CONFIGURATION OPTIMIZATION OF TRUSSES USING A MULTI HEURISTIC BASED SEARCH METHOD

Transcription

1 INTERNATIONAL JOURNAL OF OPTIMIZATION IN CIVIL ENGINEERING Int. J. Optm. Cvl Eng., 2013; 3(1): CONFIGURATION OPTIMIZATION OF TRUSSES USING A MULTI HEURISTIC BASED SEARCH METHOD A. Kaveh 1,*,, V.R. Kalatjar 2, M.H. Talebpour 2 and J. Torkamanzadeh 2 1 Department of Cvl Engneerng, Iran Unversty of Scence and Technology, Tehran, Iran 2 Department of Cvl Engneerng, Shahrood Unversty of Technology, Shahrood, Iran ABSTRACT Dfferent methods are avalable for smultaneous optmzaton of cross-secton, topology and geometry of truss structures. Snce the search space for ths problem s very large, the probablty of fallng n local optmum s consderably hgh. On the other hand, dfferent types of desgn varables (contnuous and dscrete) lead to some dffcultes n the process of optmzaton. In ths artcle, smultaneous optmzaton of cross-secton, topology and geometry of truss structures s performed by utlzng the Mult Heurstc based Search Method (MHSM) that overcome the above mentoned problem and obtans good results. The presented method performs the optmzaton by dvdng the searchng space nto fve subsectons n whch an MHSM s employed. These subsectons are named procedure slands. Some examples are then presented to scrutnze the method more carefully. Results show the capabltes of the present algorthm for optmal desgn of truss structures. Receved: 15 August 2012; Accepted: 10 January 2013 KEY WORDS: optmzaton; truss structures; mult heurstc based search method (MHSM) 1. INTRODUCTION * Correspondng author: A. Kaveh, Centre of Excellence for Fundamental Studes n Structural Engneerng, Iran Unversty of Scence and Technology, Tehran-16, Iran E-mal address: alkaveh@ust.ac.r (A. Kaveh)

2 152 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh Systems n structural engneerng usually endure the burden and transfer the forces to supports. In addton to be safe and applcable, these systems should be desgned wth less cost and materal. Therefore, optmal desgn of skeletal structures lke trusses wll be categorzed to four man dvsons: szng optmzaton, topology optmzaton, shape or geometry optmzaton and confguraton optmzaton. In the case of szng, optmzaton wll be carred out for approprate amounts of cross-sectonal areas wth fxed nodal coordnates (fxed geometry) and fxed connectvty of members (fxed topology). In applcaton of engneerng, standard sectons whch are avalable are usually used. In ths regard, optmzaton of separated sectons s developed by selectng the members from the avalable profles [1-12]. Concernng the shape or geometry, nodal coordnates of truss are adjusted for optmal desgn. Ths ssue s often consdered conjugated wth cross-sectons n related artcles. That s, the man goal n these type of optmzaton s to fnd the cross-sectons and nodal coordnates [13-17]. In topology optmzaton, best connectvty of the members s determned. Ths optmzaton s often performed by smultaneous optmzaton of cross-secton assumng the geometry of the structure to be fxed [18-25]. For confguraton optmzaton of structures the best cross-secton of members n the best confguraton and postonng of nodes are determned. In ths method, all three varables are ncluded n varables vector of the desgn. Ths type of optmzaton leads corresponds to a large desgn space and ncreases the probablty of gettng trapped n local optmum. To avod ths problem, some researchers assume the desgn varables (geometry and topology) to be constant and then manage the optmzaton process. They then select the resulted optmzed desgn as an ntal desgn for total optmzaton and repeat the process agan [24-26]. Accordng to methods whch are proposed by some other researchers, meta-heurstc based algorthms were employed to the smultaneous optmzaton of Szng, topology and geometry (confguraton) of truss structures [27-32]. Lack of cognton of constants and relatons n such algorthms and also largeness of searchng space, n some cases, result n local optmum. Parallel global optmsaton meta-heurstcs usng an asynchronous sland-model s proposed by Izzo et al for parallel computng n optmzaton [33]. In ths method, ntal populaton s dvded to smaller subsectons whch are called slands. Then, meta-heurstc based methods are assgned to each sland and optmzaton s performed. Subsequently, after some specfc repettons, durng mgraton process best desgn of each sland s substtuted to low qualty desgn n other slands. Ths process s contnued untl completon of all repettons based on determned amount for mgraton nterval. Therefore, n each meta-heurstc based method, dependence of results on relatons and parameters s decreased remarkably. On the other hand, usng the parallel processng systems s accessble as a result of parallel search technque n desgn space. Ths ablty result n an ncrease of the speed of optmzaton operaton and more sutable results are obtaned. In order to select current methods n each sland, meta-heurstc based algorthms are separately studed, ther features are specfed and the best ones are selected based on the qualty of results and the speed of optmzaton operaton. Then, the performance of the

3 CONFIGURATION OPTIMIZATION OF TRUSSES 153 selected algorthms s mproved by some addtonal alteratons. Accordngly, the Mult Heurstc based Search Method (MHSM) s desgned for optmzng the confguraton of truss structures. 2. FORMULATION OF TRUSSES CONFIGURATION PROBLEM Formulaton of trusses confguraton problem s defned as the followng: Mnmze Subject to F Ne A, a (1) 1 C1 : Truss s acceptable to the user (2) C2 : Truss s knematcally stable (3) C3 : σ j σ all (Ten), σ j σ all (Com) j=1,2,,ne (4) C4 : k max k k=1,2,,ndof (5) In equaton (1), desgn varables are the cross-sectonal areas of the members [A], and desgn varables of geometry [ξ] are defned as followng: [A]=[a 1,a 2,,a Nos ] ; a S ; =1,,Nos (6) ξ m mn ξ m (X m,y m,z m ) ξ m max, m=1,,nn (7) In equatons (1) to (7) ρ : Materals densty of member. l (ξ): Length of the th member of truss whch depends on geometrc varables ξ. a : Cross-secton area of the member. Ne: The number of members of the optmal truss. S: Lst of the avalable profles whose szng varables are selected. Nos: Number of cross-secton areas n each desgn. Nn: Number of all the geometrc varables n optmzaton problem. σ j : Stress n the jth element of structure. σ all : Allowable tensle and compressve stress. k : Nodal dsplacement of kth degree of freedom. k max : Allowable dsplacement of kth degree of freedom. ξ m max : Upper bounds of the desgn varable ξ m. ξ m mn : Lower bounds of the desgn varable ξ m. Constrant C1. Some nodes lke supports and ponts of appled load are remarkably mportant n optmzaton of trusses. In other words, n some cases t s requred to fnd the optmal confguraton as fxng some nodes. Therefore, desgn optmzaton should

4 154 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh nclude free nodes and also nodes wth fxed postons. On the other hand, to acheve a practcal desgn s an mportant ssue n truss optmzaton, and some tme t s preferred to have symmetrc nodal coordnates even f the desgn s not qute optmal. Star Graph s used as ground structure n ths artcle [19-21]. Ths ground structure avods producng long members by contactng each node to ts neghbor nodes usng addtonal members. Constrant C2. Dfferent desgns wth varous confguratons are created n optmzaton based on meta-heurstc methods, and some of them may be unstable. In ths artcle, unstableness of each desgn s nvestgated before t s analyzed. If the desgn s unstable, t s fned. In ths regard, frstly the statc formulaes are utlzed n order to control the geometrc stablty, and then f necessary, the stffness matrx of structure s employed [21]. Knematcally stable structures have symmetrc and postve defne stffness matrces [34]. Thus, a structure wll be stable f all entres of man dagonal n the decomposed stffness matrx (n the process of Cholesky method) are postve and nonzero. Constrant C3. The stress, whch s due to composton of loadng, should be n allowed regon for all the members of the truss wth optmal confguraton. Ths allowed amount s determned by code. Therefore, n optmzaton process, after controllng the stablty of structure, stress of each member of truss wll be calculated. The amount of stress constrant volaton s determned by Eq. (8). C 3 C 3 C 3 0 f all 1 f all 1 0 all 1 0 ; 1,, Ne ; 1,, Ne (8) In ths equaton, the quantty of the constrant volaton of the members wll be summed together when the number of loadng combnatons s nlc [26]. Constrant C4. After analyzng the stable truss and computng the amounts of stress, the dsplacement of actve nodes n each desgn wll be calculated. If the dsplacement of the th degree of freedom locates n the allowed range, the desgn wll not be fned. Otherwse, the quantty of the constrant volaton of dsplacement s obtaned based on Eq. (9). C 4 C C 4 0 f all 1 4 f all 1 0 ; all 1 0 1,, Ndof ; 1,, Ndof (9)

5 CONFIGURATION OPTIMIZATION OF TRUSSES 155 In these equatons, the quantty of the constrant volaton of nodal dsplacements wll be summed together when the number of loadng combnatons s nlc. 3. PROPOSED OPTIMIZATION METHODS Meta-heurstc or heurstc algorthms are ntellgent random search methods whch search the desgn space by dfferent ponts (dfferent desgn). The logc of these algorthms s n a way that they need to produce varous enhanced desgns durng the optmzaton. Even though, varous parameters n meta-heurstc methods and also the lack of nformaton about the quantty of these parameters n each optmzaton problem results n local optmum n some cases. That s, fndng correct amount of parameters and equatons n each meta-heurstc method s a dffculty of optmzaton based on these algorthms. Varous researchers tred to mprove each method by offerng dfferent solutons and also strove to decrease the mpact of parameters of related algorthms [36-40]. In ths artcle, the optmzaton of confguraton for the truss structures s performed by Mult meta-heurstc based Search Method (MHSM). Reducng the effect of parameters of meta-heurstc algorthms and ncreasng the doman of searchng of the desgn space s the specal feature of ths method. Accordng to ths method, ntal populaton s dvded to several slands. Each sland has optmzaton method wth dstnctve structure based on a meta-heurstc algorthm. Ths arrangement of acton leads to varaton n answers. The proposed MHSM method s performed n two varants of MHSM.1 and MHSM.2 [21 and 41]. (a) MHSM.1 (b) MHSM.2 Fgure 1 Two MHSMs of the Mult Heurstc based Search Method In MHSM.1, based on the mgraton rate, the best members of each sland after several specfc generatons are alternatvely transferred to other slands based on mgraton nterval. Each subpopulaton, n mgraton process, has a random destnaton whch be dentfed n each perod of mgraton. Mgraton operator, transfer a specfc percent of subpopulaton to another sland and substtute a low qualty member. Each exsted meta-heurstc method, after mgraton process, combnes mgrated and resdual

6 156 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh members n order to obtan a populaton of hgher qualty. Due to the presence of mgraton mechansm n MHSM.1, answers have remarkable dversty durng search process. Hence, each optmzaton problem s nvestgated by several methods and the desgn space s mmedately searched. Then, the best results are shared among other slands and new members are allocated to each sland. Fgure 1 shows the optmzaton process based on MHSM.1. Start Creatng ntal populaton Dstrbutng populaton among slands Formng populaton n sland () ; =1, 2,, 5 Perforng optmzaton process of sland () based on one of GA, HS, CSS, ACO and PSO methods usng Fgure 3 structures Creatng new populaton n sland () Not satsfyng mgraton nterval condton Investgatng mgraton nterval Satsfyng mgraton nterval condton Separaton of bests n each sland based on mgraton rate Selectng destnaton of mgraton for each sland, randomly Mgraton the best of each sland to destnaton Creatng new populaton n sland () Investgatng convergence and controllng number generaton steps Satsfyng fnshng condton Determnng and ntroducng best structure as optmum desgn Not satsfyng fnshng condton Fgure 1. Frst varant of Mult Heurstc Searchng Method (MHSM.1) In MHSM.2 durng the mgraton nterval, most approprate sland s recognzed n attanng to mnmum. Then, the algorthm on the best sland overcomes n the entre system. In other word, best results whch are obtaned n dfferent slands, gather n the selected sland and optmzaton process wll be contnued based on the algorthm of selected sland. Ths feature results n ncreasng the speed of the optmzaton process. End

7 CONFIGURATION OPTIMIZATION OF TRUSSES 157 Mgraton nterval n ths method s more than MHSM.1. Fgure 2 llustrates the optmzaton process based on ths method. Start Creatng ntal populaton Dstrbutng populaton among slands Formng the populaton n sland () ; =1, 2,, 5 Performng the optmzaton process of sland () based on one of GA, HS, CSS, ACO and PSO methods usng Fgure 3 structures Creatng new populaton n sland () Not satsfyng mgraton nterval condton Investgatng mgraton nterval Satsfyng mgraton nterval condton Recognzng the best sland based on mert ndex Transferrng best sland desgns to the selected sland and creatng new populaton n the selected sland Performng optmzaton process accordng to selected sland method Investgatng convergence and controllng number of generaton steps Satsfyng fnshng condton Determnng and ntroducng thebest structure as optmum desgn Not satsfyng fnshng condton Fgure 2. Second varant of the Mult Heurstc Searchng Method (MHSM.2) In ssues such as optmzaton of szng, topology and geometry that as a result of numerous desgn varables, searchng space become large and the effect of parameters of each meta-heurstc method play mportant role n the optmzaton process, desgn space wll be effectvely nvestgated takng the advantages of the MHSM methods and parallel computers, and fnally sutable answers wll be obtaned. Constant and stable state of MHSM methods results n the tendency of the optmzaton algorthm to global optmum answer. Flowchart of the structural secton of the optmzaton process for truss structure confguratons s presented n Fgure 3. End

8 158 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh Determnng each desgn based on the type of desgn varable Unstable structure Investgatng Stablty of structure based on statc equatons Stable structure Formng stffness matrx for each desgn Unstable structure Investgatng stablty of the structure based on the stffness matrx Stable structure Structural analyss to obtan members stress and dsplacement of actve node of structures Applyng nstablty penalty Evaluaton amount of Gq Fgure 3. Part of the optmzaton process of truss confguraton 3.1. Island (1) In ths artcle, optmzaton of sland 1 s performed based on the Genetc Algorthm (GA) [42]. In ths regard, optmzaton process s performed as the followng [1, 13]: Frstly, an ntal populaton s randomly formed wth bnary characters. Then, the value of the object functon and the constrant volatons are determned. In ths artcle, the proposed penalty functon wth dynamc features s used whch has a good compatblty wth the algorthms of the MHSM. C g f F( A, ). K. penalty C g nlc 4 q1 q max 0, g A, j 1 ; j 1 nk K k j Ln,... (10) In ths equaton g q (A,) s the characterstc of the constrant volaton, and C g s the representatve of the sum of all volatons that has occurred by the structure n order to resst all the load combnatons of the nlc. K s the constant of dynamc penalty, k j s a constant quantty of each mgraton range for total number of nk, and j s the counter of each mgraton nterval. Afterwards, the mert of each desgn s computed based on the object functon and the proposed penalty functon. Then, the bests are tred to be selected usng a replcaton process that s nspred by natural development rules. In ths sland, tournament method [43] s used for the selecton process. Once the selecton process s completed, the crossover operator s appled n order to produce a populaton of offsprngs. For ths purpose, unform crossover s used wth small changes [44]. Therefore, parent s strngs are selected based on the crossover rate. Then, a strng whch s called Mask, s randomly produced. Ths strng conssts of bnary bts as length of each strng. In the next step, a unform random number s produced for each bt and s compared to the amounts resulted from Eq. (11). Offsprng s bts s selected based on the Mask pattern f the random number become

9 CONFIGURATION OPTIMIZATION OF TRUSSES 159 more than the amount obtaned by Eq. (11). That s, f the amount of the bt n the Mask s equal to one, the bt of the frst offsprng wll be from the frst parent, otherwse, t s selected from the second parent. Although, whle the randomly produced number s less than the amount obtaned from Eq. (11), the bt of the offsprng s strngs are selected from more mertorous parent. P Mn Max Mn t P P P (11) T C2 C2 C2 C2 Max Mn where P C2 s the secondary rate of crossover n each generaton for each bt, PC 2 and PC 2 are, respectvely, maxmum and mnmum rate of the secondary crossover n the optmzaton process (based on the nput of user), t s the number of current generaton, and T s the total number of generatons. In ths artcle, a method s proposed whch s used dynamcally to apply the mutaton operator. Thus, frstly, the total number of makng generatons s dvded to the number of bts of each substrng n desgn varable of szng and several ntervals are formed. Then, the common operator of the mutaton s appled on all the bts n each substrng of the cross-secton area. After performng ths process n the frst nterval, the frst bt at the left-sde of each substrng of the crosssecton area becomes stablzed, and the rate of the mutaton probablty for t wll be equal to zero, and the optmzaton process wll be also contnued tll the end of the second nterval of the total number of makng generatons. Afterwards, the mutaton rate of the two bts at the left-sde becomes zero and ths process s contnued tll the last bt n the substrng of the cross-secton area. It should be mentoned that the rate of the mutaton probablty for the resdual bts n each nterval s performed utlzng the followng equaton: P m Max Max Mn t P P P (12) T m m m Max Mn In whch above equaton, P m s the mutaton rate n each nterval, Pm and Pm are, respectvely, the maxmum and mnmum amount of mutaton rate n optmzaton process (based on nput of the user), t s the number of present nterval and T s the number of all ntervals. It should be noted that bts of geometrc plannng varable s changed lke mutaton process of substrng of cross-secton, based on the numbers of bts n geometrc substrng. However, bt of topologcal desgn varable s changed durng the entre process of optmzaton wth equal probablty. Allocaton of dfferent mutaton and crossover rate for each desgn varable are some other preparatons n ths artcle for sland (1). Therefore, varables of geometry, szng and topology are not merged wth equal probablty. Ths has consderable effect n escapng from the local optmum.

10 160 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh 3.2. Island (2) In sland (2), the Harmony Search (HS) algorthm s used [45]. Accordng to ths algorthm n the process of optmzaton, each muscan substtute wth desgn varable and collecton of muscan make the vector of desgn varable. Qualty of musc s substtuted by the value of the object functon. Optmzaton process for ths algorthm s performed as follows [4]: Frst, HS parameters such as HMCR, PAR, HMS, etc. are ntalzed. Then, the ntal populaton (HM) based on HMS (number of populaton members n sland (2)) s randomly formed as a matrx. Dmenson of ths matrx s determned by desgn varables (szng, topology and geometry). 1 x1 2 x HM 1 x HMS 1 x x x HMS 2 x x x 1 N 2 N HMS N HMSN (13) In the above matrx X refers to szng, topology and geometry varables. In MHSM for the current sland n spte of general process of HS, ntal populaton wthout constrant volaton s not requred and ftness of each desgn s specfed on the bass of constrant volaton and objectve functon. In order to compute the amount of ftness for each desgn, proposed penalty functon s used accordng to Eq. (10). Optmzaton process s contnued for HM by producng a new member based on the HS rules. Vectors of the new desgn varables X = [x 1, x 2,, x N ] are made by three possble varants of HS rules and HMCR and PAR parameters. Accordngly, each amount of x 1 proportonate to the type of desgn varable, can be randomly produced, agan, or can be determned by the exsted correspondng amounts n HM. Ths step s performed by producng a unform random number between zero and one, and comparng t wth the amount of HMCR. If the random number s more than HMCR, x s determned randomly and based on the type and varable range; otherwse, the amount of x s settled by HM. Determnaton of x n HM s by PAR parameter. Therefore, a unform random number between zero and one s produced and by comparng t wth the value of the PAR, x s defned. If the random number s less than PAR, x wll be selected from the exstng correspondng value of the HM. Otherwse, x s determned based on the value of the bw and from neghborhood of correspondng values wth x at HM. Fnally, f the vector of desgn varable s better than the worst vector n HM, then the new vector wll replace the worst vector. Otherwse, HM remans wthout change. Ths artcle suggest that PAR and bw parameters should change based on the amount of mgraton the nterval as follow: PARmax PARmn PAR PARmn j j 1,... nk nk (14)

11 CONFIGURATION OPTIMIZATION OF TRUSSES 161 ln bwmn bwmax bw bwmax exp j j 1,... nk nk (15) where, the ndces max and mn refer to the maxmum and mnmum values of the related parameter. j and nk are the number of mgraton nterval and total number of mgraton nterval n the optmzaton process, respectvely. Mgraton nterval, based on amount of PAR and bw parameters, has dfferent values that ascends for PAR and exponentally descends for bw durng entre process of the optmzaton. Varyng these parameters has valuable nfluence on the optmzaton process [46]. Allocatng dfferent rates of HMCR, PAR and bw for each desgn varables (geometry, szng and topology) s another feature of the sland (2) whch s proposed n ths artcle Island (3) The Charged System Search (CSS) method s used to perform optmzaton process [47]. In the CSS method, optmzaton process s performed based on the charged partcles laws and Newton laws of moton. Thus, each vector of desgn varables s consdered as a charged partcle whch possesses electrc feld as a result of electrc charge. Each partcle s affected by electrc feld of other partcles and proportonal to the amount of electrc force of other partcles and Newton laws of moton, the partcle move n desgn space and results n new poston. The optmzaton process s performed as follows [12, 17 and 48]: Frstly, lke other heurstc methods, ntal populaton s randomly produced and other parameters of the CSS method such as the number of partcles, number of selected partcles of CMS and so on are ntalzed. Then, the ftness of each partcle s computed accordng to value of the object functon and the proposed penalty functon n Eq. (10). The magntude of the charge of each partcle (q s ) and moton probablty of partcle of s affected by the force of the partcle r, P rs, s obtaned by the followng equaton: q s P ft s ft rs best 1 0 ft ft ft r ft worst worst ft s ft best r s 1,, Charg esze ran ft else s ft r (16) (17) ft best and ft worst are, the ftness of best and worst exsted desgn n current populaton, respectvely. A small populaton whch conssts of the bests of the exsted populaton s called CMS s produced after computng the p rs and q s. Then, the resultant electrcal force actng on a partcle s computed based on followng equaton: F q s s r, rs q a r rrsprsx r X s f rrs a 3 (18)

12 162 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh F q s s r, rs rs qr PrsX r X s f rrs a 2 (19) r where a s the dameter of each partcle, r rs s the dstance between two partcles r and s that s defned accordng to poston of the partcles X r and X s. New poston of each partcle n the desgn space s determned by the followng equaton: X s, new X s, old r1 kafs r2 kvvs, old v s, new X s, new X s, old (20) (21) r 1 and r 2 are unform random number between zero and one. v s s also the velocty of the partcle s, k a and k v are, respectvely, velocty and acceleraton coeffcent whch are computed to correspond wth MHSM as: j nk j 1 nk j nk j 1 nk k a 0.5 1,... (22) k v 0.5 1,... (23) New poston of each partcle s evaluated durng the optmzaton process, provdng the amount of extng from the allowed range. Desgn varables wll then modfed based on the HS method and CMS populaton Island (4) Ant Colony Optmzaton (ACO) s used n sland 4 [49], that s performed by the followng steps [25, 50, 51]: Frstly, the amount of ntal pheromone s specfed based on the number of desgn varables and possble states of each varable. In order to calculate ft 0, the frst crosssecton area n the profle lst s ntalzed for szng varables. Addtonally, topology and geometrc varables are also calculated based on the ground structure, and then the ntal pheromone s determned accordng to followng equaton: 0 1 ft 0 (24) Then, the probablty of selecton for each type of desgn varables s evaluated based on the followng equaton [6]: j p j N kjk k1 (25)

13 CONFIGURATION OPTIMIZATION OF TRUSSES 163 where, j s the amount of exsted pheromone n the th path (state number for the consdered desgn varable) for the desgn varable number j and N s the number of possble states for the consdered desgn varable. v s zero for the geometrc and topologcal varables, and t s also calculated by Eq. (26) for szng desgn varable. 1 (26) A A refers to the selected cross-secton area of the th path. Afterwards, the amount of the varable number s determned by p j by the roulette wheel method n GA. After determnng the amount of all desgn varables, the amount of the pheromone n the selected path s dmnshed as follows:. (27) new j old j where ρ s the local update parameter assgned to a sutable value between zero and one. Then, the amount of ftness s calculated and exsted populaton desgns are sorted. Pheromone evaporaton process for all the possble paths s done based on the followng equaton: new j old r j 1 e. (28) In Eq. (28), e r s a constant referred to as the evaporaton rate. After evaporatng of pheromone, the process of depostng pheromone n the selected paths s executed as follow: j j er. r T j r k Tj k1 k (29) In the above equaton, λ r s number of the best exsted populaton and k s the number of consdered desgn n small populaton of the bests. n Eq. (29), s calculated for ant number k by the followng equaton: j k j k 1 ft k (30) Fnally, crteron of local search s nvestgated. In ths artcle, f no varaton s created n several contnual populatons for ftness of the best desgn, the local search process wll be performed, based on the HS method, n the neghbourhood of the best desgn. Allocatng evaporaton rate and dfferent local search parameters to each type of desgn varables (geometry, szng and topology) are other facltes n ths artcle.

14 164 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh 3.5. Island (5) In sland (5), Partcle Swarm Optmzaton (PSO) [52] proportonal to type of desgn varable s used n two standard states accordng to references [53, 54] and bnary state accordng to reference [24]. Ths algorthm s nfluenced by the socal behavour of the brds n searchng food. The PSO algorthm begns by producng an ntal random populaton. Partcle number (desgn) whch ntroduces brd number n the group of brds s defned by two varables X = [x 1, x 2,..., x N ] and V =[v 1, v 2,..., v N ]. X s the poston and V s velocty of the partcle number n the search space. In each step of group movement (repeat), partcle poston s changed by two amounts of P best, and R best. In order to ths, poston of each partcle (desgn) s determned n the search space utlzng the followng equatons: V k1 k k1 k k1 X V k k 1r1 pbest, X c2r2 X (31) k k R X V c (32) In the above equaton, X k s the poston of the th partcle n the teraton number k, V k s velocty of the th partcle n the teraton number k, ω s the nerta weght n the prevous step, r 1 and r 2 are the unform random numbers between zero and one, C 1 and C 2 are the acceleraton constants. P k best, s the best poston of the partcle from frst to teraton number k, R k best s the best poston of a partcle from the frst to teraton number k among all the partcles. In ths artcle, the amount of varable velocty of the partcles s controlled by defnng mnmum and maxmum velocty (v mn, v max ). In ths regard, v mn and v max, based on the coeffcent of maxmum and mnmum amount of x s defned, proportonal to the type of desgn varable [55]. In order to be compatble wth the MHSM, the parameter ω s changed based on the number of mgraton nterval as: best max mn max j j 1,... nk (33) nk ω max and ω mn are the maxmum and mnmum values of ω, respectvely. j and nk are the number and total number of mgraton nterval n the optmzaton process, respectvely. Therefore, the value of ω s lnearly altered n each mgraton, wth ntal amount of ω max and fnal amount of ω mn. Ths method n alterng the ω resultng n a balance of the local and global search n the PSO algorthm [53]. In ths artcle, acceleraton constants (C 1 and C 2 ) and ω for each type of desgn varables (geometry, Szng and topology) are dfferent. 4. NUMERICAL EXAMPLES Some common nstances n optmzaton of truss structures confguraton are subsequently evaluated n order to nvestgate the ablty of the MHSM algorthm, and

15 CONFIGURATION OPTIMIZATION OF TRUSSES 165 ther results are compared to those of some relable references. Results show that MHSM searches the space more accurately compared to other methods and leads to better results A ten-bars planar truss structure A ten-bar planar truss s nvestgated as the frst example. Fgure 4 and Table 1 llustrate the necessary nformaton for the consdered truss. Fgure 4. Intal confguraton of a ten-bar planar truss Table 1. Data for desgn of the ten-bar planar truss Desgn varables Sze varables: A ; = 1, 2,, 10 ; Geometry varables: Y1, Y3, Y5 Constrant data σ all (Ten) = σ all (Com) = ±17240 N/cm 2 = ±25 Ks ; y all = 5.04 cm = 2 n 180 n (457.2 cm) < Y1 < 1000 n (2540 cm) ; 180 n (457.2 cm) < Y3 < 1000 n (2540 cm) 180 n (457.2 cm) < Y5 < 1000 n (2540 cm) = 10 n (25.4 cm) Lst of avalable profles a S = {1.62, 1.8, 2.38, 2.62, 2.88, 3.09, 3.13, 3.38, 3.63, 3.84, 3.87, 4.18, 4.49, 4.80, 4.97, 5.12, 5.74, 7.22, 7.97, 11.5, 13.5, 13.9, 14.2, 15.5, 16.0, 18.8, 19.9, 22.0, 22.9, 26.5, 30.0, 33.5} (n 2 ) a S = {10.45, 11.61, 15.35, 16.90, 18.58, 19.94, 20.19, 21.81, 23.42, 24.77, 24.97, 26.97, 28.97, 30.97, 32.06, 33.03, 37.03, 46.58, 51.42, 74.19, 87.1, 89.68,91.61, 100.0, , , , , , , , } (cm 2 ) Loadng data P2Y = P4Y = kn = 100 Kps Materal propertes E = N/cm 2 =10 4 ks ; ρ = N/cm 3 =0.1 lb/n 3

16 166 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh Fgure 5 s resulted as plan of an optmum sx-node planar truss after executng the optmzaton process based on the MHSM. Fgure 5. Optmal geometry and topology for the ten-bar truss, based on the MHSM Results of the optmum desgn are compared to the related references n Table 2. Ths shows that the MHSM leads to more enhanced results than other references. Table 2. Fnal desgn of szng, shape and topology for the ten-bars truss Desgn Varables Rajan [26] Tang [29] Raham [30] MHSM A1 - n 2 (cm 2 ) 9.9 (63.871) 13.5 (87.1) 11.5 (74.19) 13.5 (87.1) A2 - n 2 (cm 2 ) 9.4 (60.645) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) A3 - n 2 (cm 2 ) 11.5 (74.19) 7.97 (51.42) 11.5 (74.19) 11.5 (74.19) A4 - n 2 (cm 2 ) 1.5 (9.677) 7.22 (46.58) 5.74 (37.03) 7.22 (46.58) A5 - n 2 (cm 2 ) 0.0 (0.0) 1.62 (10.45) 0.0 (0.0) 0.0 (0.0) A6 - n 2 (cm 2 ) 12.0 (77.42) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) A7 - n 2 (cm 2 ) 11.5 (74.19) 4.49 (28.97) 5.74 (37.03) 5.74 (37.03) A8 - n 2 (cm 2 ) 3.6 (23.226) 3.13 (20.19) 3.84 (24.77) 3.38 (21.81) A9 - n 2 (cm 2 ) 0.0 (0.0) 13.5 (87.1) 13.5 (87.1) 11.5 (74.19) A10 - n 2 (cm 2 ) 10.4 (67.097) 0.0 (0.0) 0.0 (0.0) 0.0 (0.0) Y1 - n (cm) (473.71) Y3 - n (cm) ( ) ( ) ( ) (1260.8) Y5 - n (cm) ( ) ( ) ( ) ( ) Weght lb (kn) (14.474) (12.516) (12.113) (12.083) σ max ks (N/cm 2 ) 15.6 (10756) 18.5 ( ) (13201) ( ) max n (cm) 1.99 (5.0546) ( ) ( ) ( )

17 CONFIGURATION OPTIMIZATION OF TRUSSES A 15-bar planar truss In ths example, a ffteen-bar planar truss s studed. The consdered truss s shown n Fgure 6 and the requred nformaton for the optmzaton process are accessble n Table 3. Fgure 6. The prmary geometry of the 15-bar truss Table 3. Data for desgn of the 15-bar planar truss Desgn varables Sze varables: A ; = 1, 2,, 15 ; Geometry varables: Y2, Y3, Y4, Y6, Y7, Y8, X2=X6, X3=X7 Constrant data σ all (Ten) = σ all (Com) = ±17240 N/cm 2 = ±25 Ks 100 n (254 cm) < X2 < 140 n (355.6 cm); 220 n (558.8 cm) < X3 < 260 n (660.4 cm) 100 n (254 cm) < Y2 < 140 n (355.6 cm); 100 n (254 cm) < Y3 < 140 n (355.6 cm) 50 n (127 cm) < Y4 < 90 n (228.6 cm); 20 n (50.8 cm) < Y6 < 20 n (50.8 cm) 20 n (50.8 cm) < Y7 < 20 n (50.8 cm); 20 n (50.8 cm) < Y8 < 60 n (152.4 cm) = 0.01 n ( cm) Lst of avalable profles a S = {0.111, 0.141, 0.174, 0.22, 0.27, 0.287, 0.347, 0.44, 0.539, 0.954, 1.081, 1.174, 1.333, 1.488, 1.764, 2.142, 2.697, 2.8, 3.131, 3.565, 3.813, 4.805, 5.952, 6.572, 7.192, 8.525, 9.3, 10.85, 13.33, 14.29, 17.17, 19.18} (n 2 ) a S = {0.716, 0.91, 1.123, 1.419, 1.742, 1.852, 2.239, 2.839, 3.477, 6.155, 6.974, 7.574, 8.600, 9.600, , , , , , 23.00, 24.6, 31.0, 38.4, 42.4, 46.4, 55.0, 60.0, 70.0, 86.0, , , } (cm 2 ) Loadng data P8Y = kn = 10 Kps Materal propertes E = N/cm 2 =10 4 ks ; ρ = N/Cm 3 =0.1 lb/n 3

18 168 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh Fgure 7 s resulted as optmum desgn by the MHSM after optmzaton. Fgure 7. Optmal geometry and topology for the 15-bar truss based on the MHSM Table 4 compares the results of the present method wth those other references. Table 4. Fnal desgn of szng, shape and topology for the 15-bars truss Desgn Varables Wu [27] Tang [29] Raham [30] MHSM A1 - n 2 (cm 2 ) (7.574) (6.974) (6.155) (6.155) A2 - n 2 (cm 2 ) (6.155) (3.477) (6.155) (3.477) A3 - n 2 (cm 2 ) ((2.839) (0.000) (0.000) (0.000) A4 - n 2 (cm 2 ) (8.600) (6.974) (6.974) (6.155) A5 - n 2 (cm 2 ) (6.155) (6.155) (3.477) (3.477) A6 - n 2 (cm 2 ) (1.123) (2.839) (3.477) (2.839) A7 - n 2 (cm 2 ) (2.839) (0.000) (0.000) (0.000) A8 - n 2 (cm 2 ) (2.839) (0.91) (0.000) (0.000) A9 - n 2 (cm 2 ) (6.974) (0.000) (0.000) (0.000) A10 - n 2 (cm 2 ) (8.600) (1.742) (2.839) (2.839) A11 - n 2 (cm 2 ) (7.574) (1.742) (1.419) (3.477) A12 - n 2 (cm 2 ) (7.574) (3.477) (0.716) (1.123) A13 - n 2 (cm 2 ) (2.239) (0.91) (2.239) (1.123) A14 - n 2 (cm 2 ) (2.239) (2.839) (3.477) (3.477) A15 - n 2 (cm 2 ) (2.839) (0.000) (0.000) (0.000) X2 - n (cm) (312.9) ( ) ( ) ( ) X3 - n (cm) ( ) ( ) ( ) ( ) Y2 - n (cm) (272.26) ( ) ( ) (336.57) Y3 - n (cm) ( ) ( ) ( ) ( ) Y4 - n (cm) ( ) Y6 - n (cm) (42.489) (27.483) (4.127) (44.671) Y7 - n (cm) (39.535) (28.27) (45.93) (25.517) Y8 - n (cm) (93.078) ( ) ( ) (98.014) Weght lb (N) ( ) ( ) ( ) ( ) σ max ks (N/cm 2 ) ( ) (17206) (17232) (17230)

19 CONFIGURATION OPTIMIZATION OF TRUSSES A 25-bar space truss In ths example, The twenty fve-bar space truss wth ten nodes shown n Fgure 8 s optmzed. Fgure 8. The prmary geometry of the 25-bar truss Table 5 shows the requred nformaton for the optmzaton of the 25-bar truss. Table 5. Data for desgn of the 25-bar truss Groupng members A1 : 1 (1,2) ; A2 : 2(1,4), 3 (2,3), 4 (1,5), 5 (2,6) A3 : 6 (2,5), 7 (2,4), 8 (1,3), 9 (1,6) ; A4 : 10 (3,6), 11 (4,5) A5 : 12 (3,4), 13 (5,6) ; A6 : 14 (3,10), 15 (6,7), 16 (4,9), 17 (5,8) A7 : 18 (3,8), 19 (4,7), 20 (6,9), 21 (5,10) ; A8 : 22 (3,7), 23 (4,8), 24 (5,9), 25 (6,10) Desgn varables Sze varables: A ; = 1, 2,, 8 Geometry varables: X4=X5=X3=X6, X8=X9=X7=X10 Y3=Y4=Y5=Y6, Y7=Y8=Y9=Y10, Z3=Z4=Z5=Z6 Constrant data σ all (Ten) = σ all (Com) = ±27850 N/cm 2 = ±40 Ks ; y all = cm = 0.35 n 20 n (50.8 cm) < X4 < 60 n (152.4 cm) ; 40 n (101.6 cm) < X8 < 80 n (203.2 cm) 40 n (101.6 cm) < Y4 < 80 n (203.2 cm) ; 100 n (254 cm) < Y8 < 140 n (355.6 cm)

20 170 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh 90 n (228.6 cm) < Z4 < 130 n (330.2 cm) ; = 0.01 n ( cm) Lst of avalable profles a S = {0.1j (j1,, 26), 2.8, 3.0, 3.2, 3.4} (n 2 ) a S = {0.645j (j1,, 26), , , , } (cm 2 ) Loadng data P2Y = P4Y = kn = 100 Kps ; P1X = kn = 1 Kps P1Y = P2Y = P1Z = P2Z = kn = 10 Kps P3X = kn = 0.5 Kps ; P6X = kn = 0.6 Kps Materal propertes E = N/cm 2 =10 4 ks ; ρ = N/Cm 3 =0.1 lb/n 3 In ths example, optmzaton process n performed n three states. In state (1), the possblty of elmnaton of each member n topologcal optmzaton process s feasble. Most of the artcles consder the topologcal optmzaton process for each group that ncreases the possblty of unsteady desgn producton due to deleton of each group. In state (2), the assumptons are consdered exactly dentcal to other references, and possblty of deleton of each group n topologcal optmzaton process s feasble. Ths leads to the creaton of optmum desgn wth more precse shapes. In state (3), accordng to optmum ponts of state (2), nodal ponts nterval for geometrc varables s changed as shown n Table 6, and then desgn space wth new ntervals are agan searched for geometrc varables. Results of optmum desgn based on MHSM are shown n Fgures 9, 10 and 11 and also n Table 7. Table 6. New nterval for geometry varable of the 25-bar truss 20 n (50.8 cm) < X4 < 60 n (152.4 cm) ; 40 n (101.6 cm) < X8 < 80 n (203.2 cm) 40 n (101.6 cm) < Y4 < 80 n (203.2 cm) ; 100 n (254 cm) < Y8 < 140 n (355.6 cm) 90 n (228.6 cm) < Z4 < 130 n (330.2 cm) Table 7. Fnal desgn of szng, shape and topology for the 25-bar truss Desgn MHSM Wu [27] Tang [29] Raham [30] Varables State 1 State 2 State 3 A1 - n 2 (cm 2 ) 0.1 (0.645) (0.645) A2 - n 2 (cm 2 ) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) (0.645) 0.1 (0.645) A3 - n 2 (cm 2 ) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) 0.8 (5.16) 0.1 (0.645) 0.1 (0.645) A4 - n 2 (cm 2 ) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) 0.8 (5.16) 0.1 (0.645) 0.1 (0.645) A5 - n 2 (cm 2 ) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) 0.8 (5.16) 0.1 (0.645) 0.1 (0.645) A6 - n 2 (cm 2 ) 1.1 (7.095) 0.9 (5.805) 0.9 (5.805) 0.6 (3.87) 0.9 (5.805) 1.0 (6.45) A7 - n 2 (cm 2 ) 1.1 (7.095) 0.9 (5.805) 0.9 (5.805) 0.6 (3.87) 0.9 (5.805) 1.0 (6.45) A8 - n 2 (cm 2 ) 1.1 (7.095) 0.9 (5.805) 0.9 (5.805) 0.6 (3.87) 0.9 (5.805) 1.0 (6.45) A9 - n 2 (cm 2 ) 1.1 (7.095) 0.9 (5.805) 0.9 (5.805) 0.6 (3.87) 0.9 (5.805) 1.0 (6.45) A10 - n 2 (cm 2 ) 0.2 (1.29) A11 - n 2 (cm 2 ) 0.2 (1.29) (0.645) A12 - n 2 (cm 2 ) 0.3 (1.935) (0.645) A13 - n 2 (cm 2 ) 0.3 (1.935)

21 CONFIGURATION OPTIMIZATION OF TRUSSES 171 A14 - n 2 (cm 2 ) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) A15 - n 2 (cm 2 ) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) A16 - n 2 (cm 2 ) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) A17 - n 2 (cm 2 ) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) 0.1 (0.645) A18 - n 2 (cm 2 ) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) (0.645) 0.1 (0.645) A19 - n 2 (cm 2 ) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) A20 - n 2 (cm 2 ) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) A21 - n 2 (cm 2 ) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) 0.2 (1.29) 0.1 (0.645) 0.1 (0.645) A22 - n 2 (cm 2 ) 0.9 (5.805) 1.0 (6.45) 1.0 (6.45) 0.9 (5.805) 1.0 (6.45) 0.9 (5.805) A23 - n 2 (cm 2 ) 0.9 (5.805) 1.0 (6.45) 1.0 (6.45) (6.45) 0.9 (5.805) A24 - n 2 (cm 2 ) 0.9 (5.805) 1.0 (6.45) 1.0 (6.45) 0.9 (5.805) 1.0 (6.45) 0.9 (5.805) A25 - n 2 (cm 2 ) 0.9 (5.805) 1.0 (6.45) 1.0 (6.45) 0.9 (5.805) 1.0 (6.45) 0.9 (5.805) X4 - n (cm) ( ) 20.0 (50.8) Y4 - n (cm) Z4 - n (cm) ( ) ( ) ( ) X8 - n (cm) 50.8 ( ) Y8 - n (cm) Weght lb (N) Max σ ks (N/cm 2 ) Max n (cm) (333.96) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (134.95) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.889) ( ) (0.889) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.889) ( ) ( ) ( ) ( ) ( ) ( ) ( ) (0.889) Fgure 9. State 1 Fgure 10. State 2 Fgure 11. State 3 Optmum desgn for 25-bar space truss based on MHSM

22 172 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh 4.4. Desgn of a 131-bar truss In order to study the ndustral applcably of the MHSM algorthm, a 131-bar truss, whch s proposed and studed for the frst tme n ths artcle, s desgned. Fgure 12 shows the geometry and ntal topology of the consdered structure. Fgure 12. Ground structure for the 131-bar truss In the case, the permssble compressve and tensle stresses and also the permssble nodal dsplacements are selected accordng to the AISC-ASD [56] as follows: when λ<c c : all( com) Fy 1 / 2 (34) 2Cc 3 8Cc 8Cc and when λ C c : 2 12 E all( com) 2 (35) 23 In the above equatons, E s the module of elastcty and F y s yeld stress of steel that s taken as 2400 kg/cm 2. λ s the slenderness rato whch s calculated for compressve member around x and y axs as: r x, y (36) l where l s the length of the member and r s the governng radus of gyraton. Maxmum slenderness rato s lmted to 300 for the members under tenson, and to 200 for the members under compresson loads. In Eqs. (34), C c s the slender rato dvdng the elastc and nelastc bucklng regons, whch s calculated as follow: C c 2 2 E F (37) y On the other hand, accordng to AISC-ASD, the allowed tensle stress s calculated by the followng equaton: all( Ten) 0.6Fy (38)

23 CONFIGURATION OPTIMIZATION OF TRUSSES 173 The permssble nodal dsplacement s also consdered as cm, accordng to AISC- ASD code [56]. In ths example, n order to make the fnal executable desgn, the nodal coordnates of the upper members n vertcal drecton, are determned based on the postve angle of each member wth respect to horzon. Ths ssue leads to enlargement of the desgn space and ncreases the desgn varables (angle desgn varable), however, n fnal confguraton, none of the upper member s slope should be negatve, and an executable desgn wll be resulted. Desgnng and optmzaton nformaton of the structural confguraton can be seen n Table 8. Table 8. Data for desgn of the 131-bar planar truss Desgn varables Sze varables: A ; = 1, 2,, 66 Geometry varables: X3 = X4 = X51 = X52, X5 = X6 = X49 = X50 X7 = X8 = X47 = X48, X9 = X10 = X45 = X46 X11 = X12 = X43 = X44, X13 = X14 = X41 = X42 X15 = X16 = X39 = X40, X17 = X18 = X37 = X38 X19 = X20 = X35 = X36, X21 = X22 = X33 = X34 X23 = X24 = X31 = X32, X25 = X26 = X29 = X30 ; X2 = X54, Y2 = Y54 Angle varables: 5 = 130, 10 = 125, 15 = 120, 20 = 115, 25 = = 105, 35 = 100, 40 = 95, 45 = 90, 50 = = 80, 60 = 75, 65 = 70 Constrant data σ all (Com) = Eq. (35), Eq. (36) ; σ all (Ten) = Eq. (40) ; all = cm 0 < < 30 =5, 10, 15,, 125, 130 ; for angel varable = cm < X2 < 1260 cm, 1240 cm < X4 < 1160 cm 1140 cm < X6 < 1060 cm, 1040 cm < X8 < 960 cm 940 cm < X10 < 860 cm, 840 cm < X12 < 760 cm 740 cm < X14 < 660 cm, 640 cm < X16 < 560 cm 540 cm < X18 < 460 cm, 440 cm < X20 < 360 cm 340 cm < X22 < 260 cm, 240 cm < X24 < 160 cm 140 cm < X26 < 60 cm, 50 cm < Y2 < 150 cm ; for geometry varable = 1 cm Lst of avalable profles S 1 = {2UNP80, 2UNP100, 2UNP120, 2UNP140, 2UNP160, 2UNP180, 2UNP200, 2UNP220, 2UNP240} S 2 = {UNP80, UNP100, UNP120, UNP140, 2UNP80, 2UNP100, 2UNP120, 2UNP140, 2UNP160} Loadng data Dstrbuted load on upper members ; w = 17.5 kn/m Materal propertes E = N/cm 2 =10 4 ks ; ρ = N/Cm 3 =0.1 lb/n 3

24 174 A. Kaveh, V.R. Kalatjar, M.H. Talebpour, and J. Torkamanzadeh Fgure 13. Optmal geometry and topology for the 131-bar truss based on the MHSM Table 9. Results of optmum desgn for the 131-bar truss Szng and topology desgn varables Geometry desgn varables A1 -- A23 UNP 80 A45 2UNP 160 X2 -- Y2 -- A2 2UNP 80 A24 UNP 80 A46 -- X Y A3 2UNP 100 A25 2UNP 140 A47 2UNP 80 X Y A4 -- A26 -- A48 UNP 80 X8 981 Y A5 -- A27 2UNP 80 A49 UNP 80 X Y A6 2UNP 80 A28 UNP 80 A50 2UNP 160 X Y A7 2UNP 100 A29 -- A51 -- X Y A8 UNP 140 A30 2UNP 160 A52 2UNP 80 X Y A9 -- A31 UNP 80 A53 -- X Y A10 2UNP 80 A32 2UNP 80 A54 UNP 80 X Y A11 UNP 80 A33 UNP 80 A55 2UNP 160 X Y A12 2UNP 80 A34 UNP 80 A56 UNP 80 X Y A13 -- A35 2UNP 160 A57 2UNP 80 X Y A14 UNP 80 A36 -- A58 UNP 80 A15 2UNP 120 A37 2UNP 80 A59 -- A16 UNP 80 A38 -- A60 2UNP 160 A17 2UNP 80 A39 UNP 80 A61 UNP 80 A18 2UNP 80 A40 2UNP 160 A62 2UNP 80 A19 UNP 80 A41 UNP 80 A63 -- A20 2UNP 120 A42 2UNP 80 A64 UNP 80 A21 -- A43 UNP 80 A65 2UNP 160 A22 2UNP 80 A44 UNP 80 A66 UNP 80 In order to attan the executve goals and beauty of the structure, desgn varables of szng, topology, geometry and slope are somehow classfed to make the fnal desgn symmetrc. Therefore, the base of structure s consdered to be the lower chord and mddle node (node 27) of the truss to make the structure crosswse symmetrc. Thus, szng desgn varable are classfed nto 66 groups so that all of them have two members except the last one whch has one correspondng to the vertcal member connectng nodes 27 and 28. In ths regard, a topologc varable s consdered for each group. On the

25 CONFIGURATION OPTIMIZATION OF TRUSSES 175 other hand, the varables of the geometry and slope are respectvely classfed nto 14 and 13 categores as shown n Table 8. The base of the varaton for slope desgn varable s the horzontal lne n the frst node of each member. Fgure 13 s resulted utlzng the MHSM. As t s llustrated n Fgure 13, the structure tends to be curved whch s referred to logcal and correct determnaton of the upper members by MHSM. Zero slopes of the upper members at the mddle of structure s due to the type and number of sectons n S 2 collecton, because the length of vertcal and oblque members at the mddle of structure, has decreased by zerong the slope. Therefore, the mddle members are prevented from becomng slender. Table 9 shows the results of the optmum desgn for desgn varables on whch bass the weght of truss s obtaned as kg. 6. CONCLUSION Usng the method of sland dstrbuton n the proposed algorthm (MHSM), answers have remarkable dversty and desgn space s more vastly searched. Ths s because of allocatng dfferent meta-heurstc methods to each sland. Hence, desgn space s wsely searched by varous algorthms and the probablty of local optmum s nearly dmnshed. Usng MHSM causes the optmzaton ssue to be studed by several meta-heurstc methods, smultaneously, and thus all advantageous of meta-heurstc algorthms are ncorporated. In meta-heurstc algorthms, due to the effect of parameters and governng relatons on the results, subsequent executons are used n whch the amount of parameters are changed to obtan better answers. Although, due to relatve parameters ndependence and governng relatons of the MHSM, ths algorthm s free of subsequent executons for not beng trapped n local optma. Therefore, t moves to global optmum wth a constant and relable rate, and the probablty of gettng trapped n local optmum s declned. Snce n the frst varant of the proposed algorthm (MHSM.1), best members of each sland are transferred to other slands, durng mgraton process, or n the second varant (MHSM.2), best members are transferred to the selected sland and substtuted wth the members of lower ftness, t s antcpated the convergence speed and average growth rate of the populaton ftness to be enhanced. Employng dfferent value for the parameters of the meta-heurstc methods based on the type of desgn varables causes each varable to be searched proportonal to the related space. Ths process postvely effects on the optmzaton process of the confguraton accordng to meta-heurstc methods.

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