Computational Mechanics

, Volume 51, Issue 2, pp 151–169 | Cite as

Generation of a cokriging metamodel using a multiparametric strategy

  • Luc Laurent
  • Pierre-Alain Boucard
  • Bruno Soulier
Original Paper


In the course of designing structural assemblies, performing a full optimization is very expensive in terms of computation time. In order or reduce this cost, we propose a multilevel model optimization approach. This paper lays the foundations of this strategy by presenting a method for constructing an approximation of an objective function. This approach consists in coupling a multiparametric mechanical strategy based on the LATIN method with a gradient-based metamodel called a cokriging metamodel. The main difficulty is to build an accurate approximation while keeping the computation cost low. Following an introduction to multiparametric and cokriging strategies, the performance of kriging and cokriging models is studied using one- and two-dimensional analytical functions; then, the performance of metamodels built from mechanical responses provided by the multiparametric strategy is analyzed based on two mechanical test examples.


Cokriging metamodel Multiparametric strategy LATIN method Multilevel optimization 


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  1. 1.
    Robinson GM, Keane AJ (1999) A case for multi-level optimisation in aeronautical design. Aeronaut J 103(1028): 481–485Google Scholar
  2. 2.
    Booker AJ, Dennis JE, Frank PD, Serafini DB, Torczon V, Trosset MW (1999) A rigorous framework for optimization of expensive functions by surrogates. Struct Multidiscip Optim 17(1): 1–13Google Scholar
  3. 3.
    Queipo NV, Haftka RT, Shyy W, Goel T, Vaidyanathan R, Tucker PK (2005) Surrogate-based analysis and optimization. Prog Aerosp Sci 41(1): 1–28CrossRefGoogle Scholar
  4. 4.
    Simpson TW, Poplinski JD, Koch PN, Allen JK (2001) Metamodels for computer-based engineering design: survey and recommendations. Eng Comput 17(2): 129–150zbMATHCrossRefGoogle Scholar
  5. 5.
    Kravanja S, Silih S, Kravanja Z (2005) The multilevel minlp optimization approach to structural synthesis: the simultaneous topology, material, standard and rounded dimension optimization. Adv Eng Softw 36(9): 568–583zbMATHCrossRefGoogle Scholar
  6. 6.
    Kravanja S, Soršak A, Kravanja Z (2003) Efficient multilevel minlp strategies for solving large combinatorial problems in engineering. Optim Eng 4(1): 97–151MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Liu B, Haftka RT, Watson LT (2004) Global-local structural optimization using response surfaces of local optimization margins. Struct Multidiscip Optim 27(5): 352–359CrossRefGoogle Scholar
  8. 8.
    Alexandrov NM, Lewis RM (2000) Analytical and computational aspects of collaborative optimization. NASA Technical Memorandum 210104Google Scholar
  9. 9.
    Chen TY, Yang CM (2005) Multidisciplinary design optimization of mechanisms. Adv Eng Softw 36(5): 301–311zbMATHCrossRefGoogle Scholar
  10. 10.
    Conceição António CA (2002) A multilevel genetic algorithm for optimization of geometrically nonlinear stiffened composite structures. Struct Multidiscip Optim 24(5): 372–386CrossRefGoogle Scholar
  11. 11.
    Bendsøe MP (1995) Optimization of structural topology, shape, and material. Springer, New YorkCrossRefGoogle Scholar
  12. 12.
    Le Riche R, Gaudin J (1998) Design of dimensionally stable composites by evolutionary optimization. Compos Struct 41(2): 97–111CrossRefGoogle Scholar
  13. 13.
    Theocaris PS, Stavroulakis GE (1998) Multilevel optimal design of composite structures including materials with negative poisson’s ratio. Struct Multidiscip Optim 15(1): 8–15Google Scholar
  14. 14.
    Keane AJ, Petruzzelli N (2000) Aircraft wing design using ga-based multi-level strategies. In: AIAA paper 2000-4937. AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Long Beach, USA, 06–08 Sep 2000. American Institute of Aeronautics and AstronauticsGoogle Scholar
  15. 15.
    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
  16. 16.
    Chattopadhyay A, McCarthy TR, Pagaldipti N (1995) Multilevel decomposition procedure for efficient design optimization of helicopter rotor blades. AIAA J 33(2): 223–230CrossRefGoogle Scholar
  17. 17.
    El-Sayed MEM, Hsiung CK (1991) Optimum structural design with parallel finite element analysis. Comput Struct 40(6): 1469–1474zbMATHCrossRefGoogle Scholar
  18. 18.
    Umesha PK, Venuraju MT, Hartmann D, Leimbach KR (2005) Optimal design of truss structures using parallel computing. Struct Multidiscip Optim 29(4): 285–297CrossRefGoogle Scholar
  19. 19.
    Dunham B, Fridshal D, Fridshal R, North JH (1963) Design by natural selection. Synthese 15(1): 254–259CrossRefGoogle Scholar
  20. 20.
    El-Beltagy MA, Keane AJ (1999) A comparison of various optimization algorithms on a multilevel problem. Eng Appl Artif Intell 12(5): 639–654CrossRefGoogle Scholar
  21. 21.
    Elby D, Averill RC, Punch WF, Goodman ED (1998) Evaluation of injection island ga performance on flywheel design optimisation. In: Proceedings of Third International Conference of Adaptive Computing in Design and Manufacture, pp 121–136Google Scholar
  22. 22.
    Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann ArborGoogle Scholar
  23. 23.
    Box GEP, Wilson KB (1951) On the experimental attainment of optimum conditions. J R Stat Soc Ser B (Methodological) 13(1): 1–45MathSciNetzbMATHGoogle Scholar
  24. 24.
    Simpson TW, Mauery TM, Korte JJ, Mistree F (1998) Multidisciplinary optimization branch. Comparison of response surface and kriging models for multidisciplinary design optimization. In: AIAA paper 98-4758. 7 th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, pp 98–4755Google Scholar
  25. 25.
    Haykin S (1994) Neural networks: a comprehensive foundation. Prentice Hall PTR, Upper Saddle RiverzbMATHGoogle Scholar
  26. 26.
    McCulloch W, Pitts W (1943) A logical calculus of the ideas immanent in nervous activity. Bull Math Biol 5(4): 115–133MathSciNetzbMATHGoogle Scholar
  27. 27.
    Hardy RL (1971) Multiquadric equations of topography and other irregular surfaces. J Geophys Res 76: 1905–1915CrossRefGoogle Scholar
  28. 28.
    Chatterjee A (2000) An introduction to the proper orthogonal decomposition. Current Science 78(7): 808–817Google Scholar
  29. 29.
    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, pp 159–168, MaiGoogle Scholar
  30. 30.
    Cressie N (1990) The origins of kriging. Mathematical Geology 22(3): 239–252MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Wackernagel H (1995) Multivariate geostatistics: an introduction with applications. Springer, BerlinzbMATHGoogle Scholar
  32. 32.
    Chung HS, Alonso JJ (2002) Using gradients to construct cokriging approximation models for high-dimensional design optimization problems. In: 40th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. CiteseerGoogle Scholar
  33. 33.
    Barthelemy JFM, Haftka RT (1993) Approximation concepts for optimum structural design: a review. Struct Multidiscip Optim 5(3): 129–144Google Scholar
  34. 34.
    Sobieszczanski-Sobieski J, Haftka RT (1997) Multidisciplinary aerospace design optimization: survey of recent developments. Struct Multidiscip Optim 14(1): 1–23Google Scholar
  35. 35.
    Ladevèze P (1999) Nonlinear computational structural mechanics: new approaches and non-incremental methods of calculation. Springer, New YorkzbMATHGoogle Scholar
  36. 36.
    Mandel J (1993) Balancing domain decomposition. Commun Num Methods Eng 9(3): 233–241MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    Le Tallec P (1994) Domain decomposition methods in computational mechanics. Comput Mech Adv 1(2): 121–220MathSciNetzbMATHGoogle Scholar
  38. 38.
    Farhat C, Roux FX (1991) A method of finite element tearing and interconnecting and its parallel solution algorithm. Int J Num Methods Eng 32(6): 1205–1227zbMATHCrossRefGoogle Scholar
  39. 39.
    Blanzé C, Champaney L, Cognard JY, Ladevèeze P (1995) A modular approach to structure assembly computations: application to contact problems. Eng Comput 13(1): 15–32Google Scholar
  40. 40.
    Ladevèze P (1985) Sur une famille d’algorithmes en mécanique des structures. Compte rendu de l’académie des Sciences 300(2): 41–44zbMATHGoogle Scholar
  41. 41.
    Champaney L, Cognard JY, Ladeveze P (1999) Modular analysis of assemblages of three-dimensional structures with unilateral contact conditions. Comput Struct 73(1-5): 249–266zbMATHCrossRefGoogle Scholar
  42. 42.
    Boucard PA, Ladevèze P (1999) Une application de la méthode latin au calcul multirésolution de structures non linéaires. Revue Européenne des Eléments Finis 8: 903–920zbMATHGoogle Scholar
  43. 43.
    Boucard PA (2001) Application of the latin method to the calculation of response surfaces. In: Proceeding of the First MIT Conference on Computational Fluid and Solid Mechanics, Cambridge, USA, vol 1, pp 78–81, JuinGoogle Scholar
  44. 44.
    Allix O, Vidal P (2002) A new multi-solution approach suitable for structural identification problems. Comput Methods Appl Mech Eng 191(25-26): 2727–2758MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Boucard PA, Champaney L (2003) A suitable computational strategy for the parametric analysis of problems with multiple contact. Int J Num Methods Eng 57(9): 1259–1281zbMATHCrossRefGoogle Scholar
  46. 46.
    Champaney L, Boucard PA, Guinard S (2008) Adaptive multi-analysis strategy for contact problems with friction. Comput Mech 42(2): 305–315zbMATHCrossRefGoogle Scholar
  47. 47.
    Soulier B, Boucard PA (2009) A multiparametric strategy for the large-scale multilevel optimization of structural assemblies. In 8th World Congress on Structural and Multidisciplinary Optimization, Lisbon, PortugalGoogle Scholar
  48. 48.
    Krige DG (1951) A statistical approach to some mine valuation and allied problems on the Witwatersrand. Master’s thesisGoogle Scholar
  49. 49.
    Matheron G (1962) Traité de géostatistique appliquée, Tome I. Memoires du Bureau de Recherches Geologiques et Minieres, vol 14Google Scholar
  50. 50.
    Matheron G (1962) Traite de Geostatistique Appliquee, Tome II: Le Krigeage. Memoires du Bureau de Recherches Geologiques et Minieres, No 24Google Scholar
  51. 51.
    Matheron G (1963) Principles of geostatistics. Econ Geol 58(8): 1246CrossRefGoogle Scholar
  52. 52.
    Sacks J, Schiller SB, Welch WJ (1989) Designs for computer experiments. Technometrics 31(1): 41–47MathSciNetCrossRefGoogle Scholar
  53. 53.
    Sacks J, Welch WJ, Mitchell TJ, Wynn HP (1989) Design and analysis of computer experiments. Stat Sci 4(4): 409–423MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. Adaptive Computation and Machine Learning, vol 1. MIT Press, CambridgeGoogle Scholar
  55. 55.
    Koehler JR, Owen AB (1996) Computer experiments. Handb Stat 13: 261–308MathSciNetCrossRefGoogle Scholar
  56. 56.
    Morris MD, Mitchell TJ, Ylvisaker D (1993) Bayesian design and analysis of computer experiments: use of derivatives in surface prediction. Technometrics 35(3): 243–255MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    Matérn B (1960) Spatial variation, Lecture notes in statistics, vol 36. Springer, BerlinGoogle Scholar
  58. 58.
    Stein ML (1999) Interpolation of spatial data: some theory for kriging. Springer, New YorkzbMATHCrossRefGoogle Scholar
  59. 59.
    Mardia KV, Marshall RJ (1984) Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71(1): 135MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Mardia KV, Watkins AJ (1989) On multimodality of the likelihood in the spatial linear model. Biometrika 76(2): 289MathSciNetzbMATHCrossRefGoogle Scholar
  61. 61.
    Sasena MJ (2002) Flexibility and efficiency enhancements for constrained global design optimization with kriging approximations. PhD thesis, University of MichiganGoogle Scholar
  62. 62.
    Warnes JJ, Ripley BD (1987) Problems with likelihood estimation of covariance functions of spatial Gaussian processes. Biometrika 74(3): 640MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    McKay MD, Conover WJ, Beckman RJ (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
  64. 64.
    Roulet V, Champaney L, Boucard P-A (2011) A parallel strategy for the multiparametric analysis of structures with large contact and friction surfaces. Adv Eng Softw 42(6): 347–358zbMATHCrossRefGoogle Scholar
  65. 65.
    Leary SJ, Bhaskar A, Keane AJ (2004) Global approximation and optimization using adjoint computational fluid dynamics codes. AIAA J 42(3): 631–641CrossRefGoogle Scholar
  66. 66.
    Giannakoglou KC, Papadimitriou DI, Kampolis IC (2006) Aerodynamic shape design using evolutionary algorithms and new gradient-assisted metamodels. Comput Methods Appl Mech Eng 195(44–47): 6312–6329zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Luc Laurent
    • 1
  • Pierre-Alain Boucard
    • 1
  • Bruno Soulier
    • 1
  1. 1.LMT-CachanENS Cachan/CNRS/Université Paris 6/PRES UniverSud ParisCachanFrance

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