Archives of Computational Methods in Engineering

, Volume 17, Issue 4, pp 351–372 | Cite as

Proper Generalized Decomposition for Multiscale and Multiphysics Problems

  • David Néron
  • Pierre Ladevèze
Original Paper


This paper is a review of the developments of the Proper Generalized Decomposition (PGD) method for the resolution, using the multiscale/multiphysics LATIN method, of the nonlinear, time-dependent problems ((visco)plasticity, damage, …) encountered in computational mechanics. PGD leads to considerable savings in terms of computing time and storage, and makes engineering problems which would otherwise be completely out of range of industrial codes accessible.


Domain Decomposition Method Proper Generalize Decomposi Reference Problem Discontinuous Galerkin Scheme Model Reduction Technique 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Akel S, Nguyen QS (1989) Determination of the limit response in cyclic plasticity. In: Proceedings of 2nd international conference on computational plasticity. Barcelone, Spain, pp 639–650 Google Scholar
  2. 2.
    Ammar A, Mokdad B, Chinesta F, Keunings R (2006) A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids. J Non-Newton Fluid Mech 139(3):153–176 zbMATHCrossRefGoogle Scholar
  3. 3.
    Ammar A, Mokdad B, Chinesta F, Keunings R (2007) A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids: Part II: Transient simulation using space-time separated representations. J Non-Newton Fluid Mech 144(2–3):98–121 zbMATHCrossRefGoogle Scholar
  4. 4.
    Beckert A (2000) Coupling fluid (CFD) and structural (FE) models using finite interpolation elements. Aerosp Sci Technol 47:13–22 CrossRefGoogle Scholar
  5. 5.
    Belytschko T, Smolinski P, Liu WK (1985) Stability of multi-time step partitioned integrators for first-order finite element systems. Comput Methods Appl Mech Eng 49(3):281–297 zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Blom FJ (1998) A monolithic fluid-structure interaction algorithm applied to the piston problem. Comput Methods Appl Mech Eng 167:369–391 zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bottasso CL (2002) Multiscale temporal integration. Comput Methods Appl Mech Eng 191(25–26):2815–2830 zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Caignot A, Ladevèze P, Néron D, Durand J-F (2010) Virtual testing for the prediction of damping in joints. Eng Comput 27(5):621–644 CrossRefGoogle Scholar
  9. 9.
    Champaney L, Cognard J-Y, Ladevèze P (1999) Modular analysis of assemblages of three-dimensional structures with unilateral contact conditions. Comput Struct 73:249–266 zbMATHCrossRefGoogle Scholar
  10. 10.
    Champaney L, Boucard P-A, Guinard S (2008) Adaptive multi-analysis strategy for contact problems with friction: application to aerospace bolted joints. Comput Mech 42(2):305–316 zbMATHCrossRefGoogle Scholar
  11. 11.
    Chatterjee A (2000) An introduction to the proper orthogonal decomposition. Curr Sci 78(7):808–817 Google Scholar
  12. 12.
    Chinesta F, Ammar A, Lemarchand F, Beauchene P, Boust F (2008) Alleviating mesh constraints: Model reduction, parallel time integration and high resolution homogenization. Comput Methods Appl Mech Eng 197:400–413 zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Cognard J-Y, Ladevèze P (1993) A large time increment approach for cyclic plasticity. Int J Plast 9:114–157 CrossRefGoogle Scholar
  14. 14.
    Combescure A, Gravouil A (2002) A numerical scheme to couple subdomains with different time-steps for predominantly linear transient analysis. Comput Methods Appl Mech Eng 191:1129–1157 zbMATHCrossRefGoogle Scholar
  15. 15.
    Comte F, Maitournam H, Burry P, Lan NTM (2006) A direct method for the solution of evolution problems. C R Mec 334(5):317–322 zbMATHGoogle Scholar
  16. 16.
    Coussy O (2004) Poromechanics. Wiley, New York Google Scholar
  17. 17.
    Cresta P, Allix O, Rey C, Guinard S (2007) Nonlinear localization strategies for domain decomposition methods in structural mechanics. Comput Methods Appl Mech Eng 196(8):1436–1446 zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Devries F, Dumontet F, Duvaut G, Léné F (1989) Homogenization and damage for composite structures. Int J Numer Methods Eng 27:285–298 zbMATHCrossRefGoogle Scholar
  19. 19.
    Dureisseix D, Farhat C (2001) A numerically scalable domain decomposition method for the solution of frictionless contact problems. Int J Numer Methods Eng 50:2643–2666 zbMATHCrossRefGoogle Scholar
  20. 20.
    Dureisseix D, Ladevèze P, Néron D, Schrefler BA (2003) A multi-time-scale strategy for multiphysics problems: application to poroelasticity. Int J Multiscale Comput Eng 1(4):387–400 CrossRefGoogle Scholar
  21. 21.
    Dureisseix D, Ladevèze P, Schrefler BA (2003) A computational strategy for multiphysics problems—application to poroelasticity. Int J Numer Methods Eng 56(10):1489–1510 zbMATHCrossRefGoogle Scholar
  22. 22.
    Farhat C, Chandesris M (2003) Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications. Int J Numer Methods Eng 58:1397–1434 zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Farhat C, Lesoinne M (2000) Two efficient staggered algorithms for the serial and parallel solution of three-dimensional nonlinear transient aeroelastic problems. Comput Methods Appl Mech Eng 182:499–515 zbMATHCrossRefGoogle Scholar
  24. 24.
    Farhat C, Lesoinne M, LeTallec P (1998) Load and motion transfer algorithms for fluid/structure interaction problems with non-matching discrete interfaces: Momentum and energy conservation, optimal discretization and application to aeroelasticity. Comput Methods Appl Mech Eng 157:95–114 zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Faucher V, Combescure A (2003) A time and space mortar method for coupling linear modal subdomains and non-linear subdomains in explicit structural dynamics. Comput Methods Appl Mech Eng 192:509–533 zbMATHCrossRefGoogle Scholar
  26. 26.
    Felippa CA, Geers TL (1988) Partitioned analysis for coupled mechanical systems. Eng Comput 5:123–133 CrossRefGoogle Scholar
  27. 27.
    Felippa CA, Park KC (1980) Staggered transient analysis procedures for coupled mechanical systems: formulation. Comput Methods Appl Mech Eng 24:61–111 zbMATHCrossRefGoogle Scholar
  28. 28.
    Felippa CA, Park KC, Farhat C (2001) Partitioned analysis of coupled mechanical systems. Comput Methods Appl Mech Eng 190:3247–3270 zbMATHCrossRefGoogle Scholar
  29. 29.
    Feyel F (2003) A multilevel finite element (FE2) to describe the response of highly non-linear structures using generalized continua. Comput Methods Appl Mech Eng 192:3233–3244 zbMATHCrossRefGoogle Scholar
  30. 30.
    Fish J, Chen W (2001) Uniformly valid multiple spatial-temporal scale modeling for wave propagation in heterogeneous media. Mech Compos Mater Struct 8:81–99 CrossRefGoogle Scholar
  31. 31.
    Fish J, Shek K, Pandheeradi M, Shephard MS (1997) Computational plasticity for composite structures based on mathematical homogenization: Theory and practice. Comput Methods Appl Mech Eng 148:53–73 zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Golub GH, Loan CFV (1996) Matrix computations, 3rd edn. Johns Hopkins University Press, Baltimore zbMATHGoogle Scholar
  33. 33.
    Gosselet P, Chiaruttini V, Rey C, Feyel F (2004) A monolithic strategy based on an hybrid domain decomposition method for multiphysic problems. Application to poroelasticity. Rev Eur Élém Finis 13(5/7):523–534 zbMATHCrossRefGoogle Scholar
  34. 34.
    Gravouil A, Combescure A (2001) Multi-time-step explicit implicit method for non-linear structural dynamics. Int J Numer Methods Eng 50:199–225 zbMATHCrossRefGoogle Scholar
  35. 35.
    Gravouil A, Combescure A (2003) Multi-time-step and two-scale domain decomposition method for non-linear structural dynamics. Int J Numer Methods Eng 58:1545–1569 zbMATHCrossRefGoogle Scholar
  36. 36.
    Guennouni T (1988) On a computational method for cycling loading: the time homogenization. Math Model Numer Anal 22(3):417–455 (in French) zbMATHMathSciNetGoogle Scholar
  37. 37.
    Guidault P, Allix O, Champaney L, Cornuault S (2008) A multiscale extended finite element method for crack propagation. Comput Methods Appl Mech Eng 197(5):381–399 zbMATHCrossRefGoogle Scholar
  38. 38.
    Gunzburger MD, Peterson JS, Shadid JN (2007) Reduced-order modeling of time-dependent pdes with multiple parameters in the boundary data. Comput Methods Appl Mech Eng 196(4–6):1030–1047 zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    Hibbitt, Karlson, Sorensen (eds) (1996) Abaqus/standard—user’s manual, vol I, pp 6.4.2–2 and 6.6.1–4 Google Scholar
  40. 40.
    Huet C (1990) Application of variational concepts to size effects in elastic heterogeneous bodies. J Mech Phys Solids 38(6):813–841 CrossRefMathSciNetGoogle Scholar
  41. 41.
    Hughes TJR (1995) Multiscale phenomena: Green’s function, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origin of stabilized methods. Comput Methods Appl Mech Eng 127:387–401 zbMATHCrossRefGoogle Scholar
  42. 42.
    Jolliffe I (1986) Principal component analysis. Springer, New York Google Scholar
  43. 43.
    Karhunen K (1943) Uber lineare methoden für wahrscheinigkeitsrechnung. Ann Acad Sci Fenn Ser A1 Math Phys 37:3–79 Google Scholar
  44. 44.
    Kouznetsova V, Geers M, Brekelmans W (2002) Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational homogenization scheme. Int J Numer Methods Eng 54:1235–1260 zbMATHCrossRefGoogle Scholar
  45. 45.
    Kunisch K, Xie L (2005) Pod-based feedback control of the burgers equation by solving the evolutionary HJB equation. Comput Math Appl 49(7–8):1113–1126 zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Ladevèze J (1985). Algorithmes adaptés aux calculs vectoriels et parallèles pour des méthodes de décomposition de domaines. In: Actes du troisième colloque tendances actuelles en calcul de structures. Pluralis, pp 893–907 Google Scholar
  47. 47.
    Ladevèze P (1985) On a family of algorithms for structural mechanics. C R Acad Sci 300(2):41–44 (in French) zbMATHMathSciNetGoogle Scholar
  48. 48.
    Ladevèze P (1989) The large time increment method for the analyse of structures with nonlinear constitutive relation described by internal variables. C R Acad Sci Paris 309(II):1095–1099 zbMATHGoogle Scholar
  49. 49.
    Ladevèze P (1991) New advances in the large time increment method. In: Ladevèze P, Zienkiewicz OC (eds) New advances in computational structural mechanics. Elsevier, Amsterdam, pp 3–21 Google Scholar
  50. 50.
    Ladevèze P (1997). A computational technique for the integrals over the time-space domain in connection with the LATIN method. Technical Report 193, LMT-Cachan (in French) Google Scholar
  51. 51.
    Ladevèze P (1999) Nonlinear computational structural mechanics—new approaches and non-incremental methods of calculation. Springer, Berlin zbMATHGoogle Scholar
  52. 52.
    Ladevèze P, Nouy A (2003) On a multiscale computational strategy with time and space homogenization for structural mechanics. Comput Methods Appl Mech Eng 192:3061–3087 zbMATHCrossRefGoogle Scholar
  53. 53.
    Ladevèze P, Loiseau O, Dureisseix D (2001) A micro-macro and parallel computational strategy for highly heterogeneous structures. Int J Numer Methods Eng 52:121–138 CrossRefGoogle Scholar
  54. 54.
    Ladevèze P, Néron D, Gosselet P (2007) On a mixed and multiscale domain decomposition method. Comput Methods Appl Mech Eng 196:1526–1540 zbMATHCrossRefGoogle Scholar
  55. 55.
    Ladevèze P, Néron D, Passieux J-C (2009) On multiscale computational mechanics with time-space homogenization. In: Fish J (ed) Multiscale methods—Bridging the scales in science and engineering, Chapter space time scale bridging methods. Oxford University Press, Oxford, pp 247–282 Google Scholar
  56. 56.
    Ladevèze P, Passieux J-C, Néron D (2010) The LATIN multiscale computational method and the proper generalized decomposition. Comput Methods Appl Mech Eng 199:1287–1296 CrossRefGoogle Scholar
  57. 57.
    Lefik M, Schrefler B (2000) Modelling of nonstationary heat conduction problems in micro-periodic composites using homogenisation theory with corrective terms. Arch Mech 52(2):203–223 zbMATHMathSciNetGoogle Scholar
  58. 58.
    Lewis RW, Schrefler BA (1998) The finite element method in the static and dynamic deformation and consolidation of porous media, 2nd edn. Wiley, New York zbMATHGoogle Scholar
  59. 59.
    Lewis RW, Schrefler BA, Simoni L (1991) Coupling versus uncoupling in soil consolidation. Int J Numer Anal Methods Geomech 15:533–548 CrossRefGoogle Scholar
  60. 60.
    Lieu T, Farhat C, Lesoinne A (2006) Reduced-order fluid/structure modeling of a complete aircraft configuration. Comput Methods Appl Mech Eng 195(41–43):5730–5742 zbMATHCrossRefGoogle Scholar
  61. 61.
    Maday Y, Ronquist EM (2004) The reduced-basis element method: application to a thermal fin problem. SIAM J Sci Comput 26(1):240–258 zbMATHCrossRefMathSciNetGoogle Scholar
  62. 62.
    Maman N, Farhat C (1995) Matching fluid and structure meshes for aeroelastic computations: a parallel approach. Comput Struct 54(4):779–785 CrossRefGoogle Scholar
  63. 63.
    Matteazzi R, Schrefler B, Vitaliani R (1996) Comparisons of partitioned solution procedures for transient coupled problems in sequential and parallel processing. In: Advances in computational structures technology. Civil-Comp Ltd, Edinburgh, pp 351–357 CrossRefGoogle Scholar
  64. 64.
    Matthies HG, Steindorf J (2003) Partitioned strong coupling algorithms for fluid-structure interaction. Comput Struct 81:805–812 CrossRefGoogle Scholar
  65. 65.
    Michler C, Hulshoff SJ, van Brummelen EH, de Borst R (2004) A monolithic approach to fluid-structure interaction. Comput Struct 33:839–848 zbMATHGoogle Scholar
  66. 66.
    Morand J-P, Ohayon R (1995) Fluid-structure interaction: applied numerical methods. Wiley, New York Google Scholar
  67. 67.
    Néron D, Dureisseix D (2008) A computational strategy for poroelastic problems with a time interface between coupled physics. Int J Numer Methods Eng 73(6):783–804 zbMATHCrossRefGoogle Scholar
  68. 68.
    Néron D, Dureisseix D (2008) A computational strategy for thermo-poroelastic structures with a time-space interface coupling. Int J Numer Methods Eng 75(9):1053–1084 zbMATHCrossRefGoogle Scholar
  69. 69.
    Néron D, Ladevèze P, Dureisseix D, Schrefler BA (2004) Accounting for nonlinear aspects in multiphysics problems: Application to poroelasticity. In: Lecture notes in computer science, vol 3039, pp 612–620 Google Scholar
  70. 70.
    Nouy A (2007) A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput Methods Appl Mech Eng 196(45–48):4521–4537 zbMATHCrossRefMathSciNetGoogle Scholar
  71. 71.
    Nouy A (2009) Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations. Arch Comput Methods Eng 16(3):251–285 CrossRefMathSciNetGoogle Scholar
  72. 72.
    Nouy A, Ladevèze P (2004) Multiscale computational strategy with time and space homogenization: a radial type approximation technique for solving micro problems. Int J Multiscale Comput Eng 170(2):557–574 CrossRefGoogle Scholar
  73. 73.
    Oden JT, Vemaganti K, Moës N (1999) Hierarchical modeling of heterogeneous solids. Comput Methods Appl Mech Eng 172:3–25 zbMATHCrossRefGoogle Scholar
  74. 74.
    Piperno S, Farhat C, Larrouturou B (1995) Partitioned procedures for the transient solution of coupled aeroelastic problems. Part I: model problem, theory and two-dimensional application. Comput Methods Appl Mech Eng 124:79–112 zbMATHCrossRefMathSciNetGoogle Scholar
  75. 75.
    Ryckelynck D (2005) A priori hyperreduction method: an adaptive approach. J Comput Phys 202:346–366 zbMATHCrossRefGoogle Scholar
  76. 76.
    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):91–128 zbMATHCrossRefMathSciNetGoogle Scholar
  77. 77.
    Sanchez-Palencia E (1974) Comportement local et macroscopique d’un type de milieux physiques hétérogènes. Int J Eng Sci 12(4):331–351 zbMATHCrossRefMathSciNetGoogle Scholar
  78. 78.
    Sanchez-Palencia E (1980) Non homogeneous media and vibration theory. Lect Notes Phys 127 Google Scholar
  79. 79.
    Turska E, Schrefler BA (1993) On convergence conditions of partitioned solution procedures for consolidation problems. Comput Methods Appl Mech Eng 106:51–63 zbMATHCrossRefMathSciNetGoogle Scholar
  80. 80.
    Turska E, Schrefler BA (1994) On consistency, stability and convergence of staggered solution procedures. Rend Mat Acc Lincei 9(5):265–271 MathSciNetGoogle Scholar
  81. 81.
    Vermeer PA, Veruijt A (1981) An accuracy condition for consolidation by finite elements. Int J Numer Anal Methods Geomech 5:1–14 zbMATHCrossRefGoogle Scholar
  82. 82.
    Violeau D, Ladeveze P, Lubineau G (2009) Micromodel-based simulations for laminated composites. Compos Sci Technol 69(9):1364–1371 CrossRefGoogle Scholar
  83. 83.
    Zohdi T, Wriggers P (2005) Introduction to computational micromechanics. Springer, Berlin zbMATHCrossRefGoogle Scholar
  84. 84.
    Zohdi T, Oden J, Rodin G (1996) Hierarchical modeling of heterogeneous bodies. Comput Methods Appl Mech Eng 138(1–4):273–298 zbMATHCrossRefMathSciNetGoogle Scholar
  85. 85.
    Zohdi TI (2004) Modeling and simulation of a class of coupled modeling and simulation of a class of coupled thermo-chemo-mechanical processes in multiphase solids. Comput Methods Appl Mech Eng 193:679–699 zbMATHCrossRefGoogle Scholar

Copyright information

© CIMNE, Barcelona, Spain 2010

Authors and Affiliations

  1. 1.ENS Cachan/CNRS/UPMC/PRES UniverSudLMT-CachanParisFrance

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