International Journal of Material Forming

, Volume 7, Issue 3, pp 275–287 | Cite as

A two-pronged approach for springback variability assessment using sparse polynomial chaos expansion and multi-level simulations

  • Jérémy Lebon
  • Guénhaël Le Quilliec
  • Piotr Breitkopf
  • Rajan Filomeno Coelho
  • Pierre Villon
Original Research


In this study, we show that stochastic analysis of metal forming process requires both a high precision and low cost numerical models in order to take into account very small perturbations on inputs (physical as well as process parameters) and to allow for numerous repeated analysis in a reasonable time. To this end, an original semi-analytical model dedicated to plain strain deep drawing based on a Bending-Under-Tension numerical model (B-U-T model) is used to accurately predict the influence of small random perturbations around a nominal solution estimated with a full scale Finite Element Model (FEM). We introduce a custom sparse variant of the Polynomial Chaos Expansion (PCE) to model the propagation of uncertainties through this model at low computational cost. Next, we apply this methodology to the deep drawing process of U-shaped metal sheet considering up to 8 random variables.


Springback variability assessment Sparse polynomial chaos expansion Semi-analytical bending-under-tension model 



This research was conducted as part of the OASIS project, supported by OSEO within the contract FUI no. F1012003Z. The first author is grateful to Innoviris for its support to BB2B project entitled multicriteria optimization with uncertainty quantification applied to the building industry. The authors also acknowledge the support of Labex MS2T.


  1. 1.
    Papeleux L, Ponthot J (2002) Finite element simulation of springback in sheet metal forming. J Mater Process Technol 125–126:785–791CrossRefGoogle Scholar
  2. 2.
    Kleiber M, Rojek J, Stochi R (2002) Reliability assessment for sheet metal forming operations. Comput Methods Appl Mech Eng 191:4511–4532CrossRefMATHGoogle Scholar
  3. 3.
    de Souza T, Rolfe B (2008) Multivariate modelling of variability in sheet metal forming. J Mater Process Technol 203:1–12CrossRefGoogle Scholar
  4. 4.
    Naceur H, Delaméziere A, Batoz J, Guo C, Knopf–Lenoir C (2004) Some improvement on the optimum process design in deep drawing using the inverse approach. J Mater Process Technol 146:250–262CrossRefGoogle Scholar
  5. 5.
    Breitkopf P, Naceur H, Villon P (2005) Moving least squares response surface approximation: formulation and metal forming applications. Comput Struct 83:1411–1428CrossRefGoogle Scholar
  6. 6.
    Kleiber M, Breitkopf P (1993) Finite elements in structural mechanics: an introduction with Pascal programs for micro-computers. Ellis HorwoodGoogle Scholar
  7. 7.
    Meinders T, Burchitz I, Bonte M, Lingbeek R (2008) Numerical product design: Springback prediction, compensation and optimization. Int J Mach Tools Manuf 48(5):499–514CrossRefGoogle Scholar
  8. 8.
    Radi B, El Hami A (2007) Reliability analysis of the metal forming process. Math Comput Model 45:431–439CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Jansson T, Nilsson L, Moshfegh R (2008) Reliability analysis of a sheet metal forming process using monte carlo analysis and metamodels. J Mater Process Technol 202:255–268CrossRefGoogle Scholar
  10. 10.
    Donglai W, Zhenshan C, Jun C (2007) Optimization and tolerance prediction of sheet metal forming process using response surface model. Comput Mater Sci 42:228–233CrossRefGoogle Scholar
  11. 11.
    Blatman G, Sudret B (2011) Adaptive sparse polynomial chaso expansion based on least angle regression. J Comput Phys 230:2345–2367CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Ghanem R, Spanos PD (1991) Spectral stochastic finite-element formulation for reliability analysis. J Eng Mech 117(10):2351–2372CrossRefGoogle Scholar
  13. 13.
    Sudret B, Der Kiureghian A (2000) Stochastic finite elements methods and reliability- a state of the art. tech. rep. Department of civil and environmental engineeringGoogle Scholar
  14. 14.
    Ghanem R, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer-Verlag, New YorkCrossRefMATHGoogle Scholar
  15. 15.
    Cameron R, Martin W (1944) Transformations of wiener integrals under translations. Ann Math 45(2):386–396CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Xiu D (2009) Fast numerical methods for stochastic computations: a review. Commun Comput Phys 5:242–272MathSciNetGoogle Scholar
  17. 17.
    Berveiller M, Sudret B, Lemaire M (2006) Stochastic finite element: a non intrusive approach by regression. Rev Eur Méc Numér 15:81–92MATHGoogle Scholar
  18. 18.
    Makinouchi A, Nakamachi E, Onate E, Wagoner R (eds) (1993) Numisheet’93 2nd international conference, numerical simulation of 3-D sheet metal forming process-verification of simulation with experiment, Riken TokyoGoogle Scholar
  19. 19.
    Hibbit D, Karlsson B, Sorensen P. ABAQUS Example problems, vol 1, version 6.7. Dassault SystèmeGoogle Scholar
  20. 20.
    Hill R (1948) A theory of the yielding and plastic flow of anisotropic metals. In: Proc R London, vol 193, pp 281–297Google Scholar
  21. 21.
    Le Quilliec G, Breitkopf P, Roelandt J, Juillard P (2013) Semi-analytical approach for plane strain sheet metal forming using a bending-under-tension numerical model. Int J Mater Form. doi: 10.1007/s12289-012-1122-7 Google Scholar
  22. 22.
    Hastie RT, Tibshirani J, Friedman G (2009) The elements of statistical learning, data mining, inference and prediction, SpringerGoogle Scholar
  23. 23.
    Hesterberg T, Choi N, Meier L, Fraley C (2008) Least angle and l 1 penalized regression: a review. Stat Surv 2:61–93CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Efron B, Hastie T, Johnstone I, Tibsirani T (2004) Least angle regression. Ann Stat 32(2):407–451CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag France 2013

Authors and Affiliations

  • Jérémy Lebon
    • 1
    • 2
  • Guénhaël Le Quilliec
    • 1
  • Piotr Breitkopf
    • 1
  • Rajan Filomeno Coelho
    • 2
  • Pierre Villon
    • 1
  1. 1.Laboratoire Roberval, UMR 7337Université de Technologie de CompiègneCompiègne cedexFrance
  2. 2.BATir Department CP 194/2Université Libre de Bruxelles (ULB)BrusselsBelgium

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