Structural and Multidisciplinary Optimization

, Volume 47, Issue 4, pp 539–553 | Cite as

A multiparametric strategy for the two step optimization of structural assemblies

  • B. SoulierEmail author
  • P.-A. Boucard
Research Paper


Generally speaking, the objective and constraint functions of a structural optimization problem are implicit with respect to the design variables; their evaluation requires finite element analyses which constitute the most expensive steps of the optimization algorithm. The work presented in this paper concerns the implementation of a two step optimization strategy which consists in optimizing first an empirical model (metamodel), then the full model. In the framework of multilevel model optimization, the computation costs are related, on the one hand, to the construction of global approximations and, on the other hand, to the optimization of the full model. Thus, many numerical simulations are required in order to perform a multilevel optimization. In this context, the objective of associating a multiparametric strategy based on the nonincremental LATIN method with the two step optimization process is to reduce these computation costs. The performance gains thus achieved will be illustrated through the optimization of structural assemblies involving contact with friction. The results obtained will show that the savings associated with the multiparametric procedure can reach a factor of 30.


Two step optimization Metamodel LATIN method Multiparametric strategy 



This work was supported by the French National Research Agency (ANR) as part of the RNTL 2005 program: Multidisciplinary Optimization Project (OMD).


  1. Barthelemy JF, Haftka R (1993) Approximation concepts for optimum structural design—a review. Struct Optim 5:129–144CrossRefGoogle Scholar
  2. Bendsoe MP (1995) Optimization of structural topology, shape and material. Springer, HeidelbergCrossRefGoogle Scholar
  3. Blanzé C, Champaney L, Cognard J, Ladevèze P (1995) A modular approach to structure assembly computations. Application to contact problems. Eng Comput 13(1):15–32Google Scholar
  4. Boucard PA, Champaney L (2003) A suitable computational strategy for the parametric analysis of problems with multiple contact. Int J Numer Methods Eng 57(9):1259–1282zbMATHCrossRefGoogle Scholar
  5. Boucard PA, Champaney L (2004) Approche multirésolution pour l’étude paramétrique d’assemblages par contact et frottement. Rev Europ Élém Finis 13:437–448CrossRefGoogle Scholar
  6. Box GEP, Hunter JS (1957) Multi-factor experimental designs for exploring response surfaces. Ann Math Stat 28:195–241MathSciNetzbMATHCrossRefGoogle Scholar
  7. Box GEP, Wilson KB (1951) On the experimental attainment of optimum conditions. J R Stat Soc, Ser B Stat Methodol 13(1):1–45MathSciNetzbMATHGoogle Scholar
  8. Braibant V, Fleury C (1985) An approximation concepts approach to shape optimal design. Comput Methods Appl Mech Eng 53:119–148CrossRefGoogle Scholar
  9. Chen TY, Yang CM (2005) Multidisciplinary design optimization of mechanisms. Adv Eng Softw 36(5):301–311zbMATHCrossRefGoogle Scholar
  10. Cressie N (1990) The origins of kriging. Math Geol 22:239–252. doi: 10.1007/BF00889887 MathSciNetzbMATHCrossRefGoogle Scholar
  11. Currin C, Mitchell T, Morris M, Ylvisaker D (1991) Bayesian prediction of deterministic functions, with applications to the design and analysis of computer experiments. J Am Stat Assoc 86(416):953–963MathSciNetCrossRefGoogle Scholar
  12. Dantzig G, Orden A, Wolfe P (1955) Generalized simplex method for minimizing a linear form under linear inequality restraints. Pac J Math 5:183–195MathSciNetzbMATHCrossRefGoogle Scholar
  13. El-Sayed ME, Hsiung CK (1991) Optimum structural design with parallel finite element analysis. Comput Struct 40(6):1469–1474zbMATHCrossRefGoogle Scholar
  14. Engels H, Becker W, Morris A (2004) Implementation of a multi-level optimisation methodology within the e-design of a blended wing body. Aerosp Sci Technol 8(2):145–153CrossRefGoogle Scholar
  15. Fleury C, Braibant V (1986) Structural optimization. A new dual method using mixed variables. Int J Numer Methods Eng 2:409–428MathSciNetCrossRefGoogle Scholar
  16. Gingold R, Monaghan J (1977) Smooth particle hydrodynamics: theory and application to non-spherical stars. Mon Not R Astron Soc 181:375–389zbMATHGoogle Scholar
  17. Goldberg D (1989) Genetic algorithms in search, optimization and machine learning. Addison WesleyGoogle Scholar
  18. Haftka R (1988) First- and second-order constraint approximations in structural optimization. Comput Mech 3:89–104zbMATHCrossRefGoogle Scholar
  19. Han SP (1977) A globally convergent method for nonlinear programming. J Optim Theory Appl 22:297–309zbMATHCrossRefGoogle Scholar
  20. Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76(8):1905–1915CrossRefGoogle Scholar
  21. Haykin S (1994) Neural networks: a comprehensive foundation, 1st edn. Prentice Hall, Upper Saddle RiverzbMATHGoogle Scholar
  22. Hilding D, Torstenfelt B, Klarbring A (2001) A computational methodology for shape optimization of structures in frictionless contact. Comput Methods Appl Mech Eng 190:4043–4060zbMATHCrossRefGoogle Scholar
  23. Keane AJ, Petruzzeli N (2000) Aircraft wing design using ga-based multi-level strategies. In: Proceedings 8th AIAA/USAF/NASA/ISSMO symposium on multidisciplinary analysis and optimization, Long Beach, USA, pp A00–40171Google Scholar
  24. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: IEEE int. conf. neural networks, vol 4, pp 1942–1948Google Scholar
  25. Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680MathSciNetzbMATHCrossRefGoogle Scholar
  26. Kravanja S, Sorsak A, Kravanja Z (2003) Efficient multilevel minlp strategies for solving large combinatorial problems in engineering. Optim Eng 4(1/2):97–151MathSciNetzbMATHCrossRefGoogle Scholar
  27. Ladevèze P (1999) Nonlinear computational structural mechanics—new approaches and non-incremental methods of calculation. Springer, BerlinzbMATHCrossRefGoogle Scholar
  28. Li W, Li Q, Steven GP, Xie YM (2005) An evolutionary shape optimization for elastic contact problems subject to multiple load cases. Comput Methods Appl Mech Eng 194:3394–3415zbMATHCrossRefGoogle Scholar
  29. Liu B, Haftka RT, Watson LT (2004) Global-local structural optimization using response surfaces of local optimization margins. Struct Multidisc Optim 27(5):352–359CrossRefGoogle Scholar
  30. McKay MD, Beckman RJ, Conover WJ (1979) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 21(2):239–245MathSciNetzbMATHGoogle Scholar
  31. Montgomery D (1997) Design and analysis of experiments. Wiley, New YorkzbMATHGoogle Scholar
  32. Nayrolles B, Touzot G, Villon P (1992) Generalizing the finite element method: diffuse approximation and diffuse elements. Comput Mech 10(5):307–318CrossRefGoogle Scholar
  33. Pedersen P (1981) The integrated approach of FEM-SLP for solving problems of optimal design. In: Optimization of distributed parameters structures, vol 1. Springer, Amsterdam, pp 757–780CrossRefGoogle Scholar
  34. Pritchard J, Adelman H (1990) Differential equation based method for accurate approximations in optimization. In: Proc. AIAA/ASME/ASCE/AHS/ASC 31st structures, structural dynamics and materials conf. (held in Long Beach, CA)Google Scholar
  35. Pritchard J, Adelman H (1991) Differential equation based method for accurate modal approximations. AIAA J 29:484–486CrossRefGoogle Scholar
  36. Rasmussen J (1998) Nonlinear programming by cumulative approximation refinement. Struct Multidisc Optim 15(1):1–7MathSciNetCrossRefGoogle Scholar
  37. Robinson GM, Keane AJ (1999) A case for multi-level optimisation in aeronautical design. Aeronaut J 103:481–485Google Scholar
  38. Sacks J, Schiller SB, Welch WJ (1989) Designs for computer experiments. Technometrics 31(1):41–47MathSciNetCrossRefGoogle Scholar
  39. Soulier B, Richard L, Hazet B, Braibant V (2003) Crashworthiness optimization using a surrogate approach by stochastic response surface. In: Gogu G, Coutellier D, Chedmail P, Ray P (eds) Recent advances in integrated design and manufacturing in mechanical engineering. Kluwer Academic, pp 159–168Google Scholar
  40. Umesha PK, Venuraju MT, Hartmann D, Leimbach KR (2005) Optimal design of truss structures using parallel computing. Struct Multidisc Optim 29:285–297CrossRefGoogle Scholar
  41. Zienkiewicz O, Campbell J (1973) Optimum structural design. Wiley, New YorkzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.LMT-Cachan (ENS Cachan/CNRS/UPMC/PRES UniverSudParis)CedexFrance

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