Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
PGD-Based Modeling of Materials, Structures and Processes

Part of the book series: ESAFORM Bookseries on Material Forming ((EBMF))

  • 659 Accesses

Abstract

Many problems in science and engineering remain intractable nowadays because of their numerical complexity, or the restrictions imposed by different requirements (real-time on deployed platforms, for instance) make them unaffordable for today’s technologies.

Imagination is more important than knowledge.

—Albert Einstein.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. E. Cancès, M. Defranceschi, W. Kutzelnigg, C. Le Bris, Y. Maday, Computational quantum chemistry: a primer, in Handbook of Numerical Analysis, vol. X (Elsevier, Amsterdam, 2003) pp. 3–270, 2003.

    Google Scholar 

  2. B.B. Bird, C.F. Curtiss, R.C. Armstrong, O. Hassager, Dynamics of polymeric liquids, in: Kinetic Theory, vol. 2 (Wiley, New York, 1987)

    Google Scholar 

  3. A. Ammar, E. Cueto, F. Chinesta, Reduction of the chemical master equation for gene regulatory networks using proper generalized decompositions. Int. J. Numer. Meth. Biomed. Eng 28(9), 960–973 (2012)

    Article  MathSciNet  Google Scholar 

  4. NSF Final Report, DDDAS Workshop 2006 (Arlington, VA, USA, 2006)

    Google Scholar 

  5. F. Darema, Engineering/Scientific and commercial applications: differences, similarities, and future evolution, in Proceedings of the Second Hellenic European Conference on Mathematics and Informatics. HERMIS, vol 1, (1994) pp. 367–374

    Google Scholar 

  6. J.T. Oden, T. Belytschko, J. Fish, T.J.R. Hughes, C. Johnson, D. Keyes, A. Laub, L. Petzold, D. Srolovitz, S.Yip, Simulation-based engineering science: revolutionizing engineering science through simulation. NSF Blue Ribbon Panel on SBES (2006)

    Google Scholar 

  7. D. Ryckelynck, F. Chinesta, E. Cueto, A. Ammar, On the a priori model reduction: overview and recent developments. State of the art reviews. Arch. Comput. Meth. Eng. 13(1), 91–128 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. R.A. Bialecki, A.J. Kassab, A. Fic, Proper orthogonal decomposition and modal analysis for acceleration of transient FEM thermal analysis. Int. J. Numer. Meth. Eng. 62, 774–797 (2005)

    Article  MATH  Google Scholar 

  9. J. Burkardt, M. Gunzburger, H-Ch. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes flows. Comput. Meth. Appl. Mech. Eng. 196, 337–355 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  10. M.D. Gunzburger, J.S. Peterson, J.N. Shadid, Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data. Comput. Meth. Appl. Mech. Eng. 196, 1030–1047 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Y. Maday, E.M. Ronquist, The reduced basis element method: application to a thermal fin problem. SIAM J. Sci. Comput. 26(1), 240–258 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  12. H.M. Park, D.H. Cho, The use of the Karhunen-Loeve decomposition for the modelling of distributed parameter systems. Chem. Eng. Sci 51, 81–98 (1996)

    Article  Google Scholar 

  13. D. Ryckelynck, L. Hermanns, F. Chinesta, E. Alarcon, An efficient a priori model reduction for boundary element models. Eng. Anal. Boundary Elem. 29, 796–801 (2005)

    Article  MATH  Google Scholar 

  14. D. Ryckelynck, A priori hyper-reduction method: an adaptive approach. J. Comput. Phys. 202, 346–366 (2005)

    Article  MATH  Google Scholar 

  15. A. Ammar, D. Ryckelynck, F. Chinesta, R. Keunings, On the reduction of kinetic theory models related to finitely extensible dumbbells. J. Nonnewton. Fluid Mech. 134, 136–147 (2006)

    Article  MATH  Google Scholar 

  16. P. Ladeveze, Nonlinear Computational Structural Mechanics (Springer, NY, 1999)

    Book  MATH  Google Scholar 

  17. 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)

    Article  MATH  Google Scholar 

  18. A. Nouy, P. Ladeveze, Multiscale computational strategy with time and space homogenization: a radial-type approximation technique for solving microproblems, Int. J. Multiscale Comput. Eng. 170(2)557–574 (2004)

    Google Scholar 

  19. F. Chinesta, R. Keunings, A. Leygue, The Proper Generalized Decomposition for Advanced Numerical Simulations. A Primer (Springerbriefs) (Springer, New York, 2014)

    Book  Google Scholar 

  20. D. Ryckelynck, Hyper-reduction of mechanical models involving internal variables. Int. J. Numer. Meth. Eng. 77(1), 75–89 (2008)

    Article  MathSciNet  Google Scholar 

  21. B. Cochelin, N. Damil, M. Potier-Ferry, The asymptotic numerical method: an efficient perturbation technique for nonlinear structural mechanics. Revue Europeenne des Elements Finis 3, 281–297 (1994)

    MATH  MathSciNet  Google Scholar 

  22. M. Barrault, Y. Maday, N.C. Nguyen, A.T. Patera, An "empirical interpolation" method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339(9), 667–672 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. S. Chaturantabut, D.C. Sorensen, Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32, 2737–2764 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. F. Chinesta, A. Ammar, E. Cueto, Proper generalized decomposition of multiscale models. Int. J. Numer. Meth. Eng. 83(8–9), 1114–1132 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Ammar, F. Chinesta, E. Cueto, Coupling finite elements and proper generalized decomposition. Int. J. Multiscale Comput. Eng. 9(1), 17–33 (2011)

    Article  Google Scholar 

  26. F. Chinesta, A. Ammar, F. Lemarchand, P. Beauchene, F. Boust, Alleviating mesh constraints: model reduction, parallel time integration and high resolution homogenization. Comput. Methods Appl. Mech. Eng. 197(5), 400–413 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  27. H. Lamari, A. Ammar, P. Cartraud, G. Legrain, F. Jacquemin, F. Chinesta, Routes for efficient computational homogenization of non-linear materials using the proper generalized decomposition. Arch. Comput. Meth. Eng. 17(4), 373–391 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. D. Gonzalez, A. Ammar, F. Chinesta, E. Cueto, Recent advances in the use of separated representations. Int. J. Numer. Meth. Eng. 81(5), 637–659 (2010)

    MATH  MathSciNet  Google Scholar 

  29. F. Chinesta, A. Ammar, E. Cueto, Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models. Arch. Comput. Meth. Eng. 17(4), 327–350 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  30. F. Chinesta, A. Ammar, A. Leygue, R. Keunings, An overview of the proper generalized decomposition with applications in computational rheology. J. Nonnewton. Fluid Mech. 166, 578–592 (2011)

    Article  MATH  Google Scholar 

  31. 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 modeling of complex fluids. J. Nonnewton. Fluid Mech. 139, 153–176 (2006)

    Article  MATH  Google Scholar 

  32. B. Mokdad, E. Pruliere, A. Ammar, F. Chinesta, On the simulation of kinetic theory models of complex fluids using the Fokker-Planck approach. Appl. Rheol. 17(2), 1–14, 26494 (2007)

    Google Scholar 

  33. 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 modeling of complex fluids (Part II). J. Nonnewton. Fluid Mech. 144, 98–121 (2007)

    Article  MATH  Google Scholar 

  34. A. Ammar, E. Pruliere, F. Chinesta, M. Laso, Reduced numerical modeling of flows involving liquid-crystalline polymeres. J. Nonnewton. Fluid Mech. 160, 140–156 (2009)

    Article  MATH  Google Scholar 

  35. A. Ammar, M. Normandin, F. Daim, D. Gonzalez, E. Cueto, F. Chinesta, Non-incremental strategies based on separated representations: applications in computational rheology. Commun. Math.Sci 8(3), 671–695 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  36. E. Pruliere, A. Ammar, N. El Kissi, F. Chinesta, Recirculating flows involving short fiber suspensions: numerical difficulties and efficient advanced micro-macro solvers, State of the art reviews. Arch. Comput. Meth. Eng 16, 1–30 (2009)

    Article  MATH  Google Scholar 

  37. B. Mokdad, A. Ammar, M. Normandin, F. Chinesta, J.R. Clermont, A fully deterministic micro-macro simulation of complex flows involving reversible network fluid models. Math. Comput. Simul. 80, 1936–1961 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  38. F. Chinesta, A. Ammar, A. Falco, M. Laso, On the reduction of stochastic kinetic theory models of complex fluids. Model. Simul. Mater. Sci. Eng. 15, 639–652 (2007)

    Article  Google Scholar 

  39. F. Chinesta, P. Ladeveze, E. Cueto, A short review in model order reduction based on proper generalized decomposition. Arch. Comput. Meth. Eng. 18, 395–404 (2011)

    Article  Google Scholar 

  40. A. Ammar, F. Chinesta, P. Joyot, The nanometric and micrometric scales of the structure and mechanics of materials revisited: an introduction to the challenges of fully deterministic numerical descriptions. Int. J. Multiscale Comput. Eng. 6(3), 191–213 (2008)

    Article  Google Scholar 

  41. F. Chinesta, A. Ammar, E. Cueto, On the use of proper generalized decompositions for solving the multidimensional chemical master equation. Eur. J. Comput. Mech. 19, 53–64 (2010)

    Google Scholar 

  42. H. Lamari, A. Ammar, A. Leygue, F. Chinesta, On the solution of the multidimensional Langer’s equation by using the proper generalized decomposition method for modeling phase transitions. Model. Simul. Mater. Sci. Eng. 20, 015007 (2012)

    Article  Google Scholar 

  43. E. Pruliere, F. Chinesta, A. Ammar, A. Leygue, A. Poitou, On the solution of the heat equation in very thin tapes. Int. J. Therm. Sci. 65, 148–157 (2013)

    Article  Google Scholar 

  44. B. Bognet, A. Leygue, F. Chinesta, A. Poitou, F. Bordeu, Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity. Comput. Methods Appl. Mech. Eng. 201, 1–12 (2012)

    Article  Google Scholar 

  45. B. Bognet, A. Leygue, F. Chinesta, On the fully 3D simulation of thermoelastic models defined in plate geometries. Eur. J. Comput. Mech. 21(1–2), 40–51 (2012)

    Google Scholar 

  46. A. Leygue, F. Chinesta, M. Beringhier, T.L. Nguyen, J.C. Grandidier, F. Pasavento, B. Schrefler, Towards a framework for non-linear thermal models in shell domains. Int. J. Numer. Meth. Heat Fluid Flow 23(1), 55–73 (2013)

    Article  Google Scholar 

  47. B. Bognet, A. Leygue, F. Chinesta, Separated representations of 3D elastic solutions in shell geometries. Adv. Model. Simul. Mater. Sci. Eng. (in press) 1(1), 1–34 (2014)

    Article  Google Scholar 

  48. A. Ammar, M. Normandin, F. Chinesta, Solving parametric complex fluids models in rheometric flows. J. Nonnewton. Fluid Mech. 165, 1588–1601 (2010)

    Article  MATH  Google Scholar 

  49. E. Pruliere, F. Chinesta, A. Ammar, On the deterministic solution of multidimensional parametric models by using the proper generalized decomposition. Math. Comput. Simul. 81, 791–810 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  50. H. Lamari, F. Chinesta, A. Ammar, E. Cueto, Non-conventional numerical strategies in the advanced simulation of materials and processes. Int. J. Mod. Manuf. Technol. 1, 49–56 (2009)

    Google Scholar 

  51. F. Chinesta, A. Leygue, F. Bordeu, J.V. Aguado, E. Cueto, D. Gonzalez, I. Alfaro, A. Ammar, A. Huerta, Parametric PGD based computational vademecum for efficient design, optimization and control. Arch. Comput. Meth. Eng. 20(1), 31–59 (2013)

    Article  MathSciNet  Google Scholar 

  52. Ch. Ghnatios, F. Chinesta, E. Cueto, A. Leygue, P. Breitkopf, P. Villon, Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: application to pultrusion. Compos. A 42, 1169–1178 (2011)

    Article  Google Scholar 

  53. A. Leygue, E. Verron, A first step towards the use of proper general decomposition method for structural optimization. Arch. Comput. Meth. Eng. 17(4), 465–472 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  54. A. Ammar, A. Huerta, A. Leygue, F. Chinesta, E. Cueto, Parametric solutions involving geometry: a step towards efficient shape optimization. Comput. Methods Appl. Mech. Eng. 268, 178–193 (2014)

    Article  MathSciNet  Google Scholar 

  55. F. Schmidt, N. Pirc, M. Mongeau, F. Chinesta, Efficient mould cooling optimization by using model reduction. Int. J. Mater. Form. 4(1), 71–82 (2011)

    Article  Google Scholar 

  56. D. Gonzalez, F. Masson, F. Poulhaon, A. Leygue, E. Cueto, F. Chinesta, Proper generalized decomposition based dynamic data-driven inverse identification. Math. Comput. Simul. 82(9), 1677–1695 (2012)

    Article  MathSciNet  Google Scholar 

  57. Ch. Ghnatios, F. Masson, A. Huerta, E. Cueto, A. Leygue, F. Chinesta, Proper generalized decomposition based dynamic data-driven control of thermal processes. Comput. Methods Appl. Mech. Eng. 213, 29–41 (2012)

    Article  Google Scholar 

  58. S. Niroomandi, I. Alfaro, E. Cueto, F. Chinesta, Real-time deformable models of non-linear tissues by model reduction techniques. Comput. Methods Programs Biomed. 91, 223–231 (2008)

    Article  Google Scholar 

  59. S. Niroomandi, I. Alfaro, E. Cueto, F. Chinesta, Model order reduction for hyperelastic materials. Int. J. Numer. Meth. Eng. 81(9), 1180–1206 (2010)

    MATH  MathSciNet  Google Scholar 

  60. S. Niroomandi, I. Alfaro, E. Cueto, F. Chinesta, Accounting for large deformations in real-time simulations of soft tissues based on reduced order models. Comput. Methods Programs Biomed. 105, 1–12 (2012)

    Article  Google Scholar 

  61. S. Niroomandi, I. Alfaro, D. Gonzalez, E. Cueto, F. Chinesta, Real time simulation of surgery by reduced order modelling and X-FEM techniques. Int. J. Numer. Meth. Biomed. Eng 28(5), 574–588 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  62. D. Gonzalez, E. Cueto, F. Chinesta, P. Diez, A. Huerta, SUPG-based stabilization of proper generalized decompositions for high-dimensional advection-diffusion equations. Int. J. Numer. Meth. Eng. 94(13), 1216–1232 (2013)

    Article  MathSciNet  Google Scholar 

  63. A. Ammar, F. Chinesta, P. Diez, A. Huerta, An error estimator for separated representations of highly multidimensional models. Comput. Methods Appl. Mech. Eng. 199, 1872–1880 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  64. A. Ammar, E. Pruliere, J. Ferec, F. Chinesta, E. Cueto, Coupling finite elements and reduced approximation bases. Eur. J. Comput. Mech. 18(5–6), 445–463 (2009)

    MATH  Google Scholar 

  65. A. Ammar, F. Chinesta, E. Cueto, M. Doblare, Proper generalized decomposition of time-multiscale models. Int. J. Numer. Meth. Eng. 90(5), 569–596 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  66. F. Poulhaon, F. Chinesta, A. Leygue, A first step towards a PGD based parallelization strategy. Eur. J. Comput. Mech. 21(3–6), 300–311 (2012)

    Google Scholar 

  67. E. Pruliere, J. Ferec, F. Chinesta, A. Ammar, An efficient reduced simulation of residual stresses in composites forming processes. Int. J. Mater. Form. 3(2), 1339–1350 (2010)

    Article  Google Scholar 

  68. A. Ammar, E. Cueto, F. Chinesta, Non-incremental PGD solution of parametric uncoupled models defined in evolving domains. Int. J. Numer. Meth. Eng. 93(8), 887–904 (2013)

    MathSciNet  Google Scholar 

  69. M. Pineda, F. Chinesta, J. Roger, M. Riera, J. Perez, F. Daim, Simulation of skin effect via separated representations. Int. J. Comput. Math. Electr. Electron. Eng. 29(4), 919–929 (2010)

    Article  MATH  Google Scholar 

  70. G. Bonithon, P. Joyot, F. Chinesta, P. Villon, Non-incremental boundary element discretization of parabolic models based on the use of proper generalized decompositions. Eng. Anal. Boundary Elem. 35(1), 2–17 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  71. C. Quesada, D. Gonzalez, I. Alfaro, F. Bordeu, A. Leygue, E. Cueto, A. Huerta, F. Chinesta. Real-time simulation on handheld devices for augmented learning in science and engineering. PLOS ONE (Submitted)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. UMR CNRS, Ecole Centrale de Nantes, Nantes, France

    Francisco Chinesta

  2. Aragon Institute of Engineering Research, Universidad de Zaragoza, Zaragoza, Spain

    Elías Cueto

Authors
  1. Francisco Chinesta
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Elías Cueto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Francisco Chinesta .

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Chinesta, F., Cueto, E. (2014). Introduction. In: PGD-Based Modeling of Materials, Structures and Processes. ESAFORM Bookseries on Material Forming. Springer, Cham. https://doi.org/10.1007/978-3-319-06182-5_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-06182-5_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-06181-8

  • Online ISBN: 978-3-319-06182-5

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation