Topology Structural Optimization Using A Hybrid of GA and ESO Methods

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1 Topology Structural Optimization Using A Hybrid of GA and ESO Methods Hiroki KAJIWARA, Graduate School of Engineering, Doshisha University Tomoyuki HIROYASU, Doshisha University, tomo@is.doshisha.ac.jp Mitsunori MIKI, Doshisha University, Tatara Miyakodani -3, Kyo-Tanabe, Kyoto To design economical structural forms, it is necessary to optimize both the topology and shape of structures. To optimize topology, we propose a hybrid of Genetic Algorithm (GA) and Evolutionary Structural Optimization (ESO). This paper describes the considerations in applying the proposed method to topology structural optimization. Through numerical examples, the proposed method showed better search ability than GA or ESO methods alone. Moreover, this hybrid method makes it possible to design a more economical structural form. Key word: Genetic Algorithm, Evolutionary Method, Topology Structural Optimization, Optimization Evolutionary Structural Optimization ESO ) ESO 2) 3) 4) 5) 6) 7) 8) ESO ESO 9) ) Genetic Algorithm:GA ) GA GA GA GA GA GA ESO ESO ESO 2 Xie ) Evolutionary Structural OptimizationESO 2. ESO α 2.2 GAESO ESO

To optimize topology, we propose a hybrid of Genetic Algorithm (GA) and Evolutionary Structural Optimization (ESO).

2 ( n Analysis result of structural layout Fig. method m Equivalent stress : Low Element number Equivalent stress : Hight Deletion rate Equivalent stress e e e-2 (nm) Removed elements Removed elements of unnecessary parts using ESO ESO 22 ESO 4.5 Design domain 2.m (2) N P.m 2) Step Step 2 Step 3 Step 4 Step 5 GA Distributed Genetic AlgorithmDGA 2) DGA GA GA GADGA ESO GA GA ESO GA ESO Step 6 Step 7 Step 8 Step 9 Step Optimal layout 3.2 Fig. 2 ESO method Evolutionary process of a structural layout using the Table Results of analysis of the optimal layout Number of elements 2 Maximum displacement [m] ( 8 ) Maximum equivalent stress [P a] ( 2 ) GAESO 3. GA GA E E = 26GP a E E = 3MP a2 Structural layout Fig. 3 coding Bit-array bityoung's modulus E =26GPa (solid element) bityoung's modulus E =3MPa (void element) Coding by bit-array representation

m 2) Step Step 2 Step 3 Step 4 Step 5 GA Distributed Genetic AlgorithmDGA 2) DGA GA GA GADGA. 2. 3. 4. ESO 5. 6. 7. 8. 9. 28 GA GA ESO GA ESO Step 6 Step 7 Step 8 Step 9 Step Optimal layout 3.2 Fig.

3 3.2.2 ()(2) ()(2)n xi F 2δ max δ g ζ (2)3σ v γ ξ γ γ =.e () (2) 2 3 ()(2) 4 Structural layout of Parent 'A' crossover point Structural layout of Parent 'B' Structural layout of Parent 'A' : solid : void (a) One-point crossover Structural layout of offspring 'A' crossover point Structural layout of offspring 'B' Structural layout of offspring 'A' Minimize : F = n x i + ζ i= ( δmax δ g ) () Structural layout of Parent 'B' : solid : void Structural layout of offspring 'B' : OR operation : AND operation x i {, } Minimize : F = x i {, } Subject to : δ max < δ g n x i + ζ i= ( δmax δ g ) ( ) σv + ξ γ Subject to : δ max < δ g (2) (b) Crossover taking structural form into consideration Fig. 4 Comparison of structural layouts according to crossover method 3.3 GA GA 4(a) 4(b)2 A B 2 A A B B A B ESO GA ESO GA 2ESO ESOGA GA 3.5 GA ESO 3) 4) Moore

offspring 'A' crossover point Structural layout of offspring 'B' Structural layout of offspring 'A' Minimize : F = n x i + ζ i= ( δmax δ g ) () Structural layout of Parent 'B' : solid : void

4 5 Moore filtering E = 26GP ae = 3MP a N Design domain Structural layout with density distribution Filtered structural layout N P.m ( 2) solid object element void Moore neighborhood 2.m (2) Fig. 5 Filtering method using Moore neighborhood Fig. 7 Cantilever problem 3.6 GA ESO 6 6 A a A a a A offspring 'a' void solid element for pulling back parent 'A' feasible area offspring 'a' GA+ESOGA GA+ESO.ESOGA5 ()() (2) Table 2 Parameters of the proposed method Population size Number of islands Number of elites Chromosome length 4 Migration rate.5 Migration interval 5 Mutation rate Deletion rate Allowable displacement.25 (/chromosome length) GA+ESO (.25.5,.75,.), GA (.) value [m] ( 8 ) Coefficient ζ. Coefficient ξ. Number of generations 4 Fig. 6 Pulling back method 2 2 Step ESO GAGA+ESOESO GA GA ESO 2 7 υ = () ESO 4.. GA+ESOGA8 88-(d)2 8-(e) GA+ESOGA 99GA+ESO. GA+ESOGA GA+ESO4

2 offspring 'a' void solid element for pulling back parent 'A' feasible area offspring 'a' GA+ESOG

5 (a) GA+ESO (.25) (b) GA+ESO (.5) -(e) (c) GA+ESO (.75) (d) GA+ESO (.) (e) GA Fig. 8 Final structural layout 4 Evaluation value GA+ESO.25 GA+ESO.5 GA+ESO.75 GA+ESO. GA Fig. Final structural layout Number of generations Fig. 9 Search ability GA+ESOGA GA+ESO.5 9 GA+ESO GA 9GA+ESO ESO 9GA+ESO. 3 GA+ESO. 3GA+ESO. Table 3 (Optimal layout) Comparison between GA+ESO (.) and ESO GA+ESO ESO Number of elements 68 2 Maximum displacement [m] ( 8 ) Fitness (2) ESO 4.2. GA+ESOGA 82 -(e)2 Evaluation value GA+ESO.25 GA+ESO.5 GA+ESO.75 GA+ESO. GA Number of generations Fig ESO Search ability GA+ESO.5 4 GA+ESO.5 4GA+ESO.5 GA GA+ESO ESO ESO

) and ESO GA+ESO ESO Number of elements 68 2 Maximum displacement [m] ( 8 ) 6.455 6.474 Fitness 69. 2. 4.2 (2) ESO 4.2. GA+ESOGA 82 -(e)2 Evaluation value 4 35 3 25 2 5 GA+ESO.

6 Table 4 (Optimal layout) Comparison between GA+ESO(.5) and ESO GA+ESO ESO Number of elements 83 2 Maximum displacement [m] ( 8 ) Decentralization of element equivalent stress ( 3 ) Fitness GA GA+ESO GA 5 GAESO ESO GA 2 ESO GA 788, 25 7) 5 Vol.5, pp ), 6 Vol.6, pp ) ESO 4 Vol.4pp ) ESO 5. Vol.5, pp ) Goldberg D.EGenetic Algorithms in search Optimization and Machine Learning Addison-Wesley Publishing Company,989. 2) Tanese RDistributed Genetic AlgorithmsProc.3rd International Conference on Genetic Algorithms, pp , ) Diaz A. and Sigmund OCheckerboard patterns in layout optimization, Structural Optimization,, pp.4-5, ) Sigmund O. and Petersson J. Numerical instabilities in topology optimization: A survey on procedures dealing with checkerboards, mesh-dependencies and local minima, Structural Optimization, 6, pp.68-75, 998. ) Xie Y.M. and Steven G.PEvolutionary Structural Optimization, Spronger-Verlag, ) Bendsϕe M.P. and Kikuchi NGenerating optimal topologies in structural design using a homogenization method, Computer Methods in Applied Mechanics and Engineering, Vol.7, pp , 988 3) (Quint Corporation) 3 Vol.3pp ) Bendsϕe M.POptimal shape design as a material distribution problem, Struct. Optimiz., Vol., pp.93-22, 989 5) 3 4 Vol.4pp ) Saito H., Kita E., and Xie Y.MThree-dimensional Structural Optimization Using Cellular Automata, Advances in Applied Mechanics (Proc ACAM25), pp.78-

EGenetic Algorithms in search Optimization and Machine Learning Addison-Wesley Publishing Company,989. 2) Tanese RDistributed Genetic AlgorithmsProc.

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