Structural and Multidisciplinary Optimization

, Volume 54, Issue 3, pp 449–468 | Cite as

Representative surrogate problems as test functions for expensive simulators in multidisciplinary design optimization of vehicle structures

  • Ramses SalaEmail author
  • Niccolò Baldanzini
  • Marco Pierini


A large variety of algorithms for multidisciplinary optimization is available, but for various industrial problem types that involve expensive function evaluations, there is still few guidance available to select efficient optimization algorithms. This is also the case for multidisciplinary vehicle design optimization problems involving, e.g., weight, crashworthiness, and vibrational comfort responses. In this paper, an approach for the development of Representative Surrogate Problems (RSPs) as synthetic test functions for a relatively complex industrial problem is presented. The work builds on existing sensitivity analysis and surrogate data generation methods to establish a novel approach to generate surrogate function sets, which are accessible (i.e. not resource demanding) and aim to generate statistically representative instances of specific classes of industrial problems. The approach is demonstrated through the construction of RSPs for multidisciplinary optimization problems that occur in the context of structural car body design. As a “proof of concept” the RSP approach is applied for the selection of suitable optimization algorithms, for several problem formulations and for a meta-optimization (i.e. an optimization of the optimization algorithm parameters) to increase optimization efficiency. The potential of the approach is demonstrated by comparing the efficiency of several optimization algorithms on an RSP and an independent simulation-based vehicle model. The results corroborate the potential of the proposed approach and significant performance gains in optimization efficiency are achieved. Although the approach is developed for the particular application presented, the approach is described in a general way, to encourage readers to use the gist of the concept.


Multidisciplinary design optimization Test problems Benchmarking Meta-optimization Vehicle design 



This work is performed in the scope of the GRESIMO and ENLIGHT projects, targeting environmentally friendly mobility solutions. The authors have been partially funded by the European Community’s 7th Framework program by means of: an ITN fellowship in the GRESIMO project as part of the People program (Marie Curie Actions) grant agreement no. 290050, and a contribution to the activities in the ENLIGHT project grant agreement no. 314567. Furthermore, the authors are thankful for the publicly available finite element vehicle models used in this work. These models have been developed by the National Crash Analysis Center (NCAC) of The George Washington University under a contract with the FHWA and NHTSA of the US DOT.


  1. Alimoradi A, Foley CM, Pezeshk S (2010) Benchmark problems in structural design and performance optimization: past, present, and future—part I. Struct Congr 2010:455–466. doi: 10.1061/41131(370)40 Google Scholar
  2. Baldanzini N, Scippa A (2004) Shape and size optimization of an engine suspension system. Proceedings of the 2004 International Conference on Noise and Vibration Engineering, ISMAGoogle Scholar
  3. Baldanzini N, Caprioli D, Pierini M (2001) Designing the dynamic behavior of an engine suspension system through genetic algorithms. J Vib Acoust 123(4):480–486CrossRefGoogle Scholar
  4. Birge B (2006) Particle swarm optimization toolbox, retrieved from MATLAB file exchange in January 2014
  5. Blumhardt R (2001) FEM-crash simulation and optimisation. Int J Veh Des 26(4):331–347CrossRefGoogle Scholar
  6. Boyd S, Vandenberghe L (2009) Convex optimization. Cambridge University Press, 2009.
  7. Buehren M (2008) Differential evolution, MATLAB file exchange
  8. Deb K, Agrawal S, Pratap A, Meyarivan T (2000) A fast elitist non-dominated sorting genetic algorithm for multi-objective optimization: NSGA-II. Lect Notes Comput Sci 1917:849–858CrossRefGoogle Scholar
  9. Duddeck F (2008) Multidisciplinary optimization of car bodies. Struct Multidiscip Optim 35(4):375–389. doi: 10.1007/s00158-007-0130-6 CrossRefGoogle Scholar
  10. Durgun I, Yildiz AR (2012) Structural design optimization of vehicle components using cuckoo search algorithm. Mater Test 54(3):185–188CrossRefGoogle Scholar
  11. Fletcher R (2010) The sequential quadratic programming method. Nonlinear Optimization. Springer Berlin Heidelberg 165–214. doi:  10.1007/978-3-642-11339-0_3
  12. Goldberg DE, Holland JH (1988) Genetic algorithms and machine learning. Mach Learn 3(2):95–99CrossRefGoogle Scholar
  13. Hallquist JO (2006) LS-DYNA theory manual. Livermore Software Technology CorporationGoogle Scholar
  14. Haug E, Scharnhorst T, DuBois P (1986) FEM-Crash, Berechnung eines Fahrzeugaufpralls. VDI-Tagung: Berechnung im Automobilbau, Würzburg, Germany, (VDI-Berichte 613), 479–505Google Scholar
  15. Hoeffding W (1948) A class of statistics with asymptotically normal distribution. Ann Math Stat 1948:293–325MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kirkpatrick S, Gelatt CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220(4598):671–680MathSciNetCrossRefzbMATHGoogle Scholar
  17. Knowles J, Hughes EJ (2005) Multiobjective optimization on a budget of 250 evaluations. In Evolutionary Multi-Criterion Optimization. Springer, Berlin Heidelberg, pp. 176–190Google Scholar
  18. Lin S (2011) NGPM A NSGA-II program in Matlab, MATLAB file exchange.
  19. Mihaylova P, Pratellesi A, Baldanzini N, Pierini M (2012) Optimization of the global static and dynamic performance of a vehicle body by means of response surface models. ASME 2012 11th Biennial Conference on Engineering Systems Design and Analysis. American Society of Mechanical EngineersGoogle Scholar
  20. NCAC Finite Element Model Archive. retrieved from January 2014, from
  21. Plischke E (2010) An effective algorithm for computing global sensitivity indices (EASI). Reliab Eng Syst Saf 95(4):354–360CrossRefGoogle Scholar
  22. Prichard D, Theiler J (1994) Generating surrogate data for time series with several simultaneously measured variables. Phys Rev Lett 73(7):951CrossRefGoogle Scholar
  23. Rastrigin LA (1974) Systems of extremal control. Theoretical Foundations of Engineering Cybernetics Series, (In Russian) Nauka, MoscowGoogle Scholar
  24. Ratto M, Pagano A (2010) Using recursive algorithms for the efficient identification of smoothing spline ANOVA models. AStA Adv Stat Anal 94(4):367–388MathSciNetCrossRefGoogle Scholar
  25. Rechenberg I (1973) Evolutionsstrategie. Stuttgart: Holzmann-Froboog. ISBN 3-7728-0373-3Google Scholar
  26. Rios LM, Sahinidis NV (2012) Derivative-free optimization: a review of algorithms and comparison of software implementations. J Glob Optim 2012:1–47MathSciNetGoogle Scholar
  27. Rodgers LJ, Nicewander WA (1988) Thirteen ways to look at the correlation coefficient. Am Stat 42(1):59–66CrossRefGoogle Scholar
  28. Rosenbrock HH (1960) An automatic method for finding the greatest or least value of a function. Comput J 3(3):175–184MathSciNetCrossRefGoogle Scholar
  29. Sala R, Pierini M, Baldanzini N (2014) Optimization efficiency in multidisciplinary vehicle design including NVH criteria. Proceedings of the Leuven Conference on Noise and Vibration Engineering (ISMA); 2014 September 15–17; Leuven, BelgiumGoogle Scholar
  30. Sala R, Pierini M, Baldanzini N (2014) The development and application of tailored test problems for meta-simulation of multidisciplinary optimization of vehicle structures. Presentation at the (XI) World Congress on Computational Mechanics; 2014 July 20–25; Barcelona, SpainGoogle Scholar
  31. Saltelli A, Annoni P, Azzini I, Campolongo F, Ratto M, Tarantola S (2010) Variance based sensitivity analysis of model output. Design and estimator for the total sensitivity index. Comput Phys Commun 181(2):259–270MathSciNetCrossRefzbMATHGoogle Scholar
  32. Schramm U, Pilkey WD (1996) Review: optimal design of structures under impact loading. Shock Vib 3:69–81CrossRefGoogle Scholar
  33. Schreiber T, Schmitz A (1996) Improved surrogate data for nonlinearity tests. Phys Rev Lett 77(4):635CrossRefGoogle Scholar
  34. Shan S, Wang GG (2010) Survey of modeling and optimization strategies to solve high-dimensional design problems with computationally-expensive black-box functions. Struct Multidiscip Optim 41(2):219–241MathSciNetCrossRefzbMATHGoogle Scholar
  35. Simpson TW, Toropov V, Balabanov V, Viana FA (2008) Design and analysis of computer experiments in multidisciplinary design optimization: a review of how far we have come or not 12th AIAA/ISSMO multidisciplinary analysis and optimization conference. 5Google Scholar
  36. Sobieszczanski-Sobieski J, Haftka RT (1997) Multidisciplinary aerospace design optimization: survey of recent developments. Struct Optim 14(1):1–23CrossRefGoogle Scholar
  37. Sobieszczanski-Sobieski J, Kodiyalam S, Yang RJ (2001) Optimization of car body under constraints of noise, vibration, and harshness (NVH), and crash. Struct Multidiscip Optim 22(4):295–306CrossRefGoogle Scholar
  38. Sobol IM (1990) On sensitivity estimation for nonlinear mathematical models. Matem Mod 2(1):112–118MathSciNetzbMATHGoogle Scholar
  39. Sobol IM (2001) Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates. Math Comput Simul 55(1–3):271–280MathSciNetCrossRefzbMATHGoogle Scholar
  40. Storm R, Price K (1997) Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. J Glob Optim 11(4):341–359Google Scholar
  41. Tang WJ, Wu QH (2009) Biologically inspired optimization: a review. Transa Inst Meas Control 31(6):495–515CrossRefGoogle Scholar
  42. Varis T, Tuovinen T (2012) Open Benchmark database for multidisciplinary optimization problems. Proc Int Conf Model Appl SimulGoogle Scholar
  43. Venema V (2003) IAAFT implementation in MATLAB retrieved January 2014 from
  44. Venkayya VB (1978) Structural optimization: a review and some recommendations. Int J Numer Methods Eng 13(2):203–228CrossRefzbMATHGoogle Scholar
  45. Wolpert DH, Macready WG (1995) No free lunch theorems for search. Technical Report SFI-TR-95-02-010, Santa Fe Institute. 10Google Scholar
  46. Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE Trans Evol Comput 1(1):67–82CrossRefGoogle Scholar
  47. Wu SR, Gu L (2012) Introduction to the explicit finite element method for nonlinear transient dynamics. WileyGoogle Scholar
  48. Yang XS (2010a) Nature-inspired metaheuristic algorithms. Luniver press, 2010.
  49. Yang XS (2010b) Engineering optimization: an introduction with metaheuristic applications. WileyGoogle Scholar
  50. Yang RJ, Tseng L, Nagy L, Cheng J (1994) Feasibility study of crash optimization. In: Gilmore BJ, Hoetzel DA, Dutta D, Eschenauer HA (eds.) Advances in design automation, ASME. DE-69–2:549–556Google Scholar
  51. Yang RJ, Gu L, Tho CH, Sobieski J (2001) Multi-disciplinary optimization of a full vehicle with high performance computing. In: Conf. of the American Inst. of Aeronautics and Astronautics, pp 688–698, AIAA Paper No. AIAA- 2001–1273Google Scholar
  52. Yildiz AR, Solanki KN (2012) Multi-objective optimization of vehicle crashworthiness using a new particle swarm based approach. Int J Adv Manuf Technol 59(1-4):367–376CrossRefGoogle Scholar
  53. Zitzler E, Deb K, Thiele L (2000) Comparison of multiobjective evolutionary algorithms: empirical results. Evol Comput 8(2):173–195CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Ramses Sala
    • 1
    Email author
  • Niccolò Baldanzini
    • 1
  • Marco Pierini
    • 1
  1. 1.Università degli Studi di FirenzeFlorenceItaly

Personalised recommendations