Optimization of Truss Structures Using Genetic Algorithms with Domain Trimming (GADT)

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1 Optmzaton of Truss Structures Usng Genetc Algorthms wth Doman Trmmng (GADT) Samer Barakat and Omar Nassf Abstract Ths paper presents evolutonary-based least-weght optmzaton procedure for desgnng truss structures. A modfed verson of Genetc Algorthm wth Doman Trmmng (GADT) s developed and presented heren. The DADT s used for solvng the nonlnear constraned optmzaton problems. In ths optmum desgn formulaton, the objectve functon s the materal weght of the truss; the desgn varables are the cross-sectons of the truss members; the constrants are the stresses n members and the dsplacements of the jonts. The constrants were handled usng non-statonary dynamcally modfed penalty functons. One classcal truss optmzaton example s presented heren to demonstrate the effcency of the GADT algorthm. The test problem ncludes a 10- bar planar truss subjected to two load condtons. The result shows that the GADT method s very effcent n fndng the best dscovered optmal solutons, whch are better of the results of other structural optmzaton methods.. Keywords Truss Structural Optmzaton, Genetc Algorthm, Doman Trmmng, Constrant Handlng. I. INTRODUCTION btang optmal desgns that satsfy multple conflctng Ocrtera, such as mnmum cost and maxmum performance, s one of the most nfluental factors n modern structural desgn. Most structural desgns are consdered constraned optmzaton problems that can be solved to dentfy the desgn values of structural performance. The optmum solutons mght be lnearly and/or nonlnearly constraned n the desgn space. In the presence of multple optma and non-smooth constrants n the desgn varable space, t s dffcult to obtan a set of optmum values usng local optmzaton technques. On the other hand, ths dffculty has geared the research towards relatvely new and nnovatve evolutonary based optmzaton technques [1] such as the Genetc Algorthm (GA) [2], Ant Colony Optmzaton (ACO) [3], Partcle Swarm Optmzer (PSO) [4], Shuffled Complex Evoluton (SCE) [5]-[7], Harmony Search [8], and Hybrd Methods [9]-[11]. These approaches are nvestgated and used n recent years for optmzng structural desgns and are proven superor to local search technques. Many structural optmzaton problems nvolve problem- S. B. Author s wth the Unversty of Sharjah, Sharjah, UAE (correspondng author phone: ; fax: ; e-mal: sbarakat@sharjah.ac.ae). T. C. Author s wth Department of Cvl Engneerng, Unversty of Tennessee, USA (e-mal: onassf@vols.utk). specfc constrants applcable to the solutons lmtng the feasble search space Compared to other constrant handlng technques the use of penalty functons s relatvely smple and easy to mplement. Ths study presents the development and mplementaton of the GADT to acheve superor optmzaton. The capabltes of the developed optmzaton tool are demonstrated on two classcal truss optmzaton problems beng challengng wth unknown global and multple local mnma. II. GENETIC ALGORITHM WITH DOMAIN TRIMMING (GADT) A. Development of the GADT Optmzaton Technque To begn GA optmzaton, a populaton of soluton alternatves Np (populaton sze) are randomly generated usng a unform probablty dstrbuton; each soluton of the GA conssts of a combnaton of varables (x1, x2, x3,, xn) whch has ts own ftness value. In cases where the optmzaton s performed to fnd the mnmum weght for a gven problem, a functon F*j(X) = total mass + penalty has to be mnmzed. Soluton alternatves that yeld low F*j(X) values for the objectve functon would have better ftness as long as they are not volatng the problem constrants. Populatons of solutons are represented by chromosomes. The desgn varables stored n the chromosome can be ether dscrete (selected from a pool of defned values) or contnuous (selected from a contnuous range of varables). In ths research, the vector of varables contans contnuous values. Once F*j(X) for every soluton j n the ntal populaton s computed, a ftness value s assgned to each soluton j usng Eq.(1), Solutons wth F*j(X) less than F*ave of the populaton are consdered unft and are elmnated by assgnng them a ftness value of zero: FF jj (XX) = F ave FF jj (XX) for FF jj (XX) < F ave 0 for FF jj (XX) F ave j = 1,2,, N p (1) where (X) s the vector of the desgn varables. The three basc operatons of a GA, reproducton, crossover, and mutaton, are used to mprove the ftness of each populaton from one generaton (teraton) to the next. The reproducton operaton selects the better ft desgns, copes them, and places them nto a matng pool allowng each to mate and reproduce. The roulette wheel selecton method s used n ths study for ts smplcty and popularty (Fg. 3). Ths method assgns for each ftness functon value a porton of unty that wll be used as the reproducton probablty, P r. If for example, the best ft soluton n a certan populaton has a ISBN:

2 P r value of 10% and the populaton sze N p =100 then the matng pool wll have approxmately 10 copes of ths soluton. Fg. 3: Pe chart of probablty of selecton of ft parents After the reproducton operaton s performed, the crossover operaton mates the selected desgns to create more ft offsprng solutons. The unform crossover operaton s used to combne genetc nformaton between two parent solutons. Unform crossover selects two parent solutons at a tme from the matng pool and swaps varables correspondng to zeros n a bnary vector known as a mask. The mask s the same length as all varable vectors and conssts of a preselected percentage of randomly arranged zeros (%c). Ths percentage has an mpact on the speed of convergence to an optmum soluton. Each mask wthn a populaton s dfferent, so the number of unque, randomly generated masks s equal to half of the number of solutons (parents) n the populaton multpled by the total number of generatons (populatons). Snce GA mmcs the natural selecton of the fttest treats through multple generatons, t s nherently vulnerable to Genetc Drft (contnuous survval of unft but lucky ndvduals, and ther genes, from one generaton to the next). The mutaton operaton s used to mnmze the effect of genetc drft and add dversty to the search space by randomly changng a varable n a desgn soluton. Durng mutaton the value of any chromosome varable may be changed to a randomly selected varable. Another counter measure of genetc drft used for contnuous varables s to use blendng technques [12], a general blendng formula would be of the form: xx nnnnnn = ββxx mmmm + (1 ββ)xx dddd (2) Where, x new = the nth varable of the offsprng chromosome of the crossover, β = random number on the nterval [0,1], x mn = the nth n the mother chromosome, and x dn = the nth n the father chromosome. One of the man advantages for the GA optmzaton s ts relatve nsenstvty to local mnma (not very susceptble to beng trapper n local mnma). However, ths may also be consdered as one of ts lmtatons: beng a low-resoluton technque especally f the populaton sze s relatvely lmted..e. convergence may sometme occur to ponts n the neghborhood of, but not exactly at, global mnma. The reason for ths s that ntally, the soluton varables are randomly selected from a pre-specfed range (doman); those varables do not change (except for mutaton, whch occurs at small probablty) but rather change place from one soluton to another. In addton, a poor choce of the soluton doman (e.g.: [1, 1000] whle the optmal value s at a value of 2) further aggravates the ssue. Blendng technques, whle actng to reduce ths dsadvantage, ntroduce a new random parameter (β), t may also have a negatve effect as t may negate the strong trats of the parents. In ths study, a new technque s developed to allevate ths nherent lmtaton of GA. The technque nvolves trmmng the doman then re-ntatng the GA so that the probablty of selectng the optmal soluton s mproved. For example, trmmng a doman from [1, 1000] to [1, 10] then re-ntatng GA would ncrease the ntal probablty of selecton of the optmal soluton by 200 tmes, when everythng beng constant. Trmmng s done as a percentage of orgnal doman sze, and contnuous trmmng then GA re-ntaton goes on untl the optmum soluton s found. One can thnk of trmmng n terms of evoluton theory as when a catastrophe occurs n nature elmnatng the majorty of the populaton leavng only the elte survvals to restart the evoluton process. Fg.4 smulates ths technque on a sngle varable chromosome (takng trmmng as 90% of doman sze): 1- After the GA converged to a certan objectve functon value, the elte 10 chromosomes (n ths case, varables) are taken regardless of whch generaton they re n, snce the range between the mnmum and maxmum varable s more than 90%, no trmmng occurs. 2- As the GA converged for the second tme, the range of values are now less than 90% and trmmng wll happen, but rather than beng centered, the trmmng range s based toward the mean by a rato: (max-mean)/(max-mn), the trmmed doman s now less than 90% of the orgnal as some of the trmmng range les outsde the orgnal doman. 3,4,5 - Because the doman has been trmmed, the chances of gettng to the global optmum are hgher, resultng n the elte values gettng closer to each other, trmmng wll stop when range of values exceeds 90% of the doman or f the GA found the optmum soluton to a satsfactory precson. Fg. 4: Illustraton of the doman trmmng method on a sngle varable (trmmng s 90%) A queston arses that how can t be sure that the optmal soluton s nsde the trmmed doman; the answer s that t doesn t guarantee that, however, the worst case scenaro s that t wll get results comparable to conventonal GA snce ISBN:

3 trmmng doesn t happen untl GA converges. Expermentng on sample problems showed that the technque gves better results than conventonal GA. Fg. 5a-d llustrates the logcal steps for the GADT technque va flowcharts. Fg. 5c: Flow chart of Roulette wheel selecton Fg. 5a: Flow chart of general GADT optmzaton Fg. 5d: Flowchart of Doman Trmmng To demonstrate the GADT algorthm performance, one of the standard test functons n optmzaton problems s consdered: the Sx-Hump Camelback functon problem (43): ff(xx, yy) = 4 2.1xx 2 + xx 4 3 xx 2 + xxxx + ( 4 + 4yy 2 )yy 2 (3) Wth boundares: x [-1.5, 1.5] and y [-2, 2], the functon takes the form shown n Fg. 6; t has sx mnma, two of them are global mnma wth a value of: located at the two ponts: ( , ) and (0.0898, ). Fg. 5b: Flow chart of Reproducton, Crossover and Mutaton Fg. 6: Sx-hump Camelback functon. a) 3D plot b) Contour plot ISBN:

4 Intally, the GA starts off wth the doman provded by the user and terates untl there s a convergence,.e. the ftness values are close to each other and the mprovement n best ftness n subsequent teratons s less than a specfed value, say, 1%. At ths stage, the GA has dentfed a number of local mnma and further teratons wll have lttle effect on fndng the global mnmum. Convergence to a seres of not necessarly most ft solutons means that the GA s approachng the vcnty of the global optmum but cannot dscover the fttest soluton. Ths may be due to elmnaton of some of the fttest solutons as they are combned wth lesser-ft solutons, or ncluson of resdual unft solutons wthn the avalable soluton doman. At ths pont, refnement of the avalable soluton doman mproves the chances of dscover the fttest soluton. GADT technque removes the most unft solutons from the doman and re-ntates GA wthn the trmmed doman. Occasonally, some fttest solutons are dsmssed as unft as result of beng combned wth lesser-ft solutons. To mnmze the probablty of ths potental ptfall, two measures are ncorporated n the GATD: (a) the ntal doman consders a large enough populaton to allow hgher probablty for dscoverng the most combnatory possbltes, and (b) the doman trmmng lmted to a hgh percentage (e.g. 90%) of the doman from the prevous step. Ths means that whle some values are clearly dentfed as unft, they are stll retaned n the doman for the subsequent GA re-ntaton. The motvaton for ths retenton t exhaust all possbltes of producng ft solutons before completely elmnatng part of the doman. The mpact on speed of dscovery of optmum solutons s obvous. However, t s justfed by the sgnfcantly mproved successful dscovery rate. The method can also be helpful to dentfy what range of varables to look for n subsequent searches. Fg.7 shows the fttest 100 overall solutons throughout the GADT; the box drawn on the contour map represents the updated (trmmed) doman, elmnatng 10% of the orgnal doman at each step. The contnuaton of doman trmmng results n dentfyng new local mnma. As more local mnma are dentfed, they begn to compete wth each other. Eventually, the trmmng technque wll exclude the values of low ftted local mnma and wll deem them as unft results. Ths helps the GA to focus more on only the top-ftted solutons ultmately enhance the dscovery rate (precson) of global optma solutons. Fg.7d shows the fnal doman after 20 doman trmmng teratons; t s noted that the doman cannot get trmmed further as the two global mnma resde on the trmmed boundares of the last step. It s worth mentonng that the fnal doman s only 3.5% of the ntal full doman n more precse (vrtually exact) optmum soluton: f(x, y) = ( % error). B. GADT Technque Robustness To ensure GADT robustness, doman trmmng s subject to further crtera preventng any potental adverse consequences. Doman trmmng wll not resume n any of the followng scenaros: (a) f the range of values that yeld ft results n the current doman s more than 90% of the prevous doman; n other words, doman trmmng wll not commence untl the feasble values get condensed nto less than 90% of the doman. And (b) f after trmmng, the GA yelds solutons worse than prevous GA ntaton. Although the latter was not encountered durng any of the algorthm testng sessons for any of the presented problems here, the crteron s set n place for potental future problem-specfc complcatons. The effect of ntal doman sze s dscussed n a subsequent secton. (a) Intal Doman User provded (c) after 10 doman trmmng (b) Frst doman trmmng (d) Fnal soluton doman Fg.7: Illustraton of the GADT algorthm appled to the Sx-hump Camelback functon at dfferent teratons of doman trmmng. C. Constrants Handlng The use of penalty functons s very popular n handlng constrants enablng the soluton of constraned problems as unconstraned. The solutons that volate any constrants are penalzed n order to characterze non-feasble solutons by hgh objectve functon values. Non-statonary (dynamc) penalty functons typcally exhbt superor performance to statonary (statc) penalty functons. In ts generalty, a nonstatonary penalty functon s defned as: f( X) = FX ( ) + pxc (, pn, epn) (4) Where F(x) s the orgnal objectve functon of the constraned optmzaton problem; p () s a dynamcally modfed penalty value, defned as: pp XX, cc pppp, ee pppp = cc pppp rr eepppp, rr 1 (5) 0, rr < 1 Where, rr s the ndvdual member s performance crtera (n a structural desgn problem, ths s often taken as the demand-to-capacty rato, or utlzaton rato to the satsfacton of the relevant desgn code); whle c pn and e pn are the penalty coeffcent and exponent, respectvely. As ther name mples, they provde means to penalze the optmzaton objectve f the rr exceeds unty. Both c pn and e pn, wth dfferent severty, wll penalze unft solutons mnmzng ther probablty of reappearng subsequent generaton and feasble soluton doman. ISBN:

5 D. Truss Structural Optmzaton The mathematcal form of the optmzaton problem for truss structure can be expressed as follows: { } T Fnd A = A, A,... A (6) 1 2 To Mnmze F= W( A) = ρ LA (7) = 1 L U Subject to g g ( A) g j = 1, 2,... m j j j n mn max and A A A = 1, 2,... n (9) Where A = the desgn varable (member cross-sectonal area), n = the number of the desgn varables, W(A) = the objectve functon (the structural weght), ρ = the materal densty, L = the member length, m = the number of nequalty constrants (g), A mn and A max n (8) are the lower and the upper bound of the th varable respectvely. The lower and upper bounds posed by Eq.(8) on the constrants nclude truss member stresses and jont dsplacements. E. Example: Cantlever 10-Bar Planar Truss Structure The GADT s tested aganst a classcal global optmzaton problem: the Cantlever 10-Bar Planar Truss Structure optmzaton problem. Ths 10-dmensonal problem has been nvestgated by many researchers, and has been wellestablshed as an optmzaton benchmark problem known for beng challengng wth unknown global and multple local mnma. A schematc of the cantlever 10-bar planar truss structure can be found n Fg.8. Table 2: Optmzaton results for the 10-bar planar truss Load Case 1 Tables 1 and 2 gve the best dscovered optmum solutons along wth the correspondng mnmum weght for the two cases 1 and 2, respectvely, benchmarked aganst optmal desgns by other publshed studes. It should be noted that the best dscovered soluton was found after teratons for case 1 and 4000 teratons for case 2; ntal populaton sze, pror to trmmng, s taken as 500 solutons. The optmal solutons found by the GADT meet all of the problem constrants and the comparsons n Tables 1 and 2 show that the GADT provdes superor results. Fg.9 shows convergence hstores for loadng cases 1 and 2. Notce that convergence plateaus after around 800 teratons; at whch pont, the trmmng has progressed such that t has lttle further effect on the GA sgnfyng the elmnaton of most or all unft values. Table 3: Optmzaton results for the 10-bar planar truss Load Case 2 Fg.8: 10-Bar planar cantlever truss model The assumed materal densty s 0.1 lb/n 3 ( kg/m 3 ), the length L s 360 n (914.4 cm) and the modulus of elastcty s 10,000 ks (68,950 MPa). The stress and deflecton lmtatons on the members are ±25.0 ks ( MPa) and ± 2.00 n (5.08 cm), respectvely. Cross-sectonal areas are allowed to vary between 0.1 n 2 and 35 n 2 ( cm 2 and cm 2 ). No member groupng s utlzed, resultng n each member havng a potentally unque crosssectonal area (A 1 to A 10 ). Ths truss optmzaton problem has 10 desgn varables and 32 (10 tenson stresses, 10 compresson stresses, and 12 dsplacements) constrants. Two loadng cases are studed: Case 1, when P 1 = 100 kps (444.8 kn) and P 2 = 0 kps and Case 2, when P 1 = 150 kps (667.2 kn) and P 2 = 50 kps (222.4 kn). ISBN:

6 Fg.9: Path to optmum soluton for load cases 1 and 2 on the 10-bar truss. Fg.11: Path to optmum soluton for both GA and GADT for load case 1. Fgs.9 and 10 compare the frst 300 teratons of the optmzaton wth and wthout doman trmmng for the loadng case 1. Note that for the shown number of teratons the dfference between the dfferent dscovered solutons s relatvely margnal. Fgs.11 and 12 compare the convergence hstores of GA wth and wthout doman trmmng for a populaton sze of 500 gong through 2000 teratons. Notce that n some subsequent teratons, the conventonal GA dentfes a hgher mnmum weght as the best dscovered soluton, as compared to a prevous teraton. Ths prmarly a result of the mutaton process, whch eventually s fltered out wth enough teratons. Ths effect s not exhbted n presence of doman trmmng snce the mutaton s unable to select the excluded/trmmed unft values. Fg.9: Frst 300 teratons of a conventonal GA optmzaton on 10-bar truss. Fg.10: Frst 300 teratons of GADT optmzaton on 10-bar truss. Fg.12: Path to optmum soluton for both GA and GADT for load case 2 Snce GA optmzaton s an evolutonary method, t reles on generatng populatons of solutons n order to fnd the optmum amongst them. To promote better dscovery of optmum results, an adequate populaton sze should be selected. Unfortunately, ths parameter s problem-specfc. The number of varables n a soluton, dversty of values and ntal doman selecton are some of the factors determnng the approprate populaton sze. The nclnaton to start wth a very large populaton sze n order to gve better dscovery chances, s sometmes counterproductve when the lowresoluton lmtatons of the GA are sgnfed. However, selectng too small of populaton sze wll result n convergence falure for the GA for lack of suffcent representatve solutons for selecton. The GADT technque provdes mtgatng to ths populaton sze effect snce ts rentaton gves a new chance for prevously undscovered varables to be selected n the new populaton. To demonstrate ths feature, the 10-bar planar truss problem was repeated wth dfferent populaton szes. Populaton szes chosen for the test were 20, 50, 100, 200, 300, 400 and 500 whle all optmzatons ran for 2000 teratons. The hstogram n Fgure 13 shows the effect of populaton sze on fndng the optmum soluton. It can be seen that ncreasng the populaton sze mproves the precson of fndng the optmum soluton, although slghtly. Populaton szes as low as 50 solutons per populaton yelded acceptable results whch are less than 0.5% away from ther counterpart wth a populaton sze of 500. However, when selectng a populaton sze of 20, the method fals and the dscovered soluton s far off from the optmum. ISBN:

7 Fg.13: Senstvty of GADT Algorthm to populaton sze. III. SUMMARY AND CONCLUSIONS A modfed verson of Genetc Algorthm wth Doman Trmmng (GADT) s developed and presented n ths study. The nnovatve technque s used to solve a least-weght optmzaton problem n the desgn of truss structures. Pror to mplementaton of the nnovatve GADT technque on the truss problem, a well-establshed Sx-Hump Camelback functon benchmark problem s used for demonstraton purposes. Then, the GADT s tested aganst a well-establshed challengng benchmark problem; t s tested aganst a classcal global optmzaton problem: the Cantlever 10-Bar Planar Truss Structure optmzaton problem. Ths benchmark problem s a 10-dmensonal optmzaton problem wth unknown local and global mnma. The GADT performed superorly n the demonstraton as compared to recently publshed optmum solutons n the lterature. In ths optmum desgn formulaton, the objectve functon s the materal weght of the supportng truss; the desgn varables the crosssectonal areas of the truss members; the constrants are the stresses n members and the dsplacements of the jonts. The GADT handles the problem-specfed constrants usng nonstatonary penalty functons method. The results show that the (GADT) method s effcent n fndng the best dscovered optmal soluton. The optmal solutons found by the GADT meet all of the problem constrants and the comparsons wth the publshed lterature show that the GADT provdes superor results. Algorthm. J Struct Eng. Amercan Socety of Cvl Engneers; 2006; 132(7): [3] Perez RE, Behdnan K. Partcle swarm approach for structural desgn optmzaton. Comput Struct. Pergamon Press, Inc.; 2007; 85(19-20): [4] Luh G-C, Ln C-Y. Structural topology optmzaton usng ant colony optmzaton algorthm. Appl Soft Comput. Elsever Scence Publshers B. V.; 2009; 9(4): [5] Barakat, Samer A., and Salah Altoubat. "Applcaton of evolutonary global optmzaton technques n the desgn of RC water tanks." Engneerng Structures 31.2 (2009): [6] Barakat, Samer, and Hsham Ibrahm. "Applcaton of shuffled complex evoluton global optmzaton technque n the desgn of truss structures."modelng, Smulaton and Appled Optmzaton (ICMSAO), th Internatonal Conference on. IEEE, [7] Barakat, Samer A. "Shuffled complex evoluton optmzer for truss structure optmzaton." Computng n Cvl and Buldng Engneerng, Proceedngs of the Internatonal Conference. Vol [8] Lee, Kang Seok, and Zong Woo Geem. "A new structural optmzaton method based on the harmony search algorthm." Computers & Structures 82.9, , [9] Kaveh, A., and S. Malakout Rad. "Hybrd genetc algorthm and partcle swarm optmzaton for the force method-based smultaneous analyss and desgn."iranan Journal of Scence and Technology, Transacton B: Engneerng 34.B1, 15-34, [10] Csébfalv, Ankó. "A hybrd meta-heurstc method for contnuous engneerng optmzaton." Cvl Engneerng 53.2, , [11] Kaveh, A., and S. Talatahar. "A hybrd partcle swarm and ant colony optmzaton for desgn of truss structures." Asan Journal of Cvl Engneerng 9.4, , [12] N. J. Radclffe, Forma Analyss and Random Respectful Recombnaton, Proceedngs of the Fourth Internatonal Conference on Genetc Algorthms Morgan Kaufmann, San Mateo, CA, 1991, [13] Schmt Jr LA, Mura H, "Approxmaton concepts for effcent structural synthess," NASA CR-2552, Washngton, DC: NASA, [14] SCHMIT JR LA, FARSHI B., Some approxmaton concepts for structural synthess, AIAA J; 12(5): [15] VENKAYYA VB., "Desgn of optmum structures." Computer &Structures 1971, 1(1 2): [16] DOBBS MW, NELSON RB., Applcaton of optmalty crtera to automated structural desgn, AIAA J; 14(10): REFERENCES [1] Lagaros ND, Papadrakaks M, Kokossalaks G. Structural optmzaton usng evolutonary algorthms. Comput Struct. 2002; 80(7-8): [2] Ballng RJ, Brggs RR, Gllman K. Multple Optmum Sze/Shape/Topology Desgns for Skeletal Structures Usng a Genetc ISBN: