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Extended-PGD Model Reduction for Nonlinear Solid Mechanics Problems Involving Many Parameters

  • P. Ladevèze
  • Ch. Paillet
  • D. Néron
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 46)

Abstract

Reduced models and especially those based on Proper Generalized Decomposition (PGD) are decision-making tools which are about to revolutionize many domains. Unfortunately, their calculation remains problematic for problems involving many parameters, for which one can invoke the “curse of dimensionality”. The paper starts with the state-of-the-art for nonlinear problems involving stochastic parameters. Then, an answer to the challenge of many parameters is given in solid mechanics with the so-called “parameter-multiscale PGD”, which is based on the Saint-Venant principle.

References

  1. 1.
    A. Ammar, B. Mokdad, F. Chinesta, R. Keunings, A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids. Part II: Transient simulation using space-time separated representations. J. Non-Newton. Fluid Mech. 144(2–3), 98–121 (2007)CrossRefzbMATHGoogle Scholar
  2. 2.
    M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. C. R. Math. 339(9), 667–672 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    M. Billaud-Friess, A. Nouy, O. Zahm, A tensor approximation method based on ideal minimal residual formulations for the solution of high-dimensional problems. ESAIM. Math. Mode. Numer. Anal. 48(6), 1777–1806 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M. Capaldo, P.-A. Guidault, D. Néron, P. Ladevèze, The Reference Point Method, a "hyperreduction" technique: application to PGD-based nonlinear model reduction. Comput. Method. Appl. Mech. Eng. 322, 483–514 (2017)Google Scholar
  5. 5.
    S. Chaturantabut, D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    F. Chinesta, P. Ladevèze, (eds.) Separated Representations and PGD-Based Model Reduction: Fundamentals and Applications, vol. CISM 554 (Springer, 2014)Google Scholar
  7. 7.
    E. de Souza Neto, D. Peric, D.R.J. Owen, Computational Methods for Plasticity, vol. 55 (2008)Google Scholar
  8. 8.
    A. Falco, W. Hackbusch, A. Nouy, Geometric Structures in Tensor Representations (2015) pp. 1–50 (Work document)Google Scholar
  9. 9.
    W. Hackbusch, Tensor Spaces and Numerical Tensor Calculus (Springer, 2012)Google Scholar
  10. 10.
    J.A. Hernandez, J. Oliver, A.E. Huespe, M.A. Caicedo, J.C. Cante, High-performance model reduction techniques in computational multiscale homogenization. Comput. Methods Appl. Mech. Eng. 276, 149–189 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    C. Heyberger, P.A. Boucard, D. Néron, A rational strategy for the resolution of parametrized problems in the PGD framework. Comput. Methods Appl. Mech. Eng. 259, 40–49 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    S. Holtz, T. Rohwedder, R. Schneider, On manifolds of tensors of fixed TT-rank. Numer. Math. 120(4), 701–731 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    P. Ladevèze, On Algorithm Family in Structural Mechanics. Comptes rendus des séances de l’Academie des sciences. Série 2, 300(2) (1985)Google Scholar
  14. 14.
    P. Ladevèze, The large time increment method for the analyse of structures with nonlinear constitutive relation described by internal variables. Comptes rendus des séances de l’Academie des sciences. Série 2, 309(2), 1095–1099 (1989) (in french)Google Scholar
  15. 15.
    P. Ladevèze, Nonlinear Computational Structural Mechanics: New Approaches and Non-incremental Methods of Calculation (Springer, New York, 1999)Google Scholar
  16. 16.
    P. Ladevèze, New variational formulations for discontinuous approximations. Technical Report, LMT Cachan, 2011, (in french)Google Scholar
  17. 17.
    P. Ladevèze, A new method for the ROM computation: the parameter-multiscale PGD, Technical report, LMT Cachan, 2016a, (in french)Google Scholar
  18. 18.
    P. Ladevèze, On reduced models in nonlinear solid mechanics. Eur. J. Mech. A/Solids 60, 227–237 (2016b)MathSciNetCrossRefGoogle Scholar
  19. 19.
    P. Ladevèze, J.C. Passieux, D. Néron, The LATIN multiscale computational method and the proper generalized decomposition. Comput. Methods Appl. Mech. Eng. 199(21–22), 1287–1296 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    P. Ladevèze, H. Riou, On Trefftz and weak Trefftz discontinuous Galerkin approaches for medium-frequency acoustics. Comput. Methods Appl. Mech. Eng. 278, 729–743 (2014)MathSciNetCrossRefGoogle Scholar
  21. 21.
    E. Monteiro, J. Yvonnet, Q.C. He, Computational homogenization for nonlinear conduction in heterogeneous materials using model reduction. Comput. Mater. Sci. 42(4), 704–712 (2008)CrossRefGoogle Scholar
  22. 22.
    D. Néron, P.A. Boucard, N. Relun, Time-space PGD for the rapid solution of 3D nonlinear parametrized problems in the many-query context. Int. J. Numer. Methods Eng. 103(4), 275–292 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    A. Nouy, A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput. Methods Appl. Mech. Eng. 199(23–24), 1603–1626 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    I.V. Oseledets, Tensor-train decomposition. SIAM J. Sci. Comput. 33(5), 2295–2317 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    C. Paillet, P. Ladevèze, D. Néron, A Parametric-multiscale PGD for problems with a lage number of parameters (2017) (In preparation)Google Scholar
  26. 26.
    A. Radermacher, S. Reese, POD-based model reduction with empirical interpolation applied to nonlinear elasticity. Int. J. Numer. Methods Eng. 107(6), 477–495 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    N. Relun, D. Néron, P.A. Boucard, A model reduction technique based on the PGD for elastic-viscoplastic computational analysis. Comput. Mech. 51(1), 83–92 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    D. Ryckelynck, Hyper-reduction of mechanical models involving internal variables. Int. J. Numer. Methods Eng. 77(1), 75–89 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2018

Authors and Affiliations

  1. 1.LMT (ENS Paris-Saclay, CNRS, Université Paris-Saclay)Cachan CedexFrance

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