International Journal of Material Forming

, Volume 7, Issue 4, pp 487–501 | Cite as

Numerical assessment of springback for the deep drawing process by level set interpolation using shape manifolds

  • Balaji Raghavan
  • Guenhael Le Quilliec
  • Piotr Breitkopf
  • Alain Rassineux
  • Jean-Marc Roelandt
  • Pierre Villon
Original Research

Abstract

In this paper, we introduce an original shape representation approach for post-springback characterization based on the automatic generation of parameterized level set functions. The central idea is the concept of the shape manifold representing the design domain in the reduced-order shape-space. Performing Proper Orthogonal Decomposition on the shapes followed by using the Diffuse Approximation allows us to efficiently reduce the problem dimensionality and to interpolate uniquely between admissible input shapes, while also determining the smallest number of parameters needed to characterize the final formed shape. We apply this methodology to the problem of springback assessment for the deep drawing operation of metal sheets.

Keywords

Shape optimization Springback Level sets Manifold learning Model reduction Forming 

References

  1. 1.
    Eggertsen PA, Mattiasson K (2009) Int J Mech Sci 51(7):547CrossRefGoogle Scholar
  2. 2.
    Teimouri R, Baseri H, Rahmani B, Bakhshi-Jooybari M (2012) Int J Mater Form 1–12. doi:10.1007/s12289-012-1117-4
  3. 3.
    D’Acquisto L, Fratini L (2001) J Manuf Process 3(1):29CrossRefGoogle Scholar
  4. 4.
    Le Quilliec G, Breitkopf P, Roelandt J, Juillard P (2013) Int J Mater Form 1–12. doi:10.1007/s12289-012-1122-7
  5. 5.
    Lebon J, Le Quilliec G, Breitkopf P, Filomeno Coelho R, Villon P (2013) Int J Mater Form1–13. doi:10.1007/s12289-013-1126-y
  6. 6.
    Zhang WH, Beckers P, Fleury C (1995) Int J Numer Methods Eng 38(13):2283CrossRefMATHGoogle Scholar
  7. 7.
    Raghavan B, Breitkopf P, Tourbier Y, Villon P (2013) Struct Multidiscip Optim doi:10.1007/s00158-013-0942-5 Google Scholar
  8. 8.
    Oshier S, Sethian JA (1988) J Comput Phys 79:12CrossRefMathSciNetGoogle Scholar
  9. 9.
    Allaire G, Jouve F, Toader AM (2004) J Comput Phys 194(1):363CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Murat F, Simon J (1976) Pre-publication du laboratoire d’analyse numerique, vol 76015. Universite de Paris, p 6Google Scholar
  11. 11.
    Moes N, Stolz C, Bernard P, Chevaugeon N (2011) Int J Numer Methods Eng 86(3):358CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Legrain G, Cartraud P, Perreard I, Moes N (2011) Int J Numer Methods Eng 86(7):915CrossRefMATHGoogle Scholar
  13. 13.
    Duvigneau R (2006) INRIA research report RR-5949Google Scholar
  14. 14.
    Xie X, Mirmehdi M (2011) Image Vis Comput 29(2–3):167CrossRefGoogle Scholar
  15. 15.
    Wang S, Lim K, Khoo B, Wang M (2007) J Comput Phys 221(1):395CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Forrester AIJ, Keane AJ (2009) Prog Aerosp Sci 45:50CrossRefGoogle Scholar
  17. 17.
    Breitkopf P, Naceur H, Rassineux A, Villon P (2005) Comput Struct 83(17–18):1411CrossRefGoogle Scholar
  18. 18.
    Willcox K, Peraire J (2002) AIAA J 40(11):2323CrossRefGoogle Scholar
  19. 19.
    Berkooz G, Holmes P, Lumley JL (1993) Annu Rev Fluid Mech 25(1):539CrossRefMathSciNetGoogle Scholar
  20. 20.
    Cordier L, El Majd BA, Favier J (2010) Int J Numer Methods Fluids 63(2):269MATHMathSciNetGoogle Scholar
  21. 21.
    Audouze C, De Vuyst F, Nair P (2009) Int J Numer Methods Eng 80(8):1025CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Raghavan B, Breitkopf P (2012) Engineering with Computers. doi:10.1007/s00366-012-0263-0
  23. 23.
    Chinesta F, Ammar A, Cueto E (2010) Int J Numer Methods Eng 83(8–9):1114CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Chinesta F, Ladeveze P, Cueto E (2011) Archives Comput Methods Eng 18(4):395CrossRefGoogle Scholar
  25. 25.
    Ghnatios C, Chinesta F, Cueto E, Leygue A, Breitkopf P, Villon P (2011) Compos A 42:1169CrossRefGoogle Scholar
  26. 26.
    Ghnatios C, Masson F, Huerta A, Cueto E, Leygue A, Chinesta F (2012) Comput Methods Appl Mech Eng 213:29CrossRefGoogle Scholar
  27. 27.
    Hamdaoui M, Le Quilliec G, Breitkopf P, Villon P (2013) Int J Mater Form. doi:10.1007/s12289-013-1132-0 Google Scholar
  28. 28.
    Raghavan B, Breitkopf P, Villon P (2010) Eur J Comput Mech 19(5–7):671Google Scholar
  29. 29.
    Raghavan B, Xiao M, Breitkopf P, Villon P (2012) Eur J Comput Mech 21(3–6):325Google Scholar
  30. 30.
    Raghavan B, Xia L, Breitkopf P, Rassineux A, Villon P (2013) Comput Methods Appl Mech Eng 265:174. doi:10.1016/j.cma.2013.06.010 CrossRefMATHMathSciNetGoogle Scholar
  31. 31.
    Nayroles B, Touzot G, Villon P (1992) Comput Mech 10(5):307CrossRefMATHGoogle Scholar
  32. 32.
    Millan D, Rosolen A, Arroyo M (2013) Int J Numer Methods Eng 93(7):685CrossRefMathSciNetGoogle Scholar
  33. 33.
    Millan D, Rosolen A, Arroyo M (2013) Comput Methods Appl Mech Eng 261–262(7):181MathSciNetGoogle Scholar
  34. 34.
    Jan S, Zolesio J (1992) Introduction to shape optimization. Springer, BerlinMATHGoogle Scholar
  35. 35.
    Schulz V (2012) A Riemannian view on shape optimization. Forschungsbericht / Universität Trier, Mathematik, Informatik. Found Comput Math 12–1. arXiv:1203.1493
  36. 36.
    Fukunaga K, Olsen D (1971) IEEE Trans Comput 20:176CrossRefMATHGoogle Scholar
  37. 37.
    Breitkopf P, Rassineux A, Touzot G, Villon P (2000) Int J Numer Methods Eng 48(3):451CrossRefMATHGoogle Scholar
  38. 38.
    Nielsen KB, Br¿nnberg N (1993) Simulation of the 2D draw bending process: numisheet benchmark. In: Makinouchi A, Nakamachi E, Onate E, Wagoner RK (eds) Proceedings of the 2nd international conference NUMISHEET ’93: Numisheet. TokyoGoogle Scholar
  39. 39.
    Hallquist JO (2006) LS-DYNA theory manual. Livermore Software Technology CorporationGoogle Scholar
  40. 40.
    Chung K, Lee M, Kim D, Kim C, Wenner ML, Barlat F (2005) Int J Plast 21(5):861MATHGoogle Scholar

Copyright information

© Springer-Verlag France 2013

Authors and Affiliations

  • Balaji Raghavan
    • 1
  • Guenhael Le Quilliec
    • 1
  • Piotr Breitkopf
    • 1
  • Alain Rassineux
    • 1
  • Jean-Marc Roelandt
    • 1
  • Pierre Villon
    • 1
  1. 1.Laboratoire Roberval UMR 7337 UTC-CNRSUniversite de Technologie de CompiegneCompiegneFrance

Personalised recommendations