Paper 67 Cvl-Comp Press, 2012 Proceedngs of the Eleventh Internatonal Conference on Computatonal Structures Technology, B.H.V. Toppng, (Edtor), Cvl-Comp Press, Strlngshre, Scotland Partcle Swarm Optmzaton for Non-Convex Problems of Sze and Shape Optmzaton of Trusses Y.J. Xu 1, M. Domaszews 2,3, D. Chamoret 2, W.H. Zhang 1 and J.G. Korvn 3,4 1 The Key Laboratory of Contemporary Desgn and Manufacturng Technology Northwestern Polytechncal Unversty, X'an Shaanx, Chna 2 M3M Laboratory Unversty of Technology of Belfort-Montbélard, Belfort, France 3 Department of Mcrosystems Engneerng (IMTEK) 4 Freburg Insttute for Advanced Studes (FRIAS) Unversty of Freburg, Germany Abstract The smultaneous sze and shape optmzaton of truss structures can represent the non-convex problems. In ths paper, an mproved partcle swarm optmzaton algorthm s developed and appled to fnd global optmum solutons. The optmsaton problem s concerned wth mnmzng the structural weght subject to the constrants on nodal dsplacements, stresses and buclng n the bars. The optmsaton varables are the nodal coordnates and the cross-sectonal areas. Also the lower and upper bounds are mposed on these two types of optmsaton varables. The classcal PSO algorthm s modfed to satsfy that all the partcles fly nsde the varable boundares. A method derved from the harmony search algorthm s used to deal wth the partcles whch fly outsde the varables boundares. The mult-stage penalty functon method s adopted wthn PSO to satsfy the constrants of the optmsaton problem and obtan the feasble optmal solutons. The classcal PSO algorthm has good exploraton abltes but wea explotaton of local optma. The nerta weght s employed to control the mpact of the prevous hstory of veloctes on the current velocty of each partcle. Thus ths parameter regulates the trade-off between global and local exploraton ablty of the swarm. A general rule s to set the ntal nerta weght to a large value n order to mae better global exploraton of the search space and then gradually decrease t to get more refned solutons. Thus a dynamc varaton of nerta weght s used n the paper by applyng a fracton multpler. Some benchmar problems of sze and shape optmsaton of truss structures are tested and the obtaned results are compared wth the nown results from the lterature. Keywords: partcle swarm optmsaton, truss structural optmsaton, constrant handlng, sze and shape optmsaton. 1
1 Introducton Over the past decade a number of optmsaton algorthms have been used extensvely n truss structural optmsaton, from the tradtonal mathematcal optmsaton algorthms to the metaheurstc algorthms. The metaheurstc algorthms do not requre conventonal mathematcal assumptons and possess better global search abltes than the conventonal optmsaton algorthms, have become more attractve n the structural optmsaton applcatons. Recently, partcle swarm optmsaton (PSO) whch s also an evolutonary algorthm was developed [1]. In PSO, each soluton to the optmsaton problem s regarded as a partcle n the search space, whch adjusts ts poston accordng to ts own flyng experence and the flyng experence of other partcles. The advantage of PSO over the other evolutonary algorthms such as the genetc algorthms s that PSO does not need complcated encodng and decodng and specal genetc operators such as mutaton and crossover but can wor drectly wth real numbers. It s therefore qute straghtforward to mplement a PSO algorthm. The PSO has ncreasngly ganed popularty n the area of structural optmsaton. Applcatons n the sze optmsaton and shape optmsaton of truss structures had been mplemented by some researchers [1-8]. Ths paper focuses on the mplementaton and applcaton of PSO for the smultaneous sze and shape optmsaton of truss structures. The paper s organzed as follows. We ntroduce frstly the basc mathematcal formulaton of the PSO followed by descrpton of the constrant handlng method. Then numercal results are obtaned and compared wth the reference results for the valdaton. Fnally, concludng remars are proposed. 2 Presentaton of smultaneous sze and shape optmsaton problem of truss structures In ths paper, smultaneous sze and shape optmsaton problems for truss structures wth contnuous desgn varables are studed. The smultaneous sze and shape optmsaton s concerned wth determnng the cross-sectonal areas of truss members and the nodal coordnates smultaneously. The objectve functon s a weght of the truss, whch s subjected to the stress and dsplacements constrants. In general, these truss structural optmsaton problems can be formulated as the follows Mnmze f ( X ) ( ) Subject to : g X 0, = 1, 2,..., m (1) where f(x) s the weght of the truss, whch s a scalar functon, and g (X) s the nequalty constrants. The varables vector X represents a set of the desgn varables (the areas of cross-sectons of the truss members and the coordnates of the nodes). It can be denoted as 2
( 1, 2,..., n, 1, 2,..., p) X = A A A R R R l u A < A < A, = 1,2,..., n l u R < R < R, = 1, 2,..., p (2) where that l u A and A are the lower and the upper bounds of the th truss member cross-sectonal area, whle l u R and R are the lower and the upper bounds of the th nodal coordnates. 3 Partcle Swarm Optmsaton 3.1 Bass of Partcle Swarm Optmsaton algorthm The PSO algorthm s a global optmsaton algorthm and descrbed as socologcally nspred. In PSO, each ndvdual of the swarm s consdered as a partcle n a mult-dmensonal space that has a poston and a velocty. These partcles fly through hyperspace and remember the best poston that they have seen. Members of a swarm remember the locaton where they had ther best success and communcate good postons to each other, then adjust ther own poston and velocty based on these good postons. Updatng the poston and velocty s done as follows ( ) ( ) V + = ωv + cr P X + c r P X (3) 1 11 2 2 g X = X + V (4) + 1 + 1 where V and X represent the current velocty and the poston of the th partcle respectvely (Note that the subscrpts and +1 refer to the recent and the next teratons respectvely); P s the best prevous poston (pbest) of the th partcle and P g s the best global poston (gbest) among all the partcles n the swarm; c 1 and c 2 represent trust parameters ndcatng how much confdence the current partcle has n tself and how much confdence t has n the swarm; r 1 and r 2 are two random numbers between 0 and 1; and ω s the nerta weght. The acceleraton constants c 1 and c 2 ndcate the stochastc acceleraton terms whch pull each partcle towards the best poston attaned by the partcle or the best poston attaned by the swarm. Low values of c 1 and c 2 allow the partcles to wander far away from the optmum regons before beng tugged bac, whle the hgh values pull the partcles toward the optmum or mae the partcles to pass through the optmum abruptly. In reference [3], the constants c 1 and c 2 are chosen equal to 2 correspondng to the optmal value for the studed problem. In the same reference, t s mentoned that the choce of these constants s problem dependent. In ths wor, c 1 = 2 and c 2 = 2 are chosen. The role of the nerta weght ω s consdered mportant for the convergence behavor of PSO algorthm. The nerta weght s employed to control the mpact of 3
the prevous hstory of veloctes on the current velocty. Thus, the parameter ω regulates the trade off between the global (wde rangng) and the local (nearby) exploraton abltes of the swarm. A proper value for the nerta weght ω provdes balance between the global and local exploraton ablty of the swarm, and, thus results n better solutons. Expermental results mply that t s preferable to ntally set the nerta to a large value, to promote global exploraton of the search space, and gradually decrease t to obtan refned solutons [1]. Thus, a dynamc varaton of nerta weght proposed n reference [2] s used n ths paper. ω s decreased dynamcally based on a fracton multpler ω as s shown below = + 1 ω ω (5) ω The basc procedure of the PSO algorthm s constructed as follows: (1) Intalze a set of partcles postons and veloctes randomly dstrbuted throughout the desgn space bounded by specfed lmts. (2) Evaluate the objectve functon values usng the desgn space postons. (3) Update the optmum partcle poston and global optmum partcle poston at current teraton. (4) Update the current velocty vector and modfy the poston of each partcle n the swarm usng Eqs. (3) and (4). (5) Repeat from step (2) untl the stop crteron s acheved. The stoppng crteron s usually defned as the number of teratons the PSO algorthm executes. 3.2 Constrants handlng strategy Most structural optmsaton problems nclude the problem-specfc constrants and the varable lmts. For the present truss structural optmsaton, the problem-specfc constrants consst of the stress, buclng and dsplacement constrants, and the varable lmts are the bounds of the truss member s cross-sectonal areas and the nodal coordnates. If a partcle fles out of the varable boundares, the soluton cannot be used even f the problem-specfc constrants are satsfed, so t s essental to mae sure that all of the partcles fly nsde the varable boundares, and then to chec whether they volate the problem-specfc constrants. Harmony search scheme: handlng the varable lmts A method ntroduced by L and Huang [7] dealng wth the partcles that fly outsde the varables boundary s used n the present study. Ths method s derved from the harmony search (HS) algorthm [9]. In the HS algorthm, the harmony memory (HM) stores the feasble vectors, whch are all n the feasble space. The harmony memory sze determnes how many vectors can be stored. A new vector s generated by selectng the components of dfferent vectors randomly n the harmony memory. Undoubtedly, the new vector does not volate the varables boundares, but t s not certan f t volates the problem-specfc constrants. When t s generated, the harmony memory wll be updated by acceptng ths new vector f t gets a better soluton and deletng the worst vector. Smlarly, the PSO algorthm stores the feasble and good vectors (partcles) n the pbest 4
swarm, le does the harmony memory n the HS algorthm. Hence, the vector (partcle) volatng the varables boundares can be generated randomly agan by such a strategy: f any component of the current partcle volates ts correspondng boundary, then t wll be replaced by selectng the correspondng component of the partcle from pbest swarm randomly. To hghlght the presentaton, a schematc dagram s gven n Fg. 1 to llustrate ths strategy. Fgure 1: Illustraton of the varable lmts handlng strategy Penalty functons method: handlng the problem-specfc constrants The most common method to handle the constrants s the use of a penalty functon. The constraned problem s transformed to an unconstraned one, by penalzng the constrants and buldng a sngle objectve functon. The optmsaton problem reduces to mnmze the objectve functon wth the penalty functon together. In ths paper, a non-statonary, mult-stage penalty functon method mplemented by Parsopoulos and Vrahats n [10] s adopted for constrants handlng n PSO. The penalty functon used n [10] s ( ) ( ) ( ) ( ), F X f X h H X X S R n = + (6) where f(x) s the orgnal objectve functon to be optmzed, h() s a penalty value whch s modfed accordng to the algorthm s current teraton number and usually set to h( ) =. H(X) s a penalty factor defned as m ( q ( X) ) ( ) = θ ( ( )) ( ) (7) H X q X q X γ = 1 where q (X) s a relatve volated functon of the constrants, whch s defned as q (X) = max {0, g (X)} (Note that g (X) s the constrant); θ ( q ( X) ) s an assgnment functon, and γ ( q ( X) ) s the power of the penalty functon. 5
Regardng the [10], the followng values are used for the penalty functon: If q ( X ) < 1, then γ ( q ( X) ) = 1; otherwse γ ( q ( X) ) = 2 ; If q ( X ) < 0.001, then θ ( q ( X) ) = 10 ; else f q ( X) 0.1, then θ ( q ( X) ) = 20 ; else f q ( X) 1, then θ ( q ( X) ) = 100 ; otherwse θ q ( X) = ( ) 300 3.3 Algorthm descrpton Based on the basc PSO algorthm and constrants handlng technology as explaned above, the pseudo-code for the present algorthm s lsted n Table 1. Set = 0; Randomly ntalze postons and veloctes of all partcles dstrbuted throughout the desgn space Calculate the objectve functon f ( X ) and penalty functon F ( X ) of the ntalzed partcle Generate local best: Set P = X Generate global best: Fnd mn F ( X ), P g s set to the poston of X mn. WHILE (the maxmum number of teratons are not met) FOR (each partcle n the swarm) Generate the velocty and update the poston of the current partcle X. Varable lmt handng: Chec whether each component of the current partcle volates ts correspondng boundary or not. If t does, select the correspondng component of the vector from pbest swarm randomly. Calculate the objectve functon f ( X ) and penalty functon ( ) F X of the current partcle. Update pbest: Compare the penalty functon value of pbest wth F ( X ). If the F ( X ) s better than the penalty functon value of pbest, set pbest to the current poston X. Update gbest: Fnd the global best poston n the swarm. If the ( ) F X s better than the penalty functon value of gbest, gbest s set to the poston of the current partcle X END FOR Set = +1 END WHILE Table 1: The pseudo-code for the present PSO algorthm 6
4 Numercal examples In ths secton, three truss structures commonly used n lterature are selected as benchmar problems to test the present PSO algorthm. The optmzed solutons are then compared to those of the exstng results. The examples gven n the smulaton studes nclude: Smultaneous sze and shape optmsaton of a 15-bar planar truss structure llustrated n Fg. 2. Smultaneous sze and shape optmsaton of a 18-bar planar truss structure llustrated n Fg. 5. Smple sze optmsaton and smultaneous sze and shape optmsaton of a 25- bar space truss structure llustrated n Fg. 8. For all these problems, a populaton of 50 ndvduals s used. The maxmum number of teratons s lmted to 1000. 4.1 15-bar planar truss structure The weght of the 15-bar planar truss shown n Fg. 2 s frstly to be mnmzed wth stress constrants. A tp load of 10ps s appled to the truss. The stress lmt s 25s n both tenson and compresson for all truss members. The modulus of elastcty s specfed as 1.0 10 4 s, and the materal densty as 0.1lb/n 3. The x and y coordnates of nodes 2, 3, 6, and 7 are allowed to vary, nodes 6 and 7 beng constraned to have the same x coordnates as nodes 2 and 3 respectvely. Nodes 4 and 8 are allowed to move only n the y drecton. Hence, there are 23 ndependent desgn varables whch nclude 15 szng varables (cross-sectonal areas of the truss members) and 8 shape varables (x 2 = x 6, x 3 = x 7, y 2, y 3, y 4, y 6, y 7, y 8 ). The bounds on the member cross-sectonal areas are 0.1n 2 and 20.0n 2. Sde constrants for the shape varables are 100n x2 140n, 220n x3 260n, 100n y2 140n, 100n y3 140n, 50n y4 90n, -20n y6 20n, -20n y7 20n, 20n y8 60n. Fgure 2: 15-bar planar truss 7
Varables Optmal cross-sectonal areas A (n 2 ) and coordnates (n) Wu [11] Hwang [12] Tang [13] Raham [14] Present study A 1 1.174 0.954 1.081 1.081 1.021 A 2 0.954 1.081 0.539 0.539 0.508 A 3 0.440 0.440 0.287 0.287 0.282 A 4 1.333 1.174 0.954 0.954 1.081 A 5 0.954 1.488 0.954 0.539 0.685 A 6 0.174 0.270 0.220 0.141 0.130 A 7 0.440 0.270 0.111 0.111 0.126 A 8 0.440 0.347 0.111 0.111 0.134 A 9 1.081 0.220 0.287 0.539 0.551 A 10 1.333 0.440 0.220 0.440 0.388 A 11 0.174 0.220 0.440 0.539 0.413 A 12 0.174 0.440 0.440 0.270 0.317 A 13 0.347 0.347 0.111 0.220 0.273 A 14 0.347 0.270 0.220 0.141 0.194 A 15 0.440 0.220 0.347 0.287 0.287 x 2 123.189 118.346 133.612 101.5775 105.3433 x 3 231.595 225.209 234.752 227.9112 247.8316 y 2 107.189 119.046 100.449 134.7986 115.8325 y 3 119.175 105.086 104.738 128.2206 104.4993 y 4 60.462 63.375 73.762 54.8630 56.3801 y 6-16.728-20.0-10.067-16.4484-18.6139 y 7 15.565-20.0-1.339-13.3007-12.7236 y 8 36.645 57.722 50.402 54.8572 49.9081 Weght (lb) 120.528 104.573 79.820 76.6854 78.739 Table 2: Comparson of optmal solutons of 15-bar planar truss wth results reported n the lterature Fg. 3 gves a convergence rate of the optmsaton procedure. It can be seen that the algorthm acheves the best solutons after 1000 teratons and t s close to the best soluton after about 100 teratons. The best vector found usng the PSO approach s lsted n Table 2 and compared wth the results obtaned usng other mathematcal methods. The correspondng objectve functon value (weght of the structure) s 78.739lb. The optmal confguraton s llustrated n Fg. 4. 8
Fgure 3: Convergence rate for the sze and shape optmsaton of 15-bar planar truss Fgure 4: Optmzed confguraton of 15-bar planar truss 4.2 18-bar planar truss structure The second example consders the 18-bar planar truss shown n Fg. 5. The optmsaton objectve s the mnmzaton of structural weght where the desgn varables are specfed as the cross-sectonal area for the truss members and the coordnates of the nodes 3, 5, 7 and 9 (bottom part of the structure). The crosssectonal areas of the members were categorzed nto four groups: (1): A 1 = A 4 = A 8 = A 12 = A 16, (2): A 2 = A 6 = A 10 = A 14 = A 18, (3): A 3 = A 7 = A 11 = A 15, and (4): A 5 = A 9 = A 13 = A 17. Thus, there are a total of 12 ndependent desgn varables that ncluded four szng and eght coordnate varables. The bounds on the member crosssectonal areas are 3.5n 2 and 18.0n 2. Sde constrants for the coordnate varables 9
are 775n x3 1225n, 525n x5 975n, 275n x7 725n, 25n x9 475n, - 225n y3, y5, y7, y9 245n. The materal densty s 0.1lb/n 3 and the modulus of elastcty s 1.0 10 4 s. The sngle loadng condton consdered n ths study s a set of vertcal loads, P = 20ps, actng on the upper nodal ponts of the truss, as llustrated n Fg. 5. The allowable stresses and the buclng constrants are mposed. In ths problem, the allowable tensle and compressve stresses are 20s. The Euler buclng compressve stress lmt for truss member used for the buclng constrants s computed as: 2 ( σ ) σ = α c b EA L where α s a constant determned from the cross-sectonal geometry and s taen to be α = 4 n ths study. E s the modulus of elastcty of the materal, and L s the member length. Fgure 5: 18-bar planar truss Fg. 6 provdes a convergence rate of the optmsaton procedure. It can be seen that the algorthm acheve the best solutons after 1000 teratons and t s close to the best soluton after about 500 teratons. The optmzed soluton found usng the presented approach s lsted n Table 3, and the correspondng objectve functon value (weght of the structure) s 4599.4lb. A comparson to other references wth respect to the cross-sectonal area of each truss member, the nodal coordnates and the fnal weght reached for the 18-bar planar truss s shown n the Table 3. The optmal confguraton s llustrated n Fg. 7. 10
Fgure 6: Convergence rate for the sze and shape optmsaton of 18-bar planar truss Varables Optmal cross-sectonal areas A (n. 2 ) and coordnates (n.) Felx [15] Rajeev [16] Yang [17] Soh [18] Yang [19] Present study A 1 11.34 12.50 12.61 12.59 12.33 12.08 A 2 19.28 16.25 18.10 17.91 17.97 17.16 A 3 10.97 8.000 5.470 5.500 5.600 7.015 A 4 5.300 4.000 3.540 3.550 3.660 4.492 x 3 994.6 891.9 914.5 909.8 907.2 912.9 y 3 162.3 145.3 183.0 184.5 184.2 178.9 x 5 747.4 610.6 647.0 640.3 643.3 641.5 y 5 102.9 118.2 147.4 147.8 149.2 127.2 x 7 482.9 385.4 414.2 410.0 413.9 406.7 y 7 33.00 72.50 100.4 97.00 102.2 79.80 x 9 221.7 184.4 200.0 200.9 202.1 196.5 y 9 17.10 23.40 31.90 32.00 30.90 20.95 Weght (lb) 5713.0 4616.8 4552.8 4531.9 4520.0 4599.4 Table 3: Comparson of optmal solutons of 18-bar planar truss wth results reported n the lterature Fgure 7: Optmzed confguraton of 18-bar planar truss 11
4.3 25-bar space truss structure The thrd example concerns the weght mnmzaton of o 25-bar transmsson tower shown n Fg. 8. The desgn varables are the cross-sectonal areas for the truss members, whch are dvded nto eght member groups, as follows: (1): A1, (2): A2- A5, (3): A6-A9, (4): A10-A11, (5): A12-A13, (6): A14-A17, (7): A18-A21 and (8): A22-A25. The range of cross-sectonal areas vares from 0.01 to 3.4 n 2. The 3 materal densty s 0.1lb n and the modulus of elastcty s 1.0 10 4 s. The stress lmts of the truss members are lsted n Table 4. All nodes n all drectons are subjected to the dsplacement lmts of ± 0.35n. The load case lsted n Table 5 s consdered. Varables Compressve stress lmtatons (s) Tensle stress lmtatons (s) A 1 35.092 40.0 A 2 11.590 40.0 A 3 17.305 40.0 A 4 35.092 40.0 A 5 35.092 40.0 A 6 6.759 40.0 A 7 6.959 40.0 A 8 11.082 40.0 Table 4: Member stress lmtatons for 25-bar truss Node Concentrated loads P X (ps) P Y (ps) P Z (ps) 1 1.0 10.0-5.0 2 0.0 10.0-5.0 3 0.5 0.0 0.0 6 0.5 0.0 0.0 Table 5: Loads for 25-bar truss Frst, the weght of ths truss s mnmzed consderng the cross-sectonal areas of bars only. The optmal soluton of sze optmsaton s gven n Table 6 and compared wth the results usng other methods. It s llustrated n Fg. 9. The convergence rate of PSO algorthm s gven n Fg. 10. It can be seen that the algorthm acheve the best solutons after 1000 teratons and t s close to the best soluton after about 100 teratons. 12
Fgure 8: 25-bar space truss Methods Optmal cross-sectonal areas Weght A 1 A 2 A 3 A 4 A 5 A 6 A 7 A 8 (lb) Zhu [20] 0.01 1.90 2.60 0.01 0.01 0.80 2.10 0.60 562.93 Erbatur [21] 0.01 1.20 3.20 0.01 1.10 0.90 0.40 3.40 493.80 Wu and Chow [22] 0.01 0.50 3.40 0.01 1.50 0.90 0.60 3.40 486.29 Perez [8] 0.01 1.02 3.40 0.01 0.01 0.64 2.04 3.40 485.33 Present study 0.01 0.53 3.40 0.01 2.13 0.99 0.30 3.40 479.60 Table 6: Comparson of optmal soluton of 25-bar truss wth results reported n the lterature Now, the same 25-bar truss s smultaneously optmzed for sze and shape. The desgn varables are concerned wth eght cross-sectonal area groups and twelve nodal coordnates (x,y,z) of the four nodes: 3, 4, 5, 6. The range of cross-sectonal areas vares from 0.01 n 2 to 3.4 n 2. The bounds for the coordnates are as follows 13
Fgure 9: Sze optmsaton of 25-bar space truss Fgure 10: Convergence rate for the sze optmsaton of 25-bar space truss ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2.5L x 3 0.5 L, 0.5L y 3 2.5 L, 16L 3 z 3 8L 3 2.5L x 4 0.5 L, 0.5L y 4 2.5 L, 16L 3 z 4 8L 3 2.5L x 5 0.5 L, 2.5L y 5 0.5 L, 16L 3 z 5 8L 3 2.5L x 6 0.5 L, 2.5L y 6 0.5 L, 16L 3 z 6 8L 3 14
The optmal soluton obtaned by the developed PSO algorthm s llustrated n Fg. 11. The mnmum weght s 280.79 lb and represents 58% of the weght obtaned by sze optmsaton (479.60 lb). Ths s evdent that the smultaneous sze and shape optmsaton gves a weght whch s smaller than a weght obtaned by the smple sze optmsaton. Fg. 11: Sze and shape optmsaton of 25-bar truss 5 Conclusons The am of ths study s to test a new developed PSO algorthm n applcaton to smultaneous sze and shape optmsaton of truss structures. The man dffculty of applyng the PSO algorthm to structural optmsaton s to satsfy the nonlnear constrants concernng the nodal dsplacements and element stresses. Also, the lmt boundares of cross-sectonal areas and nodal coordnates should be taen nto account. The classcal PSO algorthm [1] has been modfed to satsfy that all the partcles fly nsde the varable s boundares. A method derved from the Harmony Search algorthm [9] was used to deal wth the partcles whch fly outsde the varables boundares. The mult-stage penalty functon method was adopted wthn PSO to satsfy the constrants of the optmsaton problem. The frst numercal example s concerned wth a smultaneous sze and shape optmsaton of a 15-bar planar truss structure. The obtaned weght s 78.739 lb. When compared wth other four results from the lterature ([11],[12],[13],[14]) where the genetc algorthms were used for ths example, only Raham et al. [14] obtaned a smaller weght (97% of our weght). 15
In the second example, a weght of a18-bar planar truss s mnmzed subject to the stress and buclng constrants. The desgn cross-sectonal areas are grouped n four classes and there are eght varable coordnates of four bottom nodes. The comparson wth other fve results from the lterature ([15],[16],[17],[18],[19]) based on genetc algorthms shows that the best result (98% of our weght) has been obtaned by Yang and Soh [19] and the worst (124% of our weght) has been obtaned by Fex [15]. The last example concerns the smple sze and smultaneous sze-shape optmsaton of a 25-bar space transmsson tower. The varable cross-sectonal areas of truss members are classfed nto eght groups. In addton, there are twelve coordnates (x,y,z) of the nodes 3, 4, 5 and 6. A comparson wth other four results from the lterature [8, 20-22] for smple sze optmsaton shows that our result s the best. When comparng the smple sze and smultaneous sze-shape optmsaton, ths s evdent that the smultaneous approach gves the better result (the optmal weght s reduced by 58%). The developed modfed PSO algorthm has been successfully appled to smultaneous sze and shape optmsaton of truss structures. In order to mprove the effcency of PSO for structural optmsaton, the hybrdzaton strateges combnng other metaheurstc methods wth gradent-based technques and usng rapd fnte element reanalyss approach can be developed. Acnowledgements The second author gratefully acnowledges a support from the German Research Foundaton. References [1] J. Kennedy and R.C. Eberhart, Partcle swarm optmzaton, Proceedngs of the IEEE nternatonal conference on neural networs, 4, 1942-1948, 1995. [2] P. Foure and A. Groenwold, The partcle swarm optmzaton algorthm n sze and shape optmzaton, Structural and Multdscplnary Optmzaton, 23, 259-267, 2002. [3] I.C. Trelea, The partcle swarm optmzaton algorthm: convergence analyss and parameter selecton, Informaton Processng Letters, 85, 317-325, 2003. [4] G. Venter and J. Sobeszczans-Sobes, Multdscplnary optmzaton of a transport arcraft wng usng partcle swarm optmzaton, Structural and Multdscplnary Optmzaton, 26, 121-131, 2004. [5] S. He, E. Prempan and Q.H. Wu, An mproved partcle swarm optmzer for mechancal desgn optmzaton problems. Engneerng Optmzaton, 36, 585-605, 2004. [6] C. Elegbede, Structural relablty assessment based on partcles swarm optmzaton, Structural Safety, 27, 171-186, 2005. [7] L.J., L Z.B. Huang, F. Lu and Q.H. Wu, A heurstc partcle swarm optmzer for optmzaton of pn connected structures, Computers and Structures, 85, 16
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