Truss Topology Optimization Using Genetic Algorithm with Individual Identification Technique

Transcription

1 WCE 2009, July - 3, 2009, London, U.K. Truss Topology Optimiztion Using Genetic Algorithm with Individul Identifiction Technique Su Ruiyi, Gui Lingjin, Fn Zijie Abstrct Since the evlution of ech individul is bsed on the time-consuming structurl nlysis, the computtionl efficiency of truss topology optimiztion using genetic lgorithm is very low. The pper focuses on this chllenging problem. It is observed tht there re number of duplicte individuls ppering repetitively in the evolutionry process. Therefore, n individul identifiction technique is introduced to void evluting the duplicte individuls by the time-consuming structurl nlysis but by serching the evolutionry history dt to sve computing time, the computtionl complexity of this technique is deduced. The results of two truss exmples verify tht the technique cn effectively improve the efficiency of the lgorithm. Bsed on this identifiction technique, numeric experiments re implemented to study the influence of severl fctors, i.e., the popultion size, the mx genertion, nd the scle of problems, on the proportion of duplicte individuls. Results show tht the popultion size hs significnt impct on the proportion, nd tht both the mx genertion nd the scle of problems hve little influence. Keywords Genetic lgorithm, Individul identifiction, Topology optimiztion. I. ITRODUCTIO Generlly, the field of structurl optimiztion could be divided into three sub-problems, nmely sizing, shpe nd topology optimiztion. The topology optimiztion is most beneficil s it cn find out the best loding pth in the infinite topology combintions to sve the most mterils, but it is lso more intellectully chllenging thn the other two optimiztion problems becuse of its greter complexity [], [2]. Genetic lgorithm hs severl dvntges compred to the trditionl grdient bsed lgorithms, such s the powerful cpbility of deling with discrete, non-convex problems, no differentible requirement of the response functions, nd the globl convergence bility. Therefore, it is very suitble for truss topology optimiztion [3], [4]. But the low Mnuscript received Mrch 20, This work ws supported in prt by the tionl High Technology Reserch nd Development Progrm ("83" Progrm) of Chin under Grnt no. 2007AA04Z33. Su Ruiyi, Ph.D. cndidte of the Deprtment of Automotive Engineering, Tsinghu University, Beijing, CO Chin (phone: 8 (0) ; fx: 8 (0) ; e-mil: sry@tsinghu.edu.cn). Gui Lingjin, ssocite professor of the Deprtment of Automotive Engineering, Tsinghu University, Beijing, CO Chin (e-mil: gui@tsinghu.edu.cn). Fn Zijie, professor of the Deprtment of Automotive Engineering, Tsinghu University, Beijing, CO Chin (e-mil: zjfn@tsinghu.edu.cn). computtionl efficiency is the min shortcoming of truss topology optimiztion using genetic lgorithm, s the evlution of ech individul is dependent on the time-consuming structurl nlysis. There re three wys to rise the efficiency. The first one is to study the evolutionry theory to improve the convergence of genetic lgorithm, e.g., combining with grph-bsed concept [5], or developing new encoding pproch [], et l. The second is to use fster computers or utilize prllel computing technology, which cn effectively improve the computtionl efficiency wheres without reducing the scle of computtion. Becuse the stge of evlution of ech individul by structurl nlysis is the most time-consuming step in genetic lgorithm, the lst wy is to void the redundnt structurl nlysis s much s possible. This pper focuses on the third wy to improve the computtionl efficiency of truss topology optimiztion using genetic lgorithm. Becuse of the stochstic opertion of genetic lgorithm, mny kinemticlly instble or structurlly invlid individuls pper in the optimiztion process. It is wste of time to evlute these individuls. Therefore, P. Hjel [7] used two-level strtegy to generte stble structures. Both Tng [] nd Deb [8] introduced DOF (Degree Of Freedom) eqution to filter instble structures, nd utilized heuristic criteri to eliminte invlid structures. These mens reduce the number of the instble or invlid structures, thus rising the efficiency of genetic lgorithm. Furthermore, it is observed tht there re number of duplicte individuls ppering repetitively in the evolutionry process of genetic lgorithm. According to the Hollnd s Schem theorem, this phenomenon is inevitble [9], [0]. Therefore, it is better to evlute the duplicte individuls by serching the evolutionry history dt directly rther thn by the time-consuming structurl nlysis. This cn further improve the computtionl efficiency. But it is uncceptble to store ll individuls ppering in the evolutionry process becuse of the huge storge memory demnds. Menwhile, the computtionl complexity of evluting the duplicte individuls fitness by serching the evolutionry history dt is high. Therefore, n individul identifiction technique with less storge memory demnds nd lower computtionl complexity is developed. Bsed on this identifiction technique, the influence of severl fctors on the proportion of duplicte individuls is investigted. Section 2 gives the formultion of truss topology ISB: WCE 2009

2 WCE 2009, July - 3, 2009, London, U.K. optimiztion problem. Section 3 introduces the genetic lgorithm nd the individul identifiction technique in detil. Section 4 illustrtes two numeric exmples to verify the effect of the identifiction technique nd lso investigtes the influence of severl fctors on the proportion of duplicte individuls by numeric experiments. II. FORMULATIO A very importnt problem in truss topology optimiztion is the singulr solution []. Cheng et l. [2] developed n ε -relxed pproch which relxed the stress constrints to solve the singulr problem. But the ε -relxed pproch could not be gurnteed to find out the globl optiml solution. Genetic lgorithm cn not only solve the singulr problem, but lso ttin the globl optiml solution. The formultion of truss topology optimiztion using genetic lgorithm is s follows: e min W = tiρiliai st.. KUj = Pj, j=,2,, M σ ij () ti 0, i =, 2,, e [ σ ] i δk 0, k =,2,, [ δ ] k where M, e, nd is the number of loding cses, members, nd nodes respectively; [σ] nd [δ] is the llowble stress nd displcement respectively; t i is the Boolen vlue, nd ρ i, l i, A i is the density, length, nd re of the i th member respectively; K is the stiffness mtrix, U nd P is the vector of displcement nd externl force respectively. The formultion elimintes the stress constrints of bsent members utomticlly becuse of the introduction of topology vrible t, thus there is no singulr problem here. The formultion is difficult for trditionl grdient-bsed lgorithms becuse of the existence of discrete topology vribles, but esy for genetic lgorithm. Furthermore, the globl convergence bility of genetic lgorithm mkes sure it cn ttin the globl optiml solution. III. IMPLEMETATIO A. Genetic Algorithm Genetic lgorithm works with coded string of design prmeters, nmed chromosome, but not the prmeters themselves, thus the encoding pproch is key point in genetic lgorithm. In this pper, node mtrix encoding pproch [3] is utilized to hndle the topology nd sizing optimiztion of truss structure ccording to its chrcteristics of the finite element model. Becuse of the inherent sprseness of node mtrix, the sprse mtrix technique is introduced to sve the storge memory nd rise the computtionl efficiency. Following is n exmple to illustrte this sprse node mtrix encoding pproch. The truss structure shown in Fig. hs nodes nd members. The cross-sectionl re of every member is selected from set of 9 discrete vlues (5, 0, 5, 20, 25, 30, 35, 40 nd 45), which cn be denoted by 9 properties with IDs from to 9 respectively. The property encoding mtrix P is shown in Fig. b, where the element P(3, 4)=3 denotes tht the property ID of the member connecting with node 3 nd node 4 (member 4 ) is 3, which represents the cross-sectionl re of 5. The topology encoding mtrix T is shown in Fig. c, where the element T(2, 5) is equl to zero, which denotes tht the member connecting with node 2 nd node 5 is bsent in the structure () Ground structure (b) Property mtrix P (c) Topology mtrix T Fig. Exmple of encoding mtrices This node mtrix encoding method is more nturl thn the trditionl vector encoding pproch when portrying the truss structurl optimiztion problem. Different from the vector encoding, the size of node mtrix encoding is only concerned with the number of nodes, but not members in the ground structure. Consequently, it is more suitble for complicted structures with lrge numbers of members. Furthermore, this encoding pproch cn be esily extended to hndle the profile optimiztion nd mteril optimiztion problem, for the property of member cn represent not only the cross-sectionl dimensions but lso the cross-sectionl profiles nd mterils. Prents Offspring Fig. 2 Submtrix crossover Selection, crossover nd muttion re the min opertors in genetic lgorithm. In this pper, the tournment selection is utilized. Different crossover nd muttion opertors re ISB: WCE 2009

3 WCE 2009, July - 3, 2009, London, U.K. developed for topology nd property encoding mtrices. For the topology encoding mtrix, submtrix crossover nd single-point muttion re used. In the submtrix crossover, the prents re rndomly divided into four submtrices, nd then the right lower submtrices re exchnged to produce offspring, s shown in Fig. 2. In the single-point muttion, member from the ground structure is stochsticlly selected, nd then the Boolen bit of the corresponding element in the topology encoding mtrix is reversed. For the property encoding mtrix, rithmetic crossover nd Guss muttion re employed. Suppose tht the prent mtrices re A( ij ) nd B(b ij ), the formultion of rithmetic crossover nd Guss muttion is shown s (2) nd (3) respectively: ( ) ( ) = int r + ( r) b b = int ( r) + rb ij ij ij ij ij ij ij = int( + ξ ) (3) ij where r is rndom number distributed evenly between 0 nd ; ξ is normlly distributed rndom number with men 0 nd vrince 2. The clcultion of int rounds number to the nerest integer, s the property ID must be n integer. An dptive strtegy bsed on sigmoid function developed by Hngyu nd Jing et l. [4] is used to control the probbility of crossover nd muttion, s follows: pcmx pcmin + p cmin f fvg f f vg p exp c c = + fmx f (4) vg pcmx f < fvg pmmx pmmin + pmmin f fvg f f vg p exp c m = + fmx f (5) vg pmmx f < fvg where f is the higher fitness of the two crossover prent individuls; f vg nd f mx re the verge nd the mximum fitness of current popultion respectively; p cmx nd p cmin re the upper nd lower limits of crossover probbility respectively; p mmx nd p mmin re the upper nd lower boundry of muttion probbility respectively; c is constnt; here it is Moreover, the DOF eqution nd heuristic criteri ([], [8]) re utilized to filter the instble nd invlid structures. Genetic lgorithm obeys the principle of survivl of the fittest, which mens tht the higher fitness n individul owns, the more survivl chnce it hs. The fitness of n individul is evluted through the fitness function including the objective nd the penlty term. The fitness function used in this pper is shown s () f fit ( x) = M f ( x) + fp ( x) () where M is positive number lrge enough to mke sure the fitness of individul is positive; f(x) nd f p (x) is the objective function nd penlty term respectively. (2) B. Identifiction Technique In genetic lgorithm, chromosome represents nd only represents structure. Thus, the simplest wy to identify structure is to store ech chromosome ppering in the evolutionry process. But it brings two problems. The first one is tht the demnd for storge memory is huge, even uncceptble. The second problem is tht the computtionl complexity of evluting the duplicte individuls fitness by serching the evolutionry history dt is high. It is noted tht the mpping of other number systems to decimliztion is rule of correspondence between vector nd deciml number, for exmple, the mpping from binry vector to deciml number is shown s (7). bb 0 bm D (7) m m 2 D= b02 + b2 + + bm The mpping rule cn be extended to identify different individuls. Suppose tht the number of members in the ground structure is e, the Boolen vlue nd property ID of the i th member is t i nd p i respectively, p mx is the mximum of ll property IDs. Therefore, chromosome cn be seen s vector of p mx number system, whose rdix is p mx, nd the identity of the chromosome is clculted s (8). The ID mybe overflow if the scle of the problem is too lrge, but it cn be esily solved by storing the ID in string formt. e ID = t p p (8) i i i mx The identity, response vlues, nd fitness of ech individul re stored in n evolutionry history tble, whose dt structure is shown s Fig. 3. Obviously, the storge demnd of this dt structure is much less thn tht of storing the chromosome directly. ID R R 2 R m Fit Fig. 3 Dt structure of the evolutionry history tble Where ID is the identity, R i is the structurl response, nd Fit is the fitness of n individul. First, the identity of ech individul in the new popultion produced by genetic opertions, i.e., selection, crossover, nd muttion, is clculted. Then, the identity is serched in the evolutionry history tble to justify whether n individul is duplicted or not. If so, the fitness of the individul is ttined immeditely. Otherwise, the evlution of the individul is finished by the structurl nlysis. The process is repeted until ll individuls re evluted. Finlly, the evolutionry history tble is updted to include the informtion of new individuls. Following, the time complexity of this identifiction technique, s well s the direct storge of chromosomes, is deduced. Suppose the popultion size is, nd the length of current evolutionry history tble is L, then, T L= qi T (9) where q i is the proportion of non-duplicte individuls in the popultion, T is the index of current genertion. The worst time complexity of the identifiction technique ISB: WCE 2009

4 WCE 2009, July - 3, 2009, London, U.K. to serch the fitness of duplicte individul is O(T e +T 2 ), nd tht of storing the chromosome directly is O(T 2 e ). Evidently, the former is much lower thn the ltter. C. Flowchrt The flowchrt of truss topology optimiztion using genetic lgorithm with the individul identifiction technique is shown in Fig. 4. Offspring must be identified first. Only the new individuls would be evluted by structurl nlysis, the duplicte individuls re evluted by serching the evolutionry history tble. After the evlution of ll individuls, the evolutionry history tble is updted to include the informtion of new individuls P 3 2 P Fig. 5 Ground structure of the 0-br truss exmple Initition Offspring Identifiction End Y Muttion Crossover Selection Stop? ew individuls Decoding Structurl nlysis Evlution Dt updte Fig. 4 Flowchrt of genetic lgorithm Duplicte individuls Fig. Topology of the optiml solution Tble. Results of 0 independent runs Run /(+2) % % % % % % % % % % Averge % IV. EXAMPLES A. 0-Br Truss The ground structure of the 0-br truss problem is shown in Fig. 5, where =9.4m,nd P=444.5k. The elstic modulus of the mteril is 8.9GP. The xil llowble stress for ll members is 72MP nd the llowble displcement of nodes 2 nd 5 is 5.08cm. The cross-sectionl re of every member is selected from set of 32 discrete vlues in [3]. Weight is minimized by GA with prmeters s follows: popultion size 40, mximum genertion 400, crossover probbility [ ], muttion probbility [ ], nd tournment size. The weight of the optiml solution ttined is kg, which is ccordnt with [3]. The topology of the optiml solution is shown in Fig.. In order to know how mny duplicte individuls pper in the evolutionry process, 0 independent runs re implemented, nd the results re shown in tble, where nd 2 is the number of non-duplicte nd duplicte individuls in the evolutionry process respectively. The process stops when the optiml solution is ttined or the mx genertion is chieved. It shows tht the proportion of duplicte individuls of these 0 independent runs rnges between 3.40% nd 40.45%, nd 38.08% on verge, i.e., the introduction of the identifiction technique reduces structurl nlysis 38.08% on verge. In this exmple, the process of structurl nlysis tkes up 92.0% of the totl time of genertion. Therefore, the efficiency is rised by 35.24% on verge. Menwhile, the verge proportion of duplicte individuls of the 4 runs (run 2, 3, 8, 0) which ll stop t the mx genertion is higher thn tht of the other runs which stop by ttining the optiml solution. Therefore, it seems tht more genertions my induce to higher dupliction proportion in the evolutionry process. evertheless, the difference of the dupliction proportion mong the 0 runs is not significnt, which indictes tht the mx genertion is not significnt fctor to the dupliction proportion. In order to investigte the influence of the popultion size on the dupliction proportion, nother 0 independent runs re executed with the popultion size of 00. Results show tht the verge dupliction proportion is incresed to 8.29%, which indictes tht lrger popultion size will induce lrger dupliction proportion, nd tht the popultion size hs n importnt impct on the dupliction proportion. ISB: WCE 2009

5 WCE 2009, July - 3, 2009, London, U.K. B. 5-Br Truss The ground structure of the 5-br truss problem is shown in Fig. 7, where =.0m,b=0.72m nd P=89k. The elstic modulus of the mteril is 207MP. The xil llowble stress for ll members is 345MP. The cross-sectionl re of every member is selected from set of discrete vlues evenly distributed between 0.45cm 2 nd 0.3cm 2. The objective is to minimize the volume of the structure. o displcement nd symmetry constrints re considered. The popultion size is 00, the mx genertion is 500, nd the other prmeters re correspondent with the 0-br truss exmple. The optiml solution is shown in Fig. 8, where the volume is 3933cm 3, s in [5]. P P P Fig. 7 Ground structure of the 5-br truss exmple Fig. 8 Topology of the optiml solution The scle of the 5-br truss problem (7 5 ) is times of the 0-br truss problem (33 0 ). In order to investigte whether or not the scle of problems is n importnt fctor to the dupliction proportion, 0 independent runs re implemented with the popultion size 40, nd the mx genertion 00. The results show tht the verge dupliction proportion is 34.%, which is close to tht of the 0-br truss exmple. Thus, it is concluded tht the scle of problems is not n importnt fctor to the dupliction proportion. V. COCLUSIO The min shortcoming of truss topology optimiztion using genetic lgorithm is the low computtionl efficiency which is resulted by the time-consuming structurl nlysis of ll individuls. It is observed tht there re mny duplicte individuls in the evolutionry process. To rise the efficiency of genetic lgorithm, the structurl nlysis of duplicte individuls should be voided. An individul identifiction technique with less storge memory demnds nd lower time complexity compred to the direct storge of chromosomes is developed to hndle this issue. The results of two truss exmples verify the fesibility of the lgorithm, nd indicte tht the identifiction technique cn effectively improve the computtionl efficiency. Moreover, numeric experiments show tht the popultion size hs n importnt impct on the dupliction proportion, but tht the mx genertion nd the scle of problems do not. The phenomenon of duplicte individuls in genetic lgorithm is inevitble. The pper hs investigted the influence of the popultion size, the mx genertion, nd the scle of problems on the dupliction proportion. b evertheless, the influence of the selection, crossover, nd muttion on the dupliction proportion is not cler now, which should be further investigted by more numeric experiments nd theoreticl nlysis. REFERECES [] G. I. Rozvny, "Aims, scope, methods, history nd unified terminology of computer-ided topology optimiztion in structurl mechnics," Structurl nd Multidisciplinry Optimiztion, vol. 2, p , 200. [2] M. P. Bendsøe nd O. Sigmund, Topology Optimiztion - Theory, Methods nd Applictions: Springer Verlg, Berlin Heidelberg, [3] S. D. Rjn, "Sizing, shpe, nd topology design optimiztion of trusses using genetic lgorithm," Journl of Structurl Engineering, vol. 2, p , 995. [4] M. J. Jkiel, C. Chpmn, J. Dud, A. Adewuy, nd K. Sitou, "Continuum structurl topology design with genetic lgorithms," Computer Methods in Applied Mechnics nd Engineering, vol. 8, p , [5] M. Giger nd P. Ermnni, "Evolutionry truss topology optimiztion using grph-bsed prmeteriztion concept," Structurl nd Multidisciplinry Optimiztion, vol. 32, p , 200. [] W. Tng, L. Tong, nd Y. Gu, "Improved genetic lgorithm for design optimiztion of truss structures with sizing, shpe nd topology vribles," Interntionl Journl for umericl Methods in Engineering, vol. 2, p , [7] P. Hjel nd E. Lee, "Genetic lgorithms in truss topologicl optimiztion," Interntionl Journl of Solids nd Structures, vol. 32, p , 995. [8] K. Deb nd S. Gulti, "Design of truss-structures for minimum weight using genetic lgorithms," Finite Elements in Anlysis nd Design, vol. 37, p , 200. [9] Hollnd, "Adpttion in nturl nd rtificil systems," University of Michign, Ann Arbor, MI, 975. [0] Goldberg, "Genetic lgorithms in serch, optimiztion nd mchine lerning," Addison-Wesley, Reding, Msschusetts, 989. [] G. I. Rozvny, "Stress rtio nd complince bsed methods in topology optimiztion - A criticl review," Structurl nd Multidisciplinry Optimiztion, vol. 2, p. 09-9, 200. [2] G. D. Cheng nd X. Guo, "ε-relxed pproch in structurl topology optimiztion," Structurl Optimiztion, vol. 3, p , 997. [3] R. Y. Su, L. J. Gui, nd Z. J. Fn, "Topology nd sizing optimiztion of truss structures using dptive genetic lgorithm with node mtrix encoding (submitted for publiction)," the 5 th Interntionl Conference on turl Computtion, Tinjin, Chin, [4] K. Hngyu, J. Jing, nd S. Yong, "Improving Crossover nd Muttion for Adptive Genetic Algorithm (Chinese)," Computer Engineering nd Applictions, p , 200. [5] R. J. Blling, R. R. Briggs, nd K. Gillmn, "Multiple optimum size/shpe/topology designs for skeletl structures using genetic lgorithm," Journl of Structurl Engineering, vol. 32, p. 58-5, 200. ISB: WCE 2009