Computational Mechanics

, Volume 54, Issue 6, pp 1529–1539 | Cite as

Virtual charts of solutions for parametrized nonlinear equations

  • Matthieu Vitse
  • David Néron
  • Pierre-Alain Boucard
Original Paper


During many decades, engineers relied on experimental abaci to design and optimize mechanical structures. Virtual charts are based on numerical computations and aim at making the design and optimization of structures faster and cheaper as it requires less (or no) experimentation. The parametric problems are solved offline and the associated charts are used for online design. However, in this context, the cost associated to the resolution of parametric problems can be extremely prohibitive. Model reduction techniques are an answer to circumvent this issue. This paper provides the developments of an algorithm for solving nonlinear parametric problems, based on the LATIN method for the treatment of the nonlinear aspects and the PGD for the parameter dependency. A full time–space-parameter decomposition of the solution is introduced into the LATIN algorithm, numerical examples on academic problems are given so as to point out the advantages of this extension and leads on possible improvements to the strategy are given.


Nonlinear problems Reduced order modeling Proper generalized decomposition Parametric problems LATIN solver 



This work carried out under the SINAPS@ project benefited French funding managed by the National Research Agency under the program “Future Investment” (SINAPS@ Reference No. ANR-11-RSNR-0022).


  1. 1.
    Aguado J, Chinesta F, Leygue A, Cueto E, Huerta A (2013) DEIM-based PGD for parametric nonlinear model order reduction. ADMOS 2013:1–9Google Scholar
  2. 2.
    Alfaro I, González D, Bordeu F, Leygue A, Ammar A, Cueto E, Chinesta F (2014) Real-time in silico experiments on gene regulatory networks and surgery simulation on handheld devices. J Comput Surg 1(1):1CrossRefGoogle Scholar
  3. 3.
    Ammar A, Chinesta F, Cueto E, Doblaré M (2012) Proper generalized decomposition of time-multiscale models. Int J Numer Methods Eng 90:569–596CrossRefzbMATHGoogle Scholar
  4. 4.
    Barrault M, Maday Y, Nguyen NC, Patera AT (2004) An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339(9):667–672CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Boucard PA, Ladevèze P (1999) A multiple solution method for non-linear structural mechanics. Mech Eng 50(5):317–328Google Scholar
  6. 6.
    Chaturantabut S, Sorensen DC (2010) Nonlinear model reduction via discrete empirical interpolation. SIAM J Sci Comput 32(5):2737–2764CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Chinesta F, Ammar A, Cueto E (2010) Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Arch Comput Methods Eng 17(4):327–350CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Chinesta F, Ladevèze Pierre P, Cueto E (2011) A short review on model order reduction based on proper generalized decomposition. Arch Comput Methods Eng 18(4):395–404CrossRefGoogle Scholar
  9. 9.
    Chinesta F, Leygue A, Bognet B, Ghnatios C, Poulhaon F, Bordeu F, Barasinski A, Poitou A, Chatel S, Maison-Le-Poec S (2014) First steps towards an advanced simulation of composites manufacturing by automated tape placement. Int J Mater Form 7(1):81–92CrossRefGoogle Scholar
  10. 10.
    Cremonesi M, Néron D, Guidault PA, Ladevèze P (2013) A PGD-based homogenization technique for the resolution of nonlinear multiscale problems. Comput Methods Appl Mech Eng 267:275–292Google Scholar
  11. 11.
    Grepl MA, Maday Y, Nguyen NC, Patera AT (2007) Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Math Model Numer Anal 41:575– 605Google Scholar
  12. 12.
    Heyberger C, Boucard PA, Néron D (2011) Multiparametric analysis within the proper generalized decomposition framework. Comput Mech 49(3):277–289CrossRefGoogle Scholar
  13. 13.
    Heyberger C, Boucard PA, Néron D (2013) A rational strategy for the resolution of parametrized problems in the PGD framework. Comput Methods Appl Mech Eng 259:40–49CrossRefzbMATHGoogle Scholar
  14. 14.
    Jung N, Haasdonk B, Kroner D (2009) Reduced Basis Method for quadratically nonlinear transport equations. Int J Comput Sci Math 2(4):334CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Kosambi DD (1943) Statistics in function space. J Indian Math Soc 7(1):76–88zbMATHMathSciNetGoogle Scholar
  16. 16.
    Ladevèze P (1985) Sur une famille d’algorithmes en mécanique des structures. Comptes-rendus des séances de l’Académie des sciences. Série 2, Mécanique-physique, chimie, sciences de l’univers. sciences de la terre, vol. 300 (2), pp 41–44Google Scholar
  17. 17.
    Ladevèze P (1989) The large time increment method for the analyze of structures with nonlinear constitutive relation described by internal variables. CR Acad Sci Paris 309:1095–1099zbMATHGoogle Scholar
  18. 18.
    Ladevèze P (1999) Nonlinear computational structural mechanics: new approaches and non-incremental methods of calculation. Springer, BerlinCrossRefzbMATHGoogle Scholar
  19. 19.
    Lide DR (2004) CRC handbook of chemistry and physics. CRC Press, Boca RatonGoogle Scholar
  20. 20.
    Lieu T, Farhat C, Lesoinne M (2006) Reduced-order fluid/structure modeling of a complete aircraft configuration. Comput Methods Appl Mech Eng 195(41–43):5730–5742CrossRefzbMATHGoogle Scholar
  21. 21.
    Maday Y, Mula O (2013) A generalized empirical interpolation method: application of reduced basis techniques to data assimilation. Anal Numer Partial Differ Equ 1–12Google Scholar
  22. 22.
    Najah A, Cochelin B, Damil N, Potier-Ferry M (1998) A critical review on asymptotic numerical methods. Arch Comput Methods Eng 5:31–50CrossRefMathSciNetGoogle Scholar
  23. 23.
    Néron D, Dureisseix D (2008) A computational strategy for poroelastic problems with a time interface between coupled physics. Int J Numer Methods Eng 73:783–804Google Scholar
  24. 24.
    Nouy A (2010) A priori model reduction through Proper Generalized Decomposition for solving time-dependent partial differential equations. Comput Methods Appl Mech Eng 199(23–24):1603–1626Google Scholar
  25. 25.
    Patera AT, Rozza G (2007) Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. MITGoogle Scholar
  26. 26.
    Relun N, Heyberger C, Néron D, Boucard PA, Chernoualli A, Pyre A (2011) Méthode LATIN pour l’étude paramétrique de problèmes élastoviscoplastiques d’évolution quasi-statique. CSMAGoogle Scholar
  27. 27.
    Relun N, Néron D, Boucard PA (2013) A model reduction technique based on the PGD for elastic-viscoplastic computational analysis. Comput Mech 51(1):83–92CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Rosenfeld A, Kak AC (1982) Digital picture processing, vol 1. Elsevier, New YorkGoogle Scholar
  29. 29.
    Rozza G, Huynh DBP, Patera AT (2008) Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations. Arch Comput Methods Eng 15(3):229–275CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Rozza G, Veroy K (2007) On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput Methods Appl Mech Eng 196(7):1244–1260CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Ryckelynck D (2005) A priori hyperreduction method: an adaptive approach. J Comput Phys 202(1):346–366CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Ryckelynck D, Chinesta F, Cueto E, Ammar A (2006) On the a priori model reduction: overview and recent developments. Arch Comput Methods Eng 13(1):91–128CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Verdon N, Joyot P, Chinesta F, Villon P et al (2012) A PGD–ANM-based approach for fast solving nonlinear equations. ECCOMAS 2012Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Matthieu Vitse
    • 1
  • David Néron
    • 1
  • Pierre-Alain Boucard
    • 1
  1. 1.LMT Cachan (ENS Cachan/CNRS/PRES UniverSud Paris)Cachan CedexFrance

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