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A Short Review on Model Order Reduction Based on Proper Generalized Decomposition

  • Francisco Chinesta
  • Pierre Ladeveze
  • Elías Cueto
Article

Abstract

This paper revisits a new model reduction methodology based on the use of separated representations, the so called Proper Generalized Decomposition—PGD. Space and time separated representations generalize Proper Orthogonal Decompositions—POD—avoiding any a priori knowledge on the solution in contrast to the vast majority of POD based model reduction technologies as well as reduced bases approaches. Moreover, PGD allows to treat efficiently models defined in degenerated domains as well as the multidimensional models arising from multidimensional physics (quantum chemistry, kinetic theory descriptions,…) or from the standard ones when some sources of variability are introduced in the model as extra-coordinates.

Keywords

Augmented Reality Stochastic Partial Differential Equation Separate Representation Proper Generalize Decomposition Chemical Master Equation 
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.

References

  1. 1.
    Allix O, Ladevèze P, Gilleta D, Ohayon R (1989) A damage prediction method for composite structures. Int J Numer Methods Eng 27(2):271–283 zbMATHCrossRefGoogle Scholar
  2. 2.
    Allix O, Vidal P (2002) A new multi-solution approach suitable for structural identification problems. Comput Methods Appl Mech Eng 191:2727–2758 MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    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 139:153–176 zbMATHCrossRefGoogle Scholar
  4. 4.
    Ammar A, Ryckelynck D, Chinesta F, Keunings R (2006) On the reduction of kinetic theory models related to finitely extensible dumbbells. J Non-Newton Fluid Mech 134:136–147 zbMATHCrossRefGoogle Scholar
  5. 5.
    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 representation. J Non-Newton Fluid Mech 144:98–121 zbMATHCrossRefGoogle Scholar
  6. 6.
    Ammar A, Pruliere E, Chinesta F, Laso M (2009) Reduced numerical modeling of flows involving liquid-crystalline polymeres. J Non-Newton Fluid Mech 160:140–156 CrossRefGoogle Scholar
  7. 7.
    Ammar A, Pruliere E, Ferec J, Chinesta F, Cueto E (2009) Coupling finite elements and reduced approximation bases. Eur J Comput Mech 18(5–6):445–463 Google Scholar
  8. 8.
    Ammar A, Normandin M, Daim F, Gonzalez D, Cueto E, Chinesta F (2010) Non-incremental strategies based on separated representations: applications in computational rheology. Commun Math Sci 8(3):671–695 MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ammar A, Chinesta F, Falco A (2010) On the convergence of a greedy rank-one update algorithm for a class of linear systems. Arch Comput Methods Eng 17(4):473–486 MathSciNetGoogle Scholar
  10. 10.
    Ammar A, Chinesta F, Diez P, Huerta A (2010) An error estimator for separated representations of highly multidimensional models. Comput Methods Appl Mech Eng 199:1872–1880 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ammar A, Normandin M, Chinesta F (2010) Solving parametric complex fluids models in rheometric flows. J Non-Newton Fluid Mech 165:1588–1601 CrossRefGoogle Scholar
  12. 12.
    Ammar A, Chinesta F, Cueto E (2011) Coupling finite elements and proper generalized decompositions. Int J Multiscale Comput Eng 9(1):17–33 CrossRefGoogle Scholar
  13. 13.
    Ammar A, Chinesta F, Cueto E, Doblare M (2011) Proper generalized decomposition of time-multiscale models. Int J Numer Methods Eng. doi: 10.1002/nme.3331
  14. 14.
    Aubard X, Cluzel C, Guitard L, Ladevèze P (2000) Damage modeling at two scales for 4D carbon/carbon composites. Comput Struct 78(1–3):83–91 CrossRefGoogle Scholar
  15. 15.
    Beringhier M, Gueguen M, Grandidier JC (2010) Solution of strongly coupled multiphysics problems using space-time separated representations: application to thermoviscoelasticity. Arch Comput Methods Eng 17(4):393–401 MathSciNetGoogle Scholar
  16. 16.
    Blanzé C, Danwe R, Ladevèze P, Moreau J-P (1993) Une méthode pour l’étude d’assemblage de structures massives. In: Colloque National en Calcul des Structures, Hermès, pp 913–919 Google Scholar
  17. 17.
    Blanzé C, Champaney L, Cognard J-Y, Ladevèze P (1996) A modular approach to structure assembly computations—application to contact problems. Eng Comput 13(1):15 CrossRefGoogle Scholar
  18. 18.
    Bognet B, Leygue A, Chinesta F, Poitou A, Bordeu F (2011) Advanced simulation of models defined in plate geometries: 3D solutions with 2D computational complexity. Comput Methods Appl Mech Eng. doi: 10.1016/j.cma.2011.08.025
  19. 19.
    Boisse P, Ladevèze P, Rougée P (1989) A large time increment method for elastoplastic problems. Eur J Mech A, Solids 8(4):257–275 zbMATHGoogle Scholar
  20. 20.
    Boisse P, Bussy P, Ladevèze P (1990) A new approach in nonlinear mechanics—the large time increment method. Int J Numer Methods Eng 29(3):647–663 zbMATHCrossRefGoogle Scholar
  21. 21.
    Boisse P, Ladevèze P, Poss M, Rougée P (1991) A new large time increment algorithm for anisotropic plasticity. Int J Plast 7(1–2):65–77 zbMATHCrossRefGoogle Scholar
  22. 22.
    Boucard PA, Ladevèze P, Poss M, Rougée P (1997) A non-incremental approach for large displacement problems. Comput Struct 64:499–508 zbMATHCrossRefGoogle Scholar
  23. 23.
    Boucard PA, Ladevèze P (1999) A multiple solution method for non-linear structural mechanics. Mech Eng 50(5):317–328 Google Scholar
  24. 24.
    Boucard PA, Ladevèze P (1999) Une application de la méthode latin au calcul multirésolution de structures non linéaires. In: Revue Européenne des Eléments Finis, pp 903–920 Google Scholar
  25. 25.
    Boucard PA (2001) Application of the LATIN method to the calculation of response surfaces. In: 1st MIT conference on computational fluid and solid mechanics, vol 1, pp 78–81 CrossRefGoogle Scholar
  26. 26.
    Boucard PA, Derumaux M, Ladevèze P (2003) Macro-meso models for joints submitted to pyrotechnic shock. In: Computational fluid and solid mechanics, vol 1–2, pp 139–142. CrossRefGoogle Scholar
  27. 27.
    Bussy P, Rougée P, Vauchez P (1990) The large time increment method for numerical simulation of metal forming processes. In: NUMETA. Elsevier, Amsterdam, pp 102–109 Google Scholar
  28. 28.
    Caignot A, Ladevèze P, Néron D, Durand JF (2010) Virtual testing for the prediction of damping in joints. Eng Comput 27(5–6):621–644 Google Scholar
  29. 29.
    Cancès E, Ehrlacher V, Lelièvre T (2011) Convergence of a greedy algorithm for high-dimensional convex nonlinear problems. Math Models Methods Appl Sci. doi: 10.1142/S0218202511005799
  30. 30.
    Champaney L, Cognard J-Y, Dureisseix D, Ladevèze P (1997) Large scale applications on parallel computers of a mixed domain decomposition method. Comput Mech 19(4):253–263 zbMATHCrossRefGoogle Scholar
  31. 31.
    Champaney L, Cognard J-Y, Ladevèze P (1999) Modular analysis of assemblages of three-dimensional structures with unilateral contact conditions. Comput Struct 73(1–5):249–266 zbMATHCrossRefGoogle Scholar
  32. 32.
    Chevreuil M, Nouy A (2011) Model order reduction based on proper generalized decomposition for the propagation of uncertainties in structural dynamics. Int J Numer Methods Eng. doi: 10.1002/nme.3249
  33. 33.
    Chinesta F, Ammar A, Falco A, Laso M (2007) On the reduction of stochastic kinetic theory models of complex fluids. Model Simul Mater Sci Eng 15:639–652 CrossRefGoogle Scholar
  34. 34.
    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(5):400–413 MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
    Ammar A, Chinesta F, Joyot P (2008) 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 CrossRefGoogle Scholar
  36. 36.
    Chinesta F, Ammar A, Cueto E (2010) Proper generalized decomposition of multiscale models. Int J Numer Methods Eng 83(8–9):1114–1132 MathSciNetzbMATHCrossRefGoogle Scholar
  37. 37.
    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):327–350 MathSciNetGoogle Scholar
  38. 38.
    Chinesta F, Ammar A, Cueto E (2010) On the use of proper generalized decompositions for solving the multidimensional chemical master equation. Eur J Comput Mech 19:53–64 Google Scholar
  39. 39.
    Chinesta F, Ammar A, Leygue A, Keunings R (2011) An overview of the proper generalized decomposition with applications in computational rheology. J Non-Newton Fluid Mech 166:578–592 zbMATHCrossRefGoogle Scholar
  40. 40.
    Cognard J-Y (1990) Le traitement des problèmes nonlinéaires à grand nombre de degrés de liberté par la méthode à grand incrément de temps. In: Fouet J-M et al. (eds) Calcul des structures et intelligence artificielle, Pluralis, pp 211–222 Google Scholar
  41. 41.
    Cognard J-Y, Ladevèze P (1993) A large time increment approach for cyclic viscoplasticity. Int J Plast 9:141–157 zbMATHCrossRefGoogle Scholar
  42. 42.
    Cognard J-Y, Ladevèze P, Talbot P (1999) A large time increment approach for thermo-mechanical problems. Adv Eng Softw 30(9–11):583–593 CrossRefGoogle Scholar
  43. 43.
    Gonzalez D, Cueto E, Chinesta F, Debeugny L, Diez P, Huerta A (2010) Int J Mater Form 3(1):883–886 CrossRefGoogle Scholar
  44. 44.
    DeVore RA, Temlyakov VN (1996) Some remarks on greedy algorithms. Adv Comput Math 5:173–187 MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    Dumon A, Allery C, Ammar A (2011) Proper general decomposition (PGD) for the resolution of Navier-Stokes equations. J Comput Phys 230(4):1387–1407 MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    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
  47. 47.
    Dureisseix D, Ladevèze P, Schrefler BA (2003) A latin computational strategy for multiphysics problems: application to poroelasticity. Int J Numer Methods Eng 56(10):1489–1510 zbMATHCrossRefGoogle Scholar
  48. 48.
    Falco A (2010) Algorithms and numerical methods for high dimensional financial market models. Rev Econ Financ, 20:51–68 MathSciNetGoogle Scholar
  49. 49.
    Falcó A, Nouy A (2011) A proper generalized decomposition for the solution of elliptic problems in abstract form by using a functional Eckart-Young approach. J Math Anal Appl 376:469–480 MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Falco A, Nouy A Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. arXiv:1106.4424v1
  51. 51.
    Figueroa L, Süli E (2011) Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators with unbounded drift. arXiv:1103.0726
  52. 52.
    Ghnatios Ch, Chinesta F, Cueto E, Leygue A, Breitkopf P, Villon P (2011) Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: application to pultrusion. Composites, Part A, Appl Sci Manuf 42:1169–1178 CrossRefGoogle Scholar
  53. 53.
    Ghnatios Ch, Masson F, Huerta A, Cueto E, Leygue A, Chinesta F (2011) Proper generalized decomposition based dynamic data-driven control of thermal processes. Comput Methods Appl Mech Eng. Submitted Google Scholar
  54. 54.
    Gonzalez D, Ammar A, Chinesta F, Cueto E (2010) Recent advances on the use of separated representations. Int J Numer Methods Eng 81(5):637–659 MathSciNetzbMATHGoogle Scholar
  55. 55.
    Gonzalez D, Masson F, Poulhaon F, Leygue A, Cueto E, Chinesta F (2011) Proper generalized decomposition based dynamic data-driven inverse identification. Mathematics and Computers in Simulation, Submitted, 2011 Google Scholar
  56. 56.
    Bonithon G, Joyot P, Chinesta F, Villon P (2011) Non-incremental boundary element discretization of parabolic models based on the use of proper generalized decompositions. Eng Anal Bound Elem 35(1):2–17 MathSciNetCrossRefGoogle Scholar
  57. 57.
    Ladevèze P (1985) New algorithms: mechanical framework and development (in french). Technical Report 57, LMT-Cachan Google Scholar
  58. 58.
    Ladevèze P (1985) On a family of algorithms for structural mechanics. CR Acad Sci Paris 300(2):41–44 (in french) zbMATHGoogle Scholar
  59. 59.
    Ladevèze P, Rougée P (1985) Viscoplasticity under cyclic loadings: properties of the homogenized cycle. CR Acad Sci 301:891–894 zbMATHGoogle Scholar
  60. 60.
    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, 309:1095–1099 zbMATHGoogle Scholar
  61. 61.
    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
  62. 62.
    Ladevèze P, Lorong Ph (1992) A large time increment approach with domain decomposition technique for mechanical non linear problems. In: Computing methods in applied sciences and engineering INRIA, pp. 569–578 Google Scholar
  63. 63.
    Ladevèze P, Lorong Ph (1993) Formulation et stratégies “parallèles” pour l’analyse non linéaire des structures. In: Colloque national en calcul des structures. Hermès, Paris, pp 910–919 Google Scholar
  64. 64.
    Ladevèze P (1996) Mécanique non linéaire des structures. Hermès, Paris zbMATHGoogle Scholar
  65. 65.
    Ladevèze P (1997) A computational technique for the integrals over the time-space domain in connection with the LATIN method (in french). Technical Report 193, LMT-Cachan Google Scholar
  66. 66.
    Ladevèze P, Dureisseix D (1998) A 2-level and mixed domain decomposition approach for structural analysis. Contemp Math 218:246–253 Google Scholar
  67. 67.
    Ladevèze P (1999) Nonlinear computationnal structural mechanics—new approaches and non-incremental methods of calculation. Springer, Berlin Google Scholar
  68. 68.
    Ladevèze P, Cognard J-Y, Talbot P (1999) A non-incremental and adaptive computational approach in thermo-viscoplasticity. In: Bruhns OT, Stein E (eds) IUTAM symposium on micro- and macrostructural aspects of the thermoplasticity, pp 281–291 Google Scholar
  69. 69.
    Ladevèze P, Dureisseix D (1999) A new micro-macro computational strategy for structural analysis. CR Acad Sci, Ser Ii, Fascicule, B—Mec Phys Astron, 327(12):1237–1244 zbMATHGoogle Scholar
  70. 70.
    Ladevèze P, Guitard L, Champaney L, Aubard X (2000) Debond modeling for multidirectional composites. Comput Methods Appl Mech Eng 185(2–4):109–122 zbMATHCrossRefGoogle Scholar
  71. 71.
    Ladevèze P, Lemoussu H, Boucard PA (2000) A modular approach to 3-d impact computation with frictional contact. Comput Struct 78(1–3):45–51 CrossRefGoogle Scholar
  72. 72.
    Ladevèze P, Perego U (2000) Duality preserving discretization of the large time increment methods. Comput Methods Appl Mech Eng 189(1):205–232 zbMATHCrossRefGoogle Scholar
  73. 73.
    Ladevèze P, Loiseau O, Dureisseix D (2001) A micro-macro and parallel computational strategy for high heterogeneous structures. Int J Numer Methods Eng, 52(1–2):121–138 CrossRefGoogle Scholar
  74. 74.
    Ladevèze P, Nouy A (2002) A multiscale computational method with time and space homogenization. CR Mec, 330(10):683–689 zbMATHCrossRefGoogle Scholar
  75. 75.
    Ladevèze P, Nouy A (2002) Une stratégie de calcul multiéchelle avec homogénéisation en espace et en temps. CR Mec, 330:683–689 zbMATHCrossRefGoogle Scholar
  76. 76.
    Ladevèze P, Nouy A, Loiseau O (2002) A multiscale computational approach for contact problems. Comput Methods Appl Mech Eng, 191(43):4869–4891 zbMATHCrossRefGoogle Scholar
  77. 77.
    Ladevèze P, Nouy A (2003) On a multiscale computational strategy with time and space homogenization for structural mechanics. Comput Methods Appl Mech Eng, 192(28–30):3061–3087 zbMATHCrossRefGoogle Scholar
  78. 78.
    Ladevèze P (2004) Multiscale modeling and computational strategies for composites. Int J Numer Methods Eng, 60(1):233–253 zbMATHCrossRefGoogle Scholar
  79. 79.
    Ladevèze P, Néron D, Gosselet P (2007) On a mixed and multiscale domain decomposition method. Comput Methods Appl Mech Eng 96:1526–1540 CrossRefGoogle Scholar
  80. 80.
    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. Oxford University Press, Oxford, pp 247–282. chapter Space Time Scale Bridging methods Google Scholar
  81. 81.
    Ladevèze P, Passieux J-C, Néron D (2010) The latin multiscale computational method and the proper generalized decomposition. Comput Methods Appl Mech Eng, 199(21–22):1287–1296 CrossRefGoogle Scholar
  82. 82.
    Ladevèze P, Chamoin L (2011) On the verification of model reduction methods based on the proper generalized decomposition. Comput Methods Appl Mech Eng 200:2032–2047 CrossRefGoogle Scholar
  83. 83.
    Lamari H, Chinesta F, Ammar A, Cueto E (2009) Non-conventional numerical strategies in the advanced simulation of materials and processes. Int J Mod Manuf Technol, 1:49–56 Google Scholar
  84. 84.
    Lamari H, Ammar A, Cartraud P, Legrain G, Jacquemin F, Chinesta F (2010) Routes for efficient computational homogenization of non-linear materials using the proper generalized decomposition. Arch Comput Methods Eng, 17(4):373–391 MathSciNetGoogle Scholar
  85. 85.
    Lamari H, Ammar A, Leygue A, Chinesta F On the solution of the multidimensional Langerõs equation by using the proper generalized decomposition method for modeling phase transitions. Model Simul Mater Sci Eng. Submitted Google Scholar
  86. 86.
    Lemoussu H, Boucard P-A, Ladevèze P (2002) A 3d shock computational strategy for real assembly and shock attenuator. Adv Eng Softw 33(7–10):517–526 CrossRefGoogle Scholar
  87. 87.
    Leonenko GM, Phillips TN (2009) On the solution of the Fokker-Planck equation using a high-order reduced basis approximation. Comput Methods Appl Mech Eng 199(1–4):158–168 MathSciNetCrossRefGoogle Scholar
  88. 88.
    Leygue A, Verron E (2010) A first step towards the use of proper general decomposition method for structural optimization. Arch Comput Methods Eng 17(4):I465–472 MathSciNetGoogle Scholar
  89. 89.
    Le Bris C, Lelièvre T, Maday Y (2009) Results and questions on a nonlinear approximation approach for solving high-dimensional partial differential equations. Constr Approx 30:621–651 MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Mokdad B, Pruliere E, Ammar A, Chinesta F (2007) On the simulation of kinetic theory models of complex fluids using the Fokker-Planck approach. Appl Rheol, 17(2):26494, 1–4 Google Scholar
  91. 91.
    Mokdad B, Ammar A, Normandin M, Chinesta F, Clermont JR (2010) A fully deterministic micro-macro simulation of complex flows involving reversible network fluid models. Math Comput Simul 80:1936–1961 MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    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
  93. 93.
    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
  94. 94.
    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
  95. 95.
    Néron D, Ladevèze P (2010) Proper generalized decomposition for multiscale and multiphysics problems. Arch Comput Methods Eng 17(4):351–372 MathSciNetGoogle Scholar
  96. 96.
    Niroomandi S, Alfaro I, Cueto E, Chinesta F (2008) Real-time deformable models of non-linear tissues by model reduction techniques. Comput Methods Programs Biomed 91:223–231 CrossRefGoogle Scholar
  97. 97.
    Niroomandi S, Alfaro I, Cueto E, Chinesta F (2010) Model order reduction for hyperelastic materials. Int J Numer Methods Eng 81(9):1180–1206 MathSciNetzbMATHGoogle Scholar
  98. 98.
    Niroomandi S, Alfaro I, Cueto E, Chinesta F (2011) Accounting for large deformations in real-time simulations of soft tissues based on reduced order models. Comput Methods Program Biomed. doi: 10.1016/j.cmpb.2010.06.012
  99. 99.
    Niroomandi S, Alfaro I, Gonzalez D, Cueto E, Chinesta F (2011) Real time simulation of surgery by reduced order modeling and X-FEM techniques. Int J Numer Methods Biomed Eng In press Google Scholar
  100. 100.
    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
  101. 101.
    Nouy A (2007) A generalized spectral decomposition technique to solve a class of linear stochastic partial differential equations. Comput Methods Appl Mech Eng 196:4521–4537 MathSciNetzbMATHCrossRefGoogle Scholar
  102. 102.
    Nouy A (2007) Méthode de construction de bases spectrales généralisées pour l’approximation de problèmes stochastiques. Mec Ind 8(3):283–288 Google Scholar
  103. 103.
    Nouy A (2008) Generalized spectral decomposition method for solving stochastic finite element equations: invariant subspace problem and dedicated algorithms. Comput Methods Appl Mech Eng 197:4718–4736 MathSciNetzbMATHCrossRefGoogle Scholar
  104. 104.
    Nouy A, Le Maître O (2009) Generalized spectral decomposition method for stochastic non linear problems. J Comput Phys, 228(1):202–235 MathSciNetzbMATHCrossRefGoogle Scholar
  105. 105.
    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 MathSciNetCrossRefGoogle Scholar
  106. 106.
    Nouy A (2010) Proper generalized decompositions and separated representations for the numerical solution of high dimensional stochastic problems. Arch Comput Methods Eng, 17:403–434 MathSciNetGoogle Scholar
  107. 107.
    Nouy A (2010) A priori model reduction through proper generalized decomposition for solving time-dependent partial differential equations. Comput Methods Appl Mech Eng 199:1603–1626 MathSciNetCrossRefGoogle Scholar
  108. 108.
    Nouy A, Falco A Constrained tensor product approximations based on penalized best approximations. Linear Algebra Appl, oai:hal.archives-ouvertes.fr:hal-00577942
  109. 109.
    Nouy A, Chevreuil M, Safatly E (2011) Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains. Comput Methods Appl Mech Eng. doi: 10.1016/j.cma.2011.07.002
  110. 110.
    Passieux J-C, Ladevèze P, Néron D (2010) A scalable time-space multiscale domain decomposition method: adaptive time scale separation. Comput Mech 46(4):621–633 MathSciNetzbMATHCrossRefGoogle Scholar
  111. 111.
    Pineda M, Chinesta F, Roger J, Riera M, Perez J, Daim F (2010) Simulation of skin effect via separated representations. Int J Comput Math Electr Electron Eng, 29(4):919–929 zbMATHCrossRefGoogle Scholar
  112. 112.
    Pruliere E, Ammar A, El Kissi N, Chinesta F (2009) Recirculating flows involving short fiber suspensions: numerical difficulties and efficient advanced micro-macro solvers. Arch Comput Methods Eng, 16:1–30 MathSciNetzbMATHCrossRefGoogle Scholar
  113. 113.
    Pruliere E, Ferec J, Chinesta F, Ammar A (2010) An efficient reduced simulation of residual stresses in composites forming processes. Int J Mater Form, 3(2):1339–1350 CrossRefGoogle Scholar
  114. 114.
    Pruliere E, Chinesta F, Ammar A (2010) On the deterministic solution of multidimensional parametric models by using the proper generalized decomposition. Math Comput Simul 81:791–810 MathSciNetzbMATHCrossRefGoogle Scholar
  115. 115.
    Ryckelynck D, Hermanns L, Chinesta F, Alarcon E (2005) An efficient a priori model reduction for boundary element models. Eng Anal Bound Elem 29:796–801 zbMATHCrossRefGoogle Scholar
  116. 116.
    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 MathSciNetzbMATHCrossRefGoogle Scholar
  117. 117.
    Schmidt F, Pirc N, Mongeau M, Chinesta F (2011) Efficient mould cooling optimization by using model reduction. Int J Mater Form, 4(1):71–82 CrossRefGoogle Scholar
  118. 118.
    Violeau D, Ladevèze P, Lubineau G (2009) Micromodel-based simulations for laminated composites. Compos Sci Technol, 69(9):1364–1371 CrossRefGoogle Scholar

Copyright information

© CIMNE, Barcelona, Spain 2011

Authors and Affiliations

  • Francisco Chinesta
    • 1
  • Pierre Ladeveze
    • 2
  • Elías Cueto
    • 3
  1. 1.EADS Foundation Chair “Advanced Computational Manufacturing Processes”GEM, UMR CNRS - Centrale Nantes, Institut Universtaire de FranceNantes cedex 3France
  2. 2.EADS Foundation Chair “Advanced Computational Structural Mechanics”LMT-Cachan (ENS Cachan/CNRS/UPMC/PRES UniverSud Paris)Cachan cedexFrance
  3. 3.Aragón Institute of Engineering Research (I3A)Universidad de ZaragozaZaragozaSpain

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