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POD surrogates for real-time multi-parametric sheet metal forming problems

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Abstract

Our approach aims at coupling the ever increasing off-line computing power of mainframe computers with the interactive on-line possibilities of ubiquitous low computing power devices at the early design stages in order to provide insight into the design problems and to search for candidate optimal design points. In the off-line phase, the method under investigation relies on combining an optimized space-filling sampling plan on the design parameter space with extensive finite elements (FE) simulations yielding a learning set of displacement fields. The objective of this paper is the on-line phase. We provide a rigorous mathematical presentation of a family of non-intrusive, bi-level surrogates. We focus on displacement field approximation by Proper Orthogonal Decomposition (POD) combined with kriging interpolation of coefficients. The method is illustrated with two simple, easily reproduced numerical examples of quality assessment of deep-drawing process of a cylindrical cup by on-the-fly plotting forming limit diagrams (FLDs) and related quantities enabling thus to spot improved design points.

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References

  1. Forrester AIJ, Sóbester A, Keane AJ (2008) Engineering design via surrogate modelling: a practical guide, vol 1. Wiley, Hoboken

    Book  Google Scholar 

  2. Alonso D, Velazquez A, Vega J (2009) A method to generate computationally efficient reduced order models. Comput Methods Appl Mech Eng 198(33–36):2683–2691. doi:10.1016/j.cma.2009.03.012

    Article  MATH  MathSciNet  Google Scholar 

  3. Alonso D, Velazquez A, Vega J (2009) Robust reduced order modeling of heat transfer in a back step flow. Int J Heat Mass Transfer 52(5–6):1149–1157. doi:10.1016/j.ijheatmasstransfer.2008.09.011

    Article  MATH  Google Scholar 

  4. Bergmann M, Cordier L (2008) Optimal control of the cylinder wake in the laminar regime by trust-region methods and pod reduced-order models. J Comput Phys 227(16):7813–7840. doi:10.1016/j.jcp.2008.04.034, URL http://www.sciencedirect.com/science/article/pii/S0021999108002659

    Article  MATH  MathSciNet  Google Scholar 

  5. Berkooz G, Holmes P, Lumley J (1993) The proper orthogonal decomposition in the analysis of turbulent flows. Ann Rev Fluid Mech 25:539–575

    Article  MathSciNet  Google Scholar 

  6. Bui D, Hamdaoui M, De Vuyst F (2013) Pod-isat: an efficient pod-based surrogate approach with adaptive tabulation and fidelity regions for parametrized steady-state pde discrete solutions. Int J Numer Methods Eng . doi:10.1002/nme.4468

    Google Scholar 

  7. Burkardt J, Gunzburger M, Lee HC (2006) Pod and cvt-based reduced-order modeling of navier-stokes flows. Comput Methods Appl Mech Eng 196(1–3):337–355. doi:10.1016/j.cma.2006.04.004, URL http://www.sciencedirect.com/science/article/B6V29-4KJV30D-1/2/af9e08f1a1009af41a1026c5a16a3601

    Article  MATH  MathSciNet  Google Scholar 

  8. Chahlaoui Y, Gallivan K, Van Dooren P (1999) Recursive calculation of dominant singular subspaces. SIAM J Matrix Anal Appl 25

  9. Chinesta F, Ladeveze P, Cueto E (2011) A short review on model order reduction based on proper generalized decomposition. Arch Comput Methods Eng 18:395–404. doi:10.1007/s11831-011-9064-7

    Article  Google Scholar 

  10. Chinesta F, Leygue A, Bordeu F, Aguado J, Cueto E, Gonzalez D, Alfaro I, Ammar A, Huerta A (2013) Pgd-based computational vademecum for efficient design, optimization and control. Arch Comput Methods Eng 20:31–59. doi:10.1007/s11831-013-9080-x

    Article  MathSciNet  Google Scholar 

  11. Couckuyt I, Forrester A, Gorissen D, Turck FD, Dhaene T (2012) Blind kriging: implementation and performance analysis. Adv Eng Softw 49(0):1–13. doi:10.1016/j.advengsoft.2012.03.002, URL http://www.sciencedirect.com/science/article/pii/S0965997812000476

    Article  Google Scholar 

  12. Cressie N (1990) The origins of kriging. Math Geol 22:239–252. doi:10.1007/BF00889887

    Article  MATH  MathSciNet  Google Scholar 

  13. Dulong JL, Druesne F, Villon P (2007) A model reduction approach for real-time part deformation with nonlinear mechanical behavior. Int J Interact Des Manuf 1(4):229–238. doi:10.1007/s12008-007-0028-y

    Article  Google Scholar 

  14. Dunne F, Petrinic N (2005) Introduction to computational plasticity. Oxford University Press Inc, New York

    MATH  Google Scholar 

  15. Feeny B, Kappagantu R (1998) On the physical interpretation of proper orthogonal modes in vibrations. J Sound Vib 211(4):607–616. doi:10.1006/jsvi.1997.1386

    Article  Google Scholar 

  16. Ghnatios C, Chinesta F, Cueto E, Leygue A, Poitou A, Breitkopf P, Villon P (2011) Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: application to pultrusion. Compos A Appl Sci Manuf 42(9):1169–1178. doi:10.1016/j.compositesa.2011.05.001, URL http://www.sciencedirect.com/science/article/pii/

    Article  Google Scholar 

  17. Ghnatios C, Masson F, Huerta A, Leygue A, Cueto E, Chinesta F (2012) Proper generalized decomposition based dynamic data-driven control of thermal processes. Comput Methods Appl Mech Eng 213–216(0):29–41. doi:10.1016/j.cma.2011.11.018, URL http://www.sciencedirect.com/science/article/pii/S0045782511003641

    Article  Google Scholar 

  18. Grosso A, Jamali A, Locatelli M (2009) Finding maximin latin hypercube designs by iterated local search heuristics. Eur J Oper Res 197(2):541–547. doi:10.1016/j.ejor.2008.07.028

    Article  MATH  Google Scholar 

  19. den Hertog D, Stehouwer P (2002) Optimizing color picture tubes by high-cost nonlinear programming. Eur J Oper Res 140(2):197–211. doi:10.1016/S0377-2217(02)00063-2

    Article  MATH  Google Scholar 

  20. Hora P (2008) Proceedings of the 7th Int. Conf. and Workshop on Numerical Simulation of 3D Sheet Metal Forming Processes Interlaken, Switzerland

  21. Jones D (2001) A taxonomy of global optimization methods based on response surfaces. J Glob Optim 21:345–383. doi:10.1023/A:1012771025575

    Article  MATH  Google Scholar 

  22. LeGresley P, Alonso J (2000) Airfoil design optimization using reduced order models based on proper orthogonal decomposition. In: Fluids 2000 Conference and Exhibit, Denver, CO, 2545. AIAA, AIAA, Denver

  23. Li WW, Wu C (1997) Columnwise-pairwise algorithms with applications to the construction of supersaturated designs. Technometrics 39(2):171–179. Cited By (since 1996) 89

    Google Scholar 

  24. Lophaven S, Nielsen H, Søndergaard J (2002) Aspects of the matlab toolbox dace, technical report imm-rep-2002-13. Tech. rep., Technical University of Denmark, Department of Informatics and Mathematical Modelling, Lyngby, Denmark

  25. Lophaven S, Nielsen H, Søndergaard J (2002) Dace—a matlab kriging toolbox, technical report imm-tr-2002-12. Tech. rep., Technical University of Denmark, Department of Informatics and Mathematical Modelling, Lyngby, Denmark

  26. Driesse LT, Stehouwer P, Wijker J (2002) Structural mass optimization of the engine frame of the ariane 5 esc-b. In: Proceedings of the European conference on spacecraft, structures, materials and mechanical testing

  27. Ly HV, Tran HT (2001) Modeling and control of physical processes using proper orthogonal decomposition. Math Comput Model 33(1–3):223–236. doi:10.1016/S0895-7177(00)00240-5, URL http://www.sciencedirect.com/science/article/pii/S0895717700002405. Computation and control VI proceedings of the sixth Bozeman conference

    Article  MATH  Google Scholar 

  28. Schonlau M (1997) Computer experiments and global optimization. PhD thesis, Univ. of Waterloo, Waterloo, Ontario

  29. Makinouchi A (1996) Sheet metal forming simulation in industry. J Mater Proc Technol 60(1–4):19–26. doi:10.1016/0924-0136(96)02303-5. Proceedings of the 6th International Conference on Metal Forming

  30. McKay M, Beckman R, Conover W (2000) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42(1):55–61

    Article  Google Scholar 

  31. Miled B, Ryckelynck D, Cantournet S (2013) A priori hyper-reduction method for coupled viscoelastic-viscoplastic composites. Comput Struct 119(0):95–103. doi:10.1016/j.compstruc.2012.11.017, URL http://www.sciencedirect.com/science/article/pii/S0045794912002908

    Article  Google Scholar 

  32. Morris MD, Mitchell TJ (1995) Exploratory designs for computational experiments. J Stat Plan Infer 43(3):381–402. doi:10.1016/0378-3758(94)00035-T

    Article  MATH  MathSciNet  Google Scholar 

  33. My-Ha D, Lim K, Khoo B, Willcox K (2007) Real-time optimization using proper orthogonal decomposition: free surface shape prediction due to underwater bubble dynamics. Comput Fluids 36(3):499–512. doi:10.1016/j.compfluid.2006.01.016, URL http://www.sciencedirect.com/science/article/pii/S0045793006000429

    Article  MATH  Google Scholar 

  34. Niroomandi S, Alfaro I, Cueto E, Chinesta F (2008) Real-time deformable models of non-linear tissues by model reduction techniques. Comput Methods Prog Biomed 91(3):223–231. doi:10.1016/j.cmpb.2008.04.008, URL http://www.sciencedirect.com/science/article/pii/S0169260708001016

    Article  Google Scholar 

  35. Panthi S, Ramakrishnan N, Pathak K, Chouhan J (2007) An analysis of springback in sheet metal bending using finite element method (fem). J Mater Proc Technol 186(1–3):120–124. doi:10.1016/j.jmatprotec.2006.12.026

    Article  Google Scholar 

  36. Rasmussen CE, Williams CKI (2006) Gaussian processes for machine learning. The MIT Press, Cambridge

    MATH  Google Scholar 

  37. Rikards R, Auzins J (2004) Response surface method for solution of structural identification problems. Inverse Probl Sci Eng 12(1):59–70. Cited By (since 1996) 6

    Google Scholar 

  38. Ryckelynck D, Vincent F, Cantournet S (2012) Multidimensional a priori hyper-reduction of mechanical models involving internal variables. Comput Methods Appl Mech Eng 225–228(0):28–43. doi:10.1016/j.cma.2012.03.005, URL http://www.sciencedirect.com/science/article/pii/S0045782512000813

    Article  MathSciNet  Google Scholar 

  39. Sacks J, Schiller SB, Welch WJ (1989) Designs for computer experiments. Technometrics 31(1):41–47. doi:10.1080/00.401706.1989.10488474, URL http://amstat.tandfonline.com/doi/abs/10.1080/00401706.1989.10488474

    Article  MathSciNet  Google Scholar 

  40. Sirovich L (1987) Turbulence and the dynamics of coherent structures. I—coherent structures. II—symmetries and transformations. III—dynamics and scaling. Q Appl Math 45:561–571

    MATH  MathSciNet  Google Scholar 

  41. Stoughton TB, Yoon JW (2005) Sheet metal formability analysis for anisotropic materials under non-proportional loading. Int J Mech Sci 47(12):1972–2002. doi:10.1016/j.ijmecsci.2005.06.005, URL http://www.sciencedirect.com/science/article/pii/S0020740305001670

    Article  MATH  Google Scholar 

  42. Volkwein S (2004) Model reduction using proper orthogonal decomposition, lecture notes, institute of mathematics and scientific computing. University of Graz. URL http://www.uni-graz.at/imawww/volkwein/POD.pdf

  43. Wagoner RH (1989) Forming limit diagrams: concepts, methods, and applications. Tms

  44. Winton C, Pettway J, Kelley C, Howington S, Eslinger OJ (2011) Application of proper orthogonal decomposition (pod) to inverse problems in saturated groundwater flow. Adv Water Resour 34(12):1519–1526. doi:10.1016/j.advwatres.2011.09.007, URL http://www.sciencedirect.com/science/article/pii/S0309170811001746

    Article  Google Scholar 

  45. Xiao M, Breitkopf P, Filomeno Coelho R, Knopf-Lenoir C, Sidorkiewicz M, Villon P (2010) Model reduction by cpod and kriging. Struct Multidiscip Optim 41:555–574. doi:10.1007/s00158-009-0434-9

    Article  MATH  MathSciNet  Google Scholar 

  46. Marciniak Z, Duncan JL, Hu S (2002) The mechanics of sheet metal forming. Butterworth-Heinemann, Oxford

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Acknowledgments

Emmanuel Le Franois, lecturer at Roberval Laboratory is acknowledged for his advices about the finite elements method. This research was conducted as part of the OASIS project, supported by OSEO within the contract FUI no. F1012003Z. This work was carried out in the framework of the Labex MS2T, which is funded by the French Government, through the program ”Investments for the future” managed by the National Agency for Research (Reference ANR-11-IDEX-0004-02).

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Authors and Affiliations

  1. Laboratoire Roberval UMR 7337 UTC-CNRS, BP 20529, 60205, Compiegne cedex, France

    M. Hamdaoui, G. Le Quilliec, P. Breitkopf & P. Villon

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  1. M. Hamdaoui
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  3. P. Breitkopf
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  4. P. Villon
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Correspondence to M. Hamdaoui.

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Hamdaoui, M., Le Quilliec, G., Breitkopf, P. et al. POD surrogates for real-time multi-parametric sheet metal forming problems. Int J Mater Form 7, 337–358 (2014). https://doi.org/10.1007/s12289-013-1132-0

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  • DOI: https://doi.org/10.1007/s12289-013-1132-0

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