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http://dx.doi.org/10.1007/s00158-013-1025-3

http://dx.doi.org/10.1007/s00158-013-1025-3

AUTHOR'S PROOF

Metadata of the article that will be visualized in OnlineFirst

1 Article Title Airfoil shape optimization using improv ed Multiobjectiv eTerritorial Particle Swarm algorithm with the objectiv e ofimprov ing stall characteristics

2 Article Sub-Title

3 Article Copyright -Year

Springer-Verlag Berlin Heidelberg 2013(This will be the copyright line in the final PDF)

4 Journal Name Structural and Multidisciplinary Optimization

5

CorrespondingAuthor

Family Name Nejat6 Particle

7 Given Name Amir8 Suffix

9 Organization University of Tehran

10 Division School of Mechanical Engineering

11 Address Tehran, Iran

12 e-mail [email protected]

13

Author

Family Name Mirzabeygi14 Particle

15 Given Name Pooya16 Suffix





21

Author

Family Name Panahi22 Particle

23 Given Name Masoud Shariat24 Suffix





29

Schedule

Received 20 August 2012

30 Revised 13 November 2013

31 Accepted 16 November 2013

_____________________________________________________________________________________

Please note: Images will appear in color online but will be printed in black and white._____________________________________________________________________________________

AUTHOR'S PROOF

32 Abstract In this paper, a new robust optimization technique with the abil ityof solving multi-objective constrained design optimizationproblems in aerodynamics is presented. This new technique isMulti-objective Territorial Particle Swarm Optimization (MOTPSO)algorithm in which diversity is actively preserved by avoidingovercrowded clusters of particles and encouraging broaderexploration. Adaptively varying territories are formed aroundpromising individuals to prevent many of the lesser individuals frompremature clustering and encouraged them to explore newneighborhoods based on a hybrid self-social metric. Also, a newsocial interaction scheme is introduced which guided particlestowards the weighted average of their elite neighbors best foundpositions instead of their own personal bests which in turn helps theparticles to exploit the candidate local optima more effectively.The MOTPSO algorithm takes into account multiple objectivefunctions using a Pareto-Based approach. The non-dominatedsolutions found by swarm are stored in an external archive andnearest neighbor density estimator method is used to select aleader for the individual particles in the swarm. Efficiency androbustness of the proposed algorithm is demonstrated usingmultiple traditional and newly-composed optimization benchmarkfunctions and aerodynamic design problems. In final airfoil designsobtained from the Multi Objective Territorial Particle SwarmOptimization algorithm, separation point is delayed to make theairfoil less susceptible to stall in critical operating conditions and italso reveal an evident improvement over the test case airfoil acrossall objective functions presented.

33 Keywordsseparated by ' - '

Multi-Objective Territorial Particle Swarm Optimization -Aerodynamic shape optimization - Class-shape-transformationmethod - Stall prevention

34 Foot noteinformation

Part of this work has been presented at the ASME 2012International Mechanical Engineering Congress and Exposition

AUTHOR'S PROOF JrnlID 158 ArtID 1025 Proof#1 - 26/11/2013

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Struct Multidisc OptimDOI 10.1007/s00158-013-1025-3

RESEARCH PAPER1

Airfoil shape optimization using improved MultiobjectiveTerritorial Particle Swarm algorithm with the objectiveof improving stall characteristics

2

3

4

Amir Nejat Pooya Mirzabeygi Masoud Shariat Panahi

5

6

Received: 20 August 2012 / Revised: 13 November 2013 / Accepted: 16 November 20137 Springer-Verlag Berlin Heidelberg 20138

Abstract In this paper, a new robust optimization tech-9nique with the ability of solving multi-objective constrained10design optimization problems in aerodynamics is presented.11This new technique is Multi-objective Territorial Particle12Swarm Optimization (MOTPSO) algorithm in which diver-13sity is actively preserved by avoiding overcrowded clusters14of particles and encouraging broader exploration. Adap-15tively varying territories are formed around promising16individuals to prevent many of the lesser individuals from17premature clustering and encouraged them to explore new18neighborhoods based on a hybrid self-social metric. Also, a19new social interaction scheme is introduced which guided20particles towards the weighted average of their elite21neighbors best found positions instead of their own per-22sonal bests which in turn helps the particles to exploit23the candidate local optima more effectively. The MOTPSO24algorithm takes into account multiple objective functions25using a Pareto-Based approach. The non-dominated solu-26tions found by swarm are stored in an external archive and27nearest neighbor density estimator method is used to select28a leader for the individual particles in the swarm. Efficiency29and robustness of the proposed algorithm is demonstrated30

Part of this work has been presented at the ASME 2012 Interna-tional Mechanical Engineering Congress and Exposition

A. Nejat () P. Mirzabeygi M. S. PanahiSchool of Mechanical Engineering,University of Tehran, Tehran, Irane-mail: [email protected]

P. Mirzabeygie-mail: [email protected]

M. S. Panahie-mail: [email protected]

using multiple traditional and newly-composed optimiza- 31tion benchmark functions and aerodynamic design prob- 32lems. In final airfoil designs obtained from the Multi Objec- 33tive Territorial Particle Swarm Optimization algorithm, sep- 34aration point is delayed to make the airfoil less susceptible 35to stall in critical operating conditions and it also reveal 36an evident improvement over the test case airfoil across all 37objective functions presented. 38

Keywords Multi-Objective Territorial Particle Swarm 39Optimization Aerodynamic shape optimization 40Class-shape-transformation method Stall prevention 41

1 Introduction 42

Rising fuel prices and climate change present the main 43challenges for the aviation industry in the 21st century. 44These considerations will increase the pressure on the air- 45craft industry to design more fuel-efficient aircraft. Over 46the past decades, gains in fuel efficiency have mainly come 47from improvements in engine technology, reduced weight 48due to the use of composite materials, and reduction of 49the overall drag through more aerodynamically efficient air- 50frame designs (Lee and Mo 2011; Lee et al. 2009, 2010). 51Among the present solutions for improving the fuel effi- 52ciency aerodynamic shape optimization can play a key role 53in the development and evaluation of future aircraft con- 54cepts that can address both environmental and economic 55concerns (Buckley et al. 2010; Kuntawala 2011). Nowadays 56greenhouse gas emissions associated with commercial avi- 57ation is a major problem due to climate change concerns; 58burning less fuel as well as cutting the flight operating cost, 59reduces the environmental impact of the burnt fuel emission 60in the atmosphere. 61

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For the aerodynamic shape optimization, the validated62CFD solvers are typically combined with numerical opti-63mization methods, in particular gradient- and non-gradient64based methods. Perhaps the most popular approach for the65computation of the objective function gradient is the adjoint66method because its cost is virtually independent of the67number of design variables (Anderson and Venkatakrishnan681999; Jameson et al. 1998). Non-gradient based numerical69optimization methods, such as genetic algorithms, are gen-70erally not as efficient as gradient-based methods. However,71using gradient-based methods in aeronautical/aerospace72design problems causes several challenges which makes the73non-gradient based methods a suitable alternative. These74challenges as described in literature are as follows Arias-75Montano et al. (2011), Anderson (2003), Obayashi and76Tsukahara (1997), and Kroo (1997):77

1. The design space is frequently multimodal and highly78non-linear.79

2. Gradient-based methods require the knowledge of80parameters derivatives which are not easy to determine81especially in multiple dimensions and these derivatives82have to be recalculated as search continues.83

3. Gradient-based methods require a design space free of84discontinuity as the derivatives are not defined in these85regions.86

4. Gradient-based algorithms start their search from a87specific set of parameters in the design space which88is likely makes the final results biased to the start89point.90

5. Most of aerodynamic applications of gradient-based91methods reported in literature just deal with single92compromised solution for multiple objectives, while93an Evolutionary algorithm or Swarm intelligence tech-94nique like PSO give you multiple trade-off solutions95just by a single run.96

6. The complexity of the sensitivity analyses in Multidis-97ciplinary Design Optimization (MDO) increases as the98number of disciplines involved becomes larger.99

7. In MDO, trade-off solutions, or a set of them, are100searched for.101

Based on the previously indicated difficulties, designers102have been motivated to use alternative optimization tech-103niques such as Evolutionary and Swarm Intelligence Algo-104rithms. Multi-Objective Evolutionary Algorithms (MOEAs)105have gained an increasing popularity as numerical optimiza-106tion tools in aeronautical and aerospace engineering during107the last few years (Wickramasinghe et al. 2010).108

Particle Swarm Optimization (PSO) is an Swarm Intelli-109gence technique developed by Kennedy and Eberhart (1995)110that is inspired by social behavior among birds, fishes and111even humans. This algorithm benefits from several advan-112tages such as easy implementation, relatively small popu-113

lation size and low computational cost compared to other 114EAs or swarm intelligence techniques. These advantages 115have made it a rival for other EAs such as Genetic Algo- 116rithm (GA) since it was created (Eberhart and Shi 1998). It 117has been used in wide range of applications such as control 118applications, aerospace, electronics and electromagnetics, 119antenna design, power generation and power systems (Poli 1202008). 121

Although PSO is considered to be a robust algorithm 122in many applications, it fails to find a balance between 123exploration and exploitation. In dealing with multimodal 124functions, PSO cannot explore the whole region effectively 125and often suffers from the problem of premature conver- 126gence or loss of diversity (Poli et al. 2007; Kennedy 2006; 127Kennedy and Mendes 2002). Many algorithms have been 128proposed in the literature to solve this problem. These algo- 129rithms can be classified into two different categories: First, 130the algorithms that address the cause, and second, the ones 131that address effect of the problem. The researchers whose 132algorithms belong to the first category believe that the orig- 133inal PSO algorithm propagates the information among the 134swarm rapidly and ineffectively. Therefore, their proposed 135algorithms in the first category changes the way information 136propagates among the particles in the swarm by introducing 137new social interaction schemes (Mendes 2004; Kennedy and 138Mendes 2002; Mohais et al. 2005; Zhao et al. 2011; Liang 139and Suganthan 2005). On the contrary, the algorithms in the 140second category focus on diversifying the swarm whenever 141particles got stuck in local optima. They argue that no matter 142what the cause of the problem is, it can be cured by inject- 143ing diversity into the swarm whenever necessary (Xie et al. 1442002; Riget and Vesterstrm 2002; Xu and Ai 2009; Zhao 145and Suganthan 2009). 146

PSO was first extended to solve multi-objective problems 147by Moore and Chapman (1999), and since then numerous 148algorithms have been proposed by several authors. Con- 149verting a single-objective PSO to a multiobjective PSO 150(MOPSO) requires redefinition of global and personal best 151memories since the conventional definition of absolute 152global best is not useful when the goal is to find a set of 153optimal solutions. Coello Coello and Lechuga (2002) pro- 154posed a MOPSO in which the objective space is split into 155several hypercubes. A fitness value is assigned for each 156of these hypercubes. To determine the guide for the parti- 157cles a hypercube is selected by the roulette-wheel based on 158the fitness, and then a particle in the hypercube is selected 159randomly. Fieldsend et al. (2002) introduced an algorithm 160that uses a dominated tree for the selection of the leader 161for each particle in the swarm. They also use turbulence 162operator to enhance the diversity. Mostaghim and Teich 163(2003) presented -dominance on MOPSO; the purpose of 164-dominance is to limit the number of nondominated solu- 165tions stored in the archive in order to lower computational 166


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Airfoil shape optimization using improved Multiobjective

time of the algorithm and enhance convergence, and diver-167sity of the solution set. Zhang and Huang (2004) presented168a novel MOPSO with bidirectional searching strategy to169improve the diversity of the nondominated solutions. More-170over, different approaches have been introduced to select171the personal best solution based on memorizing all the172nondominated solutions visited by a particle (Branke and173Mostaghim 2006).174

In order to deal with this problem, an improved Multiob-

Q1

175

jective PSO algorithm with a territorial diversity-preserving176scheme, named MOTPSO is presented which includes two177different approaches in tackling this problem. In the first178approach, a spatial territory with some radius is assigned179to each particle to provide the algorithm with the capabil-180ity of controlling and preserving the diversity throughout181the search process. This modification, indeed, enhances the182exploratory behavior in PSO. In the second approach, a183new direction is introduced which moves particles towards184a new point that is a weighted average of its elite neigh-185bors best positions, instead of its own best experienced186position. This particular type of interaction precludes the187quick spread of information among particles and moves188the particle toward a candidate region in its neighborhood,189and encourages exploitation of nearest found so far local190optimum. Thus, it enhances the exploitative behavior of191PSO. The single-objective algorithm named Territorial Par-192ticle Swarm Optimization (TPSO) was presented by authors193Ostadmohammadi Arani et al. (2013) and its performance194was tested on several benchmark functions in comparison195to the state of art single-objective optimization algorithms196available in the literature. The focus of this paper is on197introducing this novel optimization technique in the multi-198objective optimization field, and tests its capabilities on a199airfoil shape optimization problem.200

The novel Class-Shape-Transformation (CST) method201(Kulfan 2008) is used to create a parametric description202of 2D Airfoils. This method combines an analytical class203function with a parametric shape function. The class204function describes a basic class of shapes and the shape205function describes the permutation around this basic shape.206In this way, characteristic features that are specific to a207certain class of shapes, such as the round nose and sharp208trailing edge of an airfoil, can be captured in the class209function. The location of the CST control points are then210translated to an input for the flow solver program XFOIL,211which outputs a number of relevant design characteristic212values. These are then fed into an optimization algorithm213which generates a new airfoil shape and the process is214repeated.215

In the next section our proposed Multi-Objective Territo-216rial Particle Swarm Optimization (MOTPSO) is introduced,217the constraint-handling method that is used in the study is218

presented. In Section 3, the airfoil parameterization tech- 219nique, computational flow solver and objective functions 220that are used in aerodynamic optimization are explained. In 221Section 4, the ability of the algorithm in solving Multiobjec- 222tive problems is tested on several benchmark functions and 223the results are compared to other Multiobjective algorithms. 224Moreover, aerodynamic shape optimization results are pre- 225sented and discussed. Finally we summarize our research 226in Section 5. 227

2 Optimization algorithm 228

2.1 Particle swarm optimization 229

In the original PSO algorithm (Kennedy and Eberhart 1995), 230a swarm of particles representing potential solutions move 231in the n-dimensional space of design variables. The desir- 232ability or fitness of a particle is a function of its position 233vector X which, at each search step, gets updated by the par- 234ticles velocity vector V. Each particle remembers its own 235best position P and the swarms best position G found so far. 236

In each time step t the velocity is updated and the particle 237is moved to a new position. The new position is determined 238by the sum of the previous position and the new velocity 239according to 240

X(t + 1) = X(t) + V(t + 1) (1)The update of the velocity from the previous velocity to the 241new one is determined using 242

V(t + 1) = wV(t)+1(P(t) X(t))+2(G(t) X(t))(2)

Where 1 and 2 are real numbers chosen uniformly and 243randomly in some interval, usually [0, 2]. The inertia weight 244w weighs the magnitude of the old velocity V(t) in the cal- 245culation of the new velocity V(t + 1). In the same way, 1 246and 2 determine the significance of P and G in the calcula- 247tion of V(t + 1). The choice of 1 and 2 is very important 248as they are the parameters that control and restricts the 249velocity of the particles and stability of the algorithm. 250Clerc and Kennedy (2002) proposed a PSO algorithm with 251a constriction coefficient which as they claimed restricts 252the maximum velocity of the particles. The constriction 253coefficient is given in (3). 254

= 22 2 4 (3)

Where is obtained as follows: 255

={

1 + 2 if 1 + 2 > 41 if 1 + 2 4

}(4)

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The velocity calculated from (2) is them multiplied by256constriction factor :257

V(t + 1) = (w V(t) + 1 (P(t) X(t))+2 (G(t) X(t))) (5)

2.2 Multi-objective problem concepts258

In general, the multi-objective optimization problem can be259mathematically defined as:260

minimize f(x) = [f1(x), f2(x), ..., fk(x)]Real-world optimization applications may subject to several261constraints which can be expressed as follows:262

gi(x) 0 i = 1, 2, ..., mIn which, x = [x1, x2, ..., xn] is the vector of design vari-263ables, fi : Rn Rki = 1, ..., k are the objective functions264and gi : Rn Rki = 1, ..., m are the constraint functions265of the problem.266

The objective functions in practical applications are usu-267ally conflicting; that is, the improvement in one objective268may lead to the deterioration of others. Therefore, a single269solution that can optimize all objective functions is usu-270ally meaningless, and instead, a set of trade-off solutions,271called the Pareto optimal solutions, are important (Zhou272et al. 2011).273

Some basic concepts and definitions that used in multi-274objective optimization are introduced here:275

Dominated solution: Given two design variable x, y 276Rn, x dominates y (denoted by x y) only and only if277f (xi) f (yi), i = 1, ..., k and f (x) = f (y).278

Nondominated solution: A vector of design variables279x Rn is nondominated, if there is not another y Rn280such that y x.281

Pareto-optimal: A vector of decision variables x Rn282is Pareto-optimal if it is nondominated with respect to283feasible region.284

Pareto front: The image of the Pareto-optimal solutions285in the objective space is called the Pareto front.286

2.3 Feasibility and constraint handling method287

It is important to a find a set of solutions that are close to288the optimal Pareto-front and also represents a satisfactory289diversity. However, the multi-objective problem is subject to290several constraints, and so it is of equal importance to make291sure that the final set of solutions satisfies the constraints292imposed on the problem. Therefore, the feasibility of a solu-293tion should be precisely defined. In general, if a solution294satisfies all the constraints, then it is considered to be a fea-295sible solution; otherwise, it is infeasible. The feasibility of296

a solution can be assessed mathematically by its total value 297of constraint violation using (6) 298

(x) =m

i=1max(0, gi(x)) (6)

Please note that because reaching absolute equality is dif- 299ficult, the equality constraints are treated as inequality 300constraints with the feasibility criterion . Therefore, math- 301ematically, the solution is considered to be feasible if 302(x) . 303

In multi-objective problems, a concept named constraint 304dominance is introduced which has few simple rules, and 305will replace the conventional dominance concept in the 306unconstrained multi-objective problem. These rules are 307introduced here as described in Li et al. (2009). Solution x 308is said to constraint-dominate solution y if: 309

1. Solution xis feasible and solution y is not. 3102. Solution x and solution y are both infeasible, but solu- 311

tion x has a smaller constraint violation. 3123. Solution x and solution y are feasible and solution x 313

dominates solution y in the way defined earlier. 314

2.4 Multi-Objective Territorial Particle Swarm 315Optimization (MOTPSO) 316

The solution set of a problem with multiple objectives does 317not consist of a single solution (as in global optimization). 318Instead, in multi-objective optimization, we aim to find a 319set of different solutions (the so-called Pareto optimal set), 320so the original scheme has to be modified. Unlike solv- 321ing single-objective optimization problems, the leader that 322each particle uses to update its position is completely deter- 323mined and is unique for the swarm. However, in the case 324of multi-objective optimization problems, there is a set of 325non-dominated solutions and each particle should select its 326leader from this set in order to update its position (Reyes- 327Sierra and Coello 2006). Therefore, the first modification 328that has to be made to the single-objective algorithm is to 329define an archive to store the non-dominated solutions found 330so far by the particles. This storage place which is sepa- 331rated from the original swarm is called external archive. 332The solutions in this archive are selected to be the leaders 333for individual particles in the swarm. 334

Now that non-dominated solutions are stored in an 335archive, several new issues arise: 336

1. What Mechanism should be used to select a leader 337among the external archive solutions for each individual 338particle in the swarm? 339

2. What criterion should be used to determine which non- 340dominated solutions remain in the external archive? 341

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First of all, the selection of a leader for each particle is342one of the most important factors in developing a MOPSO343algorithm; a quality criterion should be defined in order344to distinguish among the non-dominated solutions in the345archive. In this paper, the nearest neighbor density estima-346tor (Deb et al. 2002) is used as the quality measure, and it347gives us an idea of how crowded is the closest neighbors of348a given particle, in objective function space. This measure349estimates the perimeter of the cuboids formed by using the350nearest neighbors as the vertices. Then the particles in the351less crowded regions are more probable to be selected as352leaders, see Fig. 1.353

It is now clear how the global best memory or leaders,354Gi , are selected for each particle. The next concept that355needs to be modified is the personal best memory Pi ; a par-356ticle replaces its current personal best if the personal best is357being dominated by the particle or they are non-dominated358with respect to each other. The global best, personal best and359external archive concepts for minimizing two objectives are360demonstrated in Fig. 2.361

The selection of personal and global leaders in a multi-362objective problem was explained. The next step is to explain363the main mechanism that MOTPSO uses to update the364position and velocity of particles. One of the most com-365mon problems encountered by many PSO variants is the366problem of the premature clustering in the early stages of367the search process (Poli et al. 2007; Mendes 2004; Riget368and Vesterstrm 2002; Peram et al. 2003; Ben Ghalia3692008). This weakens the algorithms ability to explore the370whole domain effectively. Especially in multimodal func-371tions, the existence of many local optima makes it quite372difficult for most techniques to detect the global optimum.373As mentioned earlier, MOTPSO tackles this problem in374two different ways. The first approach deals with the effect375

Fig. 1 Nearest neighbor density estimator quality criterion for leaderselection from the archive

Fig. 2 Global best, personal best and external archive concepts fortwo minimum objectives

of the problem by controlling diversity during the pro- 376cess of searching. The second addresses the cause of the 377problem by introducing a new social interaction among par- 378ticles instead of previous self-cognitive ability of the swarm. 379These two approaches are shown to enhance the exploratory 380and exploitative ability of the original algorithm and keep 381the balance between these two. 382

In order to control the diversity in the search process, 383MOTPSO specified a territory with radius R(t) around each 384particle. If territory of one particle collides with another par- 385ticles territory, an operator, which is named collision, acts 386upon both of them according to (7) and the weaker particle 387(the particle that is being dominated) will be sent back to the 388distance 2 R(t). 389

Collisionj

i ={ (

2R(t) Xi Xj) XiXjXiXj Fj dominates Fi

0 else

(7)

Where, Fi is the target particles witness and Fj is the fit- 390ness of other particle involved in the collision. This collision 391operator ensures that weaker particles keep a predefined 392distance with the stronger ones so that the premature clus- 393tering of particles can be avoided and the weaker ones can 394search other regions of the search for candidate solutions. 395As it will be discussed later, the collision operator is simply 396added to social and personal interaction terms in the velocity 397update equation; graphical representation of this operator is 398depicted in Fig. 3. 399

It is essential for the algorithm to have higher diversity 400in the early stages of solution to ensure thorough search 401

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Fig. 3 Graphical representation of the collision operator; particles 1and 2 are dominated and sent out of the territory, particle 3 is notdominated

of the domain and to avoid premature convergence; how-402ever, the tendency for convergence increases as the solution403reaches its final stages, so territory dimensions is considered404to be large at the start and decrease to zero as the itera-405tion increases. Many approaches can be used to decrease406the territory radius, for example exponential, logarithmi-407cal and linear reduction. However, in this paper the linear408approach is used since it provides a uniform distribution of409diversity throughout the search process. Thus, the next ques-410tion is how the maximum radius for particles territories are411chosen.412

If the domain was split evenly between all the particles413in the swarm, each particles territory, which is assumed to414be sphere shaped, would have had the same radius Rmax,415see (8):416

Rmax =N

(N

i=1(XiU XiL

))/K

2(8)

In the above equation, N is the number of variables or417dimensions, K is the swarm size and XiL,X

iU are upper418

and lower bounds of ith variable or dimension, respectively.419However, assigning a large territory radius has a major420drawback. After the swarm is initialized randomly in the421domain, territory bounds are very likely to coincide with422each other and collision happens frequently. The large num-423ber of collisions at the beginning prevents the particles from424their normal search around the domain. Therefore, a coeffi-425

cient, , is defined to reduce the maximum territory radius 426according to following equation ( is usually set to be 0.5) 427

Rmax = Rmax (9)With the simultaneous use of collision operator and radius 428reduction scheme the algorithm will be equipped with a 429diversity preserving ability that not only solves the problem 430of premature diversity loss, but also enables the control of 431the diversity in different time steps of the search process. 432

In the second approach, MOTPSO moves particles to a 433new direction which is a weighted average of its elite neigh- 434bors best found position so far, instead of attracting each 435particle towards its own best position. We assign weight to 436the particles by using the inverse square of distance between 437them. Note that, for a given particle i only elite neighbors 438contribute to determine direction. In other words, only the 439particles that their best found position so far, Pj , dominates 440the current particles position, Fi , are selected to contribute 441in its direction. This argument is formulized as follows: 442

Pi =

j (Pj dominate Fi)Pj 1XiPj2

j(Pj dominate Fi)

1

XiPj2(10)

The above position P*, which is a replacement for P in the 443basic algorithm, represents a local optimum candidate near 444the particle (Ostadmohammadi Arani et al. 2013). Mov- 445ing particles toward P* enables them to search a candidate 446region in their neighborhood rather than moving necessar- 447ily toward their own and global best positions as suggested 448by original PSO. Therefore, particles will exploit the local 449optima present in their neighborhood. This enables the algo- 450rithm with local search ability and it also enhances the 451exploitation of found so far good solutions in the domain. 452

Based on the above modifications, the velocity update 453equation, (5), can be expressed as: 454

Vi (t + 1) = (W Vi (t)+ 1

(Pi (t) Xi (t)

)

+ 2 (Gi(t) Xi (t)))

+ 3

j

Collisionj

i (11)

This equation replaces the original velocity update equation, 455(2) or (5), in PSO algorithm, and (1) is used to update par- 456ticles position. Although the two velocity update equations 457seem to differ only slightly, they would make the algorithm 458perform quite differently, as the sample applications of the 459next section will demonstrate. 460

To further enhance the density the diversity of the swarm 461during the search process, a polynomial mutation opera- 462tor was chosen (Deb 2001) and it was applied for 15 % of 463the swarm particles. The mutation operator has been used 464frequently in different PSO variations (Nebro et al. 2009; 465


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Sierra and Coello 2005; Si et al. 2011) as it is described in466(12):467

Xj (t + 1) = Xj (t)+(

XjU XjL) j (12)

Where XjU and Xj

L are upper and lower bounds of the design468variables, and j is calculated from a polynomial probability469distribution, and is defined as follows:470

j ={(2rj )1/(m+1) 1 rj < 0.51 2(1 rj )1/(m+1) rj 0.5

}

(13)

The parameter rj is a random number between 0 and 1, and471m is the polynomial distribution index which helps to tune472the degree of perturbation.473

The pseudo-code below shows the way in which the474proposed MOTPSO algorithm works. First, the swarm is ini-475tialized. Then, the external archive is also initialized with476the nondominated particles from the swarm. Later, nearest477neighbor density estimator quality is calculated for all the478leaders in order to select a leader Gi for each particle of the479swarm. Next, the P* is calculated for each particle accord-480ing to (10). Knowing these two values velocity is updated481and the flight is performed for each particle. The polynomial482mutation operator is performed after the flight. Then, the483particles fitness is evaluated and its corresponding P and484the set of leaders are updated, too. Finally, the quality mea-485sure of the leaders is recalculated. This process is repeated486for a certain number of iterations.487

Pseudocode of our proposed MOTPSO

1) Initialization (Swarm and External Archive)2) Initialize P* and G for each particle3) Iteration = 04) while Iteration < MaxIteraion5) Velocity Update ((7) to (11))6) Position Update (1)7) Polynomial Mutation8) Fitness Evaluation9) Update External Archive10) Assign P* and G for each particle

from External Archive11) Iteration ++12) end while13) return External Archive

488

3 Aerodynamic shape optimization489

3.1 Airfoil parameterization490

Round nose airfoils have infinite first and second deriva-491tive at the leading edge, and therefore they are impossible492

to represent analytically. As a result, unless an appropriate 493mathematical representation is used, a large number of coor- 494dinates are required to provide a description of the airfoil. 495Using large number of coordinates has the disadvantage of 496high computational cost and difficulty in maintaining a real- 497istic geometry, and that is why using a reliable mathematical 498representation for airfoil is fundamental in aerodynamic 499optimization. A reliable mathematical representation should 500have following characteristics as described in Kulfan and 501Bussoletti (2006): 502

1. Low computational resource requirements 5032. Geometries in the design space should be smooth, 504

regular and physically acceptable. 5053. Meaningful optimum should exist in the design space. 506

Various algorithms have been proposed in the literature 507to represent airfoil geometry including polynomial repre- 508sentation, Bezier or B-Spline control point representation, 509numerically derived orthogonal basis functions, Free form 510airfoil representation, etc. (Sobieczky 1999; Hicks and 511Henne 1978; Song and Keane 2004; Samareh 2001). The 512problem with these methods is that first of all, the design 513variables convey little insight about the geometric shape. 514Moreover, they cannot adequately represent the nose and 515trailing edge of the airfoil, and thus assume the fixed nose 516and trailing edge geometry. Finally, the geometries created 517by these algorithms are not necessarily realistic. 518

Before introducing the novel parameterization technique 519developed by Kulfan (2008), it is helpful to look at the key 520design parameters in airfoil as it is depicted in Fig. 4. The 521leading edge radius, RLE affects the angle of attack (AOA) 522that the separation occurs on an airfoils nose or surface. 523The forebody shape which is characterized by maximum 524thickness and its location determines airfoils drag charac- 525teristics, and finally, closure angle, , determine whether 526the flow separates or remain attached to the aft-body of the 527airfoil (Kulfan and Bussoletti 2006). 528

A new parameterization technique named the Class- 529Shape-Transformation (CST) method was presented by 530Kulfan (2008). In two dimensions, the method combines 531two special functions, called class functions and shape 532functions, the class function, representing a specific class 533of shapes, and a shape function, which defined the deviation 534from the class function. The shape function can be defined 535in different ways but has to guarantee an analytically smooth 536geometry. In this way round-nose/sharp aft-end geometries 537can be represented exactly by simple mathematical func- 538tions. According to Kulfan (2008), the CST method has the 539following advantages: 540

1. Well behaved and produces smooth and realistic shapes 5412. This airfoil representation technique captures the entire 542

design space of smooth airfoils. 543


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Fig. 4 Axisymmetric airfoildesign parameters

3. Requires relatively few variables to represent a large544enough design space to contain optimum aerody-545namic shapes for a variety of design conditions and546constraints547

4. Allows specification of design parameters such as548leading-edge radius, boat-tail angle, airfoil closure.549

5. Every airfoil in the entire design space can be derived550from the unit shape function airfoil.551

6. Every airfoil in the design space is therefore derivable552from every other airfoil.553

Mathematically the method is defined as:554

() = CN1N2 () Su() (14)

In which = z/c, = x/c and c is equal to the chord555length. The terms CN1N2 ()and Su() represent the Class and556Shape Function respectively. The Class Function is defined557as:558

CN1N2

() = N1 (1 )N2 (15)

The values of the exponents N1 and N2 define the basic559class of geometries. Few examples of the various categories560of shapes that can be obtained using the class functions561are demonstrated in Fig. 5 as introduced in Kulfan and562Bussoletti (2006).563

In this paper only NACA type round nose and pointed aft564end airfoil is considered for optimization, and the C0.51.0 ()565Class Function is used consequently. The class function is566not changing during optimization, so the new geometries567obtained belong to the class of round nose and pointed aft568end airfoils, and therefore realistic.569

Shape functions are used to refine the base geometry; in570fact, shape function defines the design variables for opti-571mization. As suggested by Kulfan (2008), Bernstein poly-572nomials representation is used to define shape functions. In573order to do this, the shape function is first written as the574product of coefficient vectors and the Bernstein polynomial575

terms. For instance, the shape function for the upper side of 576the airfoil becomes: 577

Su() =n

i=0Ai Si() (16)

In which n is the order of selected Bernstein polynomials; 578the order of the polynomial is always equal to the number of 579control points minus one. The coefficient vector Ai is taken 580to be the design variables or control points in this problem. 581The Bernstein polynomial terms Si are typically defined as: 582

Si() =

n

i

i(1 )ni (17)

One of the remarkable features of CST is that the value 583of the shape function at x/c = 0 is directly related to the 584airfoil leading-edge nose radius RLE and the airfoil chord 585length c by the following relation: 586

S(0) = 2[RLE/c] (18)The value of the shape function at x/c = 1 is directly related 587to the airfoil boat-tail angle : 588

S(1) = tan() (19)Changes in the coefficient vector Ai will lead to a varia- 589tion in shape around the unit airfoil. Figure 6 shows the unit 590airfoil and the airfoil deformed using a different coefficient 591vector. 592

This figure demonstrates that the leading edge nose 593radius and the boat-tail angle are the same which is the result 594of the first and last values of both coefficient vectors being 595equal. 596

3.2 Computational flow software 597

The software tool XFOIL (Drela and Giles 1987) is selected 598as the computational flow solver. XFOIL is a viscous- 599inviscid iterative software code which does not require any 600prior mesh preparation. The inviscid pressure distribution 601


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Fig. 5 Various geometriesobtained by using class function

is modeled using a linear vortex strength distribution. Vis-602cous effects and the development of the laminar-turbulent603boundary layer are modeled using empirical integral bound-604ary layer theory. XFOIL provides relatively accurate results605for subsonic airfoil analysis rapidly.606

3.3 Objective functions607

A higher or more favorable Lift over Drag ratio is typically608one of the major goals in aircraft design; since a particular609aircrafts required lift is set by its weight, delivering that lift610

Fig. 6 Comparison between unit shape airfoil with coefficientvector Ai = {1, 1, 1, 1, 1, 1} and the deformed airfoil Ai ={1, 2, 0.5, 2, 0.5, 1}

with lower drag leads directly to better fuel economy, climb 611performance, and glide ratio, so the first objective function 612is 613

f1 = CD/CL at Re = 3e6,Ma = 0.1 and AOA = 5

Here the angle of attack is set to be 5, which is consid- 614ered to be a typical incidence angle during descent and 615approach flight (Wickramasinghe et al. 2010). Providing 616a reduced drag design during cruise conditions generally 617occurs at the expense of a highly aft cambered airfoil sec- 618tion which results in excessive pitching moments and has 619adverse effects on stability, so the pitching moment at angle 620which corresponds to zero-lift generation is determined as 621the second objective: 622

f2 = C2M0 at Re = 3e6,Ma = 0.1One of the main goals of this study is to introduce airfoils 623that are less susceptible to stall phenomena. It is well known 624that early flow separation or leading-edge boundary layer 625transition results in airfoils that are highly susceptible to 626stall. Therefore, it is desirable to design an airfoil in which 627the separation is delayed on the upper surface of the airfoil 628and the smooth laminar flow region would be maximized 629on the airfoil upper surface. The cause of the separation 630on an airfoil surface is the adverse pressure gradient in the 631chord-wise direction. It can be concluded that the closer 632the location of maximum adverse gradient is to the leading 633edge, the higher is the susceptibility of flow separation and 634stall. Shape Factor, H, is used to represent this concept in 635


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terms of mathematical objective. The higher the value of H636is, the stronger the adverse pressure gradient would be on637the surface of the airfoil. Thus, if the location of maximum638H is transferred closer to trailing edge the likelihood of early639flow separation and adverse pressure gradient in the leading640edge would be circumvented. In order to achieve this goal,641the objective function is taken to be the maximization of the642location of maximum Shape factor, H, on the upper surface643of the airfoil.644

f3 = xHmax at Re = 3e6,Ma = 0.1 and AOA = 5The last objective is to maximize the maximum lift coef-645ficient which occurs at the critical angle of attack (by646exceeding this angle, stall occurs and lift starts to decrease):647

f4 = C2Lmax at Re = 3e6,Ma = 0.1

4 Results and discussion648

4.1 Algorithm evaluation649

Three Algorithms have been considered to evaluate the650behavior of our MOTPSO: NSGA-II, SPEA2 (Zitzler et al.6512001), and AbYSS (Nebro et al. 2008). The benchmarking652problems chosen to evaluate the four algorithms have been653the ZDT (Zitzler et al. 2000) and DTLZ (Deb et al. 2005)654test suites. The DTLZ problems have been used with their655bi-objective formulation. For assessing the performance of656the algorithms, three quality indicators have been used:657additive epsilon indicator () (Knowles et al. 2006), spread658() (Deb et al. 2002) and hypervolume (HV) (Zitzler and659Thiele 1999). The two first indicators measure, respectively,660the convergence and the diversity of the resulting Pareto661fronts, while the last one measures both convergence and662diversity.663

The implementation of these algorithms provided by664jMetal (Durillo and Nebro 2011) is used; jMetal is a Java-665based framework for developing metaheuristics for solving666multi-objective optimization problems. All the evolutionary667algorithms use an internal population of size equal to 100.668Total numbers of function evaluations for each algorithm669are 25000 for all algorithms. MOTPSO parameterization is670given in Table 1.671

Additionally, owing to the fact that we are dealing with672the stochastic algorithms and we want to provide the results673with statistical confidence, 100 independent runs have been674performed of each experiment and statistical results are675reported in Table 2.Q2 676

Table 2 includes the epsilon, spread and hypervolume677values resulting from fronts obtained by all the algorithms.678The best results achieved by the algorithms are shown679in bold fonts, and the standard deviations are shown in680parenthesis. The additive epsilon indicator () indicates that681

Table 1 MOTSPO Parameterization for benchmark functioncomparisons

t1.1t1.2

t1.3MOTPSO Parameterization

t1.4Swarm Size: 100

t1.5Mutation polynomial (polynomial distribution index = 20)t1.6Archive Size: 100 individuals

t1.7: 0.5

t1.8Inertia weight (W): (0.9 to 0.4)

t1.9Constriction factor ( ): (3) and (4)

t1.101: Randomly chosen between 1.5 and 2.5


t1.123: Set to be equal to 1.0

MOTPSO is the algorithm that has produced the fronts clos- 682est to the true Pareto front. MOTSPO outperformed other 683algorithms in ten out of twelve benchmark functions. Also, 684the standard deviation comparison between algorithms for 685additive epsilon indicator () demonstrates the robustness 686of the proposed algorithms solution in converging to the 687true Pareto front. The results of the Spread indicator () are 688also presented in Table 2. The results show that MOTSPO is 689the algorithm that has the better diversity or distribution of 690solutions along the Pareto front by reaching the best indica- 691tor values in seven out of twelve benchmark functions. The 692success in achieving a diverse and distributed set of optimal 693solutions is due to the introduction of a novel diversity- 694preserving scheme in the algorithm and nearest neighbor 695density estimator quality criterion for leader selection from 696the archive. The last quality indicator that is considered 697in Table 2 is Hypervolume metric (HV). The * symbol 698in the table represents that the respective algorithms non- 699dominated solutions are so far from the true Pareto fronts 700that the HV values are not reliable anymore. As already 701mentioned, the HV indicator measures both convergence to 702Pareto front and diversity of the solutions, and thus provides 703an overall assessment of each algorithm. The MOTPSO has 704been the best algorithm regarding this indicator by reach- 705ing the better results in eleven out of twelve benchmark 706functions. 707

In summary, the introduction of a diversity preserving 708and new social interaction schemes, provides a diverse and 709well distributed Pareto front with improved convergence 710to the true Pareto front. This fact was illustrated by com- 711parison to different Multi-objective algorithms in various 712Benchmark functions. 713

4.2 Aerodynamic optimization results 714

As already mentioned the Bernstein coefficient vector Ai 715is taken to be the design variables in this problem. Twelve 716control points are considered for each of upper and lower 717

PooyaHighlightdelete


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F


Tabl

e2

Qua

lity

indi

cato

rsco

mpa

riso

nbe

twee

nal

gori

thm

st2

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t2.2

Prob

lem

NSG

A-I

ISP

EA

2A

bYSS

MO

TPS

O

t2.3

HV

HV

HV

HV

t2.4

ZD

T1

6.59

e-1

1.35

e-2

3.70

e-1

6.60

e-1

9.05

e

31.

49e

-16.

61e

1

8.00

e-3

1.03

e

16.

61e

-17.

46e

-41.

13e

-1

t2.5

(3.0

7e-4

)(2

.10e

-3)

(2.8

7e-2

)(2

.76e

-4)

(7.5

9e-4

)(1

.40e

-2)

(2.3

9e-

4)(1

.48e

-3)

(1.2

6e-2

)(8

.12

e-5)

(1.0

5e-4

)(1

.56e

-2)

t2.6

ZD

T2

3.26

e-1

1.34

e-2

3.81

e-1

3.25

e-1

2.10

e

21.

66e

-13.

28e

1

7.31

e-3

1.08

e

13.

28e

-15.

80e

-49.

74e

-2

t2.7

(3.6

3e-4

)(2

.67e

-3)

(3.1

4e-2

)(9

.05e

-3)

(6.5

0e-2

)(5

.83e

-2)

(2.4

e-4)

(9.7

6e-4

)(1

.72e

-2)

(1.0

5e-4

)(3

.11e

-5)

(1.7

7e-2

)

t2.8

ZD

T3

5.15

e-1

1.13

e-2

7.47

e-1

5.14

e-1

2.23

e

37.

10e

-15.

16e

-16.

10e

-37.

10e

-15.

16e

-15.

80e

4

9.74

e-2

t2.9

(3.5

7e-4

)(3

.07e

-2)

(1.4

5e-2

)(6

.60e

-4)

(6.0

1e-2

)(5

.34e

-3)

(3.5

8e-3

)(6

.2e-

4)(1

.34e

-2)

(2.2

3e-4

)(3

.11e

-5)

(1.7

7e-2

)

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46.

54e

-11.

66e

-24.

02e

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45e

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10e

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51e

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45e-

21.

28e

-16.

61e

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02e

4

1.68

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t2.1

1(4

.58e

-3)

(8.3

8e-3

)(6

.89e

-2)

(2.0

5e-2

)(6

.43e

-2)

(1.2

8e-1

)(2

.15e

-2)

(1.7

5e-2

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.02e

-2)

(3.4

2e-4

)(8

.66e

-5)

(1.8

6e-2

)

t2.1

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DT

63.

88e

-11.

47e

-23.

59e

-13.

78e

-12.

53e

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30e

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00e

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06e-

38.

93e

-24.

01e

-19.

25e

-31.

00e

0

t2.1

3(1

.73e

-3)

(2.1

1e-3

)(3

.14e

-2)

(2.8

5e-3

)(3

.83e

-3)

(2.2

9e-2

)(1

.57e

-4)

(2.9

7e-4

)(1

.04e

-2)

(9.4

4e-5

)(2

.66e

-4)

(3.6

4e-1

)

t2.1

4D

TL

Z1

4.85

e-1

8.85

e-3

4.21

e-1

4.77

e-1

1.51

e-2

3.52

e-1

4.86

e-1

5.85

e-3

1.40

e-1

4.94

e-1

2.49

e-3

1.35

e-1

t2.1

5(1

.14e

-2)

(6.7

9e-3

)(1

.30e

-1)

(4.2

6e-0

2)(2

.88e

-2)

(3.2

4e-1

)(1

.26e

-2)

(2.3

1e-3

)(1

.65e

-2)

(2.9

5e-4

)(1

.65e

-5)

(1.2

1e-2

)

t2.1

6D

TL

Z2

2.11

e-1

1.14

e-2

3.80

e-1

2.12

e-1

7.56

e-3

1.49

e-1

2.12

e-1

5.39

e-3

1.11

e-1

2.12

e-1

7.85

e-3

1.53

e-1

t2.1

7(2

.47e

-4)

(2.4

6e-3

)(3

.36e

-2)

(1.0

9e-4

)(9

.33e

-4)

(1.3

6e-2

)(5

.52e

-05)

(6.5

6e-3

)(1

.54e

-2)

(1.5

0e-4

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.04e

-3)

(9.5

2e-3

)

t2.1

8D

TL

Z3

*1.

23e

09.

31e

-1*

2.42

e0

1.07

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*1.

66e

09.

51e

-12.

12e

-16.

81e

-32.

81e

-1

t2.1

9(9

.67e

-1)

(1.5

5e-1

)(1

.60e

0)(1

.06e

-1)

(9.4

3e-1

)(1

.08e

-1)

(2.2

4e-4

)(8

.2e-

4)(1

.43e

-2)

t2.2

0D

TL

Z4

2.09

e-1

1.13

e-2

3.81

e-1

2.10

e-1

7.66

e-3

1.42

e-1

2.11

e-1

7.79

e-3

1.52

e-1

2.08

e-1

2.60

e-3

2.63

e-1

t2.2

1(2

.46e

-4)

(2.2

8e-3

)(3

.05e

-2)

(9.2

9e-5

)(9

.01e

-4)

(1.2

6e-2

)(3

.82e

-05)

(1.4

7e-3

)(2

.68e

-2)

(4.9

5e-4

)(3

.53e

-4)

(3.5

7e-2

)

t2.2

2D

TL

Z5

2.11

e-1

1.16

e-2

3.79

e-1

2.12

e-1

7.37

e-3

1.50

e-1

2.12

e-1

8.63

e-3

1.85

e-1

2.12

e-1

5.35

e-3

1.31

e-1

t2.2

3(2

.06e

-4)

(2.4

6e-3

)(2

.36e

-2)

(1.2

9e-4

)(8

.46e

-4)

(1.2

5e-2

)(5

.19e

-5)

(1.9

1e-3

)(1

.22e

-1)

(1.4

5e-4

)(2

.34e

-3)

(4.0

2e-2

)

t2.2

4D

TL

Z6

1.69

e-1

4.71

e-2

8.09

e-1

1.27

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2.98

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8.07

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1.13

e-1

9.50

e-2

2.31

e-1

2.12

e-1

5.18

e-3

1.17

e-1

t2.2

5(2

.86e

-2)

(2.5

6e-2

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-1)

(1.1

3e-2

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.25e

-2)

(6.4

0e-2

)(2

.89e

-2)

(1.5

6e-2

)(7

.63e

-2)

(3.9

2e-5

)(1

.04e

-4)

(2.3

e-2)

t2.2

6D

TL

Z7

3.33

e-1

1.08

e-2

6.26

e-1

3.33

e-1

9.27

e-3

5.45

e-1

3.34

e-1

5.51

e-3

5.19

e-1

3.34

e-1

8.38

e-4

5.19

e-1

t2.2

7(1

.56e

-4)

(2.0

2e-3

)(2

.09e

-2)

(1.7

2e-5

)(1

.17e

-3)

(1.1

4e-2

)(1

.62e

-5)

(1.4

5e-3

)(4

.94e

-2)

(2.1

2e-5

)(4

.06e

-5)

(1.5

3e-3

)

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sections of the airfoil. If all these coefficients are equal to718one, then the unit airfoil would be obtained as it can be719seen in Fig. 6. However, in this paper, the goal is to per-720form multiobjective optimization on a NACA series airfoil.721Therefore, the first step would be to adjust the coefficients722such that the corresponding curve matches the base air-723foil which is NACA 23015. In order to do so, the single724optimization TPSO has been used to minimize the average725square of distances between points on the curves (called the726mean squared error), and the resulting curve is represented727in Fig. 7.728

The first and the last of these coefficients for both729upper and lower surfaces represent airfoil leading-edge nose730radius RLE and airfoil boat-tail angle ; they were con-731sidered to be constant so that our final design solutions732demonstrate the geometric characteristics of basic airfoil733which is NACA 23015 in this case. The design variable734ranges of remaining twenty coefficients are presented in735Table 3. The algorithm parameterization and the problem736setup are given in Table 4.737

Table 5 also demonstrates the improvement over the738NACA 23015 for solutions which exhibit minimum val-739ues for particular objectives. This becomes helpful when a740designer inclined toward a particular objective rather than a741compromised solution. Additionally, it should be mentioned742that for the preferred solution in Table 5, two objective func-743tions (f3, f4) are considered to choose among the set of744non-dominated set of solutions.745

As it can be seen from Tables 5 and 6, final results746obtained using MOTPSO outperform design characteristics747of the NACA 23015 in the particular objective functions748considered, even though the NACA series solution is still749among the final non-dominated solutions.750

Fig. 7 NACA 23015 compared to the matching curve obtained usingCST method

Table 3 Design variable range t3.1

t3.2# Coefficient range Base airfoil coefficients

t3.31 (0.001,0.4) 0.1815

t3.42 (0.001,0.4) 0.3041

t3.53 (0.001,0.4) 0.3370

t3.64 (0.001,0.4) 0.0035

t3.75 (0.001,0.4) 0.2631

t3.86 (0.001,0.6) 0.4133

t3.97 (0.001,0.4) 0.1144

t3.108 (0.001,0.4) 0.0090

t3.119 (0.001,0.6) 0.4415

t3.1210 (0.001,0.4) 0.0548

t3.1311 (0.4,0.001) 0.1595t3.1412 (0.4,0.001) 0.1270t3.1513 (0.4,0.001) 0.0433t3.1614 (0.6,0.001) 0.4271t3.1715 (0.4,0.001) 0.0371t3.1816 (0.4,0.001) 0.0468t3.1917 (0.4,0.001) 0.1810t3.2018 (0.4,0.001) 0.2478t3.2119 (0.4,0.001) 0.1835t3.2220 (0.4,0.001) 0.0108

One of the main goals of this research was to introduce 751an airfoil which is less susceptible to stall problem. In order 752to accomplish this goal, one of the solutions from the Pareto 753front that had the best results in the last two objectives: 754f3 = xHmax and f4 = C2Lmax was chosen. The variation 755in airfoil geometry is shown in Fig. 8. 756

First, stall is largely sensitive to early separation on the 757upper surface of the airfoil and if it can be managed to 758transfer the location of separation from leading-edge to 759trailing-edge or in other words, if we can move the location 760

Table 4 MOTSPO Parameterization and aerodynamic optimizationproblem setup

t4.1t4.2

t4.3MOTPSO Parameterization for aerodynamic optimization

t4.4Number of objective Functions: 4

t4.5Swarm Size: 50

t4.6Number of iterations: 100

t4.7Mutation polynomial (polynomial distribution index = 20)t4.8Archive Size: 100 individuals

t4.9: 0.5

t4.10Inertia weight (W): (0.9 to 0.4)

t4.11Constriction factor ( ): (3) and (4)



t4.143: Set to be equal to 1.0

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Table 5 Optimization results

t5.2 Airfoil CL/CDCM0

xHmax CLmax

t5.3 NACA 23015 99.45 0.0080 0.1840 1.7621

t5.4 Preferred 78.040 0.0752 0.9927 2.0060

t5.5 Best Obj. 1 160.27 0.0336 0.3360 1.9401

t5.6 Best Obj. 2 131.92 0.0002 0.4151 1.5487

t5.7 Best Obj. 3 109.73 0.0918 0.9928 1.8633

t5.8 Best Obj. 4 86.20 0.0592 0.1537 2.2267

of maximum shape factor to the trailing edge, the stall char-761acteristics of the airfoil would be improved. Second, it is762desirable to obtain the highest possible lift coefficient before763entering stall regions. As it can be seen from Lift coeffi-764cient curve, Fig. 9, both of these objectives accomplished765in preferred design, and compared to the base airfoil, the766preferred solution has higher lift at critical angle of attack,767and also critical angle of attack is increased and therefore768stall is postponed. Moreover, the reduction in the lift coef-769ficient near stall region is smooth and does not experience770the sudden reduction as the base airfoil. These improved771characteristics help the airfoil to show less extreme adverse772properties of balance and control during stall.773

Figure 10 features the pressure coefficient distribution774of the preferred airfoil against the NACA23015 for the775cruise operating condition. Note that the negative pressure776coefficient is plotted in this figure. Therefore, the curve777which predominately has a negative pressure (positive pres-778sure in the figure) is the upper (suction) surface. Similarly,779the lower (pressure) surface is denoted by the curve which780is predominately experiencing positive pressure (negative781pressure in the figure). The difference between these two782curves creates the resultant lift force.783

5 Conclusions784

In this paper, a new Multi-objective Territorial Particle785Swarm Optimization (MOTPSO) was presented that dealt786with both the cause and the effect of the premature con-787vergence problem encountered in the original PSO. In the788algorithm, a new social interaction scheme was proposed789that moves particles toward a new point that is a weighted790

Table 6 Improvement over NACA 23015

t6.2 Airfoil Improvement

t6.3 Best Obj. 1 61.1 %

t6.4 Best Obj. 2 97.5 %

t6.5 Best Obj. 3 439.5 %

t6.6 Best Obj. 4 26.3 %

Fig. 8 Comparison of the preferred solution and the reference NACA23015 solution

average of their elite neighbors best experienced position. 791It was shown that this new social interaction scheme pro- 792vides the algorithms with local search ability and helps the 793particles to exploit the local candidate solutions in their 794neighborhood more effectively. Also, a new collision oper- 795ator was incorporated into the previously known territory 796concept to enhance algorithms exploration characteristics. 797Non-dominated solutions found were stored in an external 798archive and nearest neighbor density estimator was used to 799select leaders from external archive for the swarm. Both 800diversity and convergence of the algorithm was compared to 801

Fig. 9 Lift coefficient curve


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A. Nejat et al.

Fig. 10 Pressure distribution along the airfoil chord

three state of the art multi-objective optimization algorithms802in well-known benchmark function.803

The efficiency of the algorithm further examined by804implementing it on an aerodynamic shape optimization805problem using NACA 23015 as the base airfoil. Class-806Shape-Transformation (CST) method was used to represent807the geometry of the airfoil and the coefficients of its basis808functions were chosen to be the design variables. The809leading- edge nose radius and boat-tail angle was chose to810be the same as base airfoil. This study described four con-811flicting objectives for the optimization problem, and the812strength of the algorithm demonstrated such that the result-813ing solutions were significantly better than the base airfoil814in all the objective functions. Finally, a new design solution815was presented that improved the susceptibility of the base816airfoil to stall problem by delaying the separation on upper817surface of the airfoil. The suggested design had higher max-818imum lift at critical angle of attack and also stall is delayed819as the critical angle of attack was increased.820

References821

Anderson MB (2003) Genetic algorithms in aerospace design: sub-822stantial progress, tremendous potential. SVERDRUP Technology823Inc Eglin AFB FL Technical and Engineering Acquisition Support824Group825

Anderson WK, Venkatakrishnan V (1999) Aerodynamic design opti-826mization on unstructured grids with a continuous adjoint for-827mulation. Comput Fluids 28(45):443480. doi:10.1016/s0045-8287930(98)00041-3829

Arias-Montano A, Coello Coello C, Mezura-Montes E (2011) Evolu-830tionary algorithms applied to multi-objective aerodynamic shape831optimization. Comput Optim Methods Algoritm 356:211240.832doi:10.1007/978-3-642-20859-1 10833

Ben Ghalia M (2008) Particle swarm optimization with an improved834exploration-exploitation balance. 51st Midwest symposium on835circuits and systems, MWSCAS, pp 759762836

Branke J, Mostaghim S (2006) About selecting the personal best in837multi-objective particle swarm optimization. In: Parallel problem838solving from nature-PPSN IX. Springer, pp 523-532839

Buckley PH, Zhou YB, Zingg WD (2010) Airfoil optimization using 840practical aerodynamic design requirements. J Aircraft 47(5): 84113 842

Clerc M, Kennedy J (2002) The particle swarm-explosion, stabil- 843ity, and convergence in a multidimensional complex space. IEEE 844Trans Evol Comput 6(1):5873 845

Coello Coello CA, Lechuga MS (2002) MOPSO: a proposal for mul- Q3846tiple objective particle swarm optimization. In: Proceedings of the 8472002 congress on evolutionary computation. CEC02. 2002, vol. 8482. IEEE, pp 1051-1056 849

Deb K (2001) Multi-objective optimization. Multi-objective optimiza- 850tion using evolutionary algorithms, vol 13, pp 46 851

Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and eli- 852tist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol 853Comput 6(2):182197. doi:10.1109/4235.996017 854

Deb K, Thiele L, Laumanns M, Zitzler E (2005) Scalable 855test problems for evolutionary multiobjective optimization. In: 856Abraham A, Jain L, Goldberg R (eds) Advanced informa- 857tion and knowledge processing. Springer, Berlin, pp 105145. 858doi:10.1007/1-84628-137-7 6 859

Drela M, Giles MB (1987) Viscous-inviscid analysis of transonic and 860low Reynolds number airfoils. AIAA J 25(10):13471355 861

Durillo JJ, Nebro AJ (2011) jMetal: a java framework for multi- 862objective optimization. Adv Eng Softw 42:760771 863

Eberhart R, Shi Y (1998) Comparison between genetic algorithms and 864particle swarm optimization. In: Porto V, Saravanan N, Waagen 865D, Eiben A (eds) Evolutionary programming VII, vol 1447. 866Springer, Berlin, pp 611616. Lecture Notes in Computer Science. 867doi:10.1007/BFb0040812 868

Fieldsend JE, Uk EQ, Singh S (2002) A multi-objective algorithm 869based upon particle swarm optimisation, an efficient data struc- 870ture and turbulence. In: Proceedings 2002 U.K. workshop on 871computational intelligence, pp 3744 872

Hicks RM, Henne PA (1978) Wing design by numerical optimization. 873J Aircraft 15(7):407412 874

Jameson A, Martinelli L, Pierce NA (1998) Optimum aerodynamic 875design using the navierstokes equations. Theor Comput Fluid 876Dyn 10(1):213237. doi:10.1007/s001620050060 877

Kennedy J (2006) Handbook of nature-inspired and innovative com-

Q4

878puting. Swarm Intell:187219 879

Kennedy J, Eberhart R (1995) Particle swarm optimization. In: IEEE 880international conference on neural networks, 1995 proceedings, 881vol 4, pp 19421948 882

Kennedy J, Mendes R (2002) Population structure and particle swarm 883performance. In: Proceedings of the 2002 congress on evolution- 884ary computation, CEC02. 2002. IEEE, pp 16711676 885

Knowles J, Thiele L, Zitzler E (2006) A Tutorial on the perfor- 886mance assessment of stochastic multiobjective optimizers. Com- 887puter engineering and networks laboratory (TIK). Swiss Federal 888Institute of Technology (ETH), Zurich, pp 327332 889

Kroo I (1997) Multidisciplinary optimization applications in pre- 890liminary design-status and directions. In: 38th AIAA/AMSE/ 891ASCE/AHS/ASC structures, structural dynamics and materials 892conference, Kissimmee, FL, Paper No. AIAA97-1408 893

Kulfan BM (2008) Universal parametric geometry representation 894method. AIAA 45(1):17 895

Kulfan BM, Bussoletti JE (2006) Fundamental parametric geome- 896try representations for aircraft component shapes. AIAA Paper, 8976948:2006 898

Kuntawala NB (2011) Aerodynamic shape optimization of a blended- 899wing-body aircraft configuration. PhD Dissertation, University of 900Toronto 901

Lee J, Mo J (2011) Analysis of technological innovation and environ- 902mental performance improvement in aviation sector. Int J Environ 903Res Public Health 8(9):37773795 904

http://dx.doi.org/10.1016/s0045-7930(98)00041-3http://dx.doi.org/10.1016/s0045-7930(98)00041-3http://dx.doi.org/10.1007/978-3-642-20859-1_10http://dx.doi.org/10.1109/4235.996017http://dx.doi.org/10.1007/1-84628-137-7_6http://dx.doi.org/10.1007/BFb0040812http://dx.doi.org/10.1007/s001620050060PooyaHighlightchecked!

PooyaHighlightThis is a book chapter not journalJ. Kennedy. Handbook of Nature-Inspired and Innovative Computing, chapter Swarmintelligence, pages 187219. Springer, 2006.


UNCO

RRECTEDPROO

F


Lee DS, Fahey DW, Forster PM, Newton PJ, Wit RCN, Lim905LL, Owen B, Sausen R (2009) Aviation and global climate906change in the 21st century. Atmos Environ 43(2223):35203537.907doi:10.1016/j.atmosenv.2009.04.024908

Lee DS, Pitari G, Grewe V, Gierens K, Penner JE, Petzold909A, Prather MJ, Schumann U, Bais A, Berntsen T, Iachetti910D, Lim LL, Sausen R (2010) Transport impacts on atmo-911sphere and climate: aviation. Atmos Environ 44(37):46784734.912doi:10.1016/j.atmosenv.2009.06.005913

Li LD, Yu X, Li X, Guo W (2009) A modified PSO algorithm for914constrained multi-objective optimization. In: Third international915conference on network and system security. NSS09, 2009. IEEE,916pp 462467917

Liang JJ, Suganthan PN (2005) Dynamic multi-swarm particle swarm918optimizer with local search. In: Proceedings of the IEEE confer-919ence on evolutionary computation, pp 522528920

Mendes R (2004) Population topologies and their influence in par-921ticle swarm performance. PhD Dissertation, Departamento de922Informatica, Escola de Engenharia, Universidade do Minho923

Mohais AS, Mendes R, Ward C, Posthoff C (2005) Neighborhood re-924structuring in particle swarm optimization. In: AI 2005: advances925in artificial intelligence. Springer, pp 776-785926

Moore J, Chapman R (1999) Application of particle swarm to mul-927tiobjective optimization. Department of Computer Science and928Software Engineering, Auburn University929

Mostaghim S, Teich J (2003) The role of -dominance in multi objec-

Q5

930tive particle swarm optimization methods. In: The 2003 congress931on evolutionary computation. CEC03, 2003. IEEE, pp 17641771932

Nebro AJ, Luna F, Alba E, Dorronsoro B, Durillo JJ, Beham933A (2008) AbYSS: adapting scatter search to multiobjec-934tive optimization. IEEE Trans Evol Comput 12(4):439457.935doi:10.1109/tevc.2007.913109936

Nebro AJ, Durillo J, Garcia-Nieto J, Coello Coello C, Luna F, Alba E937(2009) Smpso: a new pso-based metaheuristic for multi-objective938optimization. In: IEEE symposium on computational intelligence939in miulti-criteria decision-making. mcdm09, 2009. IEEE, pp 6694073941

Obayashi S, Tsukahara T (1997) Comparison of optimization algo-942rithms for aerodynamic shape design. AIAA J 35(8):14131415943

Ostadmohammadi Arani B, Mirzabeygi P, Shariat Panahi M (2013)944An improved PSO algorithm with a territorial diversity-preserving945scheme and enhanced explorationexploitation balance. Swarm946Evol Comput 11:115947

Peram T, Veeramachaneni K, Mohan CK (2003) Fitness-distance-ratio948based particle swarm optimization. Swarm intelligence sympo-949sium, 2003 SIS 03 proceedings of the 2003 IEEE, pp 174181950

Poli R (2008) Analysis of the publications on the applica-

Q4951

tions of particle swarm optimisation. J Artif Evol Appl:110.952doi:10.1155/2008/685175953

Poli R, Kennedy J, Blackwell T (2007) Particle swarm optimization.954Swarm Intell 1(1):3357. doi:10.1007/s11721-007-0002-0955

Reyes-Sierra M, Coello CC (2006) Multi-objective particle swarm956optimizers: a survey of the state-of-the-art. Int J Comput Intell Res9572(3):287308958

Riget J, Vesterstrm JS (2002) A diversity-guided particle swarm 959optimizer-The ARPSO. Department of Computer Science, Univer- 960sity of Aarhus, Denmark 961

Samareh JA (2001) Survey of shape parameterization techniques 962for high-fidelity multidisciplinary shape optimization. AIAA J 96339(5):877884 964

Si T, Jana ND, Sil J (2011) Constrained function optimization using 965PSO with polynomial mutation. In: Swarm, evolutionary, and 966memetic computing. Springer, pp 209216 967

Sierra MR, Coello CAC (2005) Improving PSO-Based multi-objective 968optimization using crowding, mutation and -dominance. In: 969Evolutionary multi-criterion optimization. Springer, pp 505 970519 971

Sobieczky H (1999) Parametric airfoils and wings. In: Recent devel- 972opment of aerodynamic design methodologies. Springer, pp 71 97387 974

Song W, Keane AJ (2004) A study of shape parameterisation 975methods for airfoil optimisation. In: 10th AIAA/ISSMO mul- 976tidisciplinary analysis and optimization conference, pp 2031 9772038 978

Wickramasinghe UK, Carrese R, Xiaodong L (2010) Designing air- 979foils using a reference point based evolutionary many-objective 980particle swarm optimization algorithm. In: IEEE congress on 981evolutionary computation (CEC), 2010, pp 1-8, 1823 July 2010 982

Xie X-F, Zhang W-J, Yang Z-L (2002) Hybrid particle swarm opti- 983mizer with mass extinction. In: IEEE 2002 international confer- 984ence on communications, circuits and systems and West Sino 985expositions, 2002. IEEE, pp 11701173 986

Xu D, Ai X (2009) An improved diversity guided particle swarm 987optimization. In: The sixth international symposium on neural 988networks (ISNN 2009). Springer, pp 623630 989

Zhang Y, Huang S (2004) A novel multiobjective particle swarm 990optimization for buoys-arrangement design. In: Proceedings of 991the IEEE/WIC/ACM international conference on intelligent agent 992technology. IEEE Computer Society, pp 2430 993

Zhao S, Suganthan PN (2009) Diversity enhanced particle swarm opti- 994mizer for global optimization of multimodal problems. In: IEEE 995congress on evolutionary computation. CEC09, 2009. IEEE, pp 996590597 997

Zhao S-Z, Suganthan PN, Pan Q-K, Fatih Tasgetiren M (2011) 998Dynamic multi-swarm particle swarm optimizer with harmony 999search. Expert Syst Appl 38(4):37353742 1000

Zhou A, Qu B-Y, Li H, Zhao S-Z, Suganthan PN, Zhang Q (2011) 1001Multiobjective evolutionary algorithms: a survey of the state of the 1002art. Swarm Evol Comput 1(1):3249 1003

Zitzler E, Thiele L (1999) Multiobjective evolutionary algorithms: a 1004comparative case study and the strength Pareto approach. IEEE 1005Trans Evol Comput 3(4):257271. doi:10.1109/4235.797969 1006

Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolu- 1007tionary algorithms: empirical results. Evol Comput 8(2):173195. 1008doi:10.1162/106365600568202 1009

Zitzler E, Laumanns M, Thiele L (2001) SPEA2: improving the 1010strength Pareto evolutionary algorithm. Computer Engineering 1011TIK-Report, pp 121 1012

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