SHAPE OPTIMIZATION OF STRUCTURES BY MODIFIED HARMONY SEARCH

INTERNATIONAL JOURNAL OF OPTIMIZATION IN CIVIL ENGINEERING Int. J. Optm. Cvl Eng., 2011; 3:485-494 SHAPE OPTIMIZATION OF STRUCTURES BY MODIFIED HARMONY SEARCH S. Gholzadeh *,, A. Barzegar and Ch. Gheyratmand Department of Cvl Engneerng, Urma Unversty, Urma, Iran ABSTRACT The man am of the present study s to propose a modfed harmony search (MHS) algorthm for sze and shape optmzaton of structures. The standard harmony search (HS) algorthm s conceptualzed usng the muscal process of searchng for a perfect state of the harmony. It uses a stochastc random search nstead of a gradent search. The proposed MHS algorthm s desgned based on eltsm. In fact the MHS s a mult-staged verson of the HS and n each stage a new harmony memory s created usng the nformaton of the prevous stages. Numercal results reveal that the proposed algorthm s a powerful optmzaton technque wth mproved explotaton characterstcs compared wth the standard HS. Receved: 4 October 2011; Accepted: 28 December 2011 KEY WORDS: Shape optmzaton, harmony search algorthm, penalty functons, truss structure 1. INTRODUCTION Structural optmzaton s a crtcal actvty that has receved consderable attenton n the last four decades. Usually, structural optmzaton problems nvolve searchng for the mnmum of the structural weght. Ths mnmum weght desgn s subected to varous constrants on performance measures, such as stresses and dsplacements, and also restrcted by practcal mnmum cross-sectonal areas or dmensons of the structural members or components. Due to consderng these constrants the possblty of trappng n the local optma wll be larger. Optmum shape desgn of structures s one of the challengng research areas of the * Correspondng author: Department of Cvl Engneerng, Urma Unversty, Urma, Iran E-mal address: s.gholzadeh@urma.ac.r

486 S. Gholzadeh, A. Barzegar and Ch. Gheyratmand structural optmzaton feld. In ths class of optmzaton problems two types of desgn varables wth dfferent natures, ncludng szng and geometrc varables, are nvolved. The shape optmzaton problem has been dentfed as a more dffcult but more mportant task than pure szng optmzaton, snce potental savngs n materal can be far better mproved than by the latter. Most of the engneerng optmzaton algorthms are based on numercal lnear and nonlnear programmng methods that requre substantal gradent nformaton and usually seek to mprove the soluton n the neghborhood of a startng pont. These algorthms, however, reveal a lmted approach to complcated real-world optmzaton problems. If there s more than one local optmum n the problem, the result may depend on the selecton of an ntal pont, and the obtaned optmal soluton may not necessarly be the global optmum. In the last years, structural optmzaton has been studed by usng dfferent natural phenomena based meta-heurstc algorthms. The most extensvely appled meta-heurstc algorthms are Genetc Algorthm (GA) [1] Ant Colony Optmzaton (ACO) [2], Partcle Swarm Optmzaton (PSO) [3] and etc. In the feld of structural engneerng many successful applcatons of the above mentoned algorthms have been reported n the lterature [4-9]. The popularty of these algorthms s due to ths fact that for mplementaton of the meta-heurstcs the gradent of obectve functon and constrants are not requred. In other words, they use a stochastc random search strategy nstead of a gradent search so that dervatve nformaton s unnecessary. Also they are able to handle both dscrete and contnuous desgn varables and ther computer mplementaton s smple. One of the recent addtons to meta-heurstcs s the Harmony Search (HS) [10] method. The so called HS algorthm was recently developed by Lee and Geem n an analogy wth musc mprovsaton process where musc player mprovse the ptches of ther nstruments to obtan better harmony. Soluton vectors n HS algorthm, s smulated wth harmony n the musc and search plan wth artst ntatve. In comparson wth other meta-heurstcs, HS mposes fewer mathematcal requrements. These features ncrease flexblty of the HS to analyss varous engneerng optmzaton problems [11]. In ths paper, a new mult-stage HS algorthm s proposed for sze and shape optmzaton of structures. Also the exteror penalty functon method (EPFM), due to ts smplcty and ease of mplementaton, s employed n the framework of the sequental unconstraned mnmzaton technque (SUMT) [12] to handle the constrants. The proposed algorthm s denoted as modfed harmony search (MHS) algorthm. Two benchmark structural shape optmzaton problems are solved by the proposed MHS. The numercal results ndcate that the computatonal performance of the proposed MHS s better than that of the HS. 2. FORMULATION OF THE OPTIMIZATION PROBLEM The mathematcal formulaton of structural optmzaton problems toward the desgn varables, the obectve and constrant functons depend on the type of the applcaton. However, all optmzaton problems can be expressed n standard mathematcal terms, whch

SHAPE OPTIMIZATION OF STRUCTURES BY MODIFIED HARMONY SEARCH 487 n general form can be stated as follows: Mnmze F(X) Subect to g (X) 0, = 1,..., m l u X X X, = 1,..., n (1) where, X s the vector of desgn varables; F(X) s the obectve functon to be mnmzed; g (X) s the th behavoral constrants; l u X and X are the lower and the upper bounds on a typcal desgn varable X. In ths study, to transform the constraned structural optmzaton problem nto an unconstraned one the EPFM s employed. Penalty functon methods transform the basc optmzaton problem nto alternatve formulatons such that numercal solutons are sought by solvng a sequence of unconstraned mnmzaton problems. The above mentoned constraned optmzaton problem can be converted nto an unconstraned problem by constructng a functon of the followng form: m 2 Φ( X, r ) f ( X ) r [max{0, g ( X )}] (2) p p 1 whereφ, and r p are the pseudo obectve functon, and postve penalty parameter, respectvely. By choosng the mnor values for the penalty parameter, the effect of constrants n pseudo obectve functon decrease and optmzaton processes cause to mnmze obectve functon wth small amount of volated constrants, n the other sde by choosng the hgh value for penalty parameter, the effect of constrants n pseudo obectve functon ncreases and the porton of obectve functon decreases. Vanderplaats recommended that f the unconstraned mnmzaton of the pseudo obectve functon s repeated for a sequence of values of the penalty parameter, r p, the soluton may be brought to converge to that of the orgnal problem. These methods are known as sequental unconstraned mnmzaton technques (SUMT). In the present study, the EPFM s employed n the framework of the SUMT to handle the constrants. 3. MODIFIED HARMONY SEARCH ALGORITHM Harmony s defned as an attractve sound made by two or more notes beng played at the same tme. The new HS meta-heurstc algorthm was derved by adoptng the dea that exstng meta-heurstc algorthms are found n the paradgm of natural phenomena. The HS algorthm parameters that are requred to solve the optmzaton problem are also specfed n ths step: harmony memory sze (number of soluton vectors, labled as HMS), harmony memory consderng rate (HMCR), ptch adustng rate (PAR), an arbtrary dstance bandwdth for contnuous varable (bw) and termnaton crteron (maxmum number of searches). The basc steps of the HS may be mentoned as follows:

488 S. Gholzadeh, A. Barzegar and Ch. Gheyratmand Step 1. Intalze algorthm parameters: Specfcaton of each desgn varable, a possble value range n each desgn varable, HMS, HMCR, PAR, and termnaton crteron are ntalzed. Step 2. Intalze harmony memory: The harmony memory (HM) matrx s flled wth randomly generated desgns as the sze of the harmony memory (HMS). Step 3. Improvse a new harmony from the HM: New harmony vectors mprovsed from ether the ntally generated HM or the entre possble range of values. The new harmony mprovsaton progresses based on memory consderatons, ptch adustments, and randomzaton. Here, t s possble to choose the new value usng the HMCR parameter, whch vares between 0 and 1 as follows: ' 1 2 x x,x,...,x ' x ' x X HMS w.p. of HMCR w.p. of (1 HMCR) where the HMCR s the probablty of choosng one value from the sgnfcant values stored n the HM, and (1-HMCR) s the probablty of randomly choosng one practcal value not lmted to those stored n the HM. Every component of the new harmony vector s examned to determne whether t should be ptch-adusted. Ths procedure uses the PAR parameters as follows. (3) Yes No w.p. of PAR w.p. of (1 PAR) (4) The ptch adustng process s performed only after a value s chosen from the HM. The value (1-PAR) sets the rate of dong nothng. A PAR of 0.05 ndcates that the algorthm wll choose a neghborng value wth 5% HMCR probablty. If the ptch adustment appled for a desgn varable then a neghborng value wth the probablty of PAR% HMCR s taken for t as follows: ' x x ( k m) ' x x (5) n whch m s the neghborng ndex; α s the value of bw u(-1,1); bw s an arbtrary dstance bandwdth for the contnuous varable and u(-1,+1) s a unform dstrbuton between -1 and +1. Step 4. Update the Harmony Memory: If the new harmony s better than the worst vector n the HM the new soluton vector s ncluded n the HM and the exstng worst vector s excluded from the HM.

SHAPE OPTIMIZATION OF STRUCTURES BY MODIFIED HARMONY SEARCH 489 Step 5. Termnaton Crtera: Steps 3 and 4 are repeated untl the termnaton crteron s satsfed. The termnaton crteron stops the algorthm when the maxmum number of searches s reached. In order to mprove the exploraton ablty of the standard HS, the algorthm s employed n the framework of SUMT to solve the optmzaton problem and the desgn constrants are handled by EPFM. At frst by choosng a mnor r p a HM wth the sze of HMS s ntalzed and the HS s employed to acheve a prelmnary optmzaton task. The found soluton n ths manner may be nfeasble. In the next step, the harmony vectors of new HM are selected from a neghborng regon of the best soluton obtaned n the prevous process. In ths case, the best soluton s drectly transformed to the new HM and the remanng vectors are selected as the random numbers normally dstrbuted about the mentoned best soluton wth the standard devaton of 10% tmes the best soluton. After ntalzng a new HM, another optmzaton process s acheved by HS usng an ncreased r p. Ths procedure s repeated untl the algorthm converges. 4. NUMERICAL EXAMPLE In order to nvestgate the computatonal performance of the proposed MHS algorthm, two examples are presented. For all examples the HS parameters are as: HMS=10, HMCR=0.971, PAR = 0.05 and bw = 0.3. Also the maxmum numbers of teratons n each optmzaton process and the maxmum number of optmzaton processes are lmted to 100 and 5, respectvely (5000 structural analyses). All of the requred computer programs are coded n MATLAB [13] platform. 4.1. 15-bar Truss Ths problem has been nvestgated by Wu and Chow [14], Hwang and He [15], Tang et al. [16] and Raham et al. [17]. The ffteen-bar 2-D truss s shown n Fgure 1. the magntude of the vertcal load s P = 10 kps. The materal densty s 0.1 lb/n 3 and the modulus of elastcty s 10 4 ks. Fgure 1. Ffteen-bar truss

490 S. Gholzadeh, A. Barzegar and Ch. Gheyratmand In ths example there are 23 desgn varables ncludng two categores: Szng varables: A, =1,2,,15 and Geometry varables: x 2 = x 6 ; x 3 = x 7 ; y 2 ; y 3 ; y 4 ; y 6 ; y 7 ; y 8. Stress lmtaton for all elements s 25 ks. The sze varables are selected from the followng set: D = { 0.111, 0.141, 0.174, 0.220, 0.270, 0.287, 0.347, 0.440, 0.539, 0.954, 1.081, 1.174, 1.333, 1.488, 1.764, 2.142, 2.697, 2.800, 3.131, 3.565, 3.813, 4.805, 5.952, 6.572, 7.192, 8.525, 9.300, 10.850, 13.330, 14.290, 17.170, 19.180} (n. 2 ). Also sde constrants for geometry varables are as follows: 100 n. x 2 140 n.; 220 n. x 3 260 n.; 100 n. y 2 140 n.; 100 n. y 3 140 n.; 50 n. y 4 90 n.; 20 n. y 6 20 n.; 20 n. y 7 20 n.; 20 n. y 8 60 n.; The best results obtaned n ths study are compared wth those of the other works n Table 1. Desgn varables Table 1. Optmal desgn comparson for the 15-bar planner truss Wu and Chow [14] Hwang and He [15] Tang et al. [16] Raham et Present work al. [17] HS MHS A1 1.174 0.954 1.081 1.081 1.081 0.954 A2 0.954 1.081 0.539 0.539 0.954 0.539 A3 0.440 0.440 0.287 0.287 0.270 0.220 A4 1.333 1.174 0.954 0.954 0.954 0.954 A5 0.954 1.488 0.954 0.539 0.539 0.539 A6 0.174 0.270 0.220 0.141 0.270 0.220 A7 0.440 0.270 0.111 0.111 0.111 0.111 A8 0.440 0.347 0.111 0.111 0.141 0.111 A9 1.081 0.220 0.287 0.539 0.220 0.440 A10 1.333 0.440 0.220 0.440 0.220 0.347 A11 0.174 0.220 0.440 0.539 0.440 0.347 A12 0.174 0.440 0.440 0.270 0.111 0.270 A13 0.347 0.347 0.111 0.220 0.440 0.270 A14 0.347 0.270 0.220 0.141 0.287 0.220 A15 0.440 0.220 0.347 0.287 0.220 0.220 x2 123.189 118.346 133.612 101.5775 137.2764 135.5676 x3 231.595 225.209 234.752 227.9112 220.0000 245.5421 y2 107.189 119.046 100.449 134.7986 138.5269 123.1303 y3 119.175 105.086 104.738 128.2206 127.4160 120.6957 y4 60.462 63.375 73.762 54.8630 50.0000 57.9313 y6 16.728 20.0 10.067 16.4484 19.1800 5.9742 y7 15.565 20.0 1.339 16.4484 2.8000 2.9125 y8 36.645 57.722 50.402 54.8572 38.3190 56.3256 Best weght (lb) 120.52 104.573 79.820 76.6854 80.364 73.887 Number of analyses - - 8000 8000 5000 5000 The results demonstrate the computatonal advantages of the MHS wth respect to other algorthms. The geometry of the optmum structure s shown n Fgure 2.

SHAPE OPTIMIZATION OF STRUCTURES BY MODIFIED HARMONY SEARCH 491 (a) (b) Fgure 2. (a) Optmum shape of ffteen-bar planar truss (b) geometry of nodes 4 and 8 To assess the computatonal performance of the proposed MHS algorthm, 25 ndependent runs are acheved and the best, worst and mean weghts of 73.887 lb, 88.420 lb and 79.206 lb are obtaned. 4.2. 25-bar truss Ths problem has been nvestgated by Wu and Chow [14], Tang et al. [16] and Raham et al. [17]. The twenty fve-bar truss s consdered as shown n Fgure 3. Fgure 3. Twenty fve-bar space truss The materal densty s 0.1 lb/n 3 and the modulus of elastcty s 10 4 ks. Loadng data s gven n Table 2.

492 S. Gholzadeh, A. Barzegar and Ch. Gheyratmand Table 2. Loadng data for twenty fve-bar truss Node F x (kps) F y (kps) F z (kps) 1 1.0-10.0-10.0 2 0.0-10.0-10.0 3 0.5 0.0 0.0 6 0.6 0.0 0.0 There are 13 desgn varables ncludng two categores as follows: Sze varables: A 1 ; A 2 = A 3 = A 4 = A 5 ; A 6 = A 7 = A 8 = A 9 ; A 10 = A 11 ; A 12 = A 13 ; A 14 = A 15 = A 16 = A 17 ; A 18 = A 19 = A 20 = A 21 ; A 22 = A 23 = A 24 = A 25 Geometry varables: x 4 = x 5 = -x 3 = -x 6 ; x 8 = x 9 = -x 7 = -x 10 ; y 3 = y 4 = -y 5 = -y 6 ; y 7 = y 8 = -y 9 = -y 10 ; z 3 = z 4 = z 5 = z 6 Stress lmtaton for all elements s 40 ks also dsplacement constrant n all drectons s 0.35 n. The sze varables are selected from the followng set: D = {0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, 2.8, 3.0, 3.2, 3.4} (n. 2 ). Also sde constrants for geometry varables are as follows: 20 n. x 4 60 n.; 40 n. x 8 80 n.; 40 n. y 4 80 n.; 100 n. y 8 140 n.; 90 n. z 4 130 n.; The best results obtaned n ths study are compared wth those of the others n Table 3. Table 3. Optmal desgn comparson for the 25-bar space truss Desgn varables Wu and Chow [14] Tang et al. [16] Raham et al. Present work [17] HS MHS A1 0.1 0.1 0.1 0.1 0.1 A2 0.2 0.1 0.1 0.1 0.1 A3 0.1 1.1 1.1 1.0 0.1 A4 0.2 0.1 0.1 0.1 0.1 A5 0.3 0.1 0.1 0.1 0.1 A6 0.1 0.2 0.1 0.1 0.1 A7 0.2 0.2 0.2 0.1 0.1 A8 0.9 0.7 0.8 1.0 0.9 X4 41.07 35.47 33.0487 32.95 37.82 Y4 53.47 60.37 53.5663 68.185 55.485 Z4 124.6 129.07 129.9092 107.37 128.73 X8 50.8 45.06 43.7826 47.367 52.068 Y8 131.48 137.04 136.8381 136.02 139.59 Best weght (lb) 136.20 124.94 120.11 122.62 117.38 Number of analyses - 6000 8000 5000 5000

SHAPE OPTIMIZATION OF STRUCTURES BY MODIFIED HARMONY SEARCH 493 The results demonstrate the computatonal advantages of the MHS wth respect to other algorthms. The geometry of the optmum structure s shown n Fgure 4. To assess the computatonal performance of the proposed MHS algorthm, 25 ndependent runs are acheved and the best weght of 117.40 lb, the worst weght f 130.20 lb and the mean weght of 119.02 lb are obtaned. (a) (b) Fgure 4. (a) Optmum shape of truss n (a) y-z plan and (b) x-z plan 5. CONCLUSION The shape optmzaton of truss structures s tackled n ths paper usng an enhanced HS metaheurstc algorthm. In order to mprove the computatonal performance of HS the standard HS algorthm s sequentally utlzed n the framework of SUMT employng EPFM to handle the desgn constrants. The proposed meta-heurstc algorthm s termed as modfed harmony search (MHS) algorthm. Both sze and shape structural optmzaton problems are solved by the proposed algorthm and the results are compared to those of the other researchers. The numercal results ndcate that usng MHS not only better solutons can be found but also a sgnfcant reducton n computatonal effort may be acheved. For more detals, n the case of frst example, the best weght found n the lterature s 76.6854 lb spendng 8000 structural analyses whle n the present paper the best weght of 73.887 lb s obtaned after 5000 structural analyses. In the case of second example, the weght of the best structure and ts correspondng requred number of analyses reported n the lterature are 120.11 lb and 8000, respectvely whle these values n the present paper are 117.38 lb and 5000, respectvely. These results emphasze on the effcency of the proposed MHS algorthm for shape optmzaton of structures.

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