Abstract
This paper addresses single and multiobjective topology optimization of truss-like structures using genetic algorithms (GA’s). In order to improve the performance of the GA’s (despite the presence of binary topology variables) a novel approach based on kinematic stability repair (KSR) is proposed. The methodology consists of two parts, namely the creation of a number of kinematically stable individuals in the initial population (IP) and a chromosome repair procedure. The proposed method is developed for both 2D and 3D structures and is shown to produce (in the single-objective case) results which are better than, or equal to, those found in the literature, while significantly increasing the rate of convergence of the algorithm. In the multiobjective case, the proposed modifications produce superior results compared to the unmodified GA. Finally the algorithm is successfully applied to a cantilevered 3D structure.
Similar content being viewed by others
Notes
DOF = dn − m − n s , where d is the dimension, n is the number of nodes, m the number of bar members and n s the number of degrees of freedom constrained by the supports. It should be verified that DOF is not positive.
This does not include unconnected nodes.
Note that the structures conform to the Chebyshev–Grübler–Kutzbach criterion.
During collinear/planar repair, it is ensured that the number of elements removed does not lead to the node connected to one or two elements, respectively in 2D and 3D problems. This is to avoid connectivity violations, while technically satisfying the collinear/planar check.
Note that the problem of singular topologies is eliminated through the presence of the topology variable.
In the figures one of the two overlapping members is drawn below or above the other to avoid confusion. These members are connected to the nodes above or below them at the end points only.
References
Achtziger W, Stolpe M (2007) Truss topology optimization with discrete design variables’ guaranteed global optimality and benchmark examples. Struct Multidisc Optim 34:1–20
Balling R, Briggs R, Gillman K (2006) Multiple optimum size/shape/topology designs for skeletal structures using a genetic algorithm. J Struct Eng 132:1158
Ben-Tal A, Jarre F, Kočvara M, Nemirovski A, Zowe J (2000) Optimal design of trusses under a nonconvex global buckling constraint. Optim Eng 1(2):189–213
Cheng FY, Li D (1997) Multiobjective optimization design with pareto genetic algorithm. J Struct Eng 123(9):1252–1261
Coello Coello C (1999a) A comprehensive survey of evolutionary-based multiobjective optimization techniques. Knowl Inf Syst 1(3):129–156
Coello Coello C (1999b) A survey of constraint handling techniques used with evolutionary algorithms. Laboratorio Nacional de Informatica Avanzada, Veracruz, Mexico, Technical report Lania-RI-99-04
Coello Coello C, Lamont G, Van Veldhuizen D (2002) Evolutionary algorithms for solving multi-objective problems. Springer, New York
Deb K, Gulati S (2001) Design of truss-structures for minimum weight using genetic algorithms. Finite Elem Anal Des 37(5):447–465
Dorn W, Gomory R, Greenberg H (1964) Automatic design of optimal structures. J Méc 3:25–52
Eldred MS, Adams BM, Haskell K, Bohnhoff WJ, Eddy JP, Gay DM, Griffin JD, Hart WE, Hough PD, Kolda TG, Martinez-Canales ML, Swiler LP, Watson JP, Williams PJ (2007) DAKOTA, a multilevel parallel object-oriented framework for design optimization, parameter estimation, uncertainty quantification, and sensitivity analysis: Version 4.1 reference manual. Tech. Rep. SAND2006-4055, Sandia National Laboratories, Albuquerque, New Mexico
Filomeno Coelho R, Bouillard P (2005) A multicriteria evolutionary algorithm for mechanical design optimization with expert rules. Int J Numer Methods Eng 62(4):516–536
Filomeno Coelho R, Lebon J, Bouillard Ph (2010) Hierarchical stochastic metamodels based on moving least squares and polynomial chaos expansion—application to the multiobjective reliability-based optimization of 3D truss structures. Struct Multidisc Optim 43(5):707–729
Fonseca C, Fleming P et al (1993) Genetic algorithms for multiobjective optimization: formulation, discussion and generalization. In: Proceedings of the fifth international conference on genetic algorithms, vol 423, pp 416–423
Gil L, Andreu A (2001) Shape and cross-section optimisation of a truss structure. Comput Struct 79(7):681–689
Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Reading, XIII, 412 pp. DM 104.00
Gomes HM (2011) Truss optimization with dynamic constraints using a particle swarm algorithm. Expert Syst Appl 38:957–968
Hajela P, Lee E (1995) Genetic algorithms in truss topological optimization. Int J Solids Struct 32(22):3341–3357
Hajela P, Lee E, Lin C (1993) Genetic algorithms in structural topology optimization. In: Bendsøe M, Soares C (eds) Topology design of structures. Kluwer Academic, Dordrecht, pp 117–134
Huang X, Zuo Z, Xie Y (2010) Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct 88(5–6):357–364
Jin P, De-yu W (2006) Topology optimization of truss structure with fundamental frequency and frequency domain dynamic response constraints. Acta Mech Solida Sinica 19(3):231–240
Kaveh A, Kalatjari V (2003) Topology optimization of trusses using genetic algorithm, force method and graph theory. Int J Numer Methods Eng 58(5):771–791
Kawamura H, Ohmori H, Kito N (2002) Truss topology optimization by a modified genetic algorithm. Struct Multidisc Optim 23(6):467–473
Madeira J, Rodrigues H, Pina H (2006) Multiobjective topology optimization of structures using genetic algorithms with chromosome repairing. Struct Multidisc Optim 32(1):31–39
Mathakari S, Gardoni P, Agarwal P, Raich A, Haukaas T (2007) Reliability-based optimal design of electrical transmission towers using multi-objective genetic algorithms. Comput-Aided Civil Infrastruct Eng 22(4):282–292
Ohsaki M (1995) Genetic algorithm for topology optimization of trusses. Comput Struct 57(2):219–225
Papadrakakis M, Lagaros N, Plevris V (2002) Multi-objective optimization of skeletal structures under static and seismic loading conditions. Eng Optim 34(6):645–669
Pedersen N (2000) Maximization of eigenvalues using topology optimization. Struct Multidisc Optim 20:2–11
Pellegrino S (1993) Structural computations with the singular value decomposition of the equilibrium matrix. Int J Solids Struct 30(21):3025–3035
Rozvany G (1996) Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct Multidisc Optim 11(3):213–217
Rozvany G (2001) On design-dependent constraints and singular topologies. Struct Multidisc Optim 21(2):164–172
Ruiyi S, Liangjin G, Zijie F (2009) Truss topology optimization using genetic algorithm with individual identification technique. In: Ao SI, Gelman L, Hukins DW, Hunter A, Korsunsky AM (eds) Proceedings of the world congress on engineering 2009 vol II WCE ’09, 1–3 July 2009, London, UK
Ruy W, Yang Y, Kim G, Yeun Y (2001) Topology design of truss structures in a multicriteria environment. Comput-Aided Civil Infrastruct Eng 16(4):246–258
Schaffer J (1985) Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of the 1st international conference on genetic algorithms, pp 93–100
Šešok D, Belevicius R (2008) Global optimization of trusses with a modified genetic algorithm. J Civ Eng Manag 14(3):147–154
Statnikov R, Bordetsky A, Matusov J, Sobol I, Statnikov A (2009) Definition of the feasible solution set in multicriteria optimization problems with continuous, discrete, and mixed design variables. Nonlinear Anal: Theory Methods Appl 71(12):e109–e117
Su R, Wang X, Gui L, Fan Z (2011) Multi-objective topology and sizing optimization of truss structures based on adaptive multi-island search strategy. Struct Multidisc Optim 43(2):275–286
Taylor RL (2008) FEAP—a finite element analysis program. Version 8.2 user manual
Tong WH, Liu GR (2001) An optimization procedure for truss structures with discrete design variables and dynamic constraints. Comput Struct 79(2):155–162
Xie YM, Steven GP (1996) Evolutionary structural optimization for dynamic problems. Comput Struct 58(6):1067–1073
Zitzler E, Thiele L, Laumanns M, Fonseca C, Da Fonseca V (2003) Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans Evol Comput 7(2):117–132
Acknowledgement
The first author would like to thank the Fonds de la Recherche Scientifique (FNRS) for financial support of this research.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Richardson, J.N., Adriaenssens, S., Bouillard, P. et al. Multiobjective topology optimization of truss structures with kinematic stability repair. Struct Multidisc Optim 46, 513–532 (2012). https://doi.org/10.1007/s00158-012-0777-5
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-012-0777-5