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.
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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
Allix O, Vidal P (2002) A new multi-solution approach suitable for structural identification problems. Comput Methods Appl Mech Eng 191:2727–2758
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
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
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
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
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
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
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
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
Ammar A, Normandin M, Chinesta F (2010) Solving parametric complex fluids models in rheometric flows. J Non-Newton Fluid Mech 165:1588–1601
Ammar A, Chinesta F, Cueto E (2011) Coupling finite elements and proper generalized decompositions. Int J Multiscale Comput Eng 9(1):17–33
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
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
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
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
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
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
Boisse P, Ladevèze P, Rougée P (1989) A large time increment method for elastoplastic problems. Eur J Mech A, Solids 8(4):257–275
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
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
Boucard PA, Ladevèze P, Poss M, Rougée P (1997) A non-incremental approach for large displacement problems. Comput Struct 64:499–508
Boucard PA, Ladevèze P (1999) A multiple solution method for non-linear structural mechanics. Mech Eng 50(5):317–328
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
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
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.
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
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
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
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
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
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
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
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
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
Chinesta F, Ammar A, Cueto E (2010) Proper generalized decomposition of multiscale models. Int J Numer Methods Eng 83(8–9):1114–1132
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
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
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
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
Cognard J-Y, Ladevèze P (1993) A large time increment approach for cyclic viscoplasticity. Int J Plast 9:141–157
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
Gonzalez D, Cueto E, Chinesta F, Debeugny L, Diez P, Huerta A (2010) Int J Mater Form 3(1):883–886
DeVore RA, Temlyakov VN (1996) Some remarks on greedy algorithms. Adv Comput Math 5:173–187
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
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
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
Falco A (2010) Algorithms and numerical methods for high dimensional financial market models. Rev Econ Financ, 20:51–68
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
Falco A, Nouy A Proper generalized decomposition for nonlinear convex problems in tensor Banach spaces. arXiv:1106.4424v1
Figueroa L, Süli E (2011) Greedy approximation of high-dimensional Ornstein-Uhlenbeck operators with unbounded drift. arXiv:1103.0726
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
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
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
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
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
Ladevèze P (1985) New algorithms: mechanical framework and development (in french). Technical Report 57, LMT-Cachan
Ladevèze P (1985) On a family of algorithms for structural mechanics. CR Acad Sci Paris 300(2):41–44 (in french)
Ladevèze P, Rougée P (1985) Viscoplasticity under cyclic loadings: properties of the homogenized cycle. CR Acad Sci 301:891–894
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
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
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
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
Ladevèze P (1996) Mécanique non linéaire des structures. Hermès, Paris
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
Ladevèze P, Dureisseix D (1998) A 2-level and mixed domain decomposition approach for structural analysis. Contemp Math 218:246–253
Ladevèze P (1999) Nonlinear computationnal structural mechanics—new approaches and non-incremental methods of calculation. Springer, Berlin
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
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
Ladevèze P, Guitard L, Champaney L, Aubard X (2000) Debond modeling for multidirectional composites. Comput Methods Appl Mech Eng 185(2–4):109–122
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
Ladevèze P, Perego U (2000) Duality preserving discretization of the large time increment methods. Comput Methods Appl Mech Eng 189(1):205–232
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
Ladevèze P, Nouy A (2002) A multiscale computational method with time and space homogenization. CR Mec, 330(10):683–689
Ladevèze P, Nouy A (2002) Une stratégie de calcul multiéchelle avec homogénéisation en espace et en temps. CR Mec, 330:683–689
Ladevèze P, Nouy A, Loiseau O (2002) A multiscale computational approach for contact problems. Comput Methods Appl Mech Eng, 191(43):4869–4891
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
Ladevèze P (2004) Multiscale modeling and computational strategies for composites. Int J Numer Methods Eng, 60(1):233–253
Ladevèze P, Néron D, Gosselet P (2007) On a mixed and multiscale domain decomposition method. Comput Methods Appl Mech Eng 96:1526–1540
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Néron D, Ladevèze P (2010) Proper generalized decomposition for multiscale and multiphysics problems. Arch Comput Methods Eng 17(4):351–372
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
Niroomandi S, Alfaro I, Cueto E, Chinesta F (2010) Model order reduction for hyperelastic materials. Int J Numer Methods Eng 81(9):1180–1206
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
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
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
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
Nouy A (2007) Méthode de construction de bases spectrales généralisées pour l’approximation de problèmes stochastiques. Mec Ind 8(3):283–288
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
Nouy A, Le Maître O (2009) Generalized spectral decomposition method for stochastic non linear problems. J Comput Phys, 228(1):202–235
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
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
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
Nouy A, Falco A Constrained tensor product approximations based on penalized best approximations. Linear Algebra Appl, oai:hal.archives-ouvertes.fr:hal-00577942
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
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
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
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
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
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
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
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
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
Violeau D, Ladevèze P, Lubineau G (2009) Micromodel-based simulations for laminated composites. Compos Sci Technol, 69(9):1364–1371
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This work has been partially supported by the IUF—Institut Universitaire de France—, the french ANR COSINUS SIMDREAM project and the Spanish Ministry of Science and Innovation, through grants number CICYT-DPI2008-918 and DPI2011-27778-C02-01.
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Chinesta, F., Ladeveze, P. & Cueto, E. A Short Review on Model Order Reduction Based on Proper Generalized Decomposition. Arch Computat Methods Eng 18, 395–404 (2011). https://doi.org/10.1007/s11831-011-9064-7
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DOI: https://doi.org/10.1007/s11831-011-9064-7