Aerodynamic Optimization of Turbomachinery Blades Using Evolutionary Methods and ANN Based Surrogate Models 2008 Optimization and Engineering

Optim Eng (2008) 9: 239255DOI 10.1007/s11081-007-9031-1

Aerodynamic optimization of turbomachinery bladesusing evolutionary methods and ANN-based surrogatemodels

Temesgen Mengistu Wahid Ghaly

Received: 26 April 2006 / Accepted: 23 October 2007 / Published online: 1 December 2007 Springer Science+Business Media, LLC 2007

Abstract A fast, flexible, and robust simulation-based optimization scheme using anANN-surrogate model was developed, implemented, and validated. The optimizationmethod uses Genetic Algorithm (GA), which is coupled with an Artificial NeuralNetwork (ANN) that uses a back propagation algorithm. The developed optimizationscheme was successfully applied to single-point aerodynamic optimization of a tran-sonic turbine stator and multi-point optimization of a NACA65 subsonic compressorrotor in two-dimensional flow, both were represented by 2D linear cascades. Highfidelity CFD flow simulations, which solve the Reynolds-Averaged Navier-Stokesequations, were used in generating the data base used in building the ANN low fi-delity model. The optimization objective is a weighted sum of the performance ob-jectives and is penalized with the constraints; it was constructed so as to achieve abetter aerodynamic performance at the design point or over the full operating rangeby reshaping the blade profile. The latter is represented using NURBS functions,whose coefficients are used as the design variables. Parallelizing the CFD flow sim-ulations reduced the turn-around computation time at close to 100% efficiency. TheANN model was able to approximate the objective function rather accurately and toreduce the optimization computing time by ten folds. The chosen objective functionand optimization methodology result in a significant and consistent improvement inblade performance.

Keywords Artificial neural networks Genetic algorithm Computational fluiddynamics Aerodynamic design Global optimization NURBS Response surfaceapproximation

T. Mengistu W. Ghaly ()Concordia University, Montreal, Quebec, Canadae-mail: [email protected]

Present address:T. MengistuCENAERO, Gosselies, Belgium

240 T. Mengistu, W. Ghaly

1 Introduction

Aerodynamic optimization methods are becoming very attractive in todays compet-itive environment as they can reduce the design cycle time by automating the designprocess. Until recently, designers were relying mostly on manual optimization. AsComputational Fluid Dynamics (CFD) matured over the last decade and as comput-ing technology has greatly improved and has become more affordable, simulation-based optimization is becoming affordable and more popular than ever. This is due tothe fact that optimization techniques give direct control on performance parameters,even if the computational cost is at least one order of magnitude larger than the costof an analysis calculation.

Aerodynamic shape optimization allows the designer to automate the explorationof the design space to achieve a given objective. One possible design objective isto minimize flow losses, which can be measured by e.g. the total pressure loss (orentropy generation), through proper reshaping of the blade profile. Automated aero-dynamic design is accomplished by coupling a CFD flow simulation code with nu-merical optimization methods. As the aerodynamic shape optimization problem isa complex one with possibly many local minima, gradient-based methods can betrapped in a local optimum, unless the initial guess is close to the global mini-mum. For this reason, heuristic/evolutionary global algorithms such as Genetic Al-gorithm (GA) and Simulated Annealing (SA), although more computation inten-sive compared with gradient-based methods, are used to ensure reaching close tothe global minimum. These algorithms have been recently applied in turbomachin-ery design problems; examples of such algorithms are given in (Dennis et al. 1999;Wang and Damodaran 2000; Oyama et al. 2002).

In the work of Dennis et al. (1999), a combination of genetic algorithm and Se-quential Quadratic Programming (SQP) algorithms were used to optimize a two-dimensional turbine cascade. The GA followed by SQP implementation was intro-duced to reduce the total number of required function calls, keeping the global ex-ploration behavior of GA. The optimization scheme required from 220 to 675 callsto the flow analysis code.

Oyama et al. (2002) worked on 3D blade shape optimization and included themass flow rate and pressure ratio as constraints in the objective function.

The above references represent a sample of aerodynamic cascade optimization,and can be used as a useful tool to design blade cascades however they require largecomputation resources. The effort now is focused on finding a way of reducing theprohibitive computation time without compromising the solution accuracy.

Since aerodynamic design optimization problems are multi-modal and discon-tinuous in nature, gradient-based numerical optimization algorithms risk of gettingtrapped in local minima (Lai and Yuan 2002) or run into an infeasible design forwhich the flow simulation does not converge. Therefore exploratory algorithms suchas Genetic Algorithm (GA) and Simulated Annealing (SA) are more appealing forglobal exploration of the design space however, GA and SA can involve a pro-hibitively high computational cost where a large number of costly CFD simulationsare needed, which makes exploratory algorithms less appealing than gradient-basedoptimization algorithms (Lai and Yuan 2002).

Aerodynamic optimization of turbomachinery blades 241

In order to avoid this prohibitive cost of GA/SA, a low fidelity approximation ofthe objective function, using a Response Surface Approximation (RSA), can be usedso as to reduce to a minimum the number of required CFD analyses. Moreover us-ing a RSA can eliminate some of the noise in the objective function as it creates asmooth response surface thereby improving the convergence of the optimization al-gorithms. Examples of such an approximation are Artificial Neural Networks (ANN)and quadratic polynomial response surface models, which have been successfullyused in aerodynamic shape optimization as well as in other fields (Pierret et al. 2000).

In the present work, an optimization method of the evolutionary type, namely GA,and an approximation method, ANN, are first presented. They are then implementedin the aerodynamic shape optimization of 2D gas turbine blades. The optimizationobjective is to improve the blade performance, subject to some constraints, by mod-ifying the blade shape which is approximated using NURBS, whose coefficients areused as the design variables. CFD is used to generate the high fidelity data set thatis used in training and testing the ANN and in validating the optimum profile. Theoptimization methodology as well as the approximation (ANN) are demonstrated byredesigning a transonic impulse turbine stator and a subsonic axial compressor ro-tor.

2 The design methodology

2.1 Problem definition and objective function

In any optimization problem, the choice of the objective function affects the opti-mization process as well as the results. Thus a careful and well-studied identificationand formulation of that function is crucial. For example, the overall aerodynamicperformance of a compressor rotor is determined by its adiabatic efficiency, , and/ortotal pressure loss coefficient, , at design and off-design conditions, therefore onecan choose to optimize the efficiency or the total pressure loss or both; this can bedone either at the design point only or on the full operating range. With this designstrategy in mind, the objective function is constructed as a weighted sum of individ-ual objectives and is penalized with the constraints such that it can serve for singlepoint or multi-point optimization, and can be defined as follows:

Fobj(X) = Min[C1

ni=0

(1 i) + C2n

i=0

nj=0

|j i | + PT]

(1)

where stands for either adiabatic efficiency, , or (1), where is the total pres-sure loss coefficient. X is the vector of design variables, which include the backpres-sure and the shape parameters that control the blade profile. Varying the backpressurein the pre-determined range from the choke limit to the stall limit while fixing therotor speed allows for tracing a speed line, i.e. it allows for design and off-designcalculations that correspond to different mass flow rates with varying efficiency andpressure ratio.


The first term in the objective function, (1), attempts to maximize the efficiency(or minimize the total pressure loss coefficient) at the design and off-design points,while the second term would eliminate large difference in efficiency (or total pressureloss) between the design and off-design points which would tend to keep it constantand optimum over the entire operating range. The last term in the objective functionis a Penalty Term (PT) that accounts for the aerodynamic, mechanical and geometricconstraints imposed on the optimization process. The aerodynamic constraints couldinclude the exit flow angle, the spacing to chord ratio, and the stall margin.

The summation is carried out over n pre-selected points, these pre-selected pointsare the design point and off-design points. In this work, two cascade optimizationcases were considered in Sect. 5; one of them is a single-point optimization, i.e. n = 1,while the second case is a multi-point optimization where n = 4. The weights Ck ,where k = 1,2, are prescribed by the designer, they are determined such that thedifferent components of the objective function have the desired influence on the opti-mization process. Note that the current choice of the objective function, given in (1),allows for different design options depending on the values given to the Ck coeffi-cients: e.g., single or multi-point optimization for maximum efficiency or minimumtotal pressure loss.

2.2 NURBS representation

The geometric representation of the profiles is an important part in the aerodynamicshape optimization procedure. The parameters in the geometric representation ofthe blades are used as design variables in the aerodynamic optimization process.At present, the geometry is parameterized using Non-Uniform Rational B-splines(NURBS). A clear advantage of using NURBS is that one can adjust the profile lo-cally on specific regions of the blade, by modifying the weights and/or the x- andy-coordinates of the NURBS control points, without necessarily affecting the wholeblade profile. The NURBS definition and formulation are very well described in(Piegl and Tiller 1995).

An accurate geometric representation of 2D turbomachinery cascades was ob-tained using NURBS with a minimum possible number of control points rangingfrom 9 to 19 depending on the type of blade cascade, see (Ghaly and Mengistu 2003)for more detail.

In the present work, the blade profile was defined by the mean camber line anda thickness distribution; each of which was parameterized by a NURBS functionwith a number of control points ranging from nine to eleven. The position of thesecontrol points and the weights can be taken as the design variables in the optimizationprocess, see the results given in Sect. 5.

2.3 Flow simulation method

The two-dimensional turbulent transonic flow in a linear cascade is simulated usinga second-order accurate cell-vertex finite volume space discretization method on anunstructured triangular mesh. The steady state solution is reached by pseudo-timemarching the Reynolds-averaged Navier-Stokes equations using an explicit five-stage


Runge-Kutta scheme. Local time stepping and implicit residual smoothing were usedto accelerate the convergence. A non-linear blend of second and fourth order artificialviscosity was used in capturing shocks and eliminating pressure-velocity decouplingwith minimal numerical diffusion. The method of characteristics was used to imposeinflow and outflow boundary conditions. Turbulence is modeled using the Baldwin-Lomax model in the discretization of the Reynolds-averaged Navier-Stokes equations(Daneshkhah 2006).

The flow at the inlet and exit planes, which are placed at about one chord upstreamand downstream of the cascade, is subsonic. The boundary conditions at inlet aregiven by the total pressure, total temperature and the tangential velocity. At the exitplane, the ratio of exit static to inlet total pressure (referred to as the backpressure) isspecified. This set of boundary conditions allows for computing the flow at any pointon a given speed line. The latter is characterized by a fixed rotor angular speed, whichis represented by a fixed tangential velocity in the rotor relative frame and varying thebackpressure will result in changing the mass flow rate and the pressure ratio acrossthe rotor, hence will allow for moving along the speed line.

3 Optimization algorithm and surrogate model

The aerodynamic shape optimization problem is reduced to solving a constrained op-timization problem, which is transformed into an unconstrained one by penalizingthe objective function with the constraints. The optimization problem is solved us-ing Genetic Algorithm (GA) and is coupled with an Artificial Neural Network, as asurrogate model, to reduce the optimization computing time. Both GA and ANN aredescribed in this section.

3.1 Genetic algorithm

Genetic algorithms are general-purpose search algorithms based upon the principlesof evolution observed in nature. Genetic algorithms combine selection, crossover,mutation, and elitism operators with the goal of finding the best solution to a problem.They search for this optimal solution until a specified termination criterion is met(Goldberg 1985; Gen and Cheng 1997).

In the present work, the variables for the GA algorithm are real coded, wherean individual is characterized by a vector of real numbers. Two kinds of crossoveroperations are included in the real-coded GA developed in this work namely, arith-metic and heuristic crossover operators. Arithmetic crossover operator combines lin-early two parent chromosome vectors to produce two new offsprings while heuristiccrossover operator uses the fitness values of the two parent chromosomes to deter-mine the search direction and creates the new offsprings. In addition GA is naturallya population-based parallel algorithm that is best suited in a parallel computationalgorithm. A population of 32 individuals with a crossover probability of 0.80, mu-tation probability of 0.15 and elitism of 2 has been used for each generation. Theimplementation of GA is detailed in (Mengistu and Ghaly 2003).


Table 1 GA validation

Test case Global minimum

Rastrigin FunctionPresent work: 40 design variables 1014(Deb and Joshi 2002): 20 design variables between 10 and 20(Fogel and Beyer 1995): 30 design variables greater than 10

Welded-beam: 4 design variables and 5 constraintsPresent work 2.197(Ray and Liew 2003) 2.3809(Reklaitis et al. 1983) 2.38116

Speed-reducer: 7 design variables and 11 constraintsPresent work 2994.36(Luo 2004) 2994.36(Ray and Liew 2003) 2994.47

3.1.1 GA validation

The GA developed and used in this work was validated using the Rastrigin function(Deb and Joshi 2002), the welded-beam problem and the speed reducer problem (Rayand Liew 2003). These test functions vary in difficulty, in number of local minima,and in number of design variables X and constraints. They have a global extremumthat is hidden among many local extrema. The search range that was chosen for eachfunction includes several local minima.

The Rastrigin function was tested with forty design variables, xi , that are definedin the range 100.12. Table 1 gives a summary of the results obtained for the threetest problems.

The welded beam problem is about the design of a welded beam for minimum costand maximum rigidity when it must carry a certain load. The problem is described andsolved by several authors to test optimization algorithms for engineering problems(Reklaitis et al. 1983; Deb 2001; Ray and Liew 2003) who reported different resultsat different times; which implies that the problem has several local optima and needsto be solved for the global optimum. The GA implemented in this work found anoptimum cost of 2.197, satisfying all constraints. Table 1 summarizes the comparisonof the results with the literature mentioned above.

The speed reducer problem is another engineering problem that was investigatedby several authors for example, Rao (1996), Deb (2001), Ray (2003), Luo (2004). Theobjective of this problem is to find the minimum weight, subject to 11 constraints.There are seven design variables. The best-known feasible solution to this problemis 2994.36. The GA implemented in the present work has found exactly the sameoptimum solution with all constraints satisfied. Table 1 compares the literature resultswith the result obtained by the GA developed in the present work.


Fig. 1 A typical neural networks architecture

3.2 Artificial neural networks

ANN is used as a low order RSA to approximate the objective function at a relativelylow computing cost and results in reducing the computing effort by a factor of ten.A set of test cases is generated using the high fidelity CFD simulations and is usedin training and testing the ANN model, the latter is then used in the aerodynamicoptimization.

The present model is composed of a multi-layer feed-forward network with back-propagation. It is composed of three layers, an input layer, one hidden layer having41 nodes and an output layer, see Fig. 1. A sigmoid function is taken as the transferfunction between the nodes, and the weights in each connection in the network arearbitrarily initialized to one. They are updated using an optimization algorithm tominimize the error between the network output and the given training data set output.The training strategy is enhanced using genetic algorithm and simulated annealingalgorithm in the initial stage of the training which is then followed by a gradient-based method (Mengistu 2005). Results of the training and testing of this ANN modelfor the aerodynamic optimization problem is given in the context of optimizing theNACA compressor presented in the results section, see Fig. 10.

4 Numerical implementation

The aerodynamic design optimization involves four basic components: shape para-meterization using NURBS, numerical optimization using GA, response surface ap-proximation using ANN for the low fidelity calculation of the objective functions


and constraints and flow simulation using Reynolds Average Navier-Stokes solver(RANS) for the high fidelity calculation of the objective functions and constraints.The flow of calculations and information involved in building the ANN model and incarrying out the aerodynamic design optimization is displayed in Fig. 2.

As the ANN training and testing requires a pool of high fidelity flow simulationsthat are obtained by solving the RANS equations, parallel computations have beenimplemented when producing these flow simulations. This parallelization, whichis 100% efficient, results in reducing the wall clock time required for optimization bya factor that is equal to the number of processors used. Moreover, response surfaceapproximation used to evaluate the objectives and constraints, required negligibletime. The construction of the response surface needed a database of flow solutions,which was obtained using high-fidelity flow simulation.

The overall computation time includes the selection of blade geometry candidates,generation of CFD solution for the candidates, post-processing the solution, buildingthe database for response surface approximation, ANN training (Response surfacemodel construction) and optimization process using ANN. 75% of the total develop-ment time is taken by the CFD solver while building the ANN response surface takesabout 17% of the time, but it should be emphasized that selection of blade geometrycandidates must be done with utmost care so that the database contains feasible bladeprofiles, that cover reasonably well the design space, this requires about 7% of thetime. All the rest including the aerodynamic optimization takes 1% of the develop-ment time.

Fig. 2 The ANN-based aerodynamic design optimization process


5 Results and discussion

In this section, two design cases are presented. The first case given in Sect. 5.1 isintended to assess the aerodynamic optimization process while the second case isintended to assess the ANN-based aerodynamic optimization.

5.1 Redesign of a transonic impulse turbine cascade

An impulse turbine cascade is redesigned to minimize the total pressure loss coeffi-cient at a given operating point. For this cascade, the spacing to chord ratio is 0.526;the camber and thickness distributions assume a parabolic profile with maximumthickness to chord ratio of 21.45% and maximum camber to chord ratio of 21.45%,both occurring at mid-chord. The inlet flow angle is 40.63 and the ratio of exit sta-tic to inlet total pressure is 0.833. The flow is assumed inviscid and is simulated bysolving the Euler equations, which are a subset of the RANS equations described inSect. 2.3. Figure 3 shows a flow chart of the aerodynamic optimization process.

The objective function for this case is given by:Fobj(X) = Min[ + K1(m) + K2()] (2)

where Fobj is the objective function, X is the vector of design variables, is the totalpressure loss coefficient, m and are the difference between the computed andthe target mass flow rates and exit flow angles (in degree). The weights K1 and K2are user-specified penalty coefficients; they are chosen so as to have equal penalizingeffect on the objective function. In this case, they take the following values:

K1 = 1000 when |m| > 0.01, and 0 otherwise,K2 = 10 when || > 1, and 0 otherwise. (3)

The flow over this blade is transonic, where a shock is present on the blade suctionside. To reduce the total pressure loss, this shock should be weakened or eliminated,which can be accomplished by reshaping the blade profile. The latter is described

Fig. 3 The aerodynamic optimization process


Fig. 4 Convergence history forthe impulse turbine cascade

by its camber line and its tangential thickness distribution, each is represented byNURBS with 9 control points and weights; the control points and the correspondingweights are determined as described in (Ghaly and Mengistu 2003). In this case,the thickness distribution is fixed and the camber line is allowed to change. Hencethe design variables are taken to be the y-coordinates of the control points and thecorresponding weights along the camber line except those at the leading edge point;this implies a total of 16 design variables. The y-coordinates of the control points areallowed to vary from the original profile by 15%, and the weights vary between 0.5and 2.5; these ranges ensure a good coverage of the design space.

The optimization history is given in Fig. 4 where the best candidate in each GAgeneration is given. The total pressure loss is reduced from 0.0043 to 0.0026 in fivegenerations; each generation consists of 32 individuals. A single flow field analysistook approximately two minutes of CPU time (when starting from a converged solu-tion obtained for a given candidate of the previous generation) with a CFL number of4 on a mesh with 2800 points.

Figure 5 shows that the changes in shape between the original and the re-designed blades are relatively small, which would be difficult to achieve by man-ually changing the blade shape. However, the normal shock prevailing in the orig-inal blade around mid chord has been eliminated in the redesigned blade, as canbe seen in Figs. 6 and 7 that show the Mach number contours and the Machnumber distribution along the original and redesigned blades. Note that the op-timal design has a reversed curvature particularly along the suction side to al-low the flow to compress reversibly. Other researchers (Ahmadi and Ghaly 1998;Dang 1995) have also observed similar behavior.

5.2 NACA 65 subsonic compressor rotor

The optimization algorithm and the surrogate model presented earlier, were used toredesign a well-documented subsonic compressor rotor (Emery et al. 1958) for bestsustained efficiency over the full operating speed line; this is accomplished by modi-fying the rotor blade profile. The original profile is that of a NACA 65 airfoil and the


Fig. 5 The impulse turbinecascade

Fig. 6 Isentropic Machcontours

flow is assumed to be turbulent, the Reynolds number being 2.45 105. The perfor-mance measure that is included in the objective function, given in (1), is taken tobe the adiabatic efficiency, , and the weights C1 and C2 are set to 1 and 0.5, respec-tively. The Penalty Terms (PT) appearing in (1) include the optimization constraints,namely the reduced mass flow rate, the inlet and exit flow angles, while the remain-ing constraints are satisfied exactly through the imposition of the inlet and outlet flowboundary conditions for the CFD flow simulation.

The original blade profile is defined in terms of a mean camber line and a thick-ness distribution, both are parameterized using NURBS with 11 and 9 control points,respectively. The design variables controlling the blade shape are taken to be the y-coordinates of the camber line profile and the thickness distribution, excluding thethickness at the LE and TE points and the camber line point at the LE, which gives


Fig. 7 Isentropic Mach alongthe blade

a total of 17 geometric design parameters. The weights and the x-coordinates of thecontrol points are fixed during the optimization. The family of airfoils used in trainingand testing the ANN was obtained by perturbing the above 17 geometric design pa-rameters in a preset range and, to ensure a homogeneous representation of the designspace, the candidate airfoils were selected based on the Latin-Hypercube space-fillingexperiments (McKay et al. 1979). To account for the full range of the operating speedline, the flow was simulated at 4 values of backpressure that include the two limitingpoints namely the stall and choke points, and two other points.

The ANN response surface model was constructed using a family of 50 blade pro-files, 35 of these profiles were used in training the ANN model and 15 were used intesting it. For each of these profile, the flow was simulated at 4 values of backpres-sure, thus the total number of CFD runs amounted to 200 simulations that covered areasonably large range of the design space, as can be seen in Figs. 8 and 9. The CFDflow simulations were carried out in parallel (one CFD simulation per node) on fournodes of an SGI 2000 server using Message Passing Interface (MPI), which requiredabout 60 hours of wall-clock time.

The ANN model was constructed with one hidden layer containing 41 nodes, theinput layer with 18 design parameters and the output layer with 4 output variables(efficiency, reduced mass flow rate, inlet flow angle and exit flow angle) used in com-puting the individual terms appearing in the objective function, see (1). The ANNtraining took about five hours on a Pentium IV PC; the errors in the ANN trainingand testing are given in Fig. 10. The errors in the ANN approximation are given inmore detail in Fig. 11 which shows that these errors are less than 2% for about 45%of the test set.

Unfortunately, there is no rule for determining the number of nodes in the hiddenlayer. However, a good initial guess is the average of the number of input and outputnodes. This number is then increased to create an optimum-trained network. If thenumber of nodes in the hidden layer is too small underfitting occurs, where hightraining error and high generalization error occur. If the number of hidden layer nodesis too large overfitting occurs, where low training error but high generalization erroroccurs.


Fig. 8 The range of geometryexplored for the design space

Fig. 9 Speed lines of bladeprofiles used in training/testingthe ANN

Figure 9 also shows that the optimal blade is 4% more efficient than the originalone. Moreover, the optimum blade shows a better performance in terms of pressureratio (although the latter is not part of the optimization function) over the full range ofoperation; this behavior was also observed by Oyama et al. (2002). Figure 12 showsthat the optimized blade camber has significantly changed near the trailing edge, andthe maximum thickness has increased by 3.7% and has slightly moved downstream.The high camber near the trailing edge resulted in a higher flow turning at the cost ofan increase in profile loss. Figures 13 and 14, where the efficiency is plotted vs. massflow rate and vs. pressure ratio, show a 7% improvement in efficiency and about 1%increase in total pressure ratio for the same blade speed and mass flow rate.


Fig. 10 The convergencehistory of ANN training/testing

Fig. 11 The ANN modelvalidation

The performance of the optimal rotor blade, which is given in Figs. 13, 14 and 15,was evaluated by simulating the flow over the redesigned compressor blade at valuesof backpressure that are different from those used in generating the surrogate model.The fact that the performance of the optimal blade is smooth over the full operatingrange reflects the robustness and the validity of the optimization strategy, and theaccuracy of the surrogate model.

6 Conclusion

A fast, flexible, and robust simulation-based optimization scheme using an ANN-surrogate model was developed, implemented and used in the aerodynamic shapeoptimization of turbomachinery blade cascades in two-dimensional flow. The opti-mization method uses Genetic Algorithm (GA) and is combined with an ArtificialNeural Network (ANN) that uses a back propagation algorithm. GA was validated


Fig. 12 The original andoptimized blade profiles

Fig. 13 Efficiency vs. massflow rate for the NACA 65subsonic compressor (Points 1,3, 5, and 7 correspond topreselected backpressure used inthe optimization while the othersare not)

against benchmark cases and ANN was assessed for aerodynamic shape optimiza-tion. The ANN-surrogate model, used in building a low fidelity model to approximatethe optimization objective and constraints, was found to reduce the computing timeby a factor of ten.

The developed optimization scheme was successfully applied to single-point opti-mization of transonic inviscid flow in an impulse turbine where the simulation-basedoptimization scheme was assessed (ANN was not used); it was also applied to multi-point optimization of turbulent flow in a NACA65 compressor where the ANN-basedoptimization scheme was assessed. The choice of objective function and constraintswas found to be crucial for the success of the optimization process particularly for


Fig. 14 Efficiency vs. pressureratio for the NACA 65 subsoniccompressor (Points 1, 3, 5, and 7correspond to preselectedbackpressure used in theoptimization while the others arenot)

Fig. 15 Pressure ratio vs. massflow rate for the NACA 65subsonic compressor (Points 1,3, 5, and 7 correspond topreselected backpressure used inthe optimization while the othersare not)

the multi-point optimization case; it was constructed so as to achieve a better aerody-namic performance over the full operating range by reshaping the blade profile.

The test cases that were presented showed that the developed optimization algo-rithm is general and capable of handling various aerodynamic design objectives andconstraints. They also showed that the present methodology was able to reach the op-timization objective of improving the blade performance over the full operating rangewhile simultaneously satisfying the design constraints.


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Aerodynamic optimization of turbomachinery blades using evolutionary methods and ANN-based surrogate modelsAbstractIntroductionThe design methodologyProblem definition and objective functionNURBS representationFlow simulation method

Optimization algorithm and surrogate modelGenetic algorithmGA validation

Artificial neural networks

Numerical implementationResults and discussionRedesign of a transonic impulse turbine cascadeNACA 65 subsonic compressor rotor

ConclusionReferences

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Aerodynamic Optimization of Turbomachinery Blades Using Evolutionary Methods and ANN Based Surrogate Models 2008 Optimization and Engineering