Structural and Multidisciplinary Optimization

, Volume 46, Issue 4, pp 513–532 | Cite as

Multiobjective topology optimization of truss structures with kinematic stability repair

  • James N. Richardson
  • Sigrid Adriaenssens
  • Philippe Bouillard
  • Rajan Filomeno Coelho
Research Paper


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.


Truss topology optimization Multiobjective optimization Kinematic stability repair Multiobjective genetic algorithms 



The first author would like to thank the Fonds de la Recherche Scientifique (FNRS) for financial support of this research.


  1. Achtziger W, Stolpe M (2007) Truss topology optimization with discrete design variables’ guaranteed global optimality and benchmark examples. Struct Multidisc Optim 34:1–20MathSciNetCrossRefGoogle Scholar
  2. Balling R, Briggs R, Gillman K (2006) Multiple optimum size/shape/topology designs for skeletal structures using a genetic algorithm. J Struct Eng 132:1158CrossRefGoogle Scholar
  3. 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):189–213MathSciNetMATHCrossRefGoogle Scholar
  4. Cheng FY, Li D (1997) Multiobjective optimization design with pareto genetic algorithm. J Struct Eng 123(9):1252–1261CrossRefGoogle Scholar
  5. Coello Coello C (1999a) A comprehensive survey of evolutionary-based multiobjective optimization techniques. Knowl Inf Syst 1(3):129–156Google Scholar
  6. 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-99-04Google Scholar
  7. Coello Coello C, Lamont G, Van Veldhuizen D (2002) Evolutionary algorithms for solving multi-objective problems. Springer, New YorkMATHGoogle Scholar
  8. Deb K, Gulati S (2001) Design of truss-structures for minimum weight using genetic algorithms. Finite Elem Anal Des 37(5):447–465MATHCrossRefGoogle Scholar
  9. Dorn W, Gomory R, Greenberg H (1964) Automatic design of optimal structures. J Méc 3:25–52Google Scholar
  10. 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 MexicoGoogle Scholar
  11. Filomeno Coelho R, Bouillard P (2005) A multicriteria evolutionary algorithm for mechanical design optimization with expert rules. Int J Numer Methods Eng 62(4):516–536MathSciNetMATHCrossRefGoogle Scholar
  12. 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):707–729MathSciNetCrossRefGoogle Scholar
  13. 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 416–423Google Scholar
  14. Gil L, Andreu A (2001) Shape and cross-section optimisation of a truss structure. Comput Struct 79(7):681–689CrossRefGoogle Scholar
  15. Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, Reading, XIII, 412 pp. DM 104.00Google Scholar
  16. Gomes HM (2011) Truss optimization with dynamic constraints using a particle swarm algorithm. Expert Syst Appl 38:957–968CrossRefGoogle Scholar
  17. Hajela P, Lee E (1995) Genetic algorithms in truss topological optimization. Int J Solids Struct 32(22):3341–3357MathSciNetMATHCrossRefGoogle Scholar
  18. 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 117–134CrossRefGoogle Scholar
  19. Huang X, Zuo Z, Xie Y (2010) Evolutionary topological optimization of vibrating continuum structures for natural frequencies. Comput Struct 88(5–6):357–364CrossRefGoogle Scholar
  20. 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):231–240Google Scholar
  21. Kaveh A, Kalatjari V (2003) Topology optimization of trusses using genetic algorithm, force method and graph theory. Int J Numer Methods Eng 58(5):771–791MATHCrossRefGoogle Scholar
  22. Kawamura H, Ohmori H, Kito N (2002) Truss topology optimization by a modified genetic algorithm. Struct Multidisc Optim 23(6):467–473CrossRefGoogle Scholar
  23. Madeira J, Rodrigues H, Pina H (2006) Multiobjective topology optimization of structures using genetic algorithms with chromosome repairing. Struct Multidisc Optim 32(1):31–39CrossRefGoogle Scholar
  24. 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):282–292CrossRefGoogle Scholar
  25. Ohsaki M (1995) Genetic algorithm for topology optimization of trusses. Comput Struct 57(2):219–225MathSciNetMATHCrossRefGoogle Scholar
  26. Papadrakakis M, Lagaros N, Plevris V (2002) Multi-objective optimization of skeletal structures under static and seismic loading conditions. Eng Optim 34(6):645–669CrossRefGoogle Scholar
  27. Pedersen N (2000) Maximization of eigenvalues using topology optimization. Struct Multidisc Optim 20:2–11CrossRefGoogle Scholar
  28. Pellegrino S (1993) Structural computations with the singular value decomposition of the equilibrium matrix. Int J Solids Struct 30(21):3025–3035MATHCrossRefGoogle Scholar
  29. Rozvany G (1996) Difficulties in truss topology optimization with stress, local buckling and system stability constraints. Struct Multidisc Optim 11(3):213–217Google Scholar
  30. Rozvany G (2001) On design-dependent constraints and singular topologies. Struct Multidisc Optim 21(2):164–172CrossRefGoogle Scholar
  31. 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, UKGoogle Scholar
  32. 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):246–258CrossRefGoogle Scholar
  33. Schaffer J (1985) Multiple objective optimization with vector evaluated genetic algorithms. In: Proceedings of the 1st international conference on genetic algorithms, pp 93–100Google Scholar
  34. Šešok D, Belevicius R (2008) Global optimization of trusses with a modified genetic algorithm. J Civ Eng Manag 14(3):147–154CrossRefGoogle Scholar
  35. 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–e117MATHCrossRefGoogle Scholar
  36. 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):275–286CrossRefGoogle Scholar
  37. Taylor RL (2008) FEAP—a finite element analysis program. Version 8.2 user manualGoogle Scholar
  38. Tong WH, Liu GR (2001) An optimization procedure for truss structures with discrete design variables and dynamic constraints. Comput Struct 79(2):155–162CrossRefGoogle Scholar
  39. Xie YM, Steven GP (1996) Evolutionary structural optimization for dynamic problems. Comput Struct 58(6):1067–1073MATHCrossRefGoogle Scholar
  40. 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):117–132CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • James N. Richardson
    • 1
  • Sigrid Adriaenssens
    • 2
    • 3
  • Philippe Bouillard
    • 1
  • Rajan Filomeno Coelho
    • 1
  1. 1.BATir - Building, Architecture and Town Planning, Brussels School of EngineeringUniversité Libre de BruxellesBrusselsBelgium
  2. 2.MEMC - Mechanics of Materials and Construction, Faculty of Engineering SciencesVrije Universiteit BrusselBrusselsBelgium
  3. 3.Department of Civil and Environmental EngineeringPrinceton UniversityPrincetonUSA

Personalised recommendations