[American Institute of Aeronautics and Astronautics 42nd AIAA Aerospace Sciences Meeting and Exhibit...

American Institute of Aeronautics and Astronautics1

ROBUST OPTIMAL AERODYNAMIC DESIGNUSING EVOLUTIONARY METHODS AND NEURAL NETWORKS

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

ABSTRACT

Evolutionary algorithms and neural networks have been used successfully in various disciplines of aeronautical engineering including aerodynamic design. Here a new evolutionary method for multiple–objective optimization is presented. It draws upon ideas from several genetic algorithms and evolutionary methods; one of them being a relatively new member to the general class of evolutionary methods called differential evolution. The capabilities of the evolutionary method developed here are investigated using some complex test cases. Good solution accuracy and diversity are obtained in all these cases. Traditionally, aerodynamic shape optimization has focused on obtaining the best design given the requirements and flow conditions. However, the flow conditions are subject to change during operation. It is important to maintain near-optimal performance levels at these off-design operating conditions. Additionally the accuracy to which the optimal shape is manufactured depends on the available manufacturing technology and other factors such as manufacturing cost. It is imperative that the performance of the optimal design is retained when the component shape differs from the optimal shape due to manufacturing tolerances and normal wear and tear. These requirements naturally lead to the idea of robust optimal design wherein the concept of robustness to various perturbations is built into the design optimization procedure. Here we demonstrate how both evolutionary algorithms and neural networks can be used to achieve robust optimal designs. Test cases include the design of airfoils and, fins used in boiling heat transfer.

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

INTRODUCTION

Artificial neural networks have been widely used in aeronautical engineering. Recent aerodynamic applications include, for example, flow control, estimation of aerodynamic coefficients, compact functional representations of aerodynamic data for rapid interpolation, grid generation, aerodynamic design and the interpolation of wind tunnel data.1-9 Neural network applications in aeronautics are not limited to aerodynamics. Hajela and Berke10 provide a review of a variety of neural network applications in structural analysis and design.

In earlier work Rai and Madavan11-12

developed a design optimization method based on artificial neural networks. The method offers advantages over traditional optimization methods. It provides a greater level of flexibility than other methods in dealing with design in the context of both steady and unsteady flows, partial and complete data sets, combined experimental and numerical data, inclusion of various constraints and rules of thumb, and other issues that characterize the aerodynamic design process. Neural networks provide a natural framework within which a succession of numerical solutions of increasing fidelity, incorporating more realistic flow physics, can be represented and utilized for optimization. Neural networks also offer an excellent framework for multiple-objective and multi-disciplinary design optimization. Simulation tools from various disciplines can be integrated within this framework and rapid trade-off studies involving one or many disciplines can be performed. The advantages of both traditional response surface methodology13,14 (RSM) and neural networks are incorporated in this design method by employing a strategy called parameter-based partitioning of the design space. Starting from the reference design, a sequence of response surfaces based on both neural networks and polynomial fits are constructed to traverse the

42nd AIAA Aerospace Sciences Meeting and Exhibit5 - 8 January 2004, Reno, Nevada

AIAA 2004-778

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design space in search of an optimal solution. The method has been applied to aerodynamic design optimization both in the context of both steady12

and unsteady flows15-16.

Another feature of this neural network-based design system is that it can efficiently use distributed and parallel computing resources. The method lends itself to parallelism at many levels. At the coarsest level, multiple aerodynamic simulations can be performed simultaneously and independently on multiple processors. In situations where individual simulations are computationally intensive, each simulation can be partitioned across multiple processors. In addition, neural network training can be distributed over multiple processors to further accelerate the design process.

An important aspect of using a neural network (NN) to represent the functional behavior of the objective function is controlling its capacity (defined as the maximum number of training sets that can be learned without error). The generalization capability of neural networks can be poor in the absence of adequate data. In such situations optimizing the capacity of the network is critical in order to obtain reasonable generalization capability. The earlier method12 alleviated this problem by using the neural network to represent the functional behavior of the design objective with respect to only those variables that resulted in complex variations of this function. This required a partitioning of the design variables into two sets. A simple partitioning was used to demonstrate the utility of the method.12,15-16 Parameter based partitioning can be used to considerable advantage given a good partitioning of the design variables.

The results presented by Rai and Madavan12, 15-16 demonstrated both the ability of the earlier design methodology to solve some complex aerodynamic design optimization problems as well as its versatility in handling different objective functions and constraints. However, the application of this approach to even more complex optimization problems such as three-dimensional aerodynamic design, multiple objective design and multidisciplinary optimization requires a considerable reduction in both the number of optimization steps required to obtain the optimal shape and the CPU requirements for generating the training data. Rai17,18 proposed a new neural network-based design method that

does not use parameter-based partitioning of the design space. Instead, the neural network is used to model the functional dependence of the design objective on all of the design variables. Excellent generalization is achieved through a powerful training algorithm (response surfaces that are reliable both inside and outside the region of design space containing the training data). Design costs are reduced because of this improved generalization capability. Significant reductions in design costs are also achieved by exploiting synergies between the neural network model and the simulation codes that are used to generate the training data. Additionally, the method17-18 is formulated for multiple-objective design optimization. This facet of the method can be utilized in some single-objective design optimization cases to further reduce design costs. This method is briefly discussed later in this article and subsequently used for airfoil design.

Genetic and evolutionary algorithms19

have an advantage over conventional gradient-based search procedures in that they are capable of finding global optima of multi-modal functions (not guaranteed) and handling cases where the feasible region is disjoint. Genetic algorithms have been used in airfoil and wing design20-21 and compressor and turbine airfoil design. They are also finding increasing use in multiple-objective and multidisciplinary optimization22. A disadvantage of genetic algorithms is that they often require many more function evaluations than other optimization schemes to obtain the minimum. In fact they are not the preferred method when a purely local search of a smooth landscape is required. Rai23 presents an evolutionary method, based on the method of Differential Evolution24

(DE), and investigates its strengths in the context of some test problems as well as nozzle and turbine airfoil design. The results of applying a neural network-based design procedure to the same design problems are also presented in this study. It was found that DE required about an order of magnitude more computing time than the NN-based design method. Rai23 also explores the possibility of integrating design procedures based on neural networks and evolutionary algorithms; the objective being a hybrid aerodynamic design procedure that has the strengths of both the approaches. In a more recent article Madavan25

has also explored DE-based hybrid methods andcombining DE with local search methods and applied it to airfoil inverse design. The best variant


of these combined methods required 420 function evaluations for this inverse design case. In contrast, a very similar inverse design problem required 36 simulations with the neural-network based algorithm17-18. The estimated number of simulations to perform the inverse design of Madavan25 with the NN-based method is about 50, i.e., about an order of magnitude reduction in computational costs. In general where applicable, significant cost reductions can be achieved by using RSM-based methods, such as neural networks, instead of evolutionary algorithms. However, the former approach is not the preferred method for multi-modal functions and design spaces with disjoint feasible regions.

Multi-objective design optimization is an area where the cost effectiveness and utility of both methods needs to be explored. Here we develop an evolutionary approach for multiple-objective design optimization. The goal of this developmental effort is a method that requires a smaller population of parameter vectors to solve a given problem as well one that is capable of solving complex multiple-objective problems involving several Pareto fronts (global and local) and, nonlinear constraints. The results of applying the new evolutionary method to some difficult model problems involving such complexities are also presented.

Traditionally, aerodynamic shape optimization has focused on obtaining the best design given the requirements and flow conditions. However, the flow conditions are subject to change during operation. It is important to maintain near-optimal performance levels at these off-design operating conditions. Additionally the accuracy to which the optimal shape is manufactured depends on the available manufacturing technology and other factors such as manufacturing cost. It is imperative that the performance of the optimal design does not degrade appreciably when the component shape differs from the optimal shape due to manufacturing tolerances and normal wear and tear. These requirements naturally lead to the idea of robust optimal design wherein the concept of robustness to various perturbations is built into the design optimization procedure. Here we demonstrate how both evolutionary algorithms and neural networks can be used to achieve robust optimal designs. The evolutionary method is used to solve a relatively difficult problem in extended surface heat transfer wherein optimal fin

geometries are obtained for different ranges of safe operating base temperatures. This is a case where the objective of maximizing the safe operating base temperature range conflicts with the objective of maximizing fin heat transfer. It is a good example of achieving robustness in the context of changing operating conditions. Both the methods are then used to design an airfoil; the objective being reduced sensitivity of the pressure distribution to small changes in the airfoil shape. This is a relevant example of achieving robustness to manufacturing tolerances and, wear and tear. The computational costs associated with solving this problem with the two methods is also discussed.

THE MULTIPLE-OBJECTIVE EVOLUTIONARY METHOD

In this investigation we develop an evolutionary method that draws upon ideas from several genetic algorithms and evolutionary methods. One of them is a relatively new member to the general class of evolutionary methods called differential evolution24 (DE). This method is easy to use and program and it requires relatively few user-specified constants. These constants are easily determined for a wide class of problems. Fine-tuning the constants will yield the solution to the optimization problem at hand more rapidly. The method can be efficiently implemented on parallel computers and can be used for continuous, discrete and mixed discrete/continuous optimization problems. It does not require the objective function to be continuous and is noise tolerant. Additionally, the method does not require the transformation of continuous variables into binary integers.

As with other evolutionary methods and genetic algorithms, DE is a population based method for finding global optima. The three main ingredients are mutation, recombination and selection. Much of the power 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 in the population and adding a portion of this difference to a third randomly chosen parameter vector to obtain a candidate vector. The resulting magnitude of the mutation in each of the parameters is different and optimal. For example, in the case of an elliptical objective function in two dimensions, the mutation vectors would be longer in the direction of the


major axis and shorter in the direction of the minor axis. Thus, the mutation operator adapts to the particular objective function and this results in more rapid convergence to the optimal value. In addition, this approach automatically reduces the magnitude of mutation as the optimization process converges.

To describe one version of the method24, we consider the set of parameter vectors at the nth generation, Xj,n . The subscript j refers to the jth

parameter vector in a population of N parameter vectors and,

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

where x i,j,n corresponds to the parameter value in

the ith dimension in a D-dimensional problem. The initial population is assumed to be randomly distributed within the lower and upper bounds specified for each dimension. The mutation, recombination and selection operators are then applied to the population of parameter vectors as many times as required. To evolve the parameter vector Xj,n we first randomly pick three other

parameter vectors Xa,n , Xb,n andXc,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 then defined as

zi =yi if ri ≤ C

xi,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 the minimization of the objective function f(X), is given by

Xj,n + 1 =Z if f(Z) ≤ f(X

j,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 with the candidate vector if the objective function decreases by such an action.

Several modifications to the mutation, recombination and selection procedures were made by Rai23in order to enforce constraints, facilitate searches involving multiple feasible regions embedded in infeasible regions and, enrich the pool of potential candidate vectors. Although these modifications were devised for the method of differential evolution, the underlying principles can also be used with other evolutionary strategies and genetic algorithms. Here we extend this method23 to multi-objective optimization.

Abbas et. al.26 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-dominated set. The resulting candidate vector replaces a member of the population if it dominates the first selected parent (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 remove members of this set that are in close proximity. This feature improves solution diversity.

In a more recent study Madavan27

presents a different extension to DE to handle multiple objectives. This method is also a Pareto-based approach that uses non-dominated ranking and selection procedures to compute several Pareto-optimal solutions simultaneously. It combines the features of DE and the NSGA-II method of Deb et. al.28. The main difference between DE (single-objective) and this method lies in the selection process. New candidate vectors obtained from mutation and recombination are simply added to the population thus resulting in a population that is twice as large. This larger population is subjected to the non-dominated sorting and ranking procedure of Deb et al. The ranking is then used to subsequently reduce the population to its original size. Solution diversity is achieved by ascribing 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 distance metric in Abbas et al., this crowding distance metric is computed in objective space.

An important issue that both these studies (Abbas26 et al. and Madavan27) do not address is the manner in which single-objective DE achieves


its adaptability to different landscapes and, the retention of this adaptability in the context of multi-objective optimization. As explained by Price and Storn, under the assumption that the parameter vectors of a population are distributed around a level line in parameter space that represents their mean value, the set of vectors created by vector differences (in the mutation operator) are optimal. As mentioned earlier, when the contours of the objective function are elliptic, the vector differences are longer in the direction of the major axis and shorter in the direction of the minor axis. This is the ideal distribution of mutations. DE adapts to the particular landscape being optimized, is self-scaling, rotation invariant and has a zero mean. In the presence of a second minimum, the set of vector differences include ones that are optimal for each basin and also ones that facilitate the transfer of parameter vectors from one basin to another thus making DE a powerful global optimizer. However, multi-objective optimization is akin to having numerous single-objective local optima, each of which needs to be located and identified with a local population.

Consider a situation where the Pareto-optimal front is highly curved in parameter space and the parameter vectors are distributed evenly along this front but do not coincide with it. Clearly vector differences involving vectors from disparate regions of this front are not very effective mutation vectors. The parent vectors as well as the vector being considered, Xa,n, Xb,n , Xc,n and Xj,n, need to be in proximity for effective mutation and recombination. This is especially true in the final stages of optimization. In the initial stages of optimization the entire front can be considered a single entity in a basin of attraction, being approached from afar by the parameter vectors. Localization of the relevant vectors used in mutation and recombination may not be necessary at this early stage. A very simple localization strategy is used in this study. The methods of Abbas et al.26 and Madavan27 have yielded accurate Pareto-optimal fronts in some model problems without localization. This is most likely due to the presence of a population of vectors and corresponding vector differences, some of which are appropriate mutations. Additionally, in cases where the 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 the model problems considered. This is about 1/3 to 1/2 of the value normally used in single-objective DE-based optimization and may be indicative of the need for localization. Most of the Pareto-optimal solutions presented in this

study were obtained with F=2.0 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 the evolutionary process resulted in a small improvement in convergence and quality of Pareto-optimal solutions.

The extension of DE to multiple-objective optimization that is developed and used in the present study consists of the following steps:

(1) Determine the set of non-dominated parameter vectors (rank one only as in Abbas et al.26, and unlike NSGA-II).

(2) Reduce this set of potential parent vectors to improve solution diversity if the number of parameter vectors in this set exceeds a certain threshold value. The method used to perform this operation is discussed below (a combination of NSGA-II and the method of Abbas et al.)

(3) For each member of the population, chose three parent vectors from the non-dominated set,compute a candidate vector as in Eqs. 2 and 3 and, add this candidate vector to the bottom of the list of parameter vectors to create an intermediate population that is twice as large as the original (as in NSGA-II, and unlike Abbas, et al.26).

(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 Abbas et al. 26)

(5) Retain only the first N parameter vectors (like NSGA-II and unlike Abbas et al.

26)

The current method (like that of Abbas et al.26

) only requires the rank-one non-dominated vectors to be determined and is hence easier to program than NSGA-II and the computational expense for identifying the pool of parent vectors is also less than that required by NSGA-II. The selection process is similar to that of NSGA-II and is hence more elitist than that of Abbas et al.

26. Tests on

some complex multi-objective optimization problems (presented in the next section) show that the procedure described above results in a powerful multi-objective optimization tool.

Solution diversity is achieved using a combination of the methods of Abbas et al.26 and Deb et al.28 When the number of the rank-one members exceeds a certain threshold value (for


example, half the population) a distance metric is used to reduce the number of members in this set. However, instead of using a distance metric in parameter space as in Abbas et al.26, here the crowding distance metric of Deb et al.28 in objective space is utilized for this purpose. Experimentation with a method similar to that of Deb et al.28 of obtaining solution diversity did not perform as well in some Pareto-optimal problems. This is probably because the evolutionary pressure exerted by this approach is subtle and, thus not as effective in problems with Pareto-fronts with regions that are difficult to populate.

Localization in the current method is achieved in the following manner; given the parameter vector Xj,n from 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 rank one population, 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 maximum distance between parameter vectors in the population. 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 gives preference to parent vectors that are in proximity to Xj,n.

The application of the evolutionary method used in this study to shape optimization is relatively straightforward. The aerodynamic shape of interest is first parameterized using an appropriate method. The prudent selection of geometry parameters is one of the most critical aspects of any shape optimization procedure. Variations of the aerodynamic shape can be obtained by varying these parameters. Geometrical constraints imposed for various reasons, such as structural, aerodynamic (e.g., to eliminate flow separation) should be included in this parametric representation as much as possible. Additionally, the smallest number of parameters should be used to represent the aerodynamic shape. The second step involves the specification of the upper and lower bounds for the geometry parameters to be used in the search process. This step typically involves some knowledge of the aerodynamics involved and constraints such as the maximum and minimum thickness of an airfoil. The third step

involves defining an appropriate objective function (a function of the geometric parameters) to be minimized. A given engineering objective can be achieved using different objective functions; some more difficult than others to optimize. In some cases the search for an optimum can be made significantly easier by using the appropriate objective function. The final step involves using the evolutionary method to determine the optimal set of geometric parameters.

Aerodynamic shape optimization varies from simple unimodal function optimization to multi-modal function optimization, constrained optimization, optimization in cases where the search space contains disjoint regions of feasibility, and multiple-objective optimization. The performance of the evolutionary method used in this study was investigated using test cases with some of these attributes. These cases are discussed below.

Unconstrained Multiple-Objective Optimization

Several unconstrained multi-objective optimization cases are solved here. Deb29

presents a detailed discussion of these cases. They are constructed to test the ability of the optimization method to converge to the global Pareto front, compute Pareto fronts that are convex, non-convex and discontinuous, and uniformly distribute computed Pareto optimal solutions even when the distribution of such solutions along the Pareto front is non-uniform. The first five cases were formulated by Zitzler, Deb and Thiele30, and are denoted as ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6. The sixth test case was first proposed by Viennet31, 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. al30 that are solved here can be formulated as

Minimize : f1(X)

Minimize : f2 (X) = g(X)h(f1(X),g(X)) (6)

where X is a vector in n-dimensional parameter space and the functions f1(X), g(X) and h(X) are defined differently for each case. As elucidated by Deb29, both the type of problem complexity (non-convex Pareto front, discontinuous Pareto front, multiple Pareto fronts etc.) as well as the degree of complexity can be specified by appropriate choices of the functions f1(X), g(X) and h(X). The global


Pareto front for all these cases is given by g(X) =1.0.

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

f1(X) = x1

g(X) = 1 +9

n -1xi

i=2

n∑h(f1, g) =1 - f1 g

n = 30 0 ≤ xi ≤ 1

(7)

The Pareto front for this case is convex and the Pareto optimal solutions are uniformly distributed along this front. The multi-objective evolutionary algorithm developed in this study and thirty parameter vectors were used to obtain the Pareto front. Figure 1 shows the computed Pareto optimal solutions and the exact Pareto front. The agreement between the two is good. It should be noted that DE usually requires between two and 100 times as many parameter vectors as the number of variables in the problem. The optimal ratio depends on the complexity of the optimization problem. Here we obtain the solution with 30 parameter vectors for a 30 variable problem in 250 generations. This is primarily because of the powerful recombination operator that is used here. This operator has the desirable property of generating considerable diversity in the population. The computed solutions also exhibit good solution diversity, that is, the computed Pareto optimal points are nearly evenly spaced and cover the entire the front.

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

f1(X) = x1

g(X) = 1 +9

n -1xi

i=2

n∑h(f1, g) =1 - f1/g( )2

n = 30 0 ≤ xi ≤ 1

(8)

It is more complex than ZDT1 because the Pareto front is non-convex. The problem was solved using 30 parameter vectors and 250 generations. Figure 2 shows the computed Pareto optimal solutions and the exact Pareto front. As in the previous case the computed solutions are in close agreement with the exact Pareto front and exhibit

good solution diversity. The test case ZDT3 is a 30 variable problem and is defined as follows

f1(X) = x1

g(X) = 1 +9

n -1xi

i=2

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

n = 30 0 ≤ xi ≤ 1

(9)

The important characteristic of this problem is that the Pareto front is discontinuous in objective space. Sixty parameter vectors and 250 generations were used to solve this problem. Although 30 parameter vectors yielded accurate Pareto optimal solutions with good diversity, their density along the Pareto front was inadequate. As seen in Figure 3, the computed solutions are in close agreement with their exact counterparts and exhibit good solution diversity.

The test case ZDT4 has 10 variables and is a particularly difficult problem for all multi-objective optimization methods because it exhibits numerous local Pareto fronts29. The global and the next best local Pareto front are given by g(X) = 1.00 and 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/g

n = 10 0 ≤ x1 ≤ 1

-5 ≤ x2,3 10 ≤ 5

(10)

Inadequate population sizes generally yield Pareto optimal solutions from one of the local Pareto fronts. The present evolutionary method yielded the global front with a population size of about 500-600. However, it was discovered that convergence to the global front was sensitive to the choice of the user specified factor (F) used in the mutation operator. However, a population size of 1000 yielded the global front in 100 generations for a range of values of this factor. Figure 4 shows the computed optimal solutions and the exact global Pareto front. For the sake of clarity, the computed solutions were first sorted using the value of the first objective function f1 and every fifteenth


solution was plotted in Figure 4. Clearly the method yields an excellent approximation to the global Pareto front. Both, proximity to the exact front and solution diversity are superior. This computation required 12.5 CPU seconds on a single 400MHz SGI (MIPS) processor. The global Pareto-optimal solutions shown in Fig. 4, have been found to be elusive to capture in previous studies by other investigators.

Figure 5 shows the computed optimal solutions and the exact Pareto front for the test case ZDT6. This is a 10 variable problem and is characterized by a non-convex Pareto front along which the distribution of optimal solutions is non-uniform. The challenge here is to obtain a uniform distribution of computed Pareto optimal solutions so that the entire front is well represented. ZDT6 is defined as

f1(X) = 1.0 - exp -4x1( )sin6 6πx1( )

g(X) = 1 + n -1( ) x i n−1( )i=2

n∑

14

h(f1, g) =1 - f1/g( )2n = 10

0 ≤ xi ≤ 1

(11)

The computation was performed with 30 parameter vectors for 250 generations. Figure 5 shows that the computed solutions are in good agreement with the exact Pareto front, and are nearly uniformly distributed over the entire front.

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

Minimize : f1(x1, x2 ) = 0.5r2 + sin r2

Minimize : f2 (x1, x2 ) = 15.0

+ 3x1 - 2x2 + 4( )2 8 .0

+ x1 - x2 + 1( )2 27 .0

Minimize : f3(x1, x2 ) = -1.1exp -r2

+ 1.0 r2 + 1 r2 = x1

2 + x22

-3.0 ≤ x1, x2 ≤ 3.0

(12)

The Pareto front is discontinuous in both design space as well as objective space. Figure 6 shows the computed Pareto optimal solutions obtained with a population size of 100 in 200 generations. Although accurate solutions were obtained with much smaller population sizes, 100 parameter vectors were used for this problem to better represent the rather complex Pareto front. Figure 6 shows the projection of the Pareto front on the (f1, f3) plane. Again, the current evolutionary method yields optimal solutions that are diverse 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.

As mentioned earlier, the parent vectors and the vector chosen from the population for recombination (Xa,n, Xb,n , Xc,n and Xj,n in Eqs. 1, 2 and 3) need to be in proximity to recapture the essence of DE in a multi-objective setting. ZDT1, ZDT2, ZDT3, ZDT4 and ZDT6 are all characterized by Pareto-fronts that are straight lines in parameter space. The Pareto-optimal front for VNT1 in parameter space, can be closely approximated by two straight lines. None of these cases are significantly affected by localization. Although their Pareto fronts seem complicated in objective space, they are simple in parameter space.

The following problem, designated MMR1, exhibits attributes that make it suitable to illustrate the need for localization. It is defined as

Minimize : f1(x1, x2 ) = 0.5x12

+ 0.5sin2 0.5πx2( )Minimize : f2 (x1, x2 ) = 0.5 x1−1.0( )2

+ 0.5 x2 −1.0( )2-2.5 ≤ x1, x2 ≤ 2.5

(13)

Figure 7 shows the computed and exact global Pareto-optimal fronts in parameter space. The computed solutions were obtained with 100 parameter vectors in 250 generations. The Pareto front is highly curved and has a branch point thus making it a good candidate to test the ideas on localization discussed earlier. It should be noted that the Pareto-optimal solutions on the upper and lower branches with the same x1-coordinate have identical function values and hence occupy the same location on the Pareto-front in objective space. Hence, in order to adequately populate the Pareto-front in objective space, it is sufficient if the combined population (upper and lower branches) is adequate in a given segment a < x1 < b. As seen


in Fig. 8 the computed Pareto-optimal solutions in objective space are in close agreement with the exact Pareto-optimal front and, exhibit good solution diversity.

Figure 9 shows the convergence rates obtained with four variants of the current method; 1) No localization and a constant value of F=1.0, 2) Localization and a constant value of F=1.0, 3) No localization and a value of F=0.3, and 4) Localization and a reduction in the value of F from 1.0 to 0.3 after 75% of the total number of generations. The results of 20000 independent runs were averaged to obtain the data depicted in this figure. Clearly, the first three cases result in larger errors and show signs of approaching an asymptotic value. The best results are obtained with localization and a reduction in F in the last of few generations of the run. The result thus obtained is about three times more accurate than those obtained with the second and third variants of the method and about seven times more accurate than the solution obtained with the first variant of the method. Additionally, the fourth variant of the method shows a continued decrease in the error after 200 generations.

The multi-objective optimization problem presented above (MMR1) possesses both local and global Pareto-optimal fronts. Some preliminary efforts have been made to extend the method presented here to compute these fronts simultaneously. Figure 10 shows the computed local and global Pareto fronts and their exact counterparts. This computation was performed with 150 parameter vectors and required 1000 generation. The computed solutions are in close agreement with their exact counterparts and exhibit good solution diversity.

Constrained Multiple-Objective Optimization

Many engineering problems are constrained by equality and inequality constraints that can be linear or nonlinear. A novel technique of constraint satisfaction in the context of single-objective evolutionary methods was implemented for differential evolution by Rai23.This approach to constraint satisfaction is also applicable to the multi-objective differential evolution algorithm presented here. The following simple example (labeled MMR2) is used to demonstrate the constraint handling ability of the method: The feasible region consists of the interior of three nearly circular sub-regions. The optimization problem 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 - x1

j( )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

(14)

Figure 11 shows the computed Pareto front obtained with 30 parameter vectors in 250 generations, and the exact Pareto front for both the constrained and unconstrained cases. The Pareto front for the constrained case consists of three unconnected segments. Clearly the computed optimal solutions are in good agreement with the exact Pareto front and good solution diversity has been achieved. Figure 12 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 exact Pareto front.

DESIGN OPTIMIZATION OF FINS IN BOILING HEAT TRANSFER

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

Fins used in boiling heat transfer are particularly difficult to analyze from first principles because of the complex flow physics involved. Experiments show that different types of boiling


including nucleate- and transition-boiling, and film boiling occur on these fins. Often, all these boiling modes together with free-convection co-exist at a single operating 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 accurate estimates of heat load. At low fin-base temperatures (compared to the fluid boiling temperature) the entire fin is in a free-convection mode. As the base temperature increases above the boiling temperature of the fluid, high heat transfer nucleate-boiling appears at the base of the fin. As the temperature is increased further, relatively low heat transfer film boiling appears at the base, followed by high heat transfer transition-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 fin leaving the rest of the fin in a film-boiling mode. Even larger base temperatures result in the entire fin being in a film-boiling mode. Haley and Westwater32 and Westwater33 provide an excellent description of fins used in boiling heat transfer and the associated heat transfer processes. The authors also provide experimental and computational data (temperature profiles and heat load at different base temperatures) for different fin-shapes. However, the main thrust of these articles is the optimization of the shape of the fin to achieve maximum heat load for a given fin volume, that is, single-objective optimization.

Optimizing the performance of fins used in boiling heat transfer requires that the cross-section of the fin be tailored so that most of the surface area is in a nucleate- or transition-boiling mode. Considerable increases in heat load, compared to the heat loads that can be obtained with simple cylindrical spines and rectangular fins, can be achieved by using the optimal cross-sectional area distribution. These optimal shapes are typically complex and hard to manufacture. Additionally, as the nucleate- and transition-boiling regimes are pushed off the tip of the fin as the base temperature increases, there can be a sudden and large decrease in the heat load. This phenomenon is undesirable because it occurs just when a higher heat load capacity is required. Here we investigate the possibility of maximizing both the heat load and the safe operating base temperature range (SOBTR) using the multi-objective evolutionary method discussed earlier.

For the sake of simplicity, and without loss in generality, the fin shapes investigated in this optimization study are restricted to double cones

with a cylindrical base. Figure 13 shows a schematic of the fin. It is defined using four variables, the length and diameter of the cylindrical base, and the base diameter and length of the truncated cone. The length of the second cone is obtained from a fin-volume constraint. Westwater33

first suggested this particular geometry as an approximation to an optimal “turnip” shape and conducted an experimental investigation of its efficiency. Rai and Rangarajan34 provide detailed two-dimensional (axi-symmetric) finite element computations for this fin-shape.

Figure 14 shows the fin heat load as the base temperature is increased. The heat load increases to 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 is undesirable because small increases in operating base temperature result in large decreases in the amount of heat that the fin transfers to the fluid. Figure 15 shows the temperature distribution along the length of the fin. The rapid decrease in temperature in the first 10% of the fin occurs because of the small diameter of the cylindrical base. Thereafter, the temperature drop is much more gradual because of the larger cross-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 the remaining 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 (15)

where k is the thermal conductivity of the fin (assumed constant), A is the cross-sectional area of the fin at its base, dt/dx is the temperature gradient at the base, Tb is the base temperature and Tpeak-heat-load is the base temperature at which the peak heat load is obtained for a given shape. The base temperature and the fluid

(ambient) temperature were specified as 320°F

and 120°F, respectively. The fin material is

copper. Radiation losses are assumed to be negligible. The temperature variation in the fin was computed by solving the one-dimensional heat


conduction equation using an iterative finite-difference method.

Twenty parameter vectors were used in the optimization and the evolutionary process was run for 80 generations. This required computing 59113 temperature distributions, which far exceeds the 1600 (80x20) simulations that would normally be required. This is because the accurate evaluation of the SOBTR for each fin-shape requires numerous temperature-distribution computations (each with a different base temperature). Figure 16 shows the computed Pareto-optimal front (symbols) and a neural-network estimate of the Pareto-front (curve). Point A denotes the fin-shape, heat load (about 177.1

BTU/Hr.) and SOBTR (9.6°F) for the Pareto

optimal solution corresponding to maximum heat load. Point B denotes the fin shape, heat load

(about 116.8 BTU/Hr.) and SOBTR (118.8°F) for

the Pareto-optimal solution with the maximum SOBTR. Figure 16 also shows the maximum heat load computed using a single objective differential evolution method23. Clearly, the curve in Figure 16 enables one to obtain the maximum heat load possible (and the corresponding fin shape) given the required SOBTR.

The neural-network estimation of the Pareto-front shown in Figure 16 is provided for two reasons. Firstly, computations of the SOBTR contain some noise while the network estimate of the front is smooth and also provides a continuous curve relating the two objective functions. Secondly, an accurate network estimate does not require as many optimal solutions as used in Fig. 16. There is the definite possibility of reducing the cost of obtaining the Pareto-optimal front by computing just a few Pareto-optimal solutions and then subsequently utilizing the interpolative and extrapolative abilities of neural networks17-18.

Figure 17 shows the variation of heat load as a function of the base temperature for the cases denoted as points A and B in Fig. 16. It is clear that fin B allows for a wide range of safe base temperatures about the nominal value of

Tbase = 320°F (Tbase – Tambient = 200

°F) but yields

a lower heat load at the nominal value of base temperature. Fin A, on the hand yields a higher heat load at the nominal base temperature but has a smaller range of safe base temperature values. Figure 17 indicates that for a base temperature

increase (over nominal) of about 10°F, film-boiling

pervades the entire length of fin A. Figure 18 shows the shapes of fins A and B. The longer cylindrical base of fin B, results in a larger temperature drop before the majority of the fin surface is encountered. Thus, up to a point, nucleate- and transition-boiling persist on the majority of the fin surface, even though the base temperature is much higher than the temperature at which film-boiling is initiated. However, at the nominal base temperature, the temperature drop in the base of 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 optimization problem than response surface methodology and gradient based methods. This is because the discontinuity in the heat load as the base temperature is increased (Fig. 14) results in a discontinuity in the first objective function (Eq. 15) in design space. The high-heat-load Pareto optimal solutions lie in the vicinity of this discontinuity in the objective function. Thus computing the gradient and constructing standard response surfaces becomes problematic.

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

ROBUST AIRFOIL SHAPE OPTIMIZATION

Fabricating and operating complex systems involves dealing with uncertainty in the relevant variables. In the case of aircraft, flow conditions are subject to change during operation. Efficiency and engine noise may be different from the expected values because of manufacturing tolerances and normal wear and tear. Engine components may have a shorter life than expected because of manufacturing tolerances. In spite of the important effect of operating- and manufacturing-uncertainty on the performance and expected life of the component or system, traditional aerodynamic shape optimization has focused on obtaining the best design given a set of


deterministic flow conditions and component shape. Clearly it is important to both maintain near-optimal performance levels at off-design operating conditions, and, ensure that it does not degrade appreciably when the component shape differs from the optimal shape due to manufacturing tolerances and normal wear and tear. These requirements naturally lead to the idea of robust optimal design wherein the concept of robustness to various perturbations is built into the design optimization procedure.

Recognition of the importance of incorporating the probabilistic nature of the variables involved in designing and operating complex systems has led to several investigations in the recent past. Some of the basic principles of robust optimal design are discussed by Egorov et al.35. Several commonly used approaches such as maximizing the mean value of the performance metric, minimizing the deviation of this metric and, maximizing the probability that the efficiency value is no less than a prescribed value are discussed in their paper. Egorov et al.35 make the observations that a) robust design optimization is in essence 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 is substantially different from that obtained without this criterion. Various approaches to robust optimal design 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 on maximizing the probability of constraint satisfaction. Koch et al.36, provide a discussion of this and related concepts. Some of the basic steps involved in both robust optimal design as well as reliability-based optimization such as a) identifying random variables and their associated probability density functions, b) obtaining a smaller subset of key random variables, to reduce optimization costs and, c) using a Monte Carlo technique to obtain estimates of performance variability or reliability are also discussed by the authors.

Simulation based design optimization can be computationally expensive in cases where the underlying physics is complicated. Some of thecontributing factors are three-dimensionality, a

large disparity in the largest and smallest scales that are required for an accurate analysis etc. The addition of the robustness criterion can greatly increase computational requirements because of the need to estimate the variance in performance or reliability. Koch et al.36, reduce computational cost by first obtaining the optimal solution via a deterministic approach and subsequently adding the reliability requirement. In a separate article Koch et al.37 use Kriging models to compute performance variability and reliability.

Here we focus on robust optimal design. Neural network-based response surfaces are used to model the behavior of the objective function. These response surfaces are used to compute both the objective function as well its variance in design space. This approach has the advantage that once the response surface is constructed to determine the variation of the objective function in design space, its variance at a point can be computed at very little additional cost using the Monte Carlo method. Different probability density functions can be accommodated within this approach. Hybrid neural networks are used to construct the response surface.17,18 Design space is traversed using multiple response surfaces that are constructed using data specified on simplexes. The first simplex is constructed such that its centroid coincides with the initial design in design space. The optimal solution (subject to trust region constraints) obtained with this first response surfaces serves as the centroid of the second simplex and so on until the optimal solution is found. Details of this neural network-based approach can be found in the articles by Rai17-18.

The following is a simple example that illustrates the use of neural network-based RSM in robust optimal design. It consists of low-speed flow through a row of stator airfoils in an axial turbine. The flow is assumed to be inviscid and two-dimensional. Details regarding the flow, the grid generation and simulation procedures, and the airfoil parameterization method can be found elsewehere.17-18 Although eight parameters were used to parameterize the airfoil, only two of them were varied during optimization. These two parameters have very little effect on the shape of the suction side of the airfoil. Their primary effect is on the shape of the pressure side of the airfoil. To further simplify the analysis, the objective function only consists of a measure of the variance of the pressure distribution. The required pressure


loading is not included. The method, however, is not restricted to only minimizing the variance of the pressure. The objective function is defined as

Minimize : V = (pic

i=1

i=imax∑n=1

n=nmax ∑ − pin )2 (16)

In Eq. 16 imax is the number of grid points on the surface of the airfoil and nmax is the number of geometry perturbations used to compute the

variance. The quantity pic

is the pressure at the ith

point on the surface of the given geometry, and pin

is the surface pressure at the same point for the nth perturbation of this geometry. The manner in which the geometry perturbations are obtained depends on the probability density function assumed for the perturbations in the geometry. Egorov et al.35 remark that a normal distribution is commonly observed for manufacturing tolerances. Here we adopt a very simple approach and compute the perturbations by perturbing one geometry variable at a time. It is straightforward to replace this with normally distributed surface perturbations.

The perturbed airfoil shapes as well as the base airfoil used in computing the objective function (Eq. 16) are obtained using the same airfoil parameterization scheme. This assumesthat the geometry perturbations found in reality can be represented with the same number of geometric parameters that are used to represent the base airfoil. However, a finite number of geometry parameters can only yield approximate representations of the perturbations found in reality. Hence the present RSM based approach computes approximations to the objective function of Eq. 16 (even when the sample size is large). Nevertheless the current approach has practical value. The set of perturbed airfoil shapes can be greatly enhanced by adding a perturbation function to the base airfoil shape. This function can be constructed with many more independent parameters governing its behavior than the number of parameters used to generate the base airfoil shape. This results in a very flexible perturbation function. Computing the variance with this extended set of airfoils is computationally expensive. Here we use the extended set only for assessment and not during optimization.

Figure 19 shows the initial and optimal airfoils. The optimal shape was obtained in five steps (five neural network-based response surfaces) with a constraint on the minimum airfoil thickness. The optimal airfoil is more slender, has a suction surface nearly identical to the initial airfoil and, a pressure surface that is significantly different. Figure 20 shows the computed pressure distributions for the two airfoils. The optimal airfoil does not exhibit a suction side undershoot and the pressure is nearly constant in the last 75% of the axial chord. This is because sharp variations in pressure are generally sensitive to geometry variations and, are thus avoided in the optimal airfoil. The estimate of the local standard deviation of pressure as a function of the axial location along the airfoil

σ local = (pc

n=1

n=nmax∑ - pn )2

1/2

(17)

is depicted in Figure 21. Although the improvement is not very large, it is significant. Larger decreases in local variance can be expected in cases where the component shape has both large and small geometric features and, performance is affected significantly by the latter (assuming the same distribution of geometry perturbations for all regions of the airfoil).

The optimization was also performed using the single-objective version of DE described by Rai23. The objective function (Eq. 16) was computed as in the previous optimization study using neural networks. The optimal airfoil obtained with DE was nearly identical to that obtained with RSM. However, because DE requires multiple flow simulations for every objective function evaluation, computing costs are substantially higher. Figure 22 compares computational costs for DE and NN-based RSM for the case considered. DE requires about five times as much computing as RSM. Hybridization of the two methods, for robust aerodynamic design, most likely would yield a method that is less expensive than DE but retains its ability to perform a global search.

CONCLUDING REMARKS

Traditionally, aerodynamic shape optimization has focused on obtaining the best


design given the requirements and flow conditions. However, the flow conditions are subject to change during operation and, the accuracy to which the optimal shape is manufactured depends on the available manufacturing technology and other factors such as manufacturing cost. Component shapes are also subject to change because of normal wear and tear. Clearly there is a need to desensitize optimal performance to changes in operating conditions and component shape. These requirements naturally lead to the idea of robust optimal design wherein the concept of robustness to various perturbations is built into the design optimization procedure. Here both evolutionary algorithms and neural networks are used to achieve such robust optimal designs.

Building robustness into an optimal design results in a multi-objective optimization problem. An evolutionary algorithm, based on the method of differential evolution, and intended for multi-objective optimization is first presented here. The method is tested using a number of difficult model problems including ones with local Pareto-fronts and constraints. In all these cases the computed Pareto-optimal solutions closely approximated the Global Pareto-front and exhibited good solution diversity. Many of these solutions were obtained with relatively small population sizes. One of the computed solutions, ZDT4, was found elusive to capture in previous studies by other investigators. The current evolutionary method is also used compute the optimal shape of fins used in boiling heat transfer. The complex physical processes that are encountered in boiling heat transfer result in heat load capacity that can change drastically with operating conditions. The challenge here is to obtain maximum heat load given the necessary safe operating base temperature range (SOBTR). The evolutionary algorithm was used to obtain the Pareto-optimal front showing the trade-off between maximum heat load and the SOBTR. This is a good example of direct conflict between optimality and robustness during operation.

Fabricating and operating complex systems involves dealing with uncertainty in component shapes. In the case of aircraft, efficiency and engine noise may be different from the expected values because of manufacturing tolerances and normal wear and tear. Engine components may have a shorter life than expected because of manufacturing tolerances. Recognition of the importance of incorporating the probabilistic

nature of the variables involved in designing and operating complex systems has led to several investigations in the recent past. Here, a neural-network based response surface design method developed earlier is used to design a turbine blade with a pressure distribution that is relatively insensitive to geometry changes. The advantage of this method is that the response surfaces constructed to optimize a given function can also be used to the compute the variance of the function given the statistical distribution of the random variations in component shape. Thus, one can maximize performance or minimize cost given the acceptable variance or, minimize variance given an acceptable cost or level of performance, with the same response surface. Although the evolutionary approach may be necessary in cases exhibiting multi-modality and disjoint feasible regions, the network-based response surface method is found to be much quicker in the airfoil design case presented here. Application of RSM and evolutionary algorithms to robust design is in its infancy and much more research needs to be done to determine the range of applicability of these methods.

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AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, September 4-6, Atlanta, Georgia


-0.1 0.1 0.3 0.5 0.7 0.9 1.1-0.1

0.1

0.3

0.5

0.7

0.9

1.1

DE (np=30, ng=250)

ZDT1, exact paretooptimal front

F1

F 2

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

-0.1 0.1 0.3 0.5 0.7 0.9 1.1-0.1

0.1

0.3

0.5

0.7

0.9

1.1

F1

F 2

DE (np=30, ng=250)




-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1-1.0

-0.7

-0.4

-0.1

0.2

0.5

0.8

1.1

DE (np=60, ng=250)


F1

F 2


-0.1 0.1 0.3 0.5 0.7 0.9 1.1-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

F1

F 2

DE (np=1000, ng=100)


Fig. 4. Global Pareto optimal front in objective space for ZDT4.


-0.1 0.1 0.3 0.5 0.7 0.9 1.1-0.1

0.1

0.3

0.5

0.7

0.9

1.1

F1

F 2

DE (np=30, ng=250)



-1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

DE (np=100, ng=200)

VNT1, exact paretooptimal front

F1

F 3

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


-0.5 0.0 0.5 1.0 1.5 2.0-0.2

0.2

0.6

1.0

1.4

1.8

2.2

DE (np=100, ng=250)

MMR1, exact paretooptimal front in designspace

x1

x 2

Fig. 7. Global Pareto-optimal front in parameter space for MMR1.

-0.1 0.1 0.3 0.5 0.7 0.9 1.1-0.1

0.1

0.3

0.5

0.7

0.9

1.1

MMR1, exact pareto optimal front in objectivespaceDE (np=100, ng=250)

F1

F 2

Fig. 8. Global Pareto-optimal front in objective space for MMR1.


0 50 100 150 200

-4

-2

-1

10

10

10

8 x

Number of Generations

no localization, constant F (1.0)

localization, constant F (1.0)

no localization, constant F (0.3)

localization, variable F (1.0,0.3)

Avg

.Dis

tanc

efr

omP

aret

oF

ront

Fig. 9. Average distance from the global Pareto-optimal front as a function of the number of generations ( MMR1).

-0.2 0.2 0.6 1.0 1.4 1.8 2.2-0.2

0.2

0.6

1.0

1.4

1.8

2.2

F1

F 2

MMR1, exact pareto optimal front in objectivespaceDE (np=150, ng=1000)

, F-

2.9

2

Local Pareto front (F - 2.9 vs. F )2 1

Global Pareto front (F vs. F )2 1

Fig. 10. Global and local Pareto optimal fronts in objective space for MMR1.


-0.1 0.1 0.3 0.5 0.7 0.9 1.1-0.1

0.1

0.3

0.5

0.7

0.9

1.1MMR2, exact pareto optimal front in objective space (withconstraint)

DE (np=30, ng=250,constrained)

. . . Exact pareto optimal frontwithout constraint

F1

F 2

Fig. 11. Pareto optimal front in objective space for MMR2.

-0.3 0.0 0.3 0.6 0.9 1.2 1.5 1.8 2.1 2.4-0.3

0.0

0.3

0.6

0.9

1.2

1.5

MMR2, exact pareto optimal front in design space (withconstraint)

DE (np=30, ng=250,constrained)

. . . Exact pareto optimal frontwithout constraint

Constr

aint b

ound

aries

x1

x 2

Fig. 12. Pareto optimal front in parameter space for MMR2.


D2D1L2

L1

Fig. 13. Fin geometry and design variables.

0 100 200 300 4000

50

100

150

200

TBase TAmb.-( ) , R

Hea

tLoa

d(B

TU

/Hr.)

Fig. 14. Heat load at various base temperatures.


0.0 0.2 0.4 0.6 0.8 1.00

50

100

150

200

TT

Am

b.-

()

,R

x / LFig. 15. Temperature variation along the axis of the fin.

110 120 130 140 150 160 170 180 1900

25

50

75

100

125

Heat Load (BTU/Hr.)

TM

ax.

TB

ase

-(

) ,

R

DE (np=20, ng=80, nf=59113)

Neural network generalizationof DE data

Data from single objective DE (maximize heat load)

Point B

Point A

Fig. 16. Pareto optimal front in objective space for the fin showing the trade-off between heat load and safe operating base temperature range.


0 100 200 300 4000

50

100

150

200

TBase TAmb.-( ) , R

Hea

tLoa

d(B

TU

/Hr.)

Maximum temperaturerange (Point B)

Maximum heat load (Point A)

Fig. 17. Heat load as a function of base temperature for the maximum heat load and maximum safe-base-temperature range designs.

Maximum temperaturerange (Point B)

Maximum heat load (Point A)

Fig. 18. Fin geometries for the maximum heat load and maximum safe-base-temperature range designs.


baselineairfoil

optimizedairfoil

Fig. 19. Baseline and optimal airfoil shapes.

0.0 0.2 0.4 0.6 0.8 1.0

1.0

x/c

p / p

t8

baseline

optimized

Fig. 20. Surface pressure distributions for the baseline and optimal airfoils.


0.0 0.5 1.0

0.000

0.001

0.5 0.0

suction side pressure sidetrailing edge

baseline

optimized

x/c

σ loca

l

Fig. 21. Distribution of σlocalfor the baseline and optimal airfoils.

100 125 150 1754.0x10

0 25 50 75number of function evaluations

obje

ctiv

e fu

nctio

n va

lue

differential evolutionneural-network based

response surfaces

-7

1.4x10-6

Fig. 22. Comparison of airfoil design costs for the evolutionary and neural network -based design methods.

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