutionary method developed here are investigated usingsome complex model problems. Good solution accuracy anddiversity are obtained in all these cases. The method is thenapplied to robust optimal design. Applications include the designof fins used in boiling heat transfer and airfoil design.

INTRODUCTION

Fabricating and operating complex systems involves dealing with uncertainty in the relevantvariables. In the case of aircraft, flow conditions are subject to change during operation. Efficiency andengine noise may be different from the expected values because of manufacturing tolerances and normalwear and tear. Engine components may have a shorter life than expected because of manufacturingtolerances. In spite of the important effect of operating- and manufacturing-uncertainty on theperformance and expected life of the component or system, traditional aerodynamic shape optimizationhas focused on obtaining the best design given a set of deterministic flow conditions. Clearly it isimportant to both maintain near-optimal performance levels at off-design operating conditions, and,ensure that performance does not degrade appreciably when the component shape differs from theoptimal shape due to manufacturing tolerances and normal wear and tear. These requirements naturallylead to the idea of robust optimal design wherein the concept of robustness to various perturbations isbuilt into the design optimization procedure.

Senior Scientist, Information Sciences and Technology Directorate. Associate Fellow, AIAA.

10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference30 August - 1 September 2004, Albany, New York

AIAA 2004-4588

This material is declared a work of the U.S. Government and is not subject to copyright protection in the United States.


Recognition of the importance of incorporating the probabilistic nature of the variables involved indesigning and operating complex systems has led to several investigations in the recent past. Some ofthe basic principles of robust optimal design are discussed by Egorov et al.1. Several commonly usedapproaches such as maximizing the mean value of the performance metric, minimizing the deviation of thismetric and, maximizing the probability that the efficiency value is no less than a prescribed value arediscussed in their paper. Egorov et al.1 make the observations that a) robust design optimization is inessence multi-objective design optimization because of the presence of the additional objective(robustness) and, b) the addition of the robustness criterion may result in an optimal solution that issubstantially different from that obtained without this criterion. Various approaches to robust optimaldesign are also mentioned in this article.

While the discussion above focused on the effect of uncertainty in the variables on performance,their effect on constraint satisfaction is equally important from a reliability perspective. Here the focus is onmaximizing the probability of constraint satisfaction. Koch et al.2, provide a discussion of this and relatedconcepts. Some of the basic steps involved in both robust optimal design as well as reliability-basedoptimization such as a) identifying random variables and their associated probability density functions, b)reducing this set of variables to a smaller subset of key random variables, to reduce optimization costs and,c) the effective utilization of Monte Carlo techniques to obtain estimates of performance variability orreliability, are discussed by the authors.

Simulation based design optimization can be computationally expensive in cases where theunderlying physics is complicated. Some of the contributing factors are three-dimensionality, a largedisparity in the largest and smallest scales that are required for an accurate analysis etc. The addition of therobustness criterion can greatly increase computational requirements because of the need to estimatethe variance in performance or reliability. Koch et al.2, reduce computational cost by first obtaining theoptimal solution via a deterministic approach and subsequently adding the reliability requirement. In aseparate article Koch et al.3use Kriging models to compute performance variability and reliability.

The imposition of the additional requirement of robustness results in a multiple-objective

optimization problem requiring appropriate solution procedures. Typically the costs associated withmultiple-objective optimization are substantial. Efficient multiple-objective optimization procedures arecrucial to the rapid deployment of the concepts of robust design. Hence a significant portion of this articledeals with methodology for solving multiple-objective optimization problems efficiently, reliably and withlittle user intervention.

Genetic and evolutionary algorithms4 have been applied to solve numerous problems inengineering design where they have been used primarily as optimization procedures. These methodshave an advantage over conventional gradient-based search procedures because they are capable offinding global optima of multi-modal functions (not guaranteed) and searching design spaces with disjointfeasible regions. They are also robust in the presence of noisy data. Another desirable feature of thesemethods is that they can efficiently use distributed and parallel computing resources since multiplefunction evaluations (flow simulations in aerodynamics design) can be performed simultaneously andindependently on multiple processors. For these reasons genetic and evolutionary algorithms are beingused more frequently in design optimization. Examples include airfoil and wing design5 - 6 and compressorand turbine airfoil design. They are also finding increasing use in multiple-objective and multidisciplinaryoptimization.7 The references cited here represent a small sample of the literature.

A disadvantage of genetic and evolutionary algorithms is that they often require many morefunction evaluations than other optimization schemes to obtain the optimum. In fact they are not thepreferred method when a purely local search of a smooth landscape is required. Rai8 presents anevolutionary method, based on the method of Differential Evolution9 (DE), and investigates its strengthsin the context of some test problems as well as nozzle and turbine airfoil design. The results of applying aneural network-based response surface method (RSM10 – 11) to the same design problems are also


presented in this study. It was found that DE required about an order of magnitude more computing timethan the neural network-based design method. Rai8 also explores the possibility of integrating designprocedures based on neural networks and evolutionary algorithms; the objective being a hybridaerodynamic design procedure that has the strengths of both the approaches. In a more recent articleMadavan12 has also explored the possibility of combining DE with local search methods and, utilized theresulting hybrid method in airfoil inverse design. The best variant of these combined methods required420 function evaluations for this inverse design. In contrast, a very similar inverse design problem required36 simulations with the neural-network based algorithm13 - 14. The estimated number of simulations toperform the inverse design of Madavan12 with the neural network-based method is about 50, i.e., thecomputational cost is about an order of magnitude less. In general where applicable, significant costreductions can be achieved by using gradient- and RSM-based methods instead of evolutionaryalgorithms. However, the latter approach is preferred for multi-modal functions and design spaces withdisjoint feasible regions.

Multiple-objective design optimization is an area where the cost effectiveness and utility ofevolutionary algorithms (relative to local search methods) needs to be explored. Rai15 presents anevolutionary algorithm, based on the method of DE, for multiple-objective design optimization. One goalof this developmental effort was a method that required a small population of parameter vectors to solvecomplex multiple-objective problems involving several Pareto fronts (global and local) and nonlinearconstraints. Applications of this evolutionary method to some difficult model problems involving thecomplexities mentioned above are also presented in this article. The computed Pareto-optimal solutionsclosely approximate the global Pareto-front and exhibit good solution diversity. Many of these solutionswere obtained with relatively small population sizes. One of the computed solutions (referred to as ZDT4later in the text), was found elusive to capture in previous studies by other investigators. Applications ofthis method to robust design are also included in this article. The associated computational costs arecompared with those incurred with a neural network-based RSM approach. In Ref.15, unlike Koch3, neuralnetworks are used to compute performance variability.

An enhanced version of the method of Rai15 is presented here. One important goal of the currenteffort is to significantly reduce the computational costs of multi-objective optimization. The new methodhas been designed to require smaller populations than the earlier version. The number of functionevaluations required is proportionately reduced for the problems considered.

Applications of this new method to robust design are also presented here. The evolutionarymethod is first used to solve a relatively difficult problem in extended surface heat transfer wherein optimalfin geometries are obtained for different safe operating base temperatures. The objective of maximizingthe safe operating base temperature range is in direct conflict with the objective of maximizing fin heattransfer. This problem is a good example of achieving robustness in the context of changing operatingconditions. The evolutionary method is then used to design a turbine airfoil; the two objectives beingreduced sensitivity of the pressure distribution to small changes in the airfoil shape and the maximizationof the trailing edge wedge angle with the consequent increase in airfoil thickness and strength. This is arelevant example of achieving robustness to manufacturing tolerances and wear and tear in the presenceof other objectives.

The present article also explores the possibility of using neural networks to obtain estimates of thePareto optimal front. Achieving solution diversity and accurate convergence to the exact Pareto frontusually requires a significant computational effort with the evolutionary algorithm. The use of neuralnetwork estimators has the potential advantage of reducing or eliminating this effort thus reducing cost todesign. The estimating curve or surface can be used to generate any desired distribution of Paretooptimal solutions. These estimators can be used to some extent in an extrapolative mode, thus yieldingPareto optimal solutions in regions not populated by solutions obtained from the evolutionary process.This attribute can be particularly useful in cases where some regions of the Pareto optimal front are difficultto populate.


THE MULTIPLE-OBJECTIVE EVOLUTIONARY METHOD

The multiple-objective evolutionary algorithm proposed by Rai15 draws upon ideas from severalgenetic algorithms and evolutionary methods. One of them is a relatively new member to the general classof evolutionary methods called differential evolution9. This method (DE) is easy to program and use and itrequires relatively few user-specified constants. These constants are easily determined for a wide class ofproblems. Fine-tuning the constants will yield the solution to the optimization problem at hand morerapidly. The method can be efficiently implemented on parallel computers and can be used forcontinuous, discrete and mixed discrete/continuous optimization problems. It does not require theobjective function to be continuous and is noise tolerant. Additionally, the method does not require thetransformation of continuous variables into binary integers.

As with other evolutionary methods and genetic algorithms, DE is a population based method forfinding global optima. The three main ingredients are mutation, recombination and selection. Much of thepower of this method is derived from a very effective mutation operator that is simple and elegant.Mutations are obtained by computing the difference between two randomly chosen parameter vectors inthe population and adding a portion of this difference to a third randomly chosen parameter vector toobtain a candidate vector. The resulting magnitude of the mutation in each of the parameters is differentand close to optimal. For example, in the case of an elliptical objective function in two dimensions, the setof all possible mutation vectors would be longer in the direction of the major axis and shorter in thedirection of the minor axis. Thus, the mutation operator adapts to the particular objective function and thisresults in more rapid convergence to the optimal value. In addition, this approach automatically reducesthe magnitude of mutation as the optimization process converges.

To describe one version of single-objective DE9, we consider the set of parameter vectors at thenth generation,

€

Xj,n . The subscript j refers to the jth member in a population of N parameter vectors and,

€

Xj,n = x1,j,n ,x2,j,n,....xD,j,n[ ] (1)

where

€

xi,j,n corresponds to the parameter value in the ith dimension in a D-dimensional problem. The initialpopulation is assumed to be randomly distributed within the lower and upper bounds specified for eachdimension. The mutation, recombination and selection operators are then applied to the population ofparameter vectors as many times as required. To evolve the parameter vector

€

Xj,n we randomly pick threeother parameter vectors

€

Xa,n , Xb,n and

€

Xc,n such that

€

a ≠ b ≠ c ≠ j . A trial vector Y is then defined as

€

Y = Xa,n + F(Xb,n - Xc,n) (2)

where F is a user specified constant,

€

0 < F < 1( ) . The candidate vector

€

Z = z1,z2 ,....zD[ ] is obtained viaa recombination operator involving the vectors

€

Xj,n and Y and is defined as

€

Z =yi if ri ≤ Cxi,j,n if ri > C

(3)

where

€

ri is a uniformly distributed random variable

€

0 ≤ ri < 1( ) and C is a user specified constant,

€

0 < C < 1( ) .The final step in the evolution of

€

Xj,n involves the selection process and, for theminimization of the objective function f(X), is given by

€

Xj,n + 1 =Z if f(Z) ≤ f(Xj,n)

Xj,n if f(Z) > f(Xj,n)

(4)

In other words, the selection process involves a simple replacement of the original parameter vector withthe candidate vector if the objective function decreases by such an action.


Several modifications to the mutation, recombination and selection procedures were made by Rai8in order to enforce constraints, facilitate searches involving multiple feasible regions embedded ininfeasible regions and, enrich the pool of potential candidate vectors. Although these modifications weredevised for the method of differential evolution, the underlying principles can be used with otherevolutionary strategies and genetic algorithms.

Abbas et. al.16 first proposed an extension to DE (PDE) to handle multiple objectives. It is a Pareto-based approach that uses non-dominated ranking and selection procedures to compute several Pareto-optimal solutions simultaneously. The population of parameter vectors is first sorted to obtain the non-dominated set. Mutation and recombination is undertaken only among members of the non-dominatedset. The resulting candidate vector replaces a member of the population if it dominates the first selectedparent (Z replaces Xj,n if it dominates Xa,n). When the total number of parameter vectors in the non-dominated set exceeds a threshold value, a distance metric in parameter space is used to removemembers of this set that are in close proximity. This feature improves solution diversity.

In a more recent study Madavan17 presents a different extension to DE to handle multipleobjectives. This method is also a Pareto-based approach that uses non-dominated ranking and selectionprocedures to compute several Pareto-optimal solutions simultaneously. It combines the features of DEand the NSGA-II method of Deb et al.18. The main difference between DE (single-objective) and thismethod lies in the selection process. New candidate vectors obtained from mutation and recombinationare simply added to the population thus resulting in a population that is twice as large. This largerpopulation is subjected to the non-dominated sorting and ranking procedure of Deb et al. The ranking isthen used to subsequently reduce the population to its original size. Solution diversity is achieved byascribing diversity ranks to members of the last non-dominated set that contributes to the new population.Diversity ranks are based on the crowding distance metric proposed by Deb et al. Unlike the distancemetric in PDE, this crowding distance metric is computed in objective space.

An important issue that both these studies (Abbas16 et al. and Madavan17) do not address is themanner in which single-objective DE extracts valuable mutation information directly from the population ofvectors, and the retention of this ability in the context of multi-objective optimization. As explained byPrice and Storn19, under the assumption that the parameter vectors of a population are distributed arounda level line in parameter space that represents their mean value, the set of vectors created by vectordifferences (in the mutation operator) are close to optimal. For example, when the contours of theobjective function are elliptic, the set of all possible vector differences is longer in the direction of themajor axis and shorter in the direction of the minor axis. In the presence of a second populated minimumthe set of vector differences include ones that facilitate the transfer of parameter vectors from theproximity of one minimum to the proximity of the other, thus making DE a powerful global optimizer.

However, multiple-objective optimization requires the entire Pareto optimal front to be adequatelypopulated. Consider a situation where the Pareto-optimal front is highly curved in parameter space andthe parameter vectors are distributed evenly along this front but do not coincide with it. Clearly vectordifferences involving vectors from disparate regions of this front are not very effective mutation vectors.However, parameter vectors that straddle the Pareto front in parameter space and are in close proximity toeach other will yield mutations (vector differences) that are more likely to result in superior candidatevectors. The parent vectors as well as the vector being considered, Xa,n, Xb,n , Xc,n and Xj,n, need to be inproximity for effective mutation and recombination. This is especially true in the final stages ofoptimization. In the initial stages of optimization the entire front can be considered a single entity in a basinof attraction, being approached from afar by the parameter vectors. Localization of the relevant vectorsused in mutation and recombination may not be necessary at this early stage.

The methods of Abbas et al.16 and Madavan17 have yielded accurate Pareto-optimal fronts in somemodel problems without localization. This is most likely due to the presence of a population of vectors andcorresponding vector differences, some of which are appropriate mutations. Additionally, in cases wherethe Pareto-optimal front is not very curved in parameter space, localization may not be an issue. However,both the studies report that better convergence was achieved with a value of F (Eq. 2) around 0.3 for themodel problems considered. This is about 1/3 to 1/2 of the value normally used in single-objective DE-based optimization and is indicative of the need for localization. Typical values of F used with currentmethod lie between 0.6 and 1.0 for the first 75% of the total number of generations, followed by F≤0.6 for


the remaining generations. The reduction in F towards the end of the evolutionary process results in asmall improvement in convergence and quality of Pareto-optimal solutions. This improvement is to beexpected because unlike single objective DE where the magnitude of the mutation vectors approacheszero as the parameter vectors converge to a point, the mutation vectors in the case of multiple objectiveoptimization remain finite even after all the parameter vectors are located on the Pareto optimal front. Thepopulation of vectors continues to redistribute itself on the Pareto optimal front to obtain better solutiondiversity. In fact a significant number of generations may be required to obtain superior solution diversity.Pareto-optimal solutions for some of the cases presented later in the text were obtained rapidly with largevalues of F (5.0 to 10.0). The common feature in these problems was that the Pareto front was a subset ofthe boundary of the search region. The large values of F used in the first part of the evolution acceleratethe movement of the population from the interior to the relevant part of the boundary in these cases.

Rai’s15 extension of DE to multiple-objective optimization consists of the following steps:

(1) Determine the set of non-dominated parameter vectors (rank one only as in PDE16, and unlike NSGA-II).

(2) Reduce this set of potential parent vectors to improve solution diversity if the number of parametervectors in this set exceeds a certain threshold value. The method used to perform this operation isdiscussed below (as in PDE, and unlike NSGA-II).

(3) For each member of the population, chose three parent vectors from the non-dominated set, computea candidate vector as in Eqs. 2 and 3 and, add this candidate vector to the bottom of the list of parametervectors to create a population that is twice as large as the original (as in NSGA-II, and unlike PDE).

(4) Identify the non-dominated set of vectors and perform a bubble-sort so that the new set of non-dominated vectors move to the top of the list. This automatically pushes those that are no longer non-dominated down the list (different from NSGA-II and PDE).

(5) Retain only the first N parameter vectors (as in NSGA-II and unlike PDE).

This method (like PDE) only requires the rank-one non-dominated vectors to be determined. Hence iteasier to program than NSGA-II and the computational expense for identifying the pool of parent vectors isalso less than that required by NSGA-II. The selection process is similar to that of NSGA-II and is hencemore elitist than PDE. Tests on some complex multi-objective optimization problems have demonstratedthat the procedure described above is a powerful multi-objective optimization tool.15

Localization in this method is achieved in the following manner; given the parameter vector Xj,nfrom the population size of N, the parent vector Xa,n is chosen as

€

Xa,n = Xi,n if r ≤ 1- dj,i/dmax (5)

where Xi,n is randomly chosen from the non-dominated set, r is a uniformly distributed random variable (0 <r < 1), dj,i is the distance between the vectors Xi,n and Xj,n in parameter space, and dmax is the maximumdistance between parameter vectors in the population. Equation 5 states that the vector Xi,n is more likelyto be chosen for small values of dj,I (relative to dmax). The parent vectors Xb,n and Xc,n are obtained similarly.Clearly Eq. 5 does not preclude the possibility of distant vectors being chosen as parents, it merely givespreference to parent vectors that are in proximity to Xj,n. The need for localization, in order to efficientlyobtain accurate Pareto optimal solutions with DE, is demonstrated by Rai15 with a case where the shape ofthe Pareto optimal front in parameter space is complex.

Solution diversity is achieved using a combination of the methods of PDE and NSGA-II. When thenumber of the non-dominated members exceeds a certain threshold value (for example, half thepopulation) a distance metric is used to reduce the number of members in this set. However, instead ofusing a distance metric in parameter space as in PDE, the crowding distance metric of NSGA-II in objectivespace is utilized for this purpose. Experimentation with a method similar to that of NSGA-II of obtainingsolution diversity did not perform as well in some problems. The evolutionary pressure exerted by this


approach is subtle and, perhaps not as effective in cases with Pareto-fronts having regions that are difficultto populate.

Rai’s multiple-objective DE15 was modified to obtain the current method. These changes weremade with the objective of obtaining Pareto optimal fronts using very small populations. One problem thatoccurs when inadequate population sizes are used is stagnation, a situation where the population stopsevolving19. The present method resorts to a second mutation operator to maintain a healthy evolutionaryprocess. In its simplest version this involves adding a random variation to one of the coordinates of acandidate vector. The candidate vector, the coordinate to be perturbed, the magnitude of the mutationoperator and the generation of occurrence are all chosen randomly. Parameter mutations are specified as

€

pm = KR(2r -1)/2 (6)

where R is the linear magnitude of the search space in the coordinate that is being perturbed, K is a userspecified constant (typically between 0.1 to 0.25), and r is a uniformly distributed random variable (0 < r <1). The presence of the term (2r – 1) makes the method relatively insensitive to the choice of larger valuesof K. This mutation operator was found to be an effective tool in preventing stagnation and prematureconvergence in the context of small populations. Mutations involving the simultaneous perturbation ofseveral coordinates of the candidate vector can be devised using this principle.

A second change to the basic method of Rai15 involves the manner in which solution diversity isachieved. In the earlier method the non-dominated members were subjected to a second selectionprocess using the crowding distance metric of Deb et al.18 and this smaller, select population wassubsequently used to define the new population and also to create mutation vectors and perform thecrossover operation. Here the mutation vectors are computed using all of the non-dominated members ofthe population. The more select members of the non-dominated population are used for the crossoveroperation and to define the new population. Additionally the method used to select a superior subset ofthe non-dominated members has been modified. The earlier method used a combination of the methodsof Abbas et al.16 and Deb et al.18. Here, when the number of non-dominated vectors exceeds a certainthreshold value (in this study the threshold value was set to the original population size), members of thisset are eliminated in a sequential manner. In the first pass the two vectors that are closest to each other(this can be determined in either parameter space or objective space) are identified. The vector amongthese two that is further down the list of vectors is tagged for removal. The process is continued until thenumber of non-dominated members is equal to the specified threshold value.

Design optimization varies from simple unimodal function optimization to multi-modal functionoptimization, constrained optimization, optimization in cases where the search space contains disjointregions of feasibility, and multiple-objective optimization. The performance of the evolutionary methoddeveloped here was investigated using test cases with some of these attributes and, is discussed below.

Unconstrained Multiple-Objective Optimization

Several unconstrained multi-objective optimization cases are solved here. Deb20 presents adetailed discussion of these cases. They are constructed to test the ability of the optimization method toconverge to the global Pareto front and, compute Pareto fronts that are convex, non-convex anddiscontinuous. The first four cases were formulated by Zitzler, Deb and Thiele21, and are denoted asZDT1, ZDT2, ZDT3 and ZDT4. The fifth test case was first proposed by Viennet22, and is labeled as VNT1.All the ZDT test cases involve two objective functions, whereas VNT1 involves three objective functions.

The test cases of Zitzler et. Al21 that are solved here can be formulated as

€

Minimize : f1(X)Minimize : f2(X) = g(X)h(f1(X),g(X)) (7)


where X is a vector in n-dimensional parameter space and the functions f1(X), g(X) and h(X) are defineddifferently for each case. The type of problem complexity (non-convex Pareto front, discontinuous Paretofront, multiple Pareto fronts etc.) as well as the degree of complexity can be specified by appropriatechoices of the functions f1(X), g(X) and h(X). The global Pareto front for all these cases is g(X) =1.0.

The test case ZDT1 has thirty variables and is defined by the following functions

€

f1(X) = x1

g(X) = 1 + 9n -1

xii=2

n

∑

h(f1, g) = 1 - f1 g n = 30

0 ≤ xi ≤ 1 (8)

The Pareto front for this case is convex and the Pareto optimal solutions are uniformly distributed alongthis front. The multi-objective evolutionary algorithm developed in this study and ten parameter vectorswere used to obtain the Pareto front. Figure 1 shows the computed Pareto optimal solutions and theexact Pareto front. The agreement between the two is good. It should be noted that DE usually requiresbetween two and 100 times as many parameter vectors as the number of variables in the problem. Theoptimal ratio depends on the complexity of the optimization problem. Here we obtain the solution with 10parameter vectors for a 30 variable problem in 250 generations. The computed solutions also exhibit goodsolution diversity, that is, the computed Pareto optimal points are nearly evenly spaced and cover theentire front. The earlier version of the method15 required 30 parameter vectors and 250 generations toyield the Pareto optimal front for this problem (three times as many function evaluations).

The test case ZDT2 also has thirty variables and is defined by the following functions

€

f1(X) = x1

g(X) = 1 + 9n -1

xii=2

n

∑

h(f1, g) = 1 - f1/g( )2 n = 30

0 ≤ xi ≤ 1 (9)

It is more complex than ZDT1 because the Pareto front is non-convex. The problem was solved using 10parameter vectors and 250 generations. Figure 2 shows the computed Pareto optimal solutions and theexact Pareto front. As in the previous case the computed solutions are in close agreement with the exactPareto front and exhibit good solution diversity. The earlier version of the method15 required 30 parametervectors and 250 generations to yield the Pareto optimal front for this problem (three times as manyfunction evaluations). The test case ZDT3 is a 30 variable problem and is defined as follows

€

f1(X) = x1

g(X) = 1 + 9n -1

xii=2

n

∑

h(f1, g) = 1 - f1/g - f1/g( )sin 10πf1( )n = 30

0 ≤ xi ≤ 1 (10)

The important characteristic of this problem is that the Pareto front is discontinuous in objective space.Forty parameter vectors and 300 generations were used to solve this problem. Although 10 parameter


vectors yielded accurate Pareto optimal solutions, their density along the Pareto front was inadequate. Asseen in Fig. 3, the computed solutions are in close agreement with their exact counterparts and exhibitgood solution diversity. In Ref. 15 this solution was computed using 60 parameter vectors and 250generations (1.25 times as many function evaluations). A significant reduction in cost to solution is notpossible in this case because of the length and complexity of the Pareto optimal front and thesimultaneous requirement that this front be adequately populated.

The test case ZDT4 has 10 variables and is a particularly difficult problem for all multi-objectiveoptimization methods because it exhibits numerous local Pareto fronts20. The global and the next bestlocal Pareto front are given by

€

g(X) = 1.00and

€

g(X) = 1.25 , respectively. The problem is defined as

€

f1(X) = x1

g(X) = 1 + 10 n - 1( ) + xi2 - 10cos 4πxi( )( )

i=2

n

∑

h(f1, g) = 1 - f1/gn = 10 0 ≤ x1 ≤ 1 -5 ≤ x2,3K10 ≤ 5 (11)

Inadequate population sizes generally yield Pareto optimal solutions corresponding to one of the localPareto fronts. The global Pareto optimal solutions have been found to be elusive to capture in previousstudies by other investigators. Rai15 obtains the global Pareto front with 1000 parameter vectors in 100generations, a total of 100,000 function evaluations. Figure 4 shows the computed Pareto optimalsolutions obtained with the current method and, the exact global Pareto front. Six parameter vectors and3333 generations were used to compute the Pareto optimal solutions. This corresponds to about 20,000function evaluations. Both, proximity to the exact front and solution diversity are good. This computationrequired 0.68 CPU seconds on a single 400MHz SGI (MIPS) processor. The rather unusual number ofgenerations (3333) was picked to determine if the present method could yield the Pareto optimalsolutions with about one-fifth as many function evaluations as the earlier version. Clearly this objective hasbeen met. Although the smaller population size used here (6 instead of 1000 parameter vectors) requiresmore generations to converge (3333 instead of 100), the total number of function evaluations required forconvergence is reduced by a factor of five. Figure 5 shows the Pareto optimal solutions for ZDT4 obtainedin five consecutive runs. All of them converge to the global optimum thus demonstrating the reliability ofthe method.

The test case VNT1 involves three objective functions and two variables. VNT1 is defined as

€

Minimize : f1(x1, x2 ) = 0.5 x12 + x2

2

+ sin x1

2 + x22

Minimize : f2 (x1, x2 ) = 15.0 + 3x1 - 2x2 + 4( )2 8 .0 + x1 - x 2 + 1( )2 27.0

Minimize : f3(x1, x2 ) = -1.1exp- x12 + x2

2

+ 1.0 x1

2 + x22 + 1

-3.0 ≤ x1, x 2 ≤ 3.0

(12)

The Pareto front is discontinuous in both design space as well as objective space. Figure 6 shows thecomputed Pareto optimal solutions obtained with a population size of 50 in 200 generations. Althoughaccurate solutions were obtained with much smaller population sizes, 50 parameter vectors were used forthis problem to better represent the rather complex Pareto front. Figure 6 shows the projection of thePareto front on the (f1, f3) plane. Again, the current evolutionary method yields optimal solutions that arediverse and are close to the exact optimal front in spite of the complexity of this front as well as the three-dimensionality of the objective space. The earlier version of the method15 required 100 parameter vectorsand 200 generations to yield the Pareto optimal front for this problem (twice as many function evaluations).


Constrained Multiple-Objective Optimization

Many engineering problems are constrained by equality and inequality constraints that can belinear or nonlinear. A novel technique of constraint satisfaction in the context of single-objectiveevolutionary methods was implemented for differential evolution by Rai8.This approach to constraintsatisfaction is also applicable to the multi-objective differential evolution algorithm presented here. Thefollowing simple example (labeled MMR2) is used to demonstrate the constraint handling ability of themethod. The feasible region consists of the interior of three nearly circular sub-regions. The optimizationproblem is defined as:

€

Minimize f1(x1, x2) = 0.5 x12 + x2

2( )Minimize f2(x1, x2) = 0.5 x1 - 1( )2 + 0.5 x2 - 1( )2

Subject to g x1, x2( ) ≤ 0.0

g x1, x2( ) = 0.5 - exp -rj2( )j=1

3

∑

rj2 = 15.0 x1 - x1j( )2

+ 15.0 x2 - x2j( )2

x11, x2

1( ) = 0.0, 0.0( )

x12, x2

2( ) = 0.5, 0.5( )

x13, x2

3( ) = 1.0, 1.0( )-2.0 ≤ x1, x2 ≤ 2.0

(13)

Figure 7 shows the computed Pareto front obtained with 30 parameter vectors in 40 generations (3639function evaluations), and the exact Pareto front for both the constrained and unconstrained cases. ThePareto front for the constrained case consists of three unconnected segments. The computed optimalsolutions are in good agreement with the exact Pareto front and good solution diversity has beenachieved. Figure 8 shows the segmented Pareto front and the constraint boundaries in parameter space.The computed optimal solutions satisfy the constraint (lie within the circles), and are close to the exactPareto front. The earlier version of the method15 required 7500 functions evaluations to yield the Paretooptimal front for this problem (twice as many function evaluations).

Neural Network Estimates for Pareto Optimal Fronts

From the examples presented above it is clear that the current method is capable of yieldingaccurate Pareto optimal solutions with very small population sizes. Figure 4 shows the computed Paretooptimal solutions obtained for ZDT4 with 6 parameter vectors. The computed optimal solutions are welldispersed and the end points of the Pareto front are captured. However, the distribution of solutions isnot uniform. Continued evolution does result in better solution diversity. Larger population sizes alsotend to yield better solution diversity. Both of these approaches require additional function evaluations. Ithas been observed in this study that in many cases the parameter vectors converge fairly quickly to thePareto optimal front and then the population of vectors continues to redistribute itself on this front to yieldbetter solution diversity. In fact a significant number of generations may be required to obtain superiorsolution diversity. One approach to eliminate the cost involved in obtaining good solution diversity duringthe evolutionary process is to use an estimation technique to fit the data obtained from DE (or any otherevolutionary method). The response surface thus obtained can be used directly or to generate therequired distribution of Pareto optimal solutions. Here we explore the use of neural networks in obtainingaccurate, uniformly distributed Pareto optimal solutions.


Feed-forward artificial neural networks are essentially nonlinear estimators. They can be used toapproximate complex multi-dimensional functions without having to specify an underlying model. Traininga neural network to model data requires determining the connection weights that define the network.Nonlinear optimization methods are typically employed to obtain these weights. The connection weightsare not uniquely defined; many different sets of weights may yield acceptably low training error for a givennetwork architecture and dataset. This multiplicity of acceptable weight vectors can be used to advantage.One could select the neural network (or equivalently the corresponding set of weights) with the smallestvalidation error if validation data is available. In constructing response surfaces which approximate thePareto optimal front using Pareto optimal solutions as training data, this approach requires setting asidesome of these solutions as validation data. It is reasonable to expect the generalization ability of the set ofweights selected in this manner to be superior compared to the rest of the sets of weights. However, thisapproach results in very few training data when the number of optimal solutions is small as in Fig. 4.

In the absence of validation data, multiple trained neural networks can be effectively utilized bycreating a hybrid network.23 - 24 The output of the hybrid network can be defined as a simple average of theoutputs of all the trained neural networks. It can be shown23 that the sum-of-squares error (SSE) thusobtained (or integrated squared error) in modeling the function underlying the data, is a factor of N lessthan the average SSE, where N is the number of trained networks. The essential assumption that is madeto obtain this result is that the errors produced by the different networks have zero mean and are notcorrelated. When the errors produced by the different networks are correlated the SSE of the hybridnetwork continues to be less than or equal to the average SSE. However, it is not necessarily reduced bya factor of N. Note that the networks used in this ensemble average do not have to possess the samearchitecture or even be trained on the same training set.

A second and more general way of combining the outputs of different trained networks is toweight the output of each network such that error of this combined output is minimized.24 Givenweights

€

αi (i = 1,...N) , the optimal weights can be obtained by minimizing the function

€

αiα jCijj=1

N∑

i=1

N∑

(14)

subject to the constraint

€

αii=1

N∑ = 1 (15)

The matrix C in Eq. 14 is the error correlation matrix. Details of this method of creating a hybrid network arediscussed by Perrone and Cooper.24 Unlike the simple averaging technique, this method does notexplicitly require that the mean error for the networks be zero, or, that the network models be mutuallyindependent. However, practical considerations such as maintaining the full rank of C, and imposing theconstraint αi ≥ 0 to prevent data extrapolation, once again require these assumptions to be met. Hybridnetworks have been used effectively to construct response surfaces for design optimization13 - 14. In manyof the cases investigated, there was little difference in modeling error whether the simple averagingtechnique or the more general approach given in Eqs. 14-15 was used. In some cases the more generalensemble approach yielded a better model.

The creation of a hybrid network serves to reduce the variance. Effective hybridization requiresthe neural network training method to yield numerous uncorrelated network models. The nonlinearoptimization process used to train each network can be started with different random weights toaccomplish this task. Methods that improve the generalization ability of the individual networks, such asregularization and network architecture optimization can be embedded at this level.


The generalization ability of hybrid networks was tested using low-order polynomials and aGaussian function in Rai13. Good generalization (both in the region where data was available and outside ofthis region) was obtained. A feed-forward network with two hidden layers and the more general ensemblemethod of creating the hybrid network (Eqs. 14-15) were used in all these cases. These encouragingresults led to the investigation of the generalization accuracy that can be obtained for higher-orderpolynomials, polynomials combined with other functions, polynomials in multiple dimensions, and caseswhere the training data is contaminated by noise.14 Excellent generalization was obtained in all thesecases. These investigations have resulted in better training algorithms for feed-forward neural networks.

Figure 9 shows the generalization obtained for a fifth-order polynomial modeled using eighttraining points and a hybrid network consisting of ten single hidden layer neural networks. The full line wasobtained with the neural network and the dashed line (superimposed on the full line) was obtained usingthe exact function. Neural network generalization is excellent throughout the region –2 ≤ x ≤ 3 althoughtraining data is available only in the region 0 ≤ x ≤ 1. The ability of the hybrid network to extrapolate isevident in this case. Clearly this does not constitute proof that hybridization will always work as well(especially in the extrapolation mode). Note that a simple polynomial fit (fifth-order) would yield perfectgeneralization. However, it is equally important to note that the network was not supplied with theinformation that the function underlying the training data was a polynomial.

The neural network generalization shown in Fig. 9 is surprisingly accurate even in theextrapolation mode. A natural question to ask at this point is why the network generalization closelyapproximates the original function used to generate the training data, given that there are many curvesthat would fit this data. The answer lies in the fact that the given data (8 points) when interpolated using aset of polynomial basis functions (for example the Legendre polynomials) uniquely define the originalcurve, assuming a convention such as interpolating the data with the lowest-order polynomial. Hence, theneural network will reproduce the original function to the extent that it can mimic the polynomial basisfunctions. Neural networks can be made to mimic some classes of functions either through the choice ofnetwork connectivity or preprocessors that process the input data before they are fed to the input nodes.

The results of Fig. 9 indicate that it may be possible to use hybrid networks to obtain accurateestimates of a Pareto-optimal front given a few Pareto-optimal solutions. To investigate this possibility weconsider the following two-objective optimization problem (labeled MMR3):

€

Minimize : f1(X) = (xn + 1)/n( )2

n=1

D

∑

2/n( )2

n=1

D

∑

Minimize : f2(X) = xn - 1( )2

n=1

D

∑4D

(16)

where D is the dimensionality of the search space. The Pareto optimal front for this problem is given by

€

xn = (n2 + 1)x1 + (n2 - 1)(n2 - 1)x1 + (n2 + 1)

n = 2,3,....D

-1 ≤ x1 ≤ 1 (17)

This problem was formulated so that the individual objective functions mimic those found commonly inengineering optimization (contours of the first function form multi-dimensional ellipsoids in parameter


space) and, to generate a Pareto optimal front that is not a straight line (unlike ZDT1 – ZDT4). MMR3 alsoscales easily to any number of dimensions, with the ratio of the major axis to the minor axis of the ellipsoidsbeing equal to D.

In the first test case Pareto optimal solutions for MMR3 were obtained with DE for D = 2. Elevenparameter vectors were used in this computation. The Pareto optimal solutions thus obtained were sorted(increasing value of f1) and every alternate one was provided to the hybrid network as training data (sixtraining pairs). The training data are not uniformly distributed. The hybrid network consisted of ten single-hidden-layer feed-forward networks. The input to the networks was the first coordinate in parameter space(x1) and the output was the second coordinate (x2). Obtaining the estimating curve/surface in parameterspace (as opposed to objective space) permits the generation of additional Pareto optimal solutions in astraightforward manner. Figure 10 shows the Pareto-optimal front in parameter space obtained from thehybrid neural network. The training data and the exact Pareto optimal front are also provided in Fig. 10.The estimated front and the exact Pareto front are nearly identical. The non-uniformity of the Paretooptimal solutions computed using DE is not an issue because the estimating curve can be used togenerate any distribution of optimal solutions. This is an example of the savings that can be realized by notpursuing superior diversity in solutions generated by evolutionary methods.

In this computation the Pareto-optimal solutions were fully converged and thus the training datawas free of noise. Additionally the end points of the Pareto front were captured in the evolutionaryprocess. This case represents a typical application of hybrid networks to obtain an estimate of the Paretooptimal front. Hybrid networks can also be used in cases where the training data are contaminated withnoise and do not include the boundary points of the Pareto front. For moderately noisy data requiring amoderate amount of extrapolation the hybridization and training methods of Rai13 - 14 can be used togenerate the estimating curve. Extremely noisy data and situations where large extrapolations arerequired call for a more effective, specialized hybridization principle.

One such principle has been developed and applied effectively for MMR3. In the interest ofbrevity, this approach will be discussed in a companion article (Rai25). Here we provide some examples ofthis methodology. Figure 11 shows the training data obtained from DE for MMR3 with D = 2. Thecomputed solutions are not fully converged and exhibit a considerable amount of noise. The exact Paretooptimal front and the estimate obtained from the hybrid network are also shown in this figure and are nearlyidentical. Figure 12 shows the training data, the exact Pareto optimal front and the estimated front inobjective space. The ability of the hybrid network in extracting an accurate estimate of the Pareto frontfrom the noisy data is evident in both these figures. The new method of hybridization does requireadditional function evaluations. The relative costs incurred in continued refinement of the solutions usingDE and, generating estimating surfaces will be discussed in Rai25. Figure 13 shows results for a casewhere the DE search was confined to a portion of the region containing the Pareto optimal front. Thetraining data thus obtained only cover a portion of the Pareto front. The hybridization technique yields anestimated front that is once again in close agreement with the exact front both in the region of the data(interpolative mode) and far removed from the data (extrapolative mode).

Figure 14 shows the results of a similar exercise with MMR3 for the case D=4. Here three hybridnetworks, each consisting of ten individual networks, were used to generate an estimate of the Paretooptimal front. The first hybrid network was used to represent the functional relationship between x1 and x2,the second between x3 and x2 and, the third between x4 and x2. The variable x2 was used as the primaryvariable. The arc length along the Pareto front could have been used as the primary variable instead.Figure 14 shows the training data, the exact relationship between these variables and the correspondingneural network estimates. The training data were once again obtained in a restricted search space andhence do not cover the entire front. It is evident from the figure that the estimates are accurate in bothinterpolative and extrapolative modes. The three estimates can also be obtained using a single hybridnetwork in which x2 is the input and x1, x3 and x4 are obtained as the output of the individual networks.Such an approach may be necessary for more complex Pareto front topologies.


ROBUST DESIGN OPTIMIZATION OF FINS

Extended surface heat transfer, i.e., heat transfer with the aid of fins of different shapes, is ofimportance in a multitude of engineering applications ranging from transport vehicles, to electroniccomponents, industrial heat exchangers and nuclear power generation. Design optimization in thecontext of fins deals with reducing fin weight and volume for a given heat load, subject to constraintsrelated to fin shape and size.

Fins used in boiling heat transfer are particularly difficult to analyze from first principles because ofthe complex flow physics involved. Haley and Westwater26 and Westwater27 provide an excellentdescription of fins used in boiling heat transfer and the associated heat transfer processes. Theirexperiments show that different types of boiling including nucleate- and transition-boiling, and film boilingoccur on these fins. Often, all these boiling modes together with free-convection co-exist at a singleoperating point resulting in heat transfer coefficients that vary considerably along the length of the fin.Methods of analysis must take these variations in heat transfer coefficient into account to yield accurateestimates of heat load. At low fin-base temperatures (compared to the fluid boiling temperature) the entirefin is in a free-convection mode. As the base temperature increases above the boiling temperature of thefluid, high heat transfer nucleate-boiling appears at the base of the fin. As the temperature is increasedfurther, relatively low heat transfer film boiling appears at the base, followed by high heat transfertransition- and nucleate-boiling further away from the base, and free-convection at the tip of the fin.Additional increases in base temperature result in transition- and nucleate-boiling at the tip of the finleaving the rest of the fin in a film-boiling mode. Even larger base temperatures result in the entire finbeing in a film-boiling mode.

Haley and Westwater26 and Westwater27 provide experimental and computational data(temperature profiles and heat load at different base temperatures) for different fin-shapes. The mainthrust of these articles is the optimization of the shape of the fin to achieve maximum heat load for a givenfin volume, that is, single-objective optimization. Optimizing the performance of fins used in boiling heattransfer requires that the cross-section of the fin be tailored so that most of the surface area is in anucleate- or transition-boiling mode. Considerable increases in heat load, compared to the heat loads thatcan be obtained with simple cylindrical spines and rectangular fins, can be achieved by using the optimalcross-sectional area distribution. These optimal shapes are typically complex and are difficult tomanufacture. Additionally, as the nucleate- and transition-boiling regimes are pushed off the tip of the finas the base temperature increases, there can be a sudden and large decrease in the heat load. Thisphenomenon is undesirable because it occurs just when a higher heat load capacity is required. Here weinvestigate the possibility of maximizing both the heat load and the safe operating base temperature range(SOBTR) using the current multi-objective version of DE.

For the sake of simplicity, and without loss in generality, the fin shapes investigated in thisoptimization study are restricted to double cones with a cylindrical base. Figure 15 shows a schematic ofthe fin. It is defined using four variables, the length and diameter of the cylindrical base, and the basediameter and length of the truncated cone. The length of the second cone is obtained from a fin-volumeconstraint. Westwater27 first suggested this particular geometry as an approximation to an optimal “turnip”shape and conducted an experimental investigation of its efficiency. Rai and Rangarajan28 provide detailedtwo-dimensional (axi-symmetric) finite element computations for this fin-shape.

Figure 16 shows the fin heat load as the base temperature is increased. The heat load increasesto a maximum value and then abruptly decreases. Optimizing the fin shape (for a given base temperature),to maximize heat load, would yield a fin that would operate close to the peak of the heat load curve. This isundesirable because small increases in operating base temperature result in large decreases in theamount of heat that the fin transfers to the fluid. Figure 17 shows the temperature distribution along thelength of the fin. The rapid decrease in temperature in the first 10% of the fin occurs because of the smalldiameter of the cylindrical base. Thereafter, the temperature drop is more gradual because of the largercross-sectional area. Assuming that the base temperature is higher than the temperature at which film-boiling occurs, it is clear that the narrow base of the fin can be used to drop the temperature such that theremaining large portion of the fin surface is at a temperature at which nucleate- and transition-boiling occur.


The goal of the multiple-objective optimization is

€

Maximize : Heat Load = - kAdTdx

base

Maximize : SOBTR = Tpeak -heat -load - Tbase Subject to: Fin Volume = Constant

(18)

where k is the thermal conductivity of the fin, A is the cross-sectional area of the fin at its base, dt/dx is thetemperature gradient at the base, Tbase is the base temperature and Tpeak-heat-load is the base temperatureat which the peak heat load is obtained for a given shape. The base temperature and the fluid temperaturewere specified as 320°F and 120°F, respectively. The temperature variation in the fin was computed bysolving the one-dimensional heat conduction equation using an iterative finite-difference method.

Results for this multiple-objective optimization problem are provided in Ref. 15. Twenty parametervectors were used in the optimization and the evolutionary process was run for 80 generations. Thisrequired computing 59113 temperature distributions, which far exceeds the 1600 (80x20) simulationsthat would normally be required. This is because the accurate evaluation of the SOBTR for each fin-shaperequires numerous temperature-distribution computations (each with a different base temperature). Herethe optimization problem is solved with the newer version of the evolutionary method. Ten parametervectors and 80 generations were used for this optimization. The computation required 29592temperature distributions (half as many function evaluations as before). Figure 18 shows the computedPareto-optimal solutions, a neural-network estimate of the Pareto-front and, the neural network estimateobtained in Ref. 15. The two network estimates are in close proximity, with the current computationyielding a larger range and slightly larger estimates of heat load and SOBTR. Point A in Fig. 18 representsthe fin-shape, heat load (179.6 BTU/Hr.) and SOBTR (6.4°F) for the Pareto optimal solutioncorresponding to maximum heat load. Point B represents the fin shape, heat load (103.1 BTU/Hr.) andSOBTR (158.7°F) for the Pareto-optimal solution with the maximum SOBTR.

Figure 19 shows the variation of heat load as a function of the base temperature for the finsdenoted as points A and B in Fig. 18. It is clear that fin B allows for a wide range of safe base temperaturesabout the nominal value of Tbase = 320°F (Tbase – Tambient = 200°F) but yields a lower heat load at the nominalvalue of base temperature. In contrast, fin A, yields a higher heat load at the nominal base temperature buthas a smaller range of safe base temperature values. Figure 19 indicates that for a base temperatureincrease (over nominal) of about 10°F, film-boiling pervades the entire length of fin A. Figure 20 shows theshapes of fins A and B. The longer cylindrical base of fin B results in a larger temperature drop before themajority of the fin surface is encountered. Thus, up to a point, nucleate- and transition-boiling persist onthe majority of the fin surface, even though the base temperature is much higher than the temperature atwhich film-boiling is initiated. However, at the nominal base temperature, the temperature drop in the baseof the cylinder is larger than required for optimal operation and a larger portion of the fin is in a free-convection mode. This results in heat loads that are smaller than that obtained with fin A.

The multi-objective evolutionary method was found to be better suited to this fin optimizationproblem than response surface methodology and gradient based methods. This is because thediscontinuity in the heat load as the base temperature is increased (Fig. 16) results in a discontinuity in thefirst objective function (Eq. 18) in design space. The high-heat-load Pareto optimal solutions lie in thevicinity of this discontinuity in the objective function. Thus computing the gradient or constructingstandard response surfaces becomes problematic.

The fin optimization problem solved above has focused on obtaining the maximum heat load for agiven value of SOBTR. Conceptually it is similar to aerodynamic shape optimization in the context ofuncertain operating conditions. The next section focuses on design optimization in the presence ofuncertainties in airfoil shape.


ROBUST AIRFOIL SHAPE OPTIMIZATION

The accuracy to which a component is manufactured depends on the available manufacturingtechnology and other factors such as manufacturing cost. Component shapes are also subject to changebecause of wear and tear. The performance, reliability and life expectancy of the component may dependcritically on maintaining the shape of the component. Clearly there is a need to desensitize these aspectsof the component to small (and perhaps large) changes in shape.

The following is an example that illustrates the use of mutiple-objective DE in desensitizingperformance to changes in component shape. It consists of subsonic flow through a row of stator airfoils inan axial turbine. The flow is assumed to be inviscid and two-dimensional. Details regarding the gridgeneration and simulation procedures, and the airfoil parameterization method can be foundelsewehere.13-14 Although eight parameters were used to parameterize the airfoil, only two of them werevaried during optimization. These two parameters have very little effect on the shape of the suction side ofthe airfoil. Their primary effect is on the shape of the pressure side of the airfoil. The first requirement isthat the variance in pressure with small changes in airfoil shape is minimized. The second requirement isthat the wedge angle at the trailing edge is maximized (this increases the thickness of the airfoil and thusits strength). These two requirements are to be met subject to the constraint that the required flow turningangle is achieved. The two objective functions are defined as

€

Minimize : V = (pii=1

i= imax

∑n=1

n=nmax

∑ − pin)2

Minimize : θ = θref - θw

(19)

In Eq. 19 imax is the number of grid points on the surface of the airfoil, nmax is the number of geometryperturbations used to compute the variance,

€

pi is the pressure at the ith point on the surface of the givengeometry, and

€

pin is the surface pressure at the same point for the nth perturbation of this geometry. Thequantity θref is a reference angle and θw is the wedge angle. The manner in which the geometryperturbations are obtained depends on the probability density function assumed for the perturbations inthe geometry. Egorov et al.1 remark that a normal distribution is commonly observed for manufacturingtolerances. Here a simple approach is adopted; the perturbations are computed by perturbing onegeometry variable at a time. Replacing this with normally distributed surface perturbations isstraightforward.

The perturbed airfoil shapes as well as the base airfoil used in computing the objective function(Eq. 19) are obtained using the same airfoil parameterization scheme. This assumes that the geometryperturbations found in reality can be represented with the same number of geometric parameters that areused to represent the base airfoil. However, a finite number of geometry parameters can only yieldapproximate representations of the perturbations found in reality. Hence the approach computesapproximations to the objective function of Eq. 19 (even when the sample size is large). Nevertheless thecurrent approach has practical value. The set of perturbed airfoil shapes can be greatly enhanced byadding a perturbation function to the base airfoil shape. This function can be constructed with many moreindependent parameters governing its behavior than the number of parameters used to generate thebase airfoil shape. This results in a very flexible perturbation function. Computing the variance with thisextended set of airfoils is computationally expensive. Here we use the extended set only for assessmentand not during optimization.

The DE search was conducted with 10 parameter vectors and 25 generations. This required 780two-dimensional airfoil simulations. Figure 21 shows the Pareto optimal solutions obtained at the end of


this process. The computed optimal solutions exhibit a moderate amount of noise. A hybrid networkconsisting of 10 individual single-hidden-layer feed-forward networks was used to obtain an estimate ofthe Pareto optimal front using the training data from DE. The estimated Pareto front is also shown in Fig.21. Normalized values of the two parameters that govern the shape of the airfoil are shown on the twoaxes. Points A and B on the estimate represent the minimum and maximum variance solutions,respectively. Continued evolution (10 additional generations requiring 300 flow simulations) only resultedin a small improvement in the computed solutions. On the other hand, the hybrid network estimate wasobtained without requiring any additional simulations. Figure 22 shows training data and the estimate ofthe Pareto front in objective space. It is important to note that the estimating curve in this figure wasobtained by selecting uniformly spaced points on the estimating curve in Fig. 21 and subsequentlycomputing the flow over these selected shapes (and corresponding geometry perturbations).

Figure 23 shows the airfoils corresponding to the minimum and maximum variance cases (points Aand B in Fig. 21). The minimum variance case is a thinner airfoil whereas the maximum variance case isthicker (maximum wedge angle) and thus stronger. Figure 24 shows the computed pressure distributionsfor the two airfoils. The minimum variance airfoil does not exhibit a suction side undershoot and thepressure is nearly constant in the last 75% of the axial chord. This is because sharp variations in pressureare generally sensitive to geometry variations and, are thus avoided in this airfoil. The estimate of the localstandard deviation of pressure as a function of the axial location along the airfoil

€

σ local = (pn=1

n=nmax

∑ - pn )2

1/2

(20)

for the two airfoils is depicted in Figure 25. Thirty perturbations (nmax = 30) of the basic geometry and theircorresponding flow fields were computed to determine σlocal in Eq. 20. The minimum variance airfoil yieldsa distribution that is significantly lower than that obtained with the airfoil with the maximum wedge angle.However, both these airfoils (and the other airfoils on the Pareto front) perform the same function ofturning and accelerating the flow by a specified amount. The exit flow angle varied ±0.32% over the tenairfoils and, the exit Mach number varied± 0.04%.

CONCLUDING REMARKS

Traditional aerodynamic shape optimization has focused on obtaining the best design given therequirements and flow conditions. However, the flow conditions are subject to change during operationand, the accuracy to which the optimal shape is manufactured depends on the available manufacturingtechnology and other factors such as manufacturing cost. Component shapes are also subject to changebecause of normal wear and tear. Clearly there is a need to desensitize optimal performance to changes inoperating conditions and component shape. These requirements naturally lead to the idea of robustoptimal design wherein the concept of robustness to various perturbations is built into the designoptimization procedure. Here both evolutionary algorithms and neural network-based estimators are usedto achieve such robust optimal designs.

Building robustness into an optimal design results in a multi-objective optimization problem. Anevolutionary algorithm, based on the method of differential evolution, and intended for multi-objectiveoptimization is first presented here. The method is tested using a number of difficult model problemsincluding ones with constraints. In all these cases the computed Pareto-optimal solutions closelyapproximated the global Pareto-front and exhibited good solution diversity. Many of these solutions wereobtained with small population sizes. One of the computed solutions, ZDT4, was found elusive to capturein previous studies by other investigators. The current evolutionary method is also used compute theoptimal shape of fins used in boiling heat transfer. The complex physical processes that are encounteredin boiling heat transfer result in heat load capacity that can change drastically with changing operating


conditions. The challenge here is to obtain maximum heat load given the necessary safe operating basetemperature range (SOBTR). The evolutionary algorithm was used to obtain the Pareto-optimal frontshowing the trade-off between maximum heat load and the SOBTR. This is a good example of directconflict between optimality and robustness during operation.

Fabricating and operating complex systems involves dealing with uncertainty in componentshapes. In the case of aircraft, efficiency and engine noise may be different from the expected valuesbecause of manufacturing tolerances and normal wear and tear. Engine components may have a shorterlife than expected because of manufacturing tolerances. Recognition of the importance of incorporatingthe probabilistic nature of the variables involved in designing and operating complex systems has led toseveral investigations in the recent past. Here, a multi-objective evolutionary method is successfully usedto design turbine airfoils with the conflicting objectives of minimizing the sensitivity of pressuredistributions to geometry changes and maximizing blade thickness and strength. The evolutionaryalgorithm was used to obtain the Pareto-optimal front showing the trade-off between these twoobjectives. The minimum and maximum variance airfoils were found to have significantly different shapes,pressure distributions and pressure variances but resulted in the same flow turning and exit Mach number.

REFERENCES

1. Egorov, I. N., Kretinin, G. V., and Leshchenko, I. A., “How to Execute Robust Design,” AIAA Paper No.2002-5670, 9th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization,September 4-6, Atlanta, Georgia.

2. Koch, P. N., Wujek, B., Golovidov, O., “A Multi-Stage, Parallel Implementation of Probabilistic DesignOptimization in an MDO Framework,” AIAA Paper No. 2000-4805, 8th AIAA/USAF/NASA/ISSMOSymposium on Multidisciplinary Analysis and Optimization, Long Beach, California.

3. Koch, P. N., Wujek, B., Golovidov, O., and Simpson, T. W., “Facilitating probabilistic multidisciplinaryDesign Optimization Using Kriging Approximation Models,” 9th AIAA/USAF/NASA/ISSMO Symposium onMultidisciplinary Analysis and Optimization, September 4-6, Atlanta, Georgia.

4. Goldberg, D. E., Genetic algorithms in Search, Optimization and Machine Learning, Addison-Wesley,1989.

5. Obayashi, S., and Tsukahara, T., “Comparison of Optimization Algorithms for Aerodynamic ShapeOptimization,” AIAA Journal, Vol. 35, No. 8, August 1997, pp. 1413-1415.

6. Holst, T. L., and Pulliam, T. H., “Aerodynamic Shape Optimization Using a Real Number EncodedGenetic Algorithm,” AIAA Paper no. 2001-2473, AIAA 19th Applied Aerodynamics Conference.

7. Obayashi, S., and Yamaguchi, Y., “Multi-objective Genetic Algorithm for Multi-disciplinary Design ofTransonic Wing Platform,” Journal of Aircraft , Vol. 34, No. 5, 1997, pp. 690-693.

8. Rai, M. M., “Towards a Hybrid Aerodynamic Design Procedure Based on Neural Networks andEvolutionary Methods,” AIAA Paper No. 2002-3143, AIAA 20th Applied Aerodynamics Conference, St.Louis Missouri, June 24-26, 2002.

9. Price, K., and Storn, N., “Differential Evolution,” Dr. Dobb’s Journal, April 1997, pp. 18-24.

10. Myers, R. H., and Montgomery, D. C., Response Surface Methodology: Process and ProductOptimization Using Designed Experiments, John Wiley and Sons, New York, 1995.


11. Montgomery, D. C., Design and Analysis of Experiments, John Wiley and Sons, New York, 1997.

12. Madavan, N. K., “Aerodynamic Shape Optimization Using Hybridized Differential Evolution,” AIAAPaper No. 2003-3792, 21st Applied Aerodynamics Conference, Orlando, Florida, June 23-26, 2003.

13. Rai, M. M., “A Rapid Aerodynamic Design Procedure Based on Artificial Neural Networks,” AIAA PaperNo. 2001-0315, AIAA 39th Aerospace Sciences Meeting, Reno, Nevada, Jan. 8-11, 2001.

14. Rai, M. M., “Three-Dimensional Aerodynamic Design Using Artificial Neural Networks,” AIAA Paper No.2002-0987, AIAA 40th Aerospace Sciences Meeting, Reno, Nevada, Jan. 14-17, 2002.

15. Rai, M. M., “Robust Optimal Aerodynamic Design Using Evolutionary Methods and Neural Networks,”AIAA Paper No. 2004-0778, AIAA 42nd Aerospace Sciences Meeting, Reno, Nevada, Jan. 5-8, 2004.

16. Abbas, H. A., Sarker, R., and Newton, C., “PDE: A Pareto-Frontier Differential Evolution Approach forMulti-objective Optimization Problems,” Proceedings of the Congress on Evolutionary Computation,2001, Vol.2, pp. 971-978, Piscataway, New Jersey, May 2001.

17. Madavan, N. K., “Multiobjective Optimization Using a Pareto Differential Evolution Approach,”Proceedings of the Congress on Evolutionary Computation, 2002, Vol.2, pp. 1145-1150, Honolulu,Hawaii, May 2002.

18. Deb, K. Agrawal, S., Pratap, A., Meyarivan, T., “A Fast Elitist Non-Dominated Sorting GeneticAlgorithm for Multi-Objective Optimization: NSGA-II”, Proceedings of the Parallel Problem Solving fromNature VI Conference, pp. 849-858, Paris, France, September 16-20, 2000.

19. Corne, D., Dorigo, M., and Glover, D., Editors, New Ideas in Optimization, McGraw Hill , 1999.

20. Deb, K., Multi-Objective Optimization Using Evolutionary Algorithms, Wiley, 2001.

21. Zitzler, E., Deb, K., and Thiele, L., “Comparison of Multi-Objective Evolutionary Algorithms: EmpiricalResults,” Evolutionary Computation Journal, 8(2), pp. 125-148.

22. Viennet, R., “Multi-criteria Optimization Using a Genetic Algorithm for Determining the Pareto Set,”International Journal of Systems Science, 27(2), pp. 255-260.

23. Perrone, M. P., “General Averaging Results for Convex Optimization, Proceedings of the 1993Connectionist Models Summer School, M. C. Mozer et. Al. (Eds.), pp. 364-371.

24. Perrone, M. P., and Cooper, L. N., “When Networks Disagree: Ensemble Methods for Hybrid NeuralNetworks,” Artificial Neural Networks for Speech and Vision, R. J. Mammone (Ed.), 1993, pp. 126-142.

25. Rai. M. M., “Applications of Neural Networks in Design Optimization”, in preparation.

26. K. W. Haley and J. W. Westwater, “Boiling Heat Transfer from Single Fins”, Proceedings of the ThirdInternational Heat Transfer Conference, Volume 3, Page 245, 1966.27. J. W. Westwater, Paper 1, “Heat Transfer – Fundamentals and Industrial Applications”, AICHESymposium Series, 69(131), Page 1,1973.28. Rai, M. M., and Rangarajan, J., “Application of the Finite Element Method to Heat Conduction.” ProjectReport, Department of Mechanical Engineering, Indian Institute of Technology, Madras, India, 1978.


Fig. 1. Pareto optimal front in objective space for ZDT1.






Fig. 5. Pareto optimal front in objective space for ZDT4 (multiple runs).

Fig. 6. Pareto optimal front in objective space for VNT1.


Fig. 7. Pareto optimal front in objective space for MMR2.

Fig. 8. Pareto optimal front in parameter space for MMR2.


Fig. 9. Neural network generalization obtained for a fifth-order polynomial.

Fig. 10. Pareto optimal front in parameter space for MMR3 (fully converged data from DE).


Fig. 11. Pareto optimal front in design space for MMR3 (partially converged data from DE).

Fig. 12. Pareto optimal front in objective space for MMR3 (partially converged data from DE).


Fig. 13. Pareto optimal front in design space for MMR3 (restricted search domain).

Fig. 14. Pareto optimal front in design space for MMR3 (four-dimensional search space).


Fig. 15. Fin geometry and design variables.

Fig. 16. Heat load at various base temperatures.


Fig. 17. Temperature variation along the axis of the fin.

Fig. 18. Pareto optimal front in objective space for the fin showing the trade-off between heatload and safe operating base temperature range.


Fig. 19. Heat load as a function of base temperature for the maximum heat load and maximumSOBTR designs.

Fig. 20. Fin geometries for the maximum heat load and maximum SOBTR designs.


Fig. 21. Pareto optimal front in design space for the airfoil.

Fig. 22. Pareto optimal front in objective space for the airfoil.


Fig. 23. Airfoil shapes corresponding to the minimum and maximum variance cases.

Fig. 24. Surface pressure distributions corresponding to the minimum and maximum variance cases.


Fig. 25. Distribution of σlocal for the minimum and maximum variance cases.

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