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70

International Journal of

Science and Engineering Investigations vol. 1, issue 1, February 2012

Comparing the Performance of the Particle Swarm Optimizationand the Genetic Algorithm on the Geometry Design of

Longitudinal Fin

H. Azarkish1, S. Farahat

2, S.M.H. Sarvari

3

1,2Mechanical Engineering Department, University of Sistan and Baluchestan, Zahedan, Iran3Mechanical Engineering Department, Shahid Bahonar University, Kerman, Iran

([email protected], [email protected], [email protected])

Abstract-In the present work, the performance of the particleswarm optimization and the genetic algorithm compared as atypical geometry design problem. The design maximizes theheat transfer rate from a given fin volume. The analysispresumes that a linear temperature distribution along the fin.The fin profile generated using the B-spline curves andcontrolled by the change of control point coordinates. Aninverse method applied to find the appropriate fin geometryyield the linear temperature distribution along the fincorresponds to optimum design. The numbers of thepopulations, the count of iterations and time to convergencemeasure efficiency. Results show that the particle swarmoptimization is most efficient for geometry optimization.

Keywords- Genetic Algorithm; Geometry Optimization;longitudinal Fin; Particle Swarm Optimization

I. INTRODUCTION

Optimum geometry design of systems has been widelyattracted in the field of engineering. There have been manyoptimization methods for optimizing the objective function toachieve desirable plan or systems. Gradient based methodssuch as conjugate gradient and Levenberg – Marquardt orstochastic and population based optimization methods such asgenetic algorithm and particle swarm optimization [1, 2]. Eachmethod has some advantages and disadvantages. Thus therehas been some controversy recently about the performance of these algorithms.

Fin profile optimization is one of classical conjugated heattransfer problems. Bobaru and Rachakonda [3] used gradientbased method for optimized the two dimensional fin profileunder the natural convection. They conclude that, there is noguarantee to obtain the global optimum by using the gradient-based optimization methods and only a local optimum isguaranteed upon convergence. Moreover, the conductionmechanism is not very sensitive under the differential changeof the shape. Therefore, numerical evaluation of the sensitivitymatrix and gradients are more difficult. Azarkish et al. used the

B-spline curves and modified genetic algorithm for optimizedthe convective-radiative single fin profile [4] and a fin array[5]. In this method, the effect of variation of convective heat

transfer coefficient, variable conductivity along the fin, theeffect of radiation and the length of arc could be modeledeasily without needing to evaluation of gradients and fall tolocal optimum. Although, this approach could be consideredmany real conditions, however, the number of objectivefunction evaluations and the cost of numerical computation areincreased rather than gradient-based methods. Therefore, forsimple objective functions such as one dimensional fin profileoptimization this method is acceptable, however thecomputational cost would be dramatically increased for morecomplex problems such as two dimensional geometryoptimization. In conclude, it seems that a low computationalcost optimization method without needing to calculate of

gradients could be suitable for these types of problems.In the present work, the performance of particle swarm

optimization for the geometry optimization has beeninvestigated and compare with the performance of the geneticalgorithm. A single convective-radiative fin is considered assubject. Azarkish et al. [4], show that the optimum temperaturedistribution along the fin was linear in absence of volumetricheat generation. Therefore the aim of inverse problem is findan appropriated fin profile to achieve the leaner temperaturedistribution along the fin. Application of both particle swarmoptimization and genetic algorithm for this problem has beeninvestigated. First, the best value of constant parameters in theparticle swarm optimization is determined. The effect of

variation of these parameters on the convergence rate has beeninvestigated. Finally, the necessary number of population forgood convergence and corresponding number of iterations andconvergence time are compared between two optimizationalgorithms mentioned above.

II. DIRECT PROBLEM

The template Consider a longitudinal fin with variablecross sectional area at the base temperature T b which isextended into a quiescent fluid of temperature T ∞ andsurrounded by an enclosure of temperature T sur . The surface of fin is considered as diffuse and gray. The heat losses from theboundaries are assumed to be due to the radiation to the

surrounding and the natural convection to the ambient (Fig. 1).The radiation heat transfer between the base and fin surface,



International Journal of Science and Engineering Investigations, Volume 1, Issue 1, February 2012 71

www.IJSEI.com Paper ID: 10112-15

also between the different elements of fin surface areneglected. The width of fin is assumed to be very thin, in sucha way that the temperature distribution (and conduction heattransfer rate) may be regarded one dimensional along the x-axis. The energy equation and the boundary condition in this

situation can be presented as:

Figure 1. Schematic shape and orientations of the longitudinal fin withvariable cross sectional area.

)()(1)(

44

2

sur T T

k T T

k

h

dx

dy

dx

dT x y

dx

d (1)

b xT T

0(2)

0 L xdx

dT y (3)

Where y(x) is the half thickness of fin (fin profile), k is thefin thermal conductivity of fin and h is the local convectiveheat transfer coefficient calculated by the following correlation[4]:

4 / 1

2

3

4 / 1

2 / 1)(

9

5Pr3363

Pr8

H T xT g

H

k h (4)

This non-linear equation is solving with the finite volumemethod [6] to obtain the temperature distribution along the fin.

A detailed description of direct problem was briefly explainedelsewhere by the authors [4, 5].

III. INVERSE PROBLEM

In the present work, the inverse problem is consideredinstead of direct optimum geometry design of single fin. Finprofiles generated by B-spline curves [7] and controlled bymoving the coordinates of control points in x, y directions (Fig.2). The number of control points is considered to be N cp=4.The first control point is placed at the base of fin ( x = 0), thatcan move freely along the y-axis. Therefore this control point

represents the thickness of the fin base. Conversely, the last

control point is placed on the fin axis of symmetry ( y = 0),which can move freely along the x-axis in such a way that itsposition specifies the fin length. Other control points can movein xy-plane and therefore, their degree of freedom is equal to 2.

Figure 2. A schematic of fin profile generated by the B-spline curves.

The number of design variables is equal to 6 and defined as:

max4L xeps

max321,, y y y yeps

(5)

1,021

Where eps is an arbitrary small value, Lmax is maximumlength of fin and ymax is maximum amount of half thicknessof the fin base. The position of control points defined as:

)0,(:

),(:

),(:

),0(:

44

3413

24212

11

x

y x

y x

y

P

P

P

P

(6)

Therefore, the design variables are defined as the positionof control points. Each chromosome in the genetic algorithm oreach particle in particle swarm optimization represents the setof control points correspond to a fin profile. In order toevaluated the fitness of each set of control points, Directproblem (Eq(1)) is solved, the temperature distributions and thevolume of fin are obtained and compared with idealtemperature distributions and the given volume. Thus, twoerror functions have been introduced for predict the fitness of each profile:

n

i iideal

iiideal

T

T T

n E

1 ,

,

1100

1

(7)





allow

allow

V

V V E

100

2

(8)

Where n is the number of control volumes, Ti is thetemperature of each control volume obtained by solving the

direct problem and Vallow is the allowable volume of the fin.Tideal,i is the ideal temperature of each control volume definedas:

L

xT T T T

bbideal)(

(9)

If E1→0 the temperature distribution along the fin becomelinear as Eq(9). E2→0 satisfied to have a given volumeVallow. E1 and E2 are positive functions, therefore, the aim of inverse problem is minimize E1+E2.

The genetic algorithm and the particle swarm optimizationare used to minimize the amount of error functions corresponds

to the optimum fin profile. The genetic algorithm is astochastic search technique that based on the mechanism of genetics and natural selection and it is widely used in the fieldof engineering optimization problems [8]. On the other hand,Particle swarm optimization is a population-based swarmintelligence algorithm that introduced by Eberhart andKennedy in 1995. It is starts with a group of particles known asthe swarm. Each particle is function of design variables and itis improved through the algorithm by changing the position of particle on the search space. Consider the current position of particles at the moment t is given by Xi(t). The new position of particles in the next generation is expressed as:

)()()1,0(

)()()1,0()()1(

3

21

t t rand C

t t rand C t C t

ii

iiii

XG

XPVV

(10)

)1()()1( t t t iii

VXX (11)

Where Pi(t) is the local best position of particle i at themoment t and Gi(t) is the global best position of the swarm(Fig. 3). Moreover C1 is the inertia weight, C2 and C3 are theacceleration constants responsible for varying the particlevelocity towards Pi(t) and Gi(t), respectively. More detailsabout particle swarm optimization are presented in [9].

Figure 3. Three vectors for the calculation of the next displacement of aparticle: its own velocity, towards its best performance and the best

performance of its best information [9].

IV. RESULTS AND DISCUSSION

A longitudinal fin is considered at the base temperatureT b=500K and T ∞=T sur =500K . The fin is made of aluminumwith thermal conductivity of k=210 W/m.K and the surfaceemissivity of ε=0.3. The fin height is H=40cm and the

allowable volume is considered to be V allow=160 cm3 The B-spline curve with 4 control points is used to generate the finprofile. The aim of optimization is minimized E 1+E 2 to have alinear temperature distribution from T b=500K to T ∞=500K andalso V allow=160 cm

3. The acceptable error that satisfied these

conditions is E 1+E 2<0.5.

In order to investigate the effects of constant parameters C 1,C 2 and C 3 on the convergence rate of particle swarmoptimization, the case study is solved with different value of these parameters and the pairs of values C 1=0.7, C 2=1.2 andC 3=1.4 is recommended for good convergence. Fig. 4 showsthe effect of parameters C 1, C 2 and C 3 on the necessary numberof iterations for convergence. As shown, parameter C 1 is more

sensible rather than other parameters.

C1

C2

C3

I t r a t i o n

0.4 0.7 1 1.3 1.6 1.90

20

40

60

80

100

C1

, C2=1.2 , C

3=1.4

C2

, C1=0.7 , C

3=1.4

C3

, C1=0.7 , C

2=1.2

Figure 4. The effect of parameters C 1, C 2 and C 3 on the convergence rate.

Moreover, the genetic algorithm parameters are considered

as crossover rate of 0.4 and mutation rate of 0.02 and with apopulation size of 100. The elitism mechanism is applied forboth genetic algorithm and particle swarm optimization. In thecase of genetic algorithm, elitism means that the chromosomewith the best cost is kept for next generation. However, in thecase of particle swarm optimization its means that the positionof the best particle is not change for next iteration (VGi=0).In order to find the optimum location of control pointscorrespond to linear temperature distributions the inverseproblem is solved. The comparison of the temperaturedistributions obtained by particle swarm optimization, geneticalgorithm and ideal temperature distribution is shown in Fig. 5.





x (m)

T e m p e r a t u r e ( K )

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35300

350

400

450

500

PSOGAIdeal Temperature

Figure 5. The comparison of the temperature distributions obtained byparticle swarm optimization, genetic algorithm and ideal temperature

distribution along the fin.

As shown, both particle swarm optimization and geneticalgorithm could find the appropriate temperature distribution.However, the convergence time and computational cost arevery different for two cases mentioned above. Fig. 6 comparesthe variation of necessary iterations for convergence asfunction of population number for two kinds of optimizationmethods. The corresponding convergence time is presented in

Fig. 7.

Number of Population

I t r a t i o n

0 20 40 60 80 1000

20

40

60

80

100

PSOGA

Figure 6. Comparison of the necessary iterations for convergence as function of population number for two kinds of optimization methods.

As shown, the number of iterations decreases rapidly on therange of particles between 10 and 25; however the graph showsa slight decrease after 25 particles. Moreover, the minimumcorrespond convergence time occur on the range of 20 to 25particles and it is increase after 30 particles. Therefore, the

recommended number of particles is 20 to 30 for particleswarm optimization. On the other hand, the recommendednumber of chromosomes is 60 to 80 for the genetic algorithm.Therefore, the number of populations decreases about 3 timesand the convergence time decrease about 4 times in the case of particle swarm optimization rather than the genetic algorithmfor this typical problem.

Number of Population

C o n v e r g e n c e t i m e ( s

)

0 20 40 60 80 1000

20

40

60

80

100

PSOGA

Figure 7. Comparison of convergence time with respect to populationnumber for two kinds of optimization methods.

In the present work, the temperature distributions along thefin not very sensitive by changing the differential movement of control points coordinate. Therefore, the application of gradientbase method is very difficult for this problem and other similarproblems especially in conduction heat transfer problems.Moreover, Results show the successful performance of particleswarm optimization for this case rather than genetic algorithm.Therefore, the particle swarm optimization is recommend for

more applicable and complex geometry optimization problemssuch as two dimensional fin array and design of twodimensional shaped channel with conjugated heat transfer.

V. CONCLUSION

Particle swarm optimization and the genetic algorithm usedfor minimized the error functions in the inverse design of convective-radiative fin profile. Values C 1=0.7, C 2=1.2 andC 3=1.4 are recommend as constant parameters for thisapplications. It was shown the particle swarm optimization wasat least 3 times more efficient rather than genetic algorithm.





Therefore, particle swarm optimization recommended forgeometry optimization especially when the gradient basemethods failed.

REFERENCES

[1] S.M.H Sarvari, “Inverse determination of heat source distribution inconductive-radiative media with irregular geometry” Journal of Quantitative Spectroscopy and Radiative Transfer, Vol. 93, 2005, pp.383-395.

[2] S.M.H Sarvari, “Optimal geometry design of radiative enclosures usingthe genetic algorithm” Numerical Heat Transfer; Part A, Vol. 52(2), 2007, pp. 127-143.

[3] F. Bobaru and S. Rachakonda, “Boundary layer in shape optimization of convective fins using a meshfree approach” Int. J. Numer. Meth. Engng. Vol. 60, 2004, pp. 1215 – 1236.

[4] H. Azarkish, S.M.H. Sarvari and A. Behzadmehr, “Optimum geometrydesign of a longitudinal fin with volumetric heat generation under the

influences of natural convection and radiation” Energy Conversion and.Management., Vol. 51, 1010, pp. 1938 – 1946.

[5] H. Azarkish, S.M.H. Sarvari and A. Behzadmehr, “Optimum design of alongitudinal fin array with convection and radiation heat transfer using agenetic algorithm” International Journal of Thermal Science. Vol. 49,2010, pp. 2222-2229.

[6] S.V. Patankar, “Numerical Heat Transfer and Fluid Flow” McGraw-Hill,New York, 1980.

[7] L. Piegel and W. Tiller, “The NURBS Book ” second ed. Springer-Verlag Berlin, 1997.

[8] A. Osyczka, “Evolutionary algorithms for single and multicriteria designoptimization” Springer-Verlag, Berlin, 2002.

[9] M. Clerc, “Particle swarm optimization” ISTE Ltd, London, 2006.

Hassan Azarkish received the B.S. and M.S. degrees

in mechanical engineering from University of Sistan

and Baluchestan in 2006 and 2009 respectively.

He is currently the Ph.D. student of mechanical

engineering in Sistan and Baluchestan University. His

research interests are in the field of micro and macro

scale heat transfer and optimization. His current efforts are focused

on modeling and optimum design of thermal and fluidic MEMS

devices.

Said Farahat received the B.S. degree in mechanical

engineering from Sharif University of Technology in

1978 and M.S. degree in computer science from

Sharif University of technology in 1983. He is also

received the M.S. degree in mechanical engineering

from University of Concordia in 1987 and Ph.D. degree in

Mechanical Engineering, (computer automated and integrated design)

from University of Ottawa in 1999.

He is currently associate professor in mechanical engineering

department of Sistan and Baluchestan University. His research

interests are in the field of control, optimization and numerical

analysis.

Seyyed Masoud Hosseini Sarvari received the B.S.,

M.S. and Ph.D. degrees in mechanical engineering

from Shahid Bahonar University in 1992, 1995 and

2002 respectively.

He is currently associate professor in mechanical

engineering department of Shahid Bahonar University. His research

interests are in the field of inverse radiation design problems.

comparing the performance of the particle swarm optimization and the genetic algorithm on the...

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