research article a comparative analysis of nature-inspired...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 716237 15 pageshttpdxdoiorg1011552013716237

Research ArticleA Comparative Analysis of Nature-InspiredOptimization Approaches to 2D Geometric Modelling forTurbomachinery Applications

Amir Safari1 Hirpa G Lemu1 Soheil Jafari2 and Mohsen Assadi3

1 Department of Mechanical amp Structural Engineering University of Stavanger 4036 Stavanger Norway2Department of Petroleum Engineering University of Stavanger 4036 Stavanger Norway3 Centre for Sustainable Energy Solutions International Research Institute of Stavanger 4021 Stavanger Norway

Correspondence should be addressed to Amir Safari amirsafariuisno

Received 3 July 2013 Accepted 18 September 2013

Academic Editor Jung-Fa Tsai

Copyright copy 2013 Amir Safari et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A vast variety of population-based optimization techniques have been formulated in recent years for use in different engineeringapplications most of which are inspired by natural processes taking place in our environment However the mathematical andstatistical analysis of these algorithms is still lackingThis paper addresses a comparative performance analysis on some of the mostimportant nature-inspired optimization algorithms with a different basis for the complex high-dimensional curvesurface fittingproblems As a case study the point cloud of an in-hand gas turbine compressor blademeasured by touch trigger probes is optimallyfitted using B-spline curves In order to determine the optimum numberlocation of a set of BezierNURBS control points for allsegments of the airfoil profiles five dissimilar population-based evolutionary and swarm optimization techniques are employedTo comprehensively peruse and to fairly compare the obtained results parametric and nonparametric statistical evaluations asthe mathematical study are presented before designing an experiment Results illuminate a number of advantagesdisadvantagesof each optimization method for such complex geometriesrsquo parameterization from several different points of view In terms ofapplication the final appropriate parametric representation of geometries is an essential significant component of aerodynamicprofile optimization processes as well as reverse engineering purposes

1 Introduction

The turbomachinery bladesrsquo aerodynamic profile and struc-tural strength play an important role in improvement ofenergy conversion efficiency as well as the reliability andavailability of the machine in both land-based and aeroapplications From aerodynamic point of view geometry-based blade design optimization is of utmost importance forfurther enhancement of turbomachinery performance Onthe other hand a robust and precise reverse engineering ofexisting blades could alsomaintain their performance as nearas possible to the reference design In this way precise andproper geometry modeling and parameterization of currentblades are the first and sometimes the most important stepin both optimization and reverse engineeringmissions whichalso in turn need an optimization effort [1 2]

At the same time recent advances in both mathe-matical tools and optimization techniques enable furtherimprovement in geometric modeling procedure Irrespectiveof the goal of blade profile parameterization that is itsuse in inverse aerodynamic design or in its application toa direct numerical optimization there are several param-eterization methods with different impacts on the revers-ingoptimization process For free-form shapes howeverparametric curvessurfaces including Bezier [3] B-splineand nonuniform rational B-spline (NURBS) [4] are generallyused as the most suitable ones For the parameterizationof bladeairfoil shapes specifically though there exists nounique way NURBS curvessurfaces [1 5 6] and Beziercurves [7 8] can be mentioned as the significant andcommonly used methods Some other methodologies also

2 Mathematical Problems in Engineering

exist for example PARSEC method [9] and Hicks-Hennefunctions [10] for this application

As mentioned above a proper and precise curve fittingrequires an unavoidable optimization mission The nonlin-ear nature of sophisticated geometries like turbomachinerybladesrsquo shape and availability of high-speed parallel com-puters for massive computation have resulted in the useof non-gradient-based and guided random search method-ologies in curvesurface fitting problems [11ndash14] On theother hand a vast variety of population-based optimizationtechniques have been formulated in recent decades someof which are inspired by natural processes taking place inour environment Genetic algorithms (GAs) [15] particleswarm optimization (PSO) [16] and ant colony optimization(ACO) [17] are some such methods that have already beenused in various engineering optimization problems from thedevelopment of optimum control systems [18] to curve fittingand geometric modeling [11 14]

Among these optimization techniques genetic algorithmand differential evolution (DE) [19] are the most successfullyused strategies in curve fitting optimization of airfoilsrsquo geom-etry parameterized by NURBSBezier curves While GA andreal-coded genetic algorithm (RCGA) are stochastic searchand population-based optimization algorithms that mimicDarwinrsquos theory of biological evolution DE is a relativelynewer evolutionary method in engineering applications andrequires only a few user-defined parameters

PSO and invasive weed optimization (IWO) [20] areother population-based optimization methods inspired bysocial and ecological behavior The PSO algorithm is basedon the simulation of the social behavior of birds flocking fishschooling and animals herding to let them adjust to theirenvironment find rich sources of food and keep away fromother predatory animals by using information sharing andsocial cognitive intelligence IWO is also another derivative-free metaheuristic method originally formulated based onthe concept of the natural social behaviour of colonizingweeds These swarm intelligence approaches have demon-strated their high capability in parametersrsquo optimization fornonlinear multidisciplinary problems

In this paper a study on a gas turbine compressor airfoilshape parameterization and optimum curve fitting as a mainstep to blade geometry optimization has been reported Forthis purpose two most important curve parameterizationmethods that is Bezier and NURBS curves have beenapplied within the five above-mentioned population-basedoptimization loops (GA RCGA DE PSO and IWO) so thatthe best alternative solutions from different points of viewcan be discovered To achieve this goal the parametric proce-dures involving independence normality and homoscedas-ticity are firstly discussedThen the nonparametric statisticalanalysis including Quade and Friedman aligned comparisontests by applying several different post hoc procedures arebrought to perform a rigorous comparison among the perfor-mance of the optimization algorithms After that designingan experiment is done for each algorithm to finally illustratethe effectiveness of the proposed algorithms

In the rest of the article the theory behind the Bezierand nonuniform rational B-spline curves and different airfoil

geometric representation techniques are first described inSection 2 In Section 3 the problem which will be solved isclearly defined This section involves the problem statementand explanation of the formulation fitness function andflowchart considered for the optimization processes Thenthe well-organized optimization techniques for optimumcurve fitting used in this research are explained in Section 4The first subsection focuses on evolutionary optimizationtechniques (GA RCGA and DE) and the second considersswarm intelligence and colonizing techniques (PSO andIWO) Section 5 discusses the adjusted parameters for theoptimization algorithms followed by the statistical analysisexperimental results discussions and comparison Finallyconclusion and future works have been presented in the lastsection of the article

2 Airfoil Geometric ModellingTechniques and Tools

The turbomachinery blade shape parameterization addressesthe two-dimensional airfoil profile construction from eitherthe direct handling of curves of airfoil shapes or the superpo-sition of the camber line and thickness distribution aroundit Between these two different approaches of geometrydefinition however many designers have considered distinctparameterized curves for the suction surface (SS) pressuresurface (PS) leading edge (LE) and trailing edge (TE) of ablade section In this article direct handling of the curves ofthe airfoil shape has been used by dividing the profile fromthe hub to the shroud into five sections as shown in Figure 1This article focuses on only the first section that is the bladehub section (depicted in Figure 1(b))

As the first step a standard coordinate measuringmachine equipped with touch trigger probes has beenemployed to measure 2D point cloud of compressor blade in-hand (Figure 1(a) top) These measured pointsrsquo coordinateshave been used to fit the airfoil sketches using NURBSand Bezier curves To achieve this purpose after indicat-ing the digitized points of the airfoil profile and choosingthe approach for airfoil shape modeling four curves willseparately be fitted to any airfoil shape In this way theaccuracy and robustness of the created geometric model canbe managed by controlling the curve parameters like numberof BezierNURBS control points (CPs) andor curve order

Here the theory behind thesemathematicalmethods andthe characteristics of each will be briefly described

21 Bezier Curves The Bezier curve proposed by Bezierin the early 1960s [21] is defined by the vertices of apolygon which enclose the final fitted curve Bezier curvesuse the Bernstein polynomial function119861

119894119899(119906) as the blending

function [22]

119861119894119899 (119906) = (

119899

119894) 119906119894(1 minus 119906)

119899minus119894 (1)

where 119899 is the degree of the curve 119906 is the parameter of thefunction and 119894 = 0 1 119899

Mathematical Problems in Engineering 3

Segment 1suction side

Segment 4trailing edge Segment 2

pressure side

Segment 3leading edge

(a)

Tip

Hub

(b)

Figure 1 Measured points cloud for a section along the blade span and four distinct segments of the airfoil for the curve fitting problem (a)Gas turbine compressor blade sections-view from the top (b)

When this blending function is applied to the vertices ofthe polygon the Bezier curve equation is found as

119875 (119906) =

119899

sum

119894=0

(119899

119894) 119906119894(1 minus 119906)

119899minus119894119875119894 (2)

where 119875119894is the position vector of the 119894th control vertex For

a Bezier curve of degree 119899 the number of CPs that controlthe shape of the curve is (119899 + 1) In other words it canbe concluded that Bezier curves are affected by these twomain parameters the polynomial degree and the number ofCPs On the other hand (2) shows that the degree of theBezier curve is determined by the number of CPs Anotherchallenging property of a Bezier curve is that any CPs affectthe shape of the entire curveThe third important property ofa Bezier curve is its convex hull property which means thatthese curves are completely situated inside their own convexhull constructed by the polygons

22 Nonuniform Rational B-Spline (NURBS) Curves One ofthe major drawbacks of Bezier curves is that the curve degreeis directly determined by the number of CPs In additioneach CP has an impact on the shape of the whole curveThis becomes of specific concern when only some minormodifications on some specific places on the blade sectionsare requiredTherefore using B-spline curves is an alternativebecause of their local modification property

In the early 1970s Cox [23] and de Boor [24] suggestednew blending functions119873

119894119896(119906) expressed as

119873119894119896 (119906) =

(119906 minus 119905119894)119873119894119896minus1 (119906)

119905119894+119896minus1

minus 119905119894

+(119905119894+119896

minus 119906)119873119894+1119896minus1 (119906)

119905119894+119896

minus 119905119894+1

(3)

where 1198731198941(119906) is equal to 1 for 119905

119894le 119906 le 119905

119894+1and equal to zero

otherwiseBased on this equation the B-spline curve is defined as

follows

119875 (119906) =

119899

sum

119894=0

119873119894119896 (119906) 119875119894 (4)

In these equations 119899 is the number of CPs 119896 is the orderof the curve 119899 + 119896 + 1 is the number of knot values and 00

is assumed to be equal to zero The gap between neighboringknots in this specific definition is always uniformThereforethe extracting curve is called a uniform B-spline curveHowever as these curves are used for a shape optimizationprocess some knot values may be addeddeleted resulting innonuniform gaps between the knots These new generatednonperiodicnonuniform curves are also properly matchedwith computer-aided design (CAD) systems In the enda nonuniform rational B-spline curve (NURBS) which issimilar to nonuniform B-spline curves is defined by thefollowing equation

119875 (119906) =sum119899

119894=0119908119894119873119894119896 (119906) 119875119894

sum119899

119894=0119908119894119873119894119896 (119906)

(5)

where 119899 is the number of CPs 119896 is the order of curve 119896 + 119899 isthe number of knots119875

119894areCPs and119908

119894are the corresponding

weights of CPs A Bezier curve can also be considered as aspecific type of NURBS representation NURBS curves havea number of significant characteristics including

(i) numerical stability (due to the independence of poly-nomial degree from number of CPs)

(ii) local shape control or local modification property(because of the influence range of each CP)

(iii) coincidence of endpoints with firstlast CPs(iv) convex hull property (like Bezier curves)(v) more versatile modification of the created curve(vi) exact presentation of conic curves (circles ellipses

etc)

The above-mentioned characteristics are important par-ticularly when a researcher faces a real case study Forexample the last distinction can really be helpful for ourspecific problem in the case where the designer wants toconstruct two circlesellipses for trailingleading edges of theairfoil shape In the present work these two segments havealso beenmodeled by free-form curves just like the other twosegments

Depending on whether the purpose of the designer is tocreate a completely new airfoil design or tomake someminor


aerodynamic improvements to the existing blade profile aglobal geometry modeling approach or a localized parame-terization method can be used respectively Considering anumber of the important advantages for NURBS represen-tation mentioned above and because of the prospective goalof the present continuing research to slightly improve theshape of the existing turbomachinery blades for some specificobjectives in upcoming works the application of NURBScurves seems to be more reasonable Indeed their affineinvariance and local control property make them appropriatefor such geometry handlingThe comparison between Bezierand NURBS curves would be discussed further in the resultssection of the paper

3 Problem Statement andApplied Methodology

After skimming the background of BezierNURBS curvesthe optimum data fitting problem using the defined tools andthe planned approach is made clear in this section

When the digitized points are available model con-struction is continued by airfoil segmentation and CADmodeling Segmentation subdivides themeasured points intoareas for various individual features [25] and CADmodelingidentifies geometric features from the measured points andmerges the identified features into a complete CADmodel Asstated before the airfoil shape is divided into four segmentsincluding LE TE PS and SS in this work Consequently fouroptimum BezierNURBS curves will separately be fitted toany airfoil shape and then merged together by enforcing 119862

2-continuity between the segments

Concerning the NURBS curve fitting specifically variousfitting schemes have been developed so far Some researchershave identified both the CPs and the weights of a NURBScurve simultaneously byminimizing the sumof the squares ofthe distances from the measured points to the correspondingfitted curve points while others have focused only on theoptimization of the CPsrsquo location Regarding the airfoil shapemodeling a third-order NURBS curvesurface has recentlybeen employed to describe the suitable parametric geometryof 2D airfoil and 3D blades

31 Objective Function Formulation In this paper the bestCPs of both Bezier and NURBS curves among the candi-dates are searched for by using the least squares methodAlthough the proposed method can automatically determinethe appropriate CPs both number and location at thesame time these optimal CPs will only be searched in apredefined rangemdashbased on the existing knowledge about theconstraints of airfoil shape at handmdashto save time and CPUefforts Therefore the CPs of these curves can be utilizedas characteristic points This procedure clearly needs anoptimization effort due to the lack of knowledge about theexact position of the proper CPs at the first stage of curvefitting problem Hence the objective function which shouldbeminimized is the sum of distances between the fitted curveand the original scanned data points In such an optimization

Data pointsBezierNURBS fitted curveControl polygon

Figure 2 Errors between the original measured points and fittedcurve

process BezierNURBS CPs are the design variables in factthat have to be forced to the optimal locations

The above-mentioned procedure is more clearly depictedin Figure 2 and (6)

119864 =

119899

sum

119894=1

1003816100381610038161003816119862119894 minus 119875 (119906119894)1003816100381610038161003816 (6)

where 119862119894are the original data points 119899 is the number of

measured data points and 119875(119906119894) are corresponding points on

the fitted BezierNURBS curveThe airfoil (as a profile section of the compressor blade)

considered in this study consists of four BezierNURBScurves for four segments andnineCPs identify each segmentEach CP has two coordinates 119909 and 119910 and so this will leadto a total number of 72 design variables Since the optimumcurve fitting problem is separately solved for the mentionedsegments of airfoil (SS PS LE and TE) the strategy to fix theintersection points of the segments through the optimizationprocess can force the chord of the airfoil to be constant Inthis way 16 parameters are known and fixed Another 16parameters are also determined by enforcing 119862

2-continuityat the intersections of the segments As a result 40 variableswill remain for the optimization algorithms as the designparameters In addition the rational upper bounds and lowerbounds should be defined for variations of the parameters tohave an effective and efficient optimization route

All optimization techniques in this paper have been usedto locate the optimal position of the CPs for a set of openBezier andor NURBS curves according to the objectivefunction represented in (6) Based on that the objective isminimization of the sum of the errors 119864 Figure 3 depicts theflowchart of the performed optimal curve fitting procedure inthis study

4 Planned Optimization Algorithms

This section briefly discusses the procedures operators andlogic of all optimization algorithms used in this study Forthis purpose the optimization techniques are conceptuallydivided into two main categories based on their characteris-tics (1) evolutionary algorithms involving genetic algorithm


Airfoils point cloud

Guessing control points of

Generating randominitial population

Calculating fitness of control points for each airfoil segment

Optimization

Termination criteria

optimum curve

BezierNURBS curves

BezierNURBS

N

Y

Figure 3 BezierNURBS optimum curve fitting flowchart

and differential evolution and (2) swarm intelligence andcolonized algorithms including particle swarm and invasiveweed optimization methods (Figure 4)

41 Evolutionary Optimization Technique As mentionedearlier two evolutionary optimization methods customizedand specialized by the authors are used to optimally fitthe parametric curves to the airfoil point cloud As thealgorithms are based on Darwinrsquos theory of evolution theoptimization uses different mechanisms and operators tosearch for individuals that best adapt to the environment Inother words individuals with a higher level of fitness have agreater chance to survive and continue in the optimizationprocess

411 Genetic Algorithm (GA) GA was introduced by Hol-land [15] as a probabilistic global searchmethod Its operationis based on the combination and generation of DNAs andchromosomes and it mimics the metaphor of natural biolog-ical evolution GA operates on a population of individuals(potential solutions) each of which is an encoded string(chromosome) containing the decision variables (genes)

The structure of a GA is composed of an iterative proce-dure through the following key steps [26] as also depicted inFigure 4

(i) creating an initial population of airfoil shapes withconsideration of the existing experience

(ii) evaluation of the fitting performance of each profilein the population

(iii) selection of the best-fitted airfoilsrsquo segments andreproduction of a new population

(iv) application of genetic operators that is crossover andmutation

(v) iteration of steps II to IV until the maximum numberof generations is reached

The fitness value is associated with each individualexpressing the performance of any airfoil shapes created withrespect to a fitting error function that should beminimized (119864in (6)) Also the positive effect ofmutation is the preservationof genetic diversity and the fact that the local minima can beavoided Following the evaluation of the fitness of all profilesin the population the genetic operators are repeatedly appliedto produce a new population of airfoil shapes

In addition a real coding approach instead of the stan-dard GA coding strategy that is a binary pattern is also usedin this study Using this approach more adequate mutationoperators can be defined while the destructive effects ofcrossover are reduced specifically for a set of data points inthe 119909-119910 coordinate Furthermore the errors regarding theCPsrsquo locations caused by discretization are avoided [14] Thereproduction approach here is the roulette wheel methodin which the probability of choosing a certain individual isproportional to its fitness

412 Differential Evolution (DE) DE is one of the newlyemerging but well-known stochastic parallel direct searchmethods as well as a kind of population-based optimizationalgorithm DE was presented by Price and Storn in 1997[19] as an evolutionary algorithm designed for solvingcontinuous optimization problems DE can be used to findapproximate solutions for many practical problems havingobjective functions that are nondifferentiable noncontinu-ous nonlinear noisy flat and multidimensional or havingmany local minima constraints or stochasticity DE usesmutation and crossover operators and a selection method togenerate the population of the next generation [27]

As illustrated in Figure 4 DE procedure involves thefollowing steps

(i) 119873 individuals are created randomly(ii) every individual of the population (each set of CPs for

the problem in hand) undergoes mutation operationwith scale factor (SF) a positive real number typicallyless than 1

(iii) mutation vector and target vector undergo crossoveroperation with crossover rate (CR) and

(iv) trial vector is evaluated and compared with the fitnessvalue of the target vector The vector with the greater


Initializing

Initializing

Initializing

Initializing

Mutation

Velocities calculation

Recombination

New positions calculation

Selection

Reproduction

Selection

Evaluation andupdating

Mutation and crossover

Competitive exclusion

DE

PSO

GA

IWOand spatial

Evaluation

Evaluation

Figure 4 Key similarities and differences among DE PSO GA and IWO procedures

fitness value (the fitter CPsrsquo set) enters the nextgeneration

Recombination incorporates successful solutions fromthe previous generation and mutation expands the searchspace as in GA Mutation recombination and selectioncontinue until some stopping criterion is reached

The present paper proposes an alternative method byimplementing a DE algorithm for the curve fitting problemThis proposed method is compared with the result obtainedfrom the GA method in the last section of the paper

42 Swarm Intelligence Technique The theoretical bases fortwo swarm intelligence methods employed in the compara-tive study are described in this section

421 Particle Swarm Optimization (PSO) A very popularswarm intelligence algorithm introduced by Kennedy andEberhart is called particle swarm optimization [28] Thisalgorithm is based on the simulation of the social behaviorof flocks of birds school of fish and herds of animals tolet them adjust to their environment find rich sources offood and keep away from other predatory animals by usinginformation sharing and social cognitive intelligence [29]This algorithm generates a set of solutions in a multidimen-sional space randomly which represent the initial swarmThe initial swarm contains particles that move in the spaceand search for the best global position over the number ofiterations Then iterations will be continued until the bestglobal position is reached

In general the PSO algorithm consists of threemain stepsas follows (Figure 4)

(i) generating positions and velocities of particles(ii) updating the velocities(iii) updating the positions

In this way each particle refers to a point in a multi-dimensional space where its dimensions are related to the

numbers of design variables The positions of the points aswell as the particlesrsquo velocity change as the iterations whichare calculated in each step proceed In the first step position119909119894

119900and velocity V119894

119900for each point are generated randomly by

the use of upper and lower bounds of variables employing thefollowing equations

119909119894

119900= 119909min + rand (119909max minus 119909min)

V119894119900=

119909min + rand (119909max minus 119909min)

Δ119905

(7)

where 119909119894

119900and V119894119900are position and velocity of the 119894th particle

in the first step respectively 119909min and 119909max are the lower andupper bounds of the design variable respectively rand() is arandom number between 0 and 1 and Δ119905 is the time step

The first population is distributed uniformly in the spaceby this process In the second step PSO calculates newvelocities to move the particles from positions in time 119896 tonew positions in time 119896+1 In fact PSO updates the particlesrsquovelocities for transferring to the next positions For calcula-tion of the new velocities PSO needs two important valuesthe best global position of particles in the current swarm119875119892

119896 and the best position of each particle over all previous

and current steps 119875119894 These values are saved in the memoryof each particle PSO then employs these two values besidesthree coefficients 119908 119888

1 and 119888

2to calculate new velocities

for the next iteration using a random distribution functionThe three aforementioned coefficients indicate the effect ofthe current motion the particlersquos own memory and swarminfluence Recently some PSO approaches for optimizationpurposes take the effect of neighboring positions into accountin order to improve the speed of convergence to the bestglobal position In this case a term with factor 119888

3 called

neighborhood acceleration is used in order to take this effectinto consideration [30 31]


Based on the above explanations the velocity updateformula takes the following form

V119894119896+1

= 119908V119894119896+1

+ 1198881rand

(119875119894minus 119909119894

119896)

Δ119905

+ 1198882rand

(119875119892

119896minus 119909119894

119896)

Δ119905+ 1198883rand

(119871Best minus 119909119894

119896)

Δ119905

(8)

where 119908 is the inertia factor in the range from 04 to 14 V119894119896is

the velocity of the 119894th particle in current motion 1198881is the self-

confidence factor in the range from 15 to 2 1198882is the swarm

confidence factor in the range from 2 to 25 1198883is the neighbor

acceleration factor 119909119894119896is the position of the 119894th particle in the

current motion 119875119894 is the best position of the 119894th particle inthe current and all previous moves 119875119892

119896is the position of the

particle with the best global fitness at current move 119896 and119871Best is the best local neighbor position in the current move

It is worth pointing out that when the effect of neighboracceleration is considered in the velocity update (119862

3= 0) as

it has been done in this study the PSO converges noticeablyfaster This is due to more information sharing betweenparticles in this way So taking the neighbor accelerationfactor into account is a trade-off between convergence rateand computational time

Lastly using the updated velocity vectors calculated by(8) the position of each particle is changed by the followingequation

119909119894

119896+1= 119909119894

119896+ V119894119896+1

Δ119905 (9)

where 119909119894119896+1

is the new position of the 119894th particle in step 119896+1This algorithm is repeated until a stop criterion is reached

The stop criteria may be an iteration number or a specifiederror for the best global value

422 Invasive Weed Optimization (IWO) Invasive weedoptimization is an ecologically inspired optimization algo-rithm newly developed by Mehrabian and Lucas [20] Thealgorithm is a derivative-free metaheuristic method origi-nally formulated based on the concept of the natural socialbehavior of colonizing weeds [32] In their leading researchthey have shown the merits of IWO in finding the globaloptimum of different complex multidimensional optimiza-tion problems which reveals that IWO is an appropriate com-petitor for other comparatively older and well-establishedtechniques for population-based evolutionary computation

IWO is implemented in this study with a problem dimen-sion of ten and using the following procedure recursively

(i) initializing a population (randomly spreadover the 119889-dimensional problem space with random positions)

(ii) fitness evaluation for each member of the colony ofweeds to produce seeds depending on its own and thecolonyrsquos lowest and highest fitness

(iii) reproduction (based on plant fitness and number ofseeds)

(iv) spatial dispersal (ie the generated seeds are ran-domly distributed over the search space by normally

distributed randomnumberswithmean equal to zerobut varying standard deviation SD)

(v) competitive exclusion (poor plants elimination)

The flowchart of the IWO method is also illustrated inFigure 4

The variation of the SD value with generation is definedas follows

SD119868= (1 minus

119868

119868max)

119899

(SD119894minus SD119891) + SD

119891 (10)

where 119868max is the maximum number of iterations 119899 is thenonlinear modulation index and 119894 and 119891 refer to the initialand final condition respectively The SD value is decreasedgeneration by generationThe decreasing rate depends on thenonlinear modulation index value which results in groupingfitter plants and the elimination of inappropriate plantsrepresenting transformation from 119903-selection to 119896-selectionmechanism [20]

After reaching the maximum number of plants thecompetitor exclusion mechanism activates in order to elim-inate the plants with poor fitness in the generation Thismechanism is formulated so that it gives a chance to plantswith lower fitness to reproduce and if their offspring has agood fitness in the colony then they will survive The above-mentioned steps are repeated until the maximum number ofiterations is exceeded

5 Implementation and Results

This section presents several illustrations of compressorairfoil data fitting from different points of view The goalis to show the performance of the proposed optimizationmethodologies to be used in such geometric modellingproblems and to make a comparison among them In orderto have a fair comparison of the results both the initialpopulation and the stopping criteria are set to be identical forthe relevant algorithms

Before representing the experimental illustrations how-ever it is expected to compare and analyse the results in asystematic approach recently designed and proposed [33 34]For this purpose the parametric and the nonparametric sta-tistical tests are carried out besides designing an experimentin order to improve the performance evaluation process andstatistical confidence

Furthermore the experimental results are presented byapplying all optimization algorithms to the given airfoil shapeparameterization process The following figures and tablespresent the historical convergence of optimization algo-rithms during the generations differences between Bezierand NURBS curves from different points of view and theiradvantages and disadvantages static and dynamic conver-gences of the used algorithms fitting error in each case andcomputational efficiency

All coding and simulation carried out in this researchhave been implemented in MATLAB R2010b on a PC withIntel(R) Xeon(R) CPU X5650 266GHz with 2 processorsand 320GB RAM


51 Employment of Statistical Tests This section outlines thecomparisons between the results obtained from the variousalgorithms Studies conducted to prove the relative merits ofthe algorithms used for the curve fitting problem in hand arepresented

With respect to the no free lunch theorem in search andoptimization it is not possible to find one specific algorithmbeing better in behaviour for any given problems Thereforesome criteria to select an appropriate optimization algorithmin each problem are sought In this regard statistical proce-dures including the parametric tests and nonparametric testscould be utilized for a fair comparison between the resultsFor this purpose the required condition for the parametrictests is firstly described based on the published studiesTheseconditions are then checked for the obtained results in thiswork Subsequently the results of a nonparametric analysisusing several methods are represented

511 Parametric Test In order to use the parametric testit is necessary to check independence normality andhomoscedasticity procedures

For the problem in hand the independence of the resultsis obvious since they are independent runs of any algorithmwith randomly generated initial population At the sametime an observation is said to be normal when its behaviourfollows a normal or Gauss distribution with a certain averagevalue 120583 and variance 120590 and a normality test applied over asample can indicate the presence or absence of this conditionin the observed data Lastly the homoscedasticity conditionis to indicate a violation of the hypothesis of equality ofvariance [33] All the tests would get the related 119875 valueswhich indicate the dissimilarity of the samples with respectto the normal shape To do this all the given optimizationalgorithms are executed 30 times with the same initial popu-lation The results of fitness function values are presented inTable 1 For the normality check of these results theMATLABenvironment is used A low 119875 value shows a nonnormaldistribution In this paper a level of significance 120572 = 10is considered and so the 119875 value greater than 01 specifiesthe fulfilment of the normality condition In other words thismeans that there is less than a 10 probability that the sampledata does not agree with our hypothesis due to randomchance

In this study two different tests are used for numericalassessment of the normality The first one is Kolmogorov-Smirnov (K-S test) which is a distribution-free and non-parametric test but could be modified to be employed as agoodness of fit test In the special case of testing for normalityof the distribution data are standardized and comparedwith a standard normal distribution The K-S test has thisbenefit to make no assumption about the distribution of dataThe second test is Shapiro-Wilk (S-W test) that comparesthe ordered sample values with the corresponding orderstatistics from the specified distributionThe S-W test is mostcommonly used to assess a normal distribution

Table 2 shows the results of the normality tests whichhave approached to the different results Similar conclusionwas also mentioned by Garcıa et al [33] As a result the

parametric test condition is almost fulfilled with respectto the 119875 values obtained from these two different testsAccording to the S-W test however only some new results forBGA intended to achieve a better normal result are obtainedwhere the normality has not been satisfied

Finally passing these normality tests allows us to statewith 90 confidence that the data fit the normal distributionwith no significant departure from the normality

512 Nonparametric Test Since the research work deals withthe real values in this study and based on the promisingresults from the previous section the results of parametrictests seem enough for a reliable statistical analysis For moreevaluation of the all given algorithmsrsquo behaviour the useof nonparametric statistics is also considered in addition tothe parametric test The performance of any algorithm withrespect to the remaining ones is studied and based on that ithas been determined if these results offer better performancefor each one Here all results are reported with respect to thelevel of significance 120572 = 5

Tables 3 and 4 represent the 119875 values gained by applyingpost hoc method over the results of Quade and Friedmanaligned tests using several different procedures respectivelyUsing Quade test the average ranks of RCGA BGA DEPSO and IWO are 3 2 1 465 and 435 respectively Therankings for the mentioned algorithms are 5233 4713 451712730 and 10557 using Friedman aligned test Therefore DEachieves the best rank (the minimum value) among thesefive algorithms Assuming DE to be the reference algorithmall hypotheses are rejected according to Tables 3 and 4Nevertheless BGA shows a good behaviour considering its119875 values in different procedures

To compare an optional reference algorithm with othersHolm procedure driven 119875 values of the post hoc multiplecomparisons are used (Table 5) Since Holmrsquos procedurerejects those hypotheses having a 119875 valuele005 all hypothe-ses in this table are rejected (all Holm 119875 values are lessthan or equal to 005) That means that DE BGA andRCGA are better than PSO and IWO Among the first threealgorithms the best and the worst algorithms are RCGA andDE respectively Also PSO and IWOdisplay almost the sameperformance giving a little bit positive concession to PSO

Lastly contrast estimation is obtained to calculateapproximately the performance difference of the algorithmstwo by two (Table 6)This comparison shows that DE outper-forms other algorithms in terms of error rates while PSO isthe worst case in this regard

52 Designing of Experiment All the algorithms were runseveral times with the set of parameters based on the existingexperience in order to find the best possible solution

521 Comparison of NURBS and Bezier Tools To beginwith because GA-based optimization is broadly used andmore well-known than other methods employed in the paperand exhibited good performance in statistical analysis thisalgorithm has been selected to make a reliable comparisonbetween NURBS and Bezier curve fitting for airfoil shape


Table 1 The fitness values obtained from 30 independent runs

RCGA BGA DE PSO IWO1 0057 004081 0037 08259 027952 00593 00401 0037 00828 01983 00739 004049 0037 08142 035494 00537 0042 0037 18263 014855 00479 00467 0037 11282 081186 00487 004077 0037 09122 014857 0072 004054 0037 07141 014858 00514 003946 0037 06595 08669 0064 004125 0037 05901 1446410 0071 004348 0037 08109 0216911 00511 004127 0037 07419 0148512 00502 003894 0037 01108 0899413 00621 004075 0037 17911 0225514 00501 004204 0037 07631 0188915 00661 003943 0037 0245 0188916 00493 003891 0037 08892 0188917 00534 003956 0037 02679 0880718 00511 003851 0037 05432 0783719 00522 003963 0037 19922 0233520 01003 004568 0037 02032 0188921 00592 004093 0037 08074 0184222 0053 003999 0037 07006 0200823 00708 004271 0037 07102 082824 00624 003794 0037 07437 0236825 00627 004044 0037 02942 1503726 00645 004884 0037 01837 1126827 00579 004599 0037 02674 1153228 00531 004262 0037 10189 0148529 0055 004305 0037 14319 0177530 01024 004206 0037 1717 02158

Table 2 Results of normality tests

GA DE PSO IWOP values based on K-S test 01066 01027 08102 05304P values based on S-W test 00802 02507 08012 04304

parameterization The best regulations for this optimizationalgorithm involve 120 chromosomes as the initial populationselected from the whole solution space and maximumnumber of generations is set to 300 Selection method isRoulette wheel and probabilities of two-point crossover andmutation are set to 10 and 020 respectively In order tomake a fair comparison of the results the termination criteriaas well as all other parameters are set to be the same forboth curve fitting tools Finally the real-number genes fordesign variablesrsquo definition with the method of least squaresfor evaluation of curve fitting fitness were successfully usedas shown in the following illustrations

Figure 5(a) depicts optimum parameterized curves andtheir polygons for the pressure side of the airfoil using Bezierand NURBS This figure illustrates the good fitting with the

measured point cloud of this segment To limit the number ofpages other segments are not shown here but the acceptablecoincidence is the same The results in general show that forthe same conditions the NURBS curve has a better ability tobe optimally fitted on a set of data points In other words thefinal objective value that is fitting error of the curve fittingproblem by using NURBS is noticeably less than the value forthe Bezier curve fitting

Another important characteristic that should be inves-tigated is the localglobal modification property As shownin Figures 5(b) and 5(c) the shape of the entire curverepresented using Bezier tool contrary to NURBS tool isaffected by altering only one control point This helpfulcharacteristic of NURBS can be used in applications whereminormodifications to the existing bladeairfoil shape shouldbe considered during an optimization process For instancethrough the aerodynamics performance enhancement ofan airfoil the small and precise changes on the existinggeometries are typically needed so that the local modificationproperty of NURBS helps us in this regard This is exactlywhat is wanted to be done in the next stage of the currentresearch that is geometry optimization of a GT compressor


Table 3 Post hoc results by Quade comparison test (reference algorithm DE)

i Algorithms 119911 = (1198770minus 119877119894)SE P Holm Hochberg Hommel Holland Rom Finner Li

4 PSO 5522134 0 00125 0012741 0013109 0012741 00457653 IWO 5063582 0 0016667 0016952 0016667 0025321 00457652 RCGA 302449 0002491 0025 0025321 0025 0037739 00457651 BGA 1512245 0130472 005 005 005 005 005

Table 4 Post hoc results by Friedman aligned comparison test (reference algorithm DE)


4 PSO 7321863 0 00125 0012741 0013109 0012741 00073253 IWO 5384422 0 0016667 0016952 0016667 0025321 00073252 RCGA 063888 0522901 0025 0025321 0025 0037739 00073251 BGA 0175321 0860828 005 005 005 005 005

Table 5 119875 values of post hoc multiple comparisons

i Algorithms z Unadjusted P Holm P10 DE versus PSO 8981462 0 00059 DE versus IWO 8164966 0 00055568 BGA versus PSO 6531973 0 0006257 BGA versus IWO 5715476 0 00071436 RCGA versus DE 4898979 0000001 00083335 RCGA versus PSO 4082483 0000045 0014 RCGA versus IWO 3265986 0001091 001253 RCGA versus BGA 244949 0014306 00166672 BGA versus DE 244949 0014306 00251 PSO versus IWO 0816497 0414216 005

Table 6 Contrast estimation between averages of results consider-ing all pairwise comparisons

RCGA BGA DE PSO IWORCGA 0 0016 0021 minus0619 minus0226BGA minus0016 0 0005 minus0635 minus0243DE minus0021 minus0005 0 minus064 minus0248PSO 0619 0635 064 0 0392IWO 0226 0243 0248 minus0392 0

airfoilblade for better performance where designers are notgoing to create geometry which is completely different fromthe primary original shape

Table 7 shows a comparison between the effects ofNURBS and Bezier properties on some implementationcriteria The evident influence of the number of CPs on thefitting error can be seen in the table that is the error dropswhen the number of CPs increases for both mathematicaltools While the degree of the NURBS curve is a predefinedvalue independent of the number of CPs the degree of theBezier curve is increased with the rise in the number of CPsHence the fitting error in Bezier representation drops more(from 1376mm to 617mm) with an increasing number ofCPs Nevertheless as implied earlier in Section 2 of the paperwhen a curve of a complicated shape is represented by the

Bezier curve many CPs should be used This in turn resultsin a higher degree Bezier curve Higher degree functionsmay cause oscillation as well as increasing the computationalburden

Although the computational time of NURBS curve fittingis generally higher than that of Bezier curve fitting the resultsin Table 2 show that the computational time of optimizedparameterizationwith Bezier increasesmore than three timeswhen the number of CPs increases from five to ten Thisrise for NURBS representation is less than two times Thisproperty is particularly important in this case because curveswith so many CPs should be utilized due to the complexityof the current geometry Therefore independence of thecurve degree from the number of CPs in NURBS causesa better possibility of its application into the curvesurfaceparameterization of complex shapes such as turbomachineryairfoilsblades

As a result the remaining implementation is done byusing NURBS representation as the tool of geometry mod-eling in airfoil curve fitting

522 Applying Optimization Algorithms After a GA-basedcomparison of Bezier and NURBS properties and an investi-gation of their effects on airfoil geometry parameterizationthe results obtained from all the developed optimizationmethods using NURBS representation are compared here interms of static and dynamic convergences final fitness values


0 025 05 075 10

025

05

075

1

X

Y

Original point cloudNURBS curveNURBS polygon

Bezier curveBezier polygon

(a)

0 025 05 075 1

025

05

075

1

X

Y

Polygon of reference BezierReference Bezier curve

New polygon (one changed CP)New Bezier curve

(b)

0 025 05 075 1

025

05

075

1

X

Y

Polygon of reference NURBSReference NURBS curveNew polygon (one changed CP)New NURBS curve

(c)

Figure 5 Optimum curve fitted using Bezier and NURBS methods (a) and global (b) and local (c) modification properties of Bezier andNURBS respectively

Table 7 Comparison of NURBS and Bezier propertiesrsquo effect on some implementation criteria

5 control points 10 control pointsDegree of curve Fitting error [m] Comp burden [s] Degree of curve Fitting error [m] Comp burden [s]

Bezier 4 01376 60 9 00617 203NURBS 3 00518 287 3 00393 475

and computing time This is to see whether these methodscan be successfully used in optimally solving these specificcurve fitting problems To the best knowledge of the authorsthis is the first time that the DE and IWO algorithms havebeen profitably applied to aNURBS curve fitting optimizationproblem

Table 8 shows the used parameters to adjust the algo-rithms in hand for this comparison All optimization pro-cesses are terminated by the predefined number of iter-ations Based on previous experience the parameters ofthe algorithms are selected by trial-and-error Furthermore

regarding PSO and IWOalgorithmsrsquo parameters the problemdimension should be set to 10 because of the specified numberof NURBS CPs to 20 (ie forty coordinates) for all segmentsof the airfoil shape This gives 10 design variables for eachsegment

Taking the metaheuristic nature of the algorithms usedin this part into account RCGA BGA DE PSO and IWOare run several times Because of rationally convergence ofthe algorithms the final fitness values have been too close toeach other for the unique objective function and parametersin several runs However all results presented in this paper


Table 8 Evolutionary and swarm algorithmsrsquo regulations for comparison task

RCGA BGA DE PSO IWONumber of Iterations 250 250 250 250 250Populationswarmplant size 120 120 120 180 10Selection method Roulette wheel Stochastic uniform mdash mdash mdashCrossover Two-point 090 Single-point 095 05 mdash mdashMutation Uniform 035 Uniform 030 mdash mdash mdashScale factor mdash mdash 05 mdashSelf-confidence factor mdash mdash mdash 15 mdashSwarm confidence factor mdash mdash mdash 15 mdashNeighbor accel factor mdash mdash mdash 10 mdashVelocity weight at the beginning mdash mdash mdash 075 mdashVelocity weight at the end mdash mdash mdash 025 mdashNumber of plants mdash mdash mdash mdash 10Min Number of seeds mdash mdash mdash mdash 1Max Number of seeds mdash mdash mdash mdash 10Modulation index mdash mdash mdash mdash 3Standard deviation initial mdash mdash mdash mdash 10Standard deviation final mdash mdash mdash mdash 0001

Table 9 Static convergence of RCGA BGA DE PSO and IWO

Iteration number inwhich the best solution

is achieved

Final objective value(sum of errors inmillimeters)

RCGA 180 502BGA 144 408DE 135 377PSO 181 2032IWO 94 1488

Table 10 Computational efficiency (average of 30 runs)

Real-coded GA Binary GA DE PSO IWOCPU time (Sec) 240 218 216 14 23

are the mean values of those runs Moreover as addressedbefore all results have only been illustrated for one of thesegments (segment 2 according to Figure 1(a) bottom)

Figure 6 illustrates the convergence history for all givenalgorithms As shown in this figure all the algorithmsconverge to their optimal solutions reasonably by iterationsNevertheless the DE could achieve higher fitted solutionsbecause its final fitness value is better than others Table 9represents the static convergence results for the proposedalgorithms Clearly DE outperforms all other algorithmsmdashexcept IWOmdashin terms of the static convergence behaviourbecause it converges faster In other words iteration numberin which the best solution is achieved is lowest for IWO andhighest for PSO Generally evolutionary algorithms notice-ably show better performance inminimizing the fitted errorswhile it is difficult to generally make the same conclusion orits inverse about the static convergence property

0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO

Figure 6 Convergence history of all given algorithms

Additionally a comparison between the dynamic con-vergence characteristics of each of the algorithms employedhas been made in Figure 7 by plotting standard deviation offitness values over all individuals in one generation Dynamicconvergence is indeed an indication of the diversificationand robustness of the algorithms This plot shows that theproposed IWO has the smoothest dynamic convergence dueto its dual behaviour involving both mathematics-drivenand population-based procedures On the other hand GAresults show the highest fluctuations because of the mutationoperation In the case of PSO and DE solutions the standard


0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO

Figure 7 Dynamic convergence characteristics of evolutionary and swarm intelligence algorithms

1006010

100

60

10

Length along the chord ()

Thic

knes

s (

)

Measured point cloudDE-based NURBS representationOptimal control pointsrsquo position DEGA-based NURBS representationOptimal control pointsrsquo position GA

(a)

100

60

10

1006010Length along the chord ()

Thic

knes

s (

)

Measured point cloudIWO-based NURBS representationOptimal control pointsrsquo position IWOPSO-based NURBS representationOptimal control pointsrsquo position PSO

(b)

Figure 8 Airfoil shapes represented with optimal control points derived from evolutionary (a) and swarm intelligence (b) methodologies

deviation variations decrease by iterations without fluctua-tion and decays to zero in the last iterations like IWO

Finally in order to confirm the effectiveness of theplanned methods to optimize the curve fitted to the airfoilshape of GT blades 20 optimized CPs knot vectors andNURBS curves found by both evolutionary and swarmintelligence algorithms have been shown in Figure 8 Thisfigure clearly illustrates a finer NURBS curve fitting on thegiven data points by evolutionary optimization techniquesin comparison to the proposed swarmcolonized intelligenceapproaches The illustration demonstrates that DE has thebest performance in minimizing the fitting errors amongall other algorithms Also it can be seen that the IWO-basedNURBS curve shows better agreementwith the originalmeasured points in comparison to the PSO-based one Nev-ertheless it is worth mentioning that PSO implementation iseasier as the number of parameters is less in comparisonwithIWO In other words compared to IWO one of the mainadvantages of PSO is that this algorithm is easily employedwith few parameters to be adjusted

Last but not least in order to compare the computationalefficiency of all four optimization algorithms used in thisstudy the CPU time is averaged over 30 independent runs

for each method The results are shown in Table 10 It can beunderstood that swarm methods demand considerably lesscomputational effort and have huge saving time comparedto the evolutionary ones This would be crucial specificallywhen the dimensions of the fitting problem andor itscomplexity are greater than those of the problem solved inthis study (in turbomachinery for instance surface fittingin 3D geometry modeling of GT blades) Among thesemethods however the simplicity in the implementation ofPSO results in saving computational efforts compared withIWO The reason is that the functioning of IWO involvessomemathematicalstatistical procedures Nonetheless IWOcan achieve its final fitness value in less iteration than thatof the PSO (Table 9) and so it provides a huge saving incomputational costs

6 Conclusion

This paper presents the results of a study where a binary andreal-coded genetic algorithm a differential evolution a parti-cle swarm and a newly developed invasiveweed optimizationmethods were implemented using Bezier and NURBS tools


to optimally create the well-parameterized airfoil shapes onexisting turbomachinery blades The approach could be usedfor both precise reverse engineering and aerodynamic shapeoptimization purposes [35]

First the touch trigger probes are employed to measurethe point cloud of an existing gas turbine compressor bladesections After that the model construction is continued bysegmentation of the measured points to four areas ThenCAD modeling searches for the best CPs of Bezier andNURBS curves among the candidates by using the least-squares fitting technique in the objective function of theaforementioned optimization algorithms

One important conclusion is that the NURBS curvesbecause of their special properties such as local modificationand numerical stability are certainly more suitable toolsfor such complex curvesurface fitting problem in turbo-machinery than the Bezier curves Regarding the plannedoptimization algorithms the behavior of all the given evo-lutionary and swarm intelligence techniques is inclusivelyevaluated from both statistical and experimental points ofview As a result GA andDEmdashas two of themost well-knownevolutionary algorithmsmdashas well as PSO and IWOmdashas theswarm intelligence methodsmdashgenerally showed their abilityin optimization of such complex curve fitting processesNevertheless several significant behaviour differences aredetected from a comprehensive comparison of their resultsAs a result of nonparametric test it can be concluded that theevolutionary optimization algorithms obviously outperformthe swarm intelligence ones At the same time while BRCGA and DE demonstrate their superior performance interms of reaching minimum fitting error IWO has the beststatic convergence IWO also shows the finest behaviourin dynamic convergence in areas where GA results exhibitthe highest fluctuations and this is because of its mutationoperator Also swarm intelligence optimization methodsare observed to have a significantly better computationalefficiency than evolutionary algorithms

Lastly since all the results obtained from the well-organized proposed optimization strategies demonstrate thatthe final optimum fitted curves have good agreement withthe original measured points selecting one among them tobe employed in a computer-aided geometric design opti-mization process strongly depends on the final goals andpriorities

References

[1] J Grasel A Keskin M Swoboda H Przewozny and S AndreldquoA full parametric model for turbomachinery blade design andoptimisationrdquo in Proceedings of the ASME Design EngineeringTechnical Conferences and Computers and Information in Engi-neering Conference pp 907ndash914 October 2004

[2] W Song and A J Keane ldquoA study of shape parameterisationmethods for airfoil optimizationrdquo American Institute of Aero-nautics Astronautics 2006

[3] P Bezier Emploi des Machines a Commande Numerique Mas-son et Cie Paris France 1970

[4] G E FarinNURBS fromProjectiveGeometry to Practical Use AK Peters Ltd Natick Mass USA 2nd edition 1999

[5] S Pierret and C Hirsch ldquoAn integrated optimization system forturbomachinery blade shape designrdquo in Proceedings of the RTOAVT Symposium on Reduction of Military Vehicle AcquisitionTime and Cost through Advanced Modeling and Virtual Simu-lation France 2002

[6] A Oyama M-S Liou and S Obayashi ldquoTransonic axial flowblade shape optimization using evolutionary algorithm andthree-dimensional Navier-stokes solverrdquo AIAA 2002-56422002

[7] F Sieverding B Ribi M Casey and M Meyer ldquoDesign ofindustrial axial compressor blade sections for optimal range andperformancerdquo Journal of Turbomachinery vol 126 no 2 pp323ndash331 2004

[8] D Buche G Guidati and P Stoll ldquoAutomated design optimiza-tion of compressor blades for stationary large-scale turboma-chineryrdquo in Proceedings of the 2003ASMETurbo Expo pp 1249ndash1257 Atlanta Ga USA June 2003

[9] A Shahrokhi and A Jahangirian ldquoA surrogate assisted evolu-tionary optimization method with application to the transonicairfoil designrdquo Engineering Optimization vol 42 no 6 pp 497ndash515 2010

[10] A Kumar Robust design methodologies application to compres-sor blades [PhD thesis] University of Southampton 2006

[11] D I S Adi S M Shamsuddin and A Ali ldquoParticle swarm opti-mization for NURBS curve fittingrdquo in Proceedings of the 6thInternational Conference on Computer Graphics Imaging andVisualization New Advances and Trends (CGIV rsquo09) pp 259ndash263 August 2009

[12] Z Jing F Shaowei and C Hanguo ldquoOptimized NURBS curveand surface fitting using simulated annealingrdquo in Proceedings ofthe International Symposium on Computational Intelligence andDesign (ISCID rsquo09) pp 324ndash329 December 2009

[13] P Pandunata and S M H Shamsuddin ldquoDifferential Evolutionoptimization for Bezier curve fittingrdquo in Proceedings of the 7thInternational Conference on Computer Graphics Imaging andVisualization (CGIV rsquo10) pp 68ndash72 August 2010

[14] F Yoshimoto T Harada and Y Yoshimoto ldquoData fitting witha spline using a real-coded genetic algorithmrdquo Computer AidedDesign vol 35 no 8 pp 751ndash760 2003

[15] J H HollandAdaptation in Natural and Artificial Systems Uni-versity of Michigan Press Ann Arbor Mich USA 1975

[16] Y Shi and R C Eberhart ldquoEmpirical study of particle swarmoptimizationrdquo in Proceedings of the International Conference onEvolutionary Computation pp 1945ndash1950 1999

[17] M Dorigo Optimization learning and natural algorithms[PhD thesis] Politecnico di Milano Milano Italy 1992

[18] M Montazeri and A Safari ldquoTuning of fuzzy fuel controller foraero-engine thrust regulation and safety considerations usinggenetic algorithmrdquo Aerospace Science and Technology vol 15no 3 pp 183ndash192 2011

[19] K Price and R Storn ldquoDifferential evolutionrdquo Dr DobbrsquosJournal pp 18ndash24 1997

[20] A RMehrabian and C Lucas ldquoA novel numerical optimizationalgorithm inspired from weed colonizationrdquo Ecological Infor-matics vol 1 no 4 pp 355ndash366 2006

[21] P BezierTheMathematical Basis of the UNISURF CAD SystemButterworths London UK 1986

[22] K Lee Principles of CADCAMCAE Systems Addison-WesleyLongman 1999

[23] M G Cox ldquoThe numerical evaluation of B-splinerdquo Journal ofthe Institute of Mathematics and Its Applications vol 15 pp 95ndash108 1972


[24] C de Boor ldquoOn calculating with B-splinesrdquo Journal of Approxi-mation Theory vol 6 pp 50ndash62 1972

[25] W Ma and J-P Kruth ldquoNURBS curve and surface fitting forreverse engineeringrdquo International Journal of Advanced Manu-facturing Technology vol 14 no 12 pp 918ndash927 1998

[26] D Goldberg Genetic Algorithms in Search Optimization andMachine Learning Addison-Wesley 1989

[27] U K Chakraborty Advances in Differential Evolution Springer2008

[28] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquo inProceedings of the IEEE International Conference on NeuralNetworks pp 1942ndash1948 December 1995

[29] R C Eberhart and Y Shi ldquoParticle swarm optimization deve-lopments applications and resourcesrdquo in Proceedings of theIEEE Conference 2001

[30] H T FeiGao ldquoParticle swarm optimization an efficient methodfor tracing periodic orbits and controlling chaosrdquo in Proceedingsof the International Conference on Complex Systems and Appli-cations Watam Press 2006

[31] M Montazeri S Jafari and M R Ilkhani ldquoApplication of par-ticle swarm optimization in gas turbine engine fuel controllergain tuningrdquo Engineering Optimization vol 44 no 2 pp 225ndash240 2012

[32] A RMallahzadehHOraizi andZDavoodi-Rad ldquoApplicationof the invasive weed optimization technique for antenna con-figurationsrdquo Progress in Electromagnetics Research vol 79 pp137ndash150 2008

[33] S Garcıa DMolina M Lozano and F Herrera ldquoA study on theuse of non-parametric tests for analyzing the evolutionaryalgorithmsrsquo behaviour a case study on the CECrsquo2005 SpecialSession on Real Parameter Optimizationrdquo Journal of Heuristicsvol 15 no 6 pp 617ndash644 2009

[34] J Derrac S Garcıa D Molina and F Herrera ldquoA practicaltutorial on the use of nonparametric statistical tests as amethodology for comparing evolutionary and swarm intelli-gence algorithmsrdquo Swarm and Evolutionary Computation vol1 no 1 pp 3ndash18 2011

[35] A Safari G Lemu and M Assadi ldquoA novel combination of ad-aptive tools for turbomachinery airfoil shape optimizationusing a real-coded genetic algorithmrdquo in Proceedings of ASMETurbo Expo San Antonio Tex USA 2013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society



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Stochastic AnalysisInternational Journal of


exist for example PARSEC method [9] and Hicks-Hennefunctions [10] for this application

As mentioned above a proper and precise curve fittingrequires an unavoidable optimization mission The nonlin-ear nature of sophisticated geometries like turbomachinerybladesrsquo shape and availability of high-speed parallel com-puters for massive computation have resulted in the useof non-gradient-based and guided random search method-ologies in curvesurface fitting problems [11ndash14] On theother hand a vast variety of population-based optimizationtechniques have been formulated in recent decades someof which are inspired by natural processes taking place inour environment Genetic algorithms (GAs) [15] particleswarm optimization (PSO) [16] and ant colony optimization(ACO) [17] are some such methods that have already beenused in various engineering optimization problems from thedevelopment of optimum control systems [18] to curve fittingand geometric modeling [11 14]

Among these optimization techniques genetic algorithmand differential evolution (DE) [19] are the most successfullyused strategies in curve fitting optimization of airfoilsrsquo geom-etry parameterized by NURBSBezier curves While GA andreal-coded genetic algorithm (RCGA) are stochastic searchand population-based optimization algorithms that mimicDarwinrsquos theory of biological evolution DE is a relativelynewer evolutionary method in engineering applications andrequires only a few user-defined parameters

PSO and invasive weed optimization (IWO) [20] areother population-based optimization methods inspired bysocial and ecological behavior The PSO algorithm is basedon the simulation of the social behavior of birds flocking fishschooling and animals herding to let them adjust to theirenvironment find rich sources of food and keep away fromother predatory animals by using information sharing andsocial cognitive intelligence IWO is also another derivative-free metaheuristic method originally formulated based onthe concept of the natural social behaviour of colonizingweeds These swarm intelligence approaches have demon-strated their high capability in parametersrsquo optimization fornonlinear multidisciplinary problems

In this paper a study on a gas turbine compressor airfoilshape parameterization and optimum curve fitting as a mainstep to blade geometry optimization has been reported Forthis purpose two most important curve parameterizationmethods that is Bezier and NURBS curves have beenapplied within the five above-mentioned population-basedoptimization loops (GA RCGA DE PSO and IWO) so thatthe best alternative solutions from different points of viewcan be discovered To achieve this goal the parametric proce-dures involving independence normality and homoscedas-ticity are firstly discussedThen the nonparametric statisticalanalysis including Quade and Friedman aligned comparisontests by applying several different post hoc procedures arebrought to perform a rigorous comparison among the perfor-mance of the optimization algorithms After that designingan experiment is done for each algorithm to finally illustratethe effectiveness of the proposed algorithms

In the rest of the article the theory behind the Bezierand nonuniform rational B-spline curves and different airfoil

geometric representation techniques are first described inSection 2 In Section 3 the problem which will be solved isclearly defined This section involves the problem statementand explanation of the formulation fitness function andflowchart considered for the optimization processes Thenthe well-organized optimization techniques for optimumcurve fitting used in this research are explained in Section 4The first subsection focuses on evolutionary optimizationtechniques (GA RCGA and DE) and the second considersswarm intelligence and colonizing techniques (PSO andIWO) Section 5 discusses the adjusted parameters for theoptimization algorithms followed by the statistical analysisexperimental results discussions and comparison Finallyconclusion and future works have been presented in the lastsection of the article

2 Airfoil Geometric ModellingTechniques and Tools

The turbomachinery blade shape parameterization addressesthe two-dimensional airfoil profile construction from eitherthe direct handling of curves of airfoil shapes or the superpo-sition of the camber line and thickness distribution aroundit Between these two different approaches of geometrydefinition however many designers have considered distinctparameterized curves for the suction surface (SS) pressuresurface (PS) leading edge (LE) and trailing edge (TE) of ablade section In this article direct handling of the curves ofthe airfoil shape has been used by dividing the profile fromthe hub to the shroud into five sections as shown in Figure 1This article focuses on only the first section that is the bladehub section (depicted in Figure 1(b))

As the first step a standard coordinate measuringmachine equipped with touch trigger probes has beenemployed to measure 2D point cloud of compressor blade in-hand (Figure 1(a) top) These measured pointsrsquo coordinateshave been used to fit the airfoil sketches using NURBSand Bezier curves To achieve this purpose after indicat-ing the digitized points of the airfoil profile and choosingthe approach for airfoil shape modeling four curves willseparately be fitted to any airfoil shape In this way theaccuracy and robustness of the created geometric model canbe managed by controlling the curve parameters like numberof BezierNURBS control points (CPs) andor curve order

Here the theory behind thesemathematicalmethods andthe characteristics of each will be briefly described

21 Bezier Curves The Bezier curve proposed by Bezierin the early 1960s [21] is defined by the vertices of apolygon which enclose the final fitted curve Bezier curvesuse the Bernstein polynomial function119861

119894119899(119906) as the blending

function [22]

119861119894119899 (119906) = (

119899

119894) 119906119894(1 minus 119906)

119899minus119894 (1)

where 119899 is the degree of the curve 119906 is the parameter of thefunction and 119894 = 0 1 119899




pressure side


(a)

Tip

Hub

(b)



119875 (119906) =

119899

sum

119894=0

(119899

119894) 119906119894(1 minus 119906)

119899minus119894119875119894 (2)





119894119896(119906) expressed as

119873119894119896 (119906) =

(119906 minus 119905119894)119873119894119896minus1 (119906)

119905119894+119896minus1

minus 119905119894

+(119905119894+119896

minus 119906)119873119894+1119896minus1 (119906)

119905119894+119896

minus 119905119894+1

(3)


119894le 119906 le 119905



follows

119875 (119906) =

119899

sum

119894=0

119873119894119896 (119906) 119875119894 (4)



119875 (119906) =sum119899

119894=0119908119894119873119894119896 (119906) 119875119894

sum119899

119894=0119908119894119873119894119896 (119906)

(5)








etc)















119864 =

119899

sum

119894=1

1003816100381610038161003816119862119894 minus 119875 (119906119894)1003816100381610038161003816 (6)














Optimization


optimum curve

BezierNURBS curves

BezierNURBS

N

Y




















Initializing

Initializing

Initializing

Initializing

Mutation


Recombination


Selection

Reproduction

Selection




DE

PSO

GA

IWOand spatial

Evaluation

Evaluation














119909119894


V119894119900=


Δ119905

(7)

where 119909119894






1 and 119888



3 called




V119894119896+1

= 119908V119894119896+1

+ 1198881rand

(119875119894minus 119909119894

119896)

Δ119905

+ 1198882rand

(119875119892

119896minus 119909119894

119896)

Δ119905+ 1198883rand

(119871Best minus 119909119894

119896)

Δ119905

(8)










3= 0) as



119909119894

119896+1= 119909119894

119896+ V119894119896+1

Δ119905 (9)

where 119909119894119896+1













SD119868= (1 minus

119868

119868max)

119899

(SD119894minus SD119891) + SD

119891 (10)



































4 PSO 5522134 0 00125 0012741 0013109 0012741 00457653 IWO 5063582 0 0016667 0016952 0016667 0025321 00457652 RCGA 302449 0002491 0025 0025321 0025 0037739 00457651 BGA 1512245 0130472 005 005 005 005 005



4 PSO 7321863 0 00125 0012741 0013109 0012741 00073253 IWO 5384422 0 0016667 0016952 0016667 0025321 00073252 RCGA 063888 0522901 0025 0025321 0025 0037739 00073251 BGA 0175321 0860828 005 005 005 005 005












0 025 05 075 10

025

05

075

1

X

Y



(a)

0 025 05 075 1

025

05

075

1

X

Y



(b)

0 025 05 075 1

025

05

075

1

X

Y


(c)




Bezier 4 01376 60 9 00617 203NURBS 3 00518 287 3 00393 475










is achieved







0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO




0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












pressure side


(a)

Tip

Hub

(b)



119875 (119906) =

119899

sum

119894=0

(119899

119894) 119906119894(1 minus 119906)

119899minus119894119875119894 (2)





119894119896(119906) expressed as

119873119894119896 (119906) =

(119906 minus 119905119894)119873119894119896minus1 (119906)

119905119894+119896minus1

minus 119905119894

+(119905119894+119896

minus 119906)119873119894+1119896minus1 (119906)

119905119894+119896

minus 119905119894+1

(3)


119894le 119906 le 119905



follows

119875 (119906) =

119899

sum

119894=0

119873119894119896 (119906) 119875119894 (4)



119875 (119906) =sum119899

119894=0119908119894119873119894119896 (119906) 119875119894

sum119899

119894=0119908119894119873119894119896 (119906)

(5)








etc)















119864 =

119899

sum

119894=1

1003816100381610038161003816119862119894 minus 119875 (119906119894)1003816100381610038161003816 (6)














Optimization


optimum curve

BezierNURBS curves

BezierNURBS

N

Y




















Initializing

Initializing

Initializing

Initializing

Mutation


Recombination


Selection

Reproduction

Selection




DE

PSO

GA

IWOand spatial

Evaluation

Evaluation














119909119894


V119894119900=


Δ119905

(7)

where 119909119894






1 and 119888



3 called




V119894119896+1

= 119908V119894119896+1

+ 1198881rand

(119875119894minus 119909119894

119896)

Δ119905

+ 1198882rand

(119875119892

119896minus 119909119894

119896)

Δ119905+ 1198883rand

(119871Best minus 119909119894

119896)

Δ119905

(8)










3= 0) as



119909119894

119896+1= 119909119894

119896+ V119894119896+1

Δ119905 (9)

where 119909119894119896+1













SD119868= (1 minus

119868

119868max)

119899

(SD119894minus SD119891) + SD

119891 (10)



































4 PSO 5522134 0 00125 0012741 0013109 0012741 00457653 IWO 5063582 0 0016667 0016952 0016667 0025321 00457652 RCGA 302449 0002491 0025 0025321 0025 0037739 00457651 BGA 1512245 0130472 005 005 005 005 005



4 PSO 7321863 0 00125 0012741 0013109 0012741 00073253 IWO 5384422 0 0016667 0016952 0016667 0025321 00073252 RCGA 063888 0522901 0025 0025321 0025 0037739 00073251 BGA 0175321 0860828 005 005 005 005 005












0 025 05 075 10

025

05

075

1

X

Y



(a)

0 025 05 075 1

025

05

075

1

X

Y



(b)

0 025 05 075 1

025

05

075

1

X

Y


(c)




Bezier 4 01376 60 9 00617 203NURBS 3 00518 287 3 00393 475










is achieved







0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO




0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















119864 =

119899

sum

119894=1

1003816100381610038161003816119862119894 minus 119875 (119906119894)1003816100381610038161003816 (6)














Optimization


optimum curve

BezierNURBS curves

BezierNURBS

N

Y




















Initializing

Initializing

Initializing

Initializing

Mutation


Recombination


Selection

Reproduction

Selection




DE

PSO

GA

IWOand spatial

Evaluation

Evaluation














119909119894


V119894119900=


Δ119905

(7)

where 119909119894






1 and 119888



3 called




V119894119896+1

= 119908V119894119896+1

+ 1198881rand

(119875119894minus 119909119894

119896)

Δ119905

+ 1198882rand

(119875119892

119896minus 119909119894

119896)

Δ119905+ 1198883rand

(119871Best minus 119909119894

119896)

Δ119905

(8)










3= 0) as



119909119894

119896+1= 119909119894

119896+ V119894119896+1

Δ119905 (9)

where 119909119894119896+1













SD119868= (1 minus

119868

119868max)

119899

(SD119894minus SD119891) + SD

119891 (10)



































4 PSO 5522134 0 00125 0012741 0013109 0012741 00457653 IWO 5063582 0 0016667 0016952 0016667 0025321 00457652 RCGA 302449 0002491 0025 0025321 0025 0037739 00457651 BGA 1512245 0130472 005 005 005 005 005



4 PSO 7321863 0 00125 0012741 0013109 0012741 00073253 IWO 5384422 0 0016667 0016952 0016667 0025321 00073252 RCGA 063888 0522901 0025 0025321 0025 0037739 00073251 BGA 0175321 0860828 005 005 005 005 005












0 025 05 075 10

025

05

075

1

X

Y



(a)

0 025 05 075 1

025

05

075

1

X

Y



(b)

0 025 05 075 1

025

05

075

1

X

Y


(c)




Bezier 4 01376 60 9 00617 203NURBS 3 00518 287 3 00393 475










is achieved







0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO




0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Optimization


optimum curve

BezierNURBS curves

BezierNURBS

N

Y




















Initializing

Initializing

Initializing

Initializing

Mutation


Recombination


Selection

Reproduction

Selection




DE

PSO

GA

IWOand spatial

Evaluation

Evaluation














119909119894


V119894119900=


Δ119905

(7)

where 119909119894






1 and 119888



3 called




V119894119896+1

= 119908V119894119896+1

+ 1198881rand

(119875119894minus 119909119894

119896)

Δ119905

+ 1198882rand

(119875119892

119896minus 119909119894

119896)

Δ119905+ 1198883rand

(119871Best minus 119909119894

119896)

Δ119905

(8)










3= 0) as



119909119894

119896+1= 119909119894

119896+ V119894119896+1

Δ119905 (9)

where 119909119894119896+1













SD119868= (1 minus

119868

119868max)

119899

(SD119894minus SD119891) + SD

119891 (10)



































4 PSO 5522134 0 00125 0012741 0013109 0012741 00457653 IWO 5063582 0 0016667 0016952 0016667 0025321 00457652 RCGA 302449 0002491 0025 0025321 0025 0037739 00457651 BGA 1512245 0130472 005 005 005 005 005



4 PSO 7321863 0 00125 0012741 0013109 0012741 00073253 IWO 5384422 0 0016667 0016952 0016667 0025321 00073252 RCGA 063888 0522901 0025 0025321 0025 0037739 00073251 BGA 0175321 0860828 005 005 005 005 005












0 025 05 075 10

025

05

075

1

X

Y



(a)

0 025 05 075 1

025

05

075

1

X

Y



(b)

0 025 05 075 1

025

05

075

1

X

Y


(c)




Bezier 4 01376 60 9 00617 203NURBS 3 00518 287 3 00393 475










is achieved







0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO




0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Initializing

Initializing

Initializing

Initializing

Mutation


Recombination


Selection

Reproduction

Selection




DE

PSO

GA

IWOand spatial

Evaluation

Evaluation














119909119894


V119894119900=


Δ119905

(7)

where 119909119894






1 and 119888



3 called




V119894119896+1

= 119908V119894119896+1

+ 1198881rand

(119875119894minus 119909119894

119896)

Δ119905

+ 1198882rand

(119875119892

119896minus 119909119894

119896)

Δ119905+ 1198883rand

(119871Best minus 119909119894

119896)

Δ119905

(8)










3= 0) as



119909119894

119896+1= 119909119894

119896+ V119894119896+1

Δ119905 (9)

where 119909119894119896+1













SD119868= (1 minus

119868

119868max)

119899

(SD119894minus SD119891) + SD

119891 (10)



































4 PSO 5522134 0 00125 0012741 0013109 0012741 00457653 IWO 5063582 0 0016667 0016952 0016667 0025321 00457652 RCGA 302449 0002491 0025 0025321 0025 0037739 00457651 BGA 1512245 0130472 005 005 005 005 005



4 PSO 7321863 0 00125 0012741 0013109 0012741 00073253 IWO 5384422 0 0016667 0016952 0016667 0025321 00073252 RCGA 063888 0522901 0025 0025321 0025 0037739 00073251 BGA 0175321 0860828 005 005 005 005 005












0 025 05 075 10

025

05

075

1

X

Y



(a)

0 025 05 075 1

025

05

075

1

X

Y



(b)

0 025 05 075 1

025

05

075

1

X

Y


(c)




Bezier 4 01376 60 9 00617 203NURBS 3 00518 287 3 00393 475










is achieved







0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO




0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











V119894119896+1

= 119908V119894119896+1

+ 1198881rand

(119875119894minus 119909119894

119896)

Δ119905

+ 1198882rand

(119875119892

119896minus 119909119894

119896)

Δ119905+ 1198883rand

(119871Best minus 119909119894

119896)

Δ119905

(8)










3= 0) as



119909119894

119896+1= 119909119894

119896+ V119894119896+1

Δ119905 (9)

where 119909119894119896+1













SD119868= (1 minus

119868

119868max)

119899

(SD119894minus SD119891) + SD

119891 (10)



































4 PSO 5522134 0 00125 0012741 0013109 0012741 00457653 IWO 5063582 0 0016667 0016952 0016667 0025321 00457652 RCGA 302449 0002491 0025 0025321 0025 0037739 00457651 BGA 1512245 0130472 005 005 005 005 005



4 PSO 7321863 0 00125 0012741 0013109 0012741 00073253 IWO 5384422 0 0016667 0016952 0016667 0025321 00073252 RCGA 063888 0522901 0025 0025321 0025 0037739 00073251 BGA 0175321 0860828 005 005 005 005 005












0 025 05 075 10

025

05

075

1

X

Y



(a)

0 025 05 075 1

025

05

075

1

X

Y



(b)

0 025 05 075 1

025

05

075

1

X

Y


(c)




Bezier 4 01376 60 9 00617 203NURBS 3 00518 287 3 00393 475










is achieved







0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO




0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




































4 PSO 5522134 0 00125 0012741 0013109 0012741 00457653 IWO 5063582 0 0016667 0016952 0016667 0025321 00457652 RCGA 302449 0002491 0025 0025321 0025 0037739 00457651 BGA 1512245 0130472 005 005 005 005 005



4 PSO 7321863 0 00125 0012741 0013109 0012741 00073253 IWO 5384422 0 0016667 0016952 0016667 0025321 00073252 RCGA 063888 0522901 0025 0025321 0025 0037739 00073251 BGA 0175321 0860828 005 005 005 005 005












0 025 05 075 10

025

05

075

1

X

Y



(a)

0 025 05 075 1

025

05

075

1

X

Y



(b)

0 025 05 075 1

025

05

075

1

X

Y


(c)




Bezier 4 01376 60 9 00617 203NURBS 3 00518 287 3 00393 475










is achieved







0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO




0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 025 05 075 10

025

05

075

1

X

Y



(a)

0 025 05 075 1

025

05

075

1

X

Y



(b)

0 025 05 075 1

025

05

075

1

X

Y


(c)




Bezier 4 01376 60 9 00617 203NURBS 3 00518 287 3 00393 475










is achieved







0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO




0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














is achieved







0 50 100 150 200 2500

005

01

015

02

025

03

035

04

045

05

Iteration

Fitn

ess

DERCGABGA

IWOPSO




0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0 50 100 150 200 2500

010203040506070809

1

IterationsSt

anda

rd d

evia

tion

GADE

IWOPSO


1006010

100

60

10


Thic

knes

s (

)


(a)

100

60

10


Thic

knes

s (

)


(b)






6 Conclusion







References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a comparative analysis of nature-inspired...

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