Research Article Accelerated Particle Swarm for … · index permitted by the code of the practice. ... Canonical Particle Swarm Optimization (PSO) ... algorithm was coded in the

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 649857 6 pageshttpdxdoiorg1011552013649857

Research ArticleAccelerated Particle Swarm for Optimum Design ofFrame Structures

S Talatahari1 E Khalili2 and S M Alavizadeh3

1 Marand Faculty of Engineering University of Tabriz Tabriz 51666-14766 Iran2Department of Engineering Islamic Azad University Ahar Branch Ahar 54516 Iran3Department of Structural Engineering Islamic Azad University Shabestar BranchShabestar 57168-14758 Iran

Correspondence should be addressed to S Talatahari talataharitabrizuacir

Received 29 November 2012 Revised 29 December 2012 Accepted 30 December 2012

Academic Editor Fei Kang

Copyright copy 2013 S Talatahari 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

Accelerated particle swarm optimization (APSO) is developed for finding optimum design of frame structures APSO shows someextra advantages in convergence for global searchThemodifications on standard PSO effectively accelerate the convergence rate ofthe algorithm and improve the performance of the algorithm in finding better optimum solutions The performance of the APSOalgorithm is also validated by solving two frame structure problems

1 Introduction

Optimum design of frame structures are inclined to deter-mine suitable sections for elements that fulfill all designrequirements while having the lowest possible cost In thisissue optimization provides engineers with a variety of tech-niques to deal with these problems [1] These techniques canbe categorized as two general groups classical methods andmetaheuristic approaches [2] Classical methods are oftenbased on mathematical programming and many of meta-heuristic methods make use of the ideas from nature and donot suffer the discrepancies of mathematical programming[3ndash8]

Particle swarmoptimization (PSO) one ofmeta-heuristicalgorithms is based on the simulation of the social behaviorof bird flocking and fish schooling PSO is the most suc-cessful swarm intelligence inspired optimization algorithmsHowever the local search capability of PSO is poor [9] sincepremature convergence occurs often In order to overcomethese disadvantages of PSO many improvements have beenproposed Shi and Eberhart [10] introduced a fuzzy system toadapt the inertia weight for three benchmark test functionsLiu et al [11] proposed center particle swarm optimization

by adding a center particle into PSO to improve the perfor-mance An improved quantum-behaved PSO was proposedby Xi et al [12] Jiao et al [13] proposed the dynamic inertiaweight PSO by defining a dynamic inertia weight to decreasethe inertia factor in the velocity update equation of theoriginal PSO Yang et al [14] proposed another dynamicinertia weight to modify the velocity update formula in amethod called modified particle swarm optimization withdynamic adaptation

A number of studies have applied the PSO and improvedit to be used in the field of structural engineering [15ndash21]In this study we developed an improved PSO so-calledaccelerated particle swarm optimization (APSO) [22] to findoptimum design of frame structures The resulted methodis then tested by some numerical examples to estimate itspotential for solving structural optimization problems

2 Statement of StructuralOptimization Problem

Optimumdesign of structures includes finding optimum sec-tions for members that minimizes the structural weight 119882

2 Mathematical Problems in Engineering

This minimum design should also satisfy inequality con-straints that limit design variables and structural responsesThus the optimal design of a structure is formulated as [23]

minimize 119882(119909) =

119899

sum

119894=1

120574119894sdot 119860119894sdot 119897119894

subject to 119892min le 119892119894(119909) le 119892max 119894 = 1 2 3 119898

(1)

where 119882(119909) is the weight of the structure 119899 and 119898 are thenumber ofmembersmaking up the structure and the numberof total constraints respectively max and min denote upperand lower bounds respectively 119892(119909) denotes the con-straints considered for the structure containing interactionconstraints as well as the lateral and interstory displacementsas follows

The maximum lateral displacement

119892Δ

=Δ119879

119867minus 119877 ge 0 (2)

The interstory displacements

119892119889

119895=

119889119895

ℎ119895

minus 119877119868ge 0 119895 = 1 2 119899119904 (3)

where Δ119879is the maximum lateral displacement 119867 is the

height of the frame structure 119877 is the maximum drift index119889119895is the inter-story drift ℎ

119895is the story height of the 119895th floor

119899119904 is the total number of stories 119877119868is the inter-story drift

index permitted by the code of the practiceLRFD interaction formula constraints (AISC 2001 [24

Equation H1-1ab])

119892119868

119894=

119875119906

2120601119888119875119899

+(119872119906119909

120601119887119872119899119909

+

119872119906119910

120601119887119872119899119910

)minus1 ge 0 for119875119906

120601119888119875119899

lt02

119892119868

119894=

119875119906

120601119888119875119899

+8

9(

119872119906119909

120601119887119872119899119909

+

119872119906119910

120601119887119872119899119910

)minus1 ge 0 for119875119906

120601119888119875119899

ge02

(4)

where 119875119906is the required strength (tension or compression)

119875119899is the nominal axial strength (tension or compression)

120601119888is the resistance factor (120601

119888= 09 for tension 120601

119888= 085

for compression) 119872119906119909

and 119872119906119910

are the required flexuralstrengths in the119909 and119910directions respectively119872

119899119909and119872

119899119910

are the nominal flexural strengths in the 119909 and 119910 directions(for two-dimensional structures 119872

119899119910= 0) 120601

119887is the flexural

resistance reduction factor (120601119887= 090)

For the proposed method it is essential to transform theconstrained optimization problem to an unconstraint one Adetailed review of some constraint-handling approaches ispresented in [25] In this study a modified penalty functionmethod is utilized for handling the design constraints whichis calculated using the following formulas [2]

119892119894le 119900 997904rArr Φ

(119894)

119892= 0

119892119894gt 119900 997904rArr Φ

(119894)

119892= 119892119894

(5)

The objective function that determines the fitness of eachparticle is defined as

Mer119896 = 1205761sdot 119882119896

+ 1205762sdot (sumΦ

(119894)

119892)1205763

(6)

where Mer is the merit function to be minimized 1205761 1205762

and 1205763are the coefficients of merit function Φ(119894)

119892denotes

the summation of penalties In this study 1205761and 1205762are set

to 1 and 119882 (the weight of structure) respectively while thevalue of 120576

3is taken as 085 in order to achieve a feasible

solution [26] Before calculating Φ(119894)

119892 we first determine the

weight of the structures generated by the particles and ifit becomes smaller than the so far best solution then Φ

(119894)

119892

will be calculated otherwise the structural analysis doesnot perform This methodology will decrease the requiredcomputational costs considerably

3 Canonical Particle SwarmOptimization (PSO)

The PSO algorithm inspired by social behavior simulation[27 28] is a population-based optimization algorithm whichinvolves a number of particles that move through the searchspace and their positions are updated based on the bestpositions of individual particles (called 119909

lowast

119894) and the best of

the swarm (called 119892lowast) in each iteration This matter is shown

mathematically as the following equations

119907119905+1

119894= 119908 sdot 119907

119905

119894+ 120572 sdot rand

1(119909lowast

119894minus 119909119905

119894) + 120573 sdot rand

2(119892lowast

119894minus 119909119905

119894)

(7)

119909119905+1

119894= 119909119905

119894+ 120584119905+1

119894 (8)

where119909119894and 119907119894represent the current position and the velocity

of the 119894th particle respectively rand1and rand

2represent

random numbers between 0 and 1 119909lowast119894is the best position

visited by each particle itself 119892lowast corresponds to the globalbest position in the swarm up to iteration 119896 120572 and 120573 representcognitive and social parameters respectively According toKennedy and Eberhart [27] these two constants are set to 2 inorder to make the average velocity change coefficient close toone 119882 is a weighting factor (inertia weight) which controlsthe trade-off between the global exploration and the localexploitation abilities of the flying particles A larger inertiaweight makes the global exploration easier while a smallerinertia weight tends to facilitate local exploitationThe inertiaweight can be reduced linearly from 09 to 04 during theoptimization process [29]

4 Accelerated Particle Swam Optimization

The standard PSO uses both the current global best 119892lowast andthe individual best 119909

lowast The reason of using the individualbest is primarily to increase the diversity in the qualitysolutions however this diversity can be simulated using somerandomness Subsequently there is no compelling reason forusing the individual best unless the optimization problem ofinterest is highly nonlinear and multimodal [22]

Mathematical Problems in Engineering 3

8 at 304 m

304 m

12592 kN(2831 kips)

8743 kN(1905 kips)

6054 kN(1361 kips)

7264 kN(1633 kips)

4839 kN(1088 kips)

363 kN(0816 kips)

121 kN(0272 kips)

242 kN(0544 kips)

119860 4448 kN (100 kips) downwardload is applied at each connection

(10prime )

(10prime )

8

8

4 4

4 4

7

3 3

7

3 3

6

2 2

6

2 2

1 1

1 1

5

5

Figure 1 Topology of the 1-bay 8-story frame

A simplified version which could accelerate the conver-gence of the algorithm is to use the global best only Thus inthe APSO [22] the velocity vector is generated by a simplerformula as

119907119905+1

119894= 119907119905

119894+ 120572 sdot rand119899 (119905) + 120573 sdot (119892

lowast

minus 119909119905

119894) (9)

where rand119899 is drawn from 119873(0 1) to replace the secondterm The update of the position is simply like (8) In orderto increase the convergence even further we can also writethe update of the location in a single step as

119909119905+1

119894= (1 minus 120573) 119909

119905

119894+ 120573119892lowast

+ 120572119903 (10)

Table 1 Optimal design comparison for the 1-bay 8-story frame

Element group Optimal W-shaped sections This studyGA [31] ACO [32] IACO [26]

1 W18 times 35 W16 times 26 W21 times 44 W21 times 442 W18 times 35 W18 times 40 W18 times 35 W16 times 263 W18 times 35 W18 times 35 W18 times 35 W14 times 224 W18 times 26 W14 times 22 W12 times 22 W12 times 165 W18 times 46 W21 times 50 W18 times 40 W18 times 356 W16 times 31 W16 times 26 W16 times 26 W18 times 357 W16 times 26 W16 times 26 W16 times 26 W18 times 358 W12 times 16 W12 times 14 W12 times 14 W16 times 26Weight (kN) 3283 3168 3105 3091


Element group Optimal W-shaped sections This studyPSO [18] HBB-BC [33] ICA [34]

1 W33 times 118 W24 times 117 W24 times 117 W27 times 1292 W33 times 263 W21 times 132 W21 times 147 W21 times 1473 W24 times 76 W12 times 95 W27 times 84 W16 times 774 W36 times 256 W18 times 119 W27 times 114 W27 times 1145 W21 times 73 W21 times 93 W14 times 74 W14 times 746 W18 times 86 W18 times 97 W18 times 86 W30 times 997 W18 times 65 W18 times 76 W12 times 96 W12 times 728 W21 times 68 W18 times 65 W24 times 68 W12 times 799 W18 times 60 W18 times 60 W10 times 39 W8 times 2410 W18 times 65 W10 times 39 W12 times 40 W14 times 4311 W21 times 44 W21 times 48 W21 times 44 W21 times 44Weight (kN) 49668 43454 41746 41150

This simpler version will give the same order of conver-gence [30] Typically 120572 = 01 119871ndash05 119871 where 119871 is the scaleof each variable while 120573 = 02ndash07 is sufficient for mostapplications It is worth pointing out that the velocity does notappear in (10) and there is no need to deal with initializationof velocity vectors Therefore the APSO is much simplerComparing withmany PSO variants the APSO uses only twoparameters and the mechanism is simple to understand Afurther improvement to the accelerated PSO is to reduce therandomness as iterations proceedThismeans that we can useamonotonically decreasing function In our implementationwe use [30]

120572 = 07119905

(11)

where 119905 isin [0 119905max] and 119905max is the maximum number ofiterations

5 Numerical Examples

This section presents the numerical examples to evaluatethe capability of the new algorithm in finding the optimaldesign of the steel structures The final results are comparedto the solutions of other methods to show the efficiency ofthe present approach The proposed algorithm is coded inMatlab and structures are analyzed using the direct stiffness


1110

1110

119

1110

1110

119

1110

1110

119

118

118

117

118

118

117

118

118

117

116

116

115

116

116

115

116

116

115

114

114

113

114

114

113

114

114

113

11112

111

11112

111

111121

1

1

3

3

3

5

5

5

7

7

7

9

9

9

2

11

1

1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 = 50kNm

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

14 at35m

4m

3 at 5m

(342 kipsft)

(115ft)

(131ft)

(164ft)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)


methodThe steel members used for the design consist of 267W-shaped sections from the AISC database

51 1-Bay 8-Story Frame Figure 1 shows the configuration ofthe 1-bay 8-story framed structure and applied loads Severalresearchers have developed design procedures for this frameCamp et al [31] used a genetic algorithm Kaveh and Shojaee

0 50 100 150 200 250400

500

600

700

800

900

1000

Iteration

Wei

ght

Current best resultSo far best result

Figure 3 The convergence history for the 3-bay 15-story frame

[32] utilized ACO and Kaveh and Talatahari [26] applied animproved ACO to solve this problem

TheAPSOalgorithm found the optimalweight of the one-bay eight-story frame to be 3091 kN which is the best onecompared to the othermethod Table 1 lists the optimal valuesof the eight design variables obtained by this research andcompares them with other results

52 Design of a 3-Bay 15-Story Frame The configuration andapplied loads of a 3-bay 15-story frame structure is shownin Figure 2 The sway of the top story is limited to 235 cm(925 in) The material has a modulus of elasticity equal to119864 = 200GPa and a yield stress of 119865

119910= 2482MPa

The effective length factors of the members are calculatedas 119870119909

ge 0 for a sway-permitted frame and the out-of-planeeffective length factor is specified as 119870

119910= 10 Each column

is considered as non-braced along its length and the non-braced length for each beammember is specified as one-fifthof the span length

The optimum design of the frame obtained by usingAPSO has the minimum weight of 41150 kN The optimumdesigns for PSO [18] HBB-BC [33] and ICA [34] had theweights of 49668 kN 43454 kN and 41746 kN respectivelyTable 2 summarizes the optimal results for these differentalgorithms Clearly it can be seen that the present algorithmcan find the better design Figure 3 provides the convergencehistory for this example obtained by the APSO

6 Conclusions

The APSO algorithm as an improved meta-heuristic algo-rithm is developed to solve frame structural optimiza-tion problems Optimization software based on the APSOalgorithm was coded in the Matlab using object-orientedtechnology A methodology to handle the constraints is alsodeveloped in a way that we first determine the weight ofthe structures generated by the particles and if they becomesmaller than the so far best solution then the structural


analyses are performed Two test problems were studiedusing the optimization program to show the efficiency ofthe algorithm The comparison of the results of the newalgorithm with those of other algorithms shows that theAPSO algorithm provides results as good as or better thanother algorithms and can be used effectively for solvingengineering problems

References

[1] A Kaveh and S Talatahari ldquoOptimal design of skeletal struc-tures via the charged system search algorithmrdquo Structural andMultidisciplinary Optimization vol 41 no 6 pp 893ndash911 2010

[2] A Kaveh B F Azar and S Talatahari ldquoAnt colony optimizationfor design of space trussesrdquo International Journal of SpaceStructures vol 23 no 3 pp 167ndash181 2008

[3] S Talatahari M Kheirollahi C Farahmandpour and A HGandomi ldquoOptimum design of truss structures using multistage particle swarm optimizationrdquoNeural Computing amp Appli-cations 2012

[4] S Chen Y Zheng C Cattani and W Wang ldquoModeling ofbiological intelligence for SCM systemoptimizationrdquoComputa-tional and Mathematical Methods in Medicine vol 2012 ArticleID 769702 10 pages 2012

[5] S Chen Y Wang and C Cattani ldquoKey issues in modeling ofcomplex 3D structures from video sequencesrdquo MathematicalProblems in Engineering vol 2012 Article ID 856523 17 pages2012

[6] S ChenWHuang C Cattani andG Altieri ldquoTraffic dynamicson complex networks a surveyrdquo Mathematical Problems inEngineering vol 2012 Article ID 732698 23 pages 2012

[7] F Kang J Li and ZMa ldquoRosenbrock artificial bee colony algo-rithm for accurate global optimization of numerical functionsrdquoInformation Sciences vol 181 no 16 pp 3508ndash3531 2011

[8] F Kang J Li and Z Ma ldquoAn artificial bee colony algorithmfor locating the critical slip surface in slope stability analysisrdquoEngineering Optimization vol 45 no 2 pp 207ndash223 2013

[9] P Angeline ldquoEvolutionary optimization versus particle swarmoptimization philosophy and performance differencerdquo in Pro-ceedings of the Evolutionary Programming Conference SanDiego Calif USA 1998

[10] Y Shi and R C Eberhart ldquoFuzzy adaptive particle swarm opti-mizationrdquo in Proceedings of the Congress on Evolutionary Com-putation pp 101ndash106 May 2001

[11] Y Liu Z Qin Z Shi and J Lu ldquoCenter particle swarm optimi-zationrdquo Neurocomputing vol 70 no 4-6 pp 672ndash679 2007

[12] M Xi J Sun andW Xu ldquoAn improved quantum-behaved par-ticle swarm optimization algorithm with weighted mean bestpositionrdquo Applied Mathematics and Computation vol 205 no2 pp 751ndash759 2008

[13] B Jiao Z Lian and X S Gu ldquoA dynamic inertia weight particleswarm optimization algorithmrdquo Chaos Solitons and Fractalsvol 37 no 3 pp 698ndash705 2008

[14] X Yang J Yuan J Yuan and H Mao ldquoA modified particleswarm optimizer with dynamic adaptationrdquoAppliedMathemat-ics and Computation vol 189 no 2 pp 1205ndash1213 2007

[15] A Kaveh and S Talatahari ldquoA particle swarm ant colony opti-mization for truss structures with discrete variablesrdquo Journal ofConstructional Steel Research vol 65 no 8-9 pp 1558ndash15682009

[16] A Kaveh and S Talatahari ldquoParticle swarm optimizer ant col-ony strategy and harmony search scheme hybridized for opti-mization of truss structuresrdquo Computers and Structures vol 87no 5-6 pp 267ndash283 2009

[17] P C Fourie and A A Groenwold ldquoThe particle swarm optimi-zation algorithm in size and shape optimizationrdquo Structural andMultidisciplinary Optimization vol 23 no 4 pp 259ndash267 2002

[18] A Kaveh and S Talatahari ldquoHybrid algorithm of harmonysearch particle swarm and ant colony for structural design opti-mizationrdquo in Studies in Computational Intelligence vol 239 pp159ndash198 Springer Berlin Heidelberg 2009

[19] A Hadidi A Kaveh B Farahnadazar S Talatahari and CFarahmandpour ldquoAn efficient hybrid algorithm based on parti-cle swarm and simulated annealing for optimal design of spacetrussesrdquo International Journal of Optimization in Civil Engineer-ing vol 1 no 3 pp 377ndash395 2011

[20] J F Schutte and A A Groenwold ldquoSizing design of truss struc-tures using particle swarmsrdquo Structural and MultidisciplinaryOptimization vol 25 no 4 pp 261ndash269 2003

[21] F Kang J Li and Q Xu ldquoDamage detection based on improvedparticle swarm optimization using vibration datardquo Applied SoftComputing vol 12 no 8 pp 2329ndash2335 2012

[22] X S Yang Nature-Inspired Metaheuristic Algorithms LuniverPress 2nd edition 2010

[23] A Kaveh and S Talatahari ldquoCharged system search for optimaldesign of frame structuresrdquo Applied Soft Computing vol 12 no1 pp 382ndash393 2012

[24] American Institute of Steel Construction (AISC) Manual ofSteel Construction Load Resistance Factor Design AISC Chica-go Ill USA 3rd edition 2001

[25] C A C Coello ldquoTheoretical and numerical constraint-han-dling techniques used with evolutionary algorithms a surveyof the state of the artrdquo Computer Methods in Applied Mechanicsand Engineering vol 191 no 11-12 pp 1245ndash1287 2002

[26] A Kaveh and S Talatahari ldquoAn improved ant colony optimi-zation for the design of planar steel framesrdquo Engineering Struc-tures vol 32 no 3 pp 864ndash873 2010

[27] J Kennedy and R C Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks vol 4 pp 1942ndash1948 December 1995

[28] R C Eberhart and J Kennedy ldquoA new optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium onMicroMachine and Human Science pp 39ndash43 NagoyaJapan October 1995

[29] R C Eberhart and Y Shi ldquoComparing inertia weights and con-striction factors in particle swarm optimizationrdquo in Proceedingsof the Congress on Evolutionary Computation (CEC rsquo00) pp 84ndash88 July 2000

[30] A H Gandomi G J Yun X S Yang and S Talatahari ldquoCom-bination of chaos and accelerated particle swarm optimizationrdquoCommunications in Nonlinear Science and Numerical Simula-tions vol 18 pp 327ndash340 2013

[31] C V Camp S Pezeshk and G Cao ldquoOptimized design of two-dimensional structures using a genetic algorithmrdquo Journal ofStructural Engineering vol 124 no 5 pp 551ndash559 1998

[32] A Kaveh and S Shojaee ldquoOptimal design of skeletal structuresusing ant colony optimizationrdquo International Journal forNumer-ical Methods in Engineering vol 70 no 5 pp 563ndash581 2007

[33] A Kaveh and S Talatahari ldquoA discrete Big BangmdashBig Crunchalgorithm for optimal design of skeletal structuresrdquo AsianJournal of Civil Engineering vol 11 no 1 pp 103ndash122 2010


[34] A Kaveh and S Talatahari ldquoOptimum design of skeletal struc-tures using imperialist competitive algorithmrdquo Computers andStructures vol 88 no 21-22 pp 1220ndash1229 2010

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This minimum design should also satisfy inequality con-straints that limit design variables and structural responsesThus the optimal design of a structure is formulated as [23]

minimize 119882(119909) =

119899

sum

119894=1

120574119894sdot 119860119894sdot 119897119894

subject to 119892min le 119892119894(119909) le 119892max 119894 = 1 2 3 119898

(1)

where 119882(119909) is the weight of the structure 119899 and 119898 are thenumber ofmembersmaking up the structure and the numberof total constraints respectively max and min denote upperand lower bounds respectively 119892(119909) denotes the con-straints considered for the structure containing interactionconstraints as well as the lateral and interstory displacementsas follows

The maximum lateral displacement

119892Δ

=Δ119879

119867minus 119877 ge 0 (2)

The interstory displacements

119892119889

119895=

119889119895

ℎ119895

minus 119877119868ge 0 119895 = 1 2 119899119904 (3)

where Δ119879is the maximum lateral displacement 119867 is the

height of the frame structure 119877 is the maximum drift index119889119895is the inter-story drift ℎ

119895is the story height of the 119895th floor

119899119904 is the total number of stories 119877119868is the inter-story drift

index permitted by the code of the practiceLRFD interaction formula constraints (AISC 2001 [24

Equation H1-1ab])

119892119868

119894=

119875119906

2120601119888119875119899

+(119872119906119909

120601119887119872119899119909

+

119872119906119910

120601119887119872119899119910

)minus1 ge 0 for119875119906

120601119888119875119899

lt02

119892119868

119894=

119875119906

120601119888119875119899

+8

9(

119872119906119909

120601119887119872119899119909

+

119872119906119910

120601119887119872119899119910

)minus1 ge 0 for119875119906

120601119888119875119899

ge02

(4)

where 119875119906is the required strength (tension or compression)

119875119899is the nominal axial strength (tension or compression)

120601119888is the resistance factor (120601

119888= 09 for tension 120601

119888= 085

for compression) 119872119906119909

and 119872119906119910

are the required flexuralstrengths in the119909 and119910directions respectively119872

119899119909and119872

119899119910

are the nominal flexural strengths in the 119909 and 119910 directions(for two-dimensional structures 119872

119899119910= 0) 120601

119887is the flexural

resistance reduction factor (120601119887= 090)

For the proposed method it is essential to transform theconstrained optimization problem to an unconstraint one Adetailed review of some constraint-handling approaches ispresented in [25] In this study a modified penalty functionmethod is utilized for handling the design constraints whichis calculated using the following formulas [2]

119892119894le 119900 997904rArr Φ

(119894)

119892= 0

119892119894gt 119900 997904rArr Φ

(119894)

119892= 119892119894

(5)

The objective function that determines the fitness of eachparticle is defined as

Mer119896 = 1205761sdot 119882119896

+ 1205762sdot (sumΦ

(119894)

119892)1205763

(6)

where Mer is the merit function to be minimized 1205761 1205762

and 1205763are the coefficients of merit function Φ(119894)

119892denotes

the summation of penalties In this study 1205761and 1205762are set

to 1 and 119882 (the weight of structure) respectively while thevalue of 120576

3is taken as 085 in order to achieve a feasible

solution [26] Before calculating Φ(119894)

119892 we first determine the

weight of the structures generated by the particles and ifit becomes smaller than the so far best solution then Φ

(119894)

119892

will be calculated otherwise the structural analysis doesnot perform This methodology will decrease the requiredcomputational costs considerably

3 Canonical Particle SwarmOptimization (PSO)

The PSO algorithm inspired by social behavior simulation[27 28] is a population-based optimization algorithm whichinvolves a number of particles that move through the searchspace and their positions are updated based on the bestpositions of individual particles (called 119909

lowast

119894) and the best of

the swarm (called 119892lowast) in each iteration This matter is shown

mathematically as the following equations

119907119905+1

119894= 119908 sdot 119907

119905

119894+ 120572 sdot rand

1(119909lowast

119894minus 119909119905

119894) + 120573 sdot rand

2(119892lowast

119894minus 119909119905

119894)

(7)

119909119905+1

119894= 119909119905

119894+ 120584119905+1

119894 (8)

where119909119894and 119907119894represent the current position and the velocity

of the 119894th particle respectively rand1and rand

2represent

random numbers between 0 and 1 119909lowast119894is the best position

visited by each particle itself 119892lowast corresponds to the globalbest position in the swarm up to iteration 119896 120572 and 120573 representcognitive and social parameters respectively According toKennedy and Eberhart [27] these two constants are set to 2 inorder to make the average velocity change coefficient close toone 119882 is a weighting factor (inertia weight) which controlsthe trade-off between the global exploration and the localexploitation abilities of the flying particles A larger inertiaweight makes the global exploration easier while a smallerinertia weight tends to facilitate local exploitationThe inertiaweight can be reduced linearly from 09 to 04 during theoptimization process [29]

4 Accelerated Particle Swam Optimization

The standard PSO uses both the current global best 119892lowast andthe individual best 119909

lowast The reason of using the individualbest is primarily to increase the diversity in the qualitysolutions however this diversity can be simulated using somerandomness Subsequently there is no compelling reason forusing the individual best unless the optimization problem ofinterest is highly nonlinear and multimodal [22]


8 at 304 m

304 m

12592 kN(2831 kips)

8743 kN(1905 kips)

6054 kN(1361 kips)

7264 kN(1633 kips)

4839 kN(1088 kips)

363 kN(0816 kips)

121 kN(0272 kips)

242 kN(0544 kips)


(10prime )

(10prime )

8

8

4 4

4 4

7

3 3

7

3 3

6

2 2

6

2 2

1 1

1 1

5

5



119907119905+1

119894= 119907119905

119894+ 120572 sdot rand119899 (119905) + 120573 sdot (119892

lowast

minus 119909119905

119894) (9)


119909119905+1

119894= (1 minus 120573) 119909

119905

119894+ 120573119892lowast

+ 120572119903 (10)








120572 = 07119905

(11)





1110

1110

119

1110

1110

119

1110

1110

119

118

118

117

118

118

117

118

118

117

116

116

115

116

116

115

116

116

115

114

114

113

114

114

113

114

114

113

11112

111

11112

111

111121

1

1

3

3

3

5

5

5

7

7

7

9

9

9

2

11

1

1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 = 50kNm

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

14 at35m

4m

3 at 5m

(342 kipsft)

(115ft)

(131ft)

(164ft)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)




0 50 100 150 200 250400

500

600

700

800

900

1000

Iteration

Wei

ght






119910= 2482MPa






6 Conclusions




References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










8 at 304 m

304 m

12592 kN(2831 kips)

8743 kN(1905 kips)

6054 kN(1361 kips)

7264 kN(1633 kips)

4839 kN(1088 kips)

363 kN(0816 kips)

121 kN(0272 kips)

242 kN(0544 kips)


(10prime )

(10prime )

8

8

4 4

4 4

7

3 3

7

3 3

6

2 2

6

2 2

1 1

1 1

5

5



119907119905+1

119894= 119907119905

119894+ 120572 sdot rand119899 (119905) + 120573 sdot (119892

lowast

minus 119909119905

119894) (9)


119909119905+1

119894= (1 minus 120573) 119909

119905

119894+ 120573119892lowast

+ 120572119903 (10)








120572 = 07119905

(11)





1110

1110

119

1110

1110

119

1110

1110

119

118

118

117

118

118

117

118

118

117

116

116

115

116

116

115

116

116

115

114

114

113

114

114

113

114

114

113

11112

111

11112

111

111121

1

1

3

3

3

5

5

5

7

7

7

9

9

9

2

11

1

1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 = 50kNm

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

14 at35m

4m

3 at 5m

(342 kipsft)

(115ft)

(131ft)

(164ft)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)




0 50 100 150 200 250400

500

600

700

800

900

1000

Iteration

Wei

ght






119910= 2482MPa






6 Conclusions




References











































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1110

1110

119

1110

1110

119

1110

1110

119

118

118

117

118

118

117

118

118

117

116

116

115

116

116

115

116

116

115

114

114

113

114

114

113

114

114

113

11112

111

11112

111

111121

1

1

3

3

3

5

5

5

7

7

7

9

9

9

2

11

1

1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 1199081 1199081

1199081 = 50kNm

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

30kN

14 at35m

4m

3 at 5m

(342 kipsft)

(115ft)

(131ft)

(164ft)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)

(675 kips)




0 50 100 150 200 250400

500

600

700

800

900

1000

Iteration

Wei

ght






119910= 2482MPa






6 Conclusions




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
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Accelerated Particle Swarm for … · index permitted by the code of the practice. ... Canonical Particle Swarm Optimization (PSO) ... algorithm was coded in the