The numerical modelling of the behaviour of materials at the microstructural scale has been greatly developed over the last two decades. Unfortunately, conventional resolution methods cannot simulate polycrystalline aggregates beyond tens of loading cycles, and they do not remain quantitative due to the plasticity behaviour. This work presents the development of a numerical solver for the resolution of the Finite Element modelling of polycrystalline aggregates subjected to cyclic mechanical loading. The method is based on two concepts. The first one consists in maintaining a constant stiffness matrix. The second uses a time/space model reduction method. In order to analyse the applicability and the performance of the use of a space–time separated representation, the simulations are carried out on a three-dimensional polycrystalline aggregate under cyclic loading. Different numbers of elements per grain and two time increments per cycle are investigated. The results show a significant CPU time saving while maintaining good precision. Moreover, increasing the number of elements and the number of time increments per cycle, the model reduction method is faster than the standard solver.
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Mohamed Aziz Nasri 1; Camille Robert 1; Amine Ammar 1; Saber El Arem 1; Franck Morel 1
@article{CRMECA_2018__346_2_132_0,
author = {Mohamed Aziz Nasri and Camille Robert and Amine Ammar and Saber El Arem and Franck Morel},
title = {Proper {Generalized} {Decomposition} {(PGD)} for the numerical simulation of polycrystalline aggregates under cyclic loading},
journal = {Comptes Rendus. M\'ecanique},
pages = {132--151},
publisher = {Elsevier},
volume = {346},
number = {2},
year = {2018},
doi = {10.1016/j.crme.2017.11.009},
language = {en},
}
TY - JOUR AU - Mohamed Aziz Nasri AU - Camille Robert AU - Amine Ammar AU - Saber El Arem AU - Franck Morel TI - Proper Generalized Decomposition (PGD) for the numerical simulation of polycrystalline aggregates under cyclic loading JO - Comptes Rendus. Mécanique PY - 2018 SP - 132 EP - 151 VL - 346 IS - 2 PB - Elsevier DO - 10.1016/j.crme.2017.11.009 LA - en ID - CRMECA_2018__346_2_132_0 ER -
%0 Journal Article %A Mohamed Aziz Nasri %A Camille Robert %A Amine Ammar %A Saber El Arem %A Franck Morel %T Proper Generalized Decomposition (PGD) for the numerical simulation of polycrystalline aggregates under cyclic loading %J Comptes Rendus. Mécanique %D 2018 %P 132-151 %V 346 %N 2 %I Elsevier %R 10.1016/j.crme.2017.11.009 %G en %F CRMECA_2018__346_2_132_0
Mohamed Aziz Nasri; Camille Robert; Amine Ammar; Saber El Arem; Franck Morel. Proper Generalized Decomposition (PGD) for the numerical simulation of polycrystalline aggregates under cyclic loading. Comptes Rendus. Mécanique, Volume 346 (2018) no. 2, pp. 132-151. doi : 10.1016/j.crme.2017.11.009. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.11.009/
[1] Influence d'accidents géométriques et du mode de chargement sur le comportement en fatigue à grand nombre de cycles d'un acier inoxydable austénitique 316L, École nationale supérieure des arts et métiers, Paris, 2014 (PhD thesis)
[2] Non-linear Finite Element Analysis of Solids and Structures, vol. 1, John Wiley and Sons Ltd., 2000
[3] (1965), pp. 507-510
[4] The Finite Element Method in Structural and Continuum Mechanics, McGraw-Hill, New York, 1967
[5] A numerical method for computing the overall response of nonlinear composites with complex microstructure, Comput. Methods Appl. Mech. Eng., Volume 157 (1998), pp. 69-94
[6] Experimental and numerical study of the evolution of stored and dissipated energies in a medium carbon steel under cyclic loading, Mech. Mater., Volume 60 (2013), pp. 93-106
[7] N-site modelling of a 3d viscoplastic polycrystal using fast Fourier transform, Acta Mater., Volume 49 (2001), pp. 2723-2737
[8] A comparison between different numerical methods for the modelling of polycrystalline materials with an elastic–viscoplastic behaviour, Comput. Mater. Sci., Volume 103 (2015), pp. 134-144
[9] Micro-mechanical modelling of high cycle fatigue behaviour of metals under multiaxial loads, Mech. Mater., Volume 55 (2012), pp. 112-129
[10] Sur une famille d'algorithmes en mécanique des structures, C. R. Acad. Sci. Paris, Ser. II, Volume 300 (1985), pp. 41-44
[11] La méthode à grand incrément pour l'analyse de structures à comportement non linéaire décrit par variables internes, C. R. Acad. Sci. Paris, Ser. II, Volume 309 (1989) no. 11, pp. 1095-1099
[12] Modélisation des phénomènes d'endommagement dus aux contacts à faible amplitude de débatement, École nationale des ponts et chaussées, Paris, 1997 (PhD thesis)
[13] Détermination de la réponse asymptotique d'une structure anélastique soumise à un chargement thermomécanique cyclique, École polytechnique, 2003 (PhD thesis)
[14] Determination of the asymptotic response of a structure under cyclic thermomechanical loading, C. R. Mecanique, Volume 330 (2002), pp. 703-708
[15] An efficient cycles jump technique for viscoplastic structures calculations involving large number of cycles, Barcelona, Spain (1989), pp. 591-602
[16] Modèles à grand nombre de variables internes et méthodes numériques associées, École nationale supérieure des mines de Paris, 1993 (PhD thesis)
[17] A New Approach in Inelastic Analysis of Structures, CADLM, 1990
[18] Simplified dynamical analysis of inelastic structures, Nucl. Eng. Des., Volume 92 (1986), pp. 89-103
[19] Theory of stress concentration for shear-strained prismatic bodies with arbitrary non-linear stress–strain law, J. Appl. Mech., Volume 28 (1961), pp. 544-551
[20] Reduced order modelling method via proper orthogonal decomposition (POD) for flow around an oscillating cylinder, J. Fluids Struct., Volume 26 (2010) no. 2, pp. 292-311
[21] Réduction de dynamique par Décomposition Orthogonale aux Valeurs Propres (POD), Optimisation et Contrôle des Ecoulements et des Transferts, École de printemps OCET, 2006
[22] Matrix Computations, Johns Hopkins University Press, Baltimore, MD, USA, 1990
[23] POD and CVT-based reduced-order modelling of Navier–Stokes flows, Comput. Methods Appl. Mech. Eng., Volume 196 (2006) no. 1–3, pp. 337-355
[24] A priori hyperreduction method: an adaptive approach, Comput. Methods Appl. Mech. Eng., Volume 202 (2005), pp. 346-366
[25] On the reduction of kinetic theory models related to finitely extensible dumbbells, J. Non-Newton. Fluid Mech., Volume 134 (2006), pp. 136-147
[26] A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modelling of complex fluids, J. Non-Newton. Fluid Mech., Volume 139 (2006), pp. 153-176
[27] An error estimator for separated representations of highly multidimensional models, Comput. Methods Appl. Mech. Eng., Volume 199 (2010), pp. 1872-1880
[28] Proper general decomposition (PGD) for the resolution of Navier—Stokes equations, J. Comput. Phys., Volume 230 (2011), pp. 1387-1407
[29] Simplified methods for the steady state inelastic analysis of cyclically loaded structures, Comput. Methods Appl. Mech. Eng., Volume 223–224 (2012), pp. 186-198
[30] The proper generalized decomposition: a powerful tool for model reduction, Int. J. Mater. Form., Volume 3 (2010), pp. 89-102
[31] Proper generalized decomposition of multiscale models, Int. J. Numer. Methods Eng., Volume 89 (2010), pp. 1114-1132
[32] Nonlinear Computational Structural Mechanics—New Approaches and Non-incremental Methods of Calculation, Springer Verlag, 1999
[33] On a multiscale computational strategy with time and space homogenization for structural mechanics, Comput. Methods Appl. Mech. Eng., Volume 192 (2003), pp. 3061-3087
[34] Multiscale computational strategy with time and space homogenization: a radial-type approximation technique for solving micro problems, Int. J. Multiscale Comput. Eng. (2004), p. 170
[35] A model reduction technique based on the PGD for elastic-viscoplastic computational analysis, Comput. Mech., Volume 51 (2013), pp. 83-92
[36] European journal of mechanics a/solids, Comput. Mech., Volume 60 (2016), pp. 227-237
[37] 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., Volume 144 (2007), pp. 98-121
[38] 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., Volume 6 (2008) no. 3, pp. 191-213
[39] On the use of proper generalized decompositions for solving the multidimensional chemical master equation, Eur. J. Comput. Mech., Volume 19 (2010) no. 1, pp. 53-64
[40] Recent advances and new challenges in the use of the proper generalized decomposition for solving multidimensional models, Arch. Comput. Methods Eng., Volume 17 (2010), pp. 327-350
[41] On the deterministic solution of multidimensional parametric models using the proper generalized decomposition, Math. Comput. Simul., Volume 81 (2010), pp. 791-810
[42] Non incremental strategies based on separated representations: applications in computational rheology, Commun. Math. Sci., Volume 8 (2010) no. 3, pp. 671-695
[43] Hyper-reduced predictions for lifetime assessment of elasto-plastic structures, Mecanica, Volume 51 (2016), pp. 309-317
[44] Model reduction in elastoplasticity: proper orthogonal decomposition combined with adaptive sub-structuring, Comput. Mech., Volume 54 (2014), pp. 677-687
[45] The proper generalized decomposition as a space–time integrator for elasto-plastic problems, C. R. Mecanique, Volume 344 (2016), pp. 759-768
[46] The Finite Element Method, vols. I–II, Butterworth-Heinemann, 2000
[47] A brief survey of convergence results for quasi-Newton methods, SIAM-AMS Proceedings, vol. 9, 1976, pp. 185-199
[48] Finite Element Procedures, Prentice-Hall, 1996
[49] The solution of nonlinear finite element equations, Int. J. Numer. Methods Eng., Volume 14 (1979), pp. 1613-1626
[50] Some practical procedures for the solution of nonlinear finite element equations, Comput. Methods Appl. Mech. Eng., Volume 22 (1980), pp. 59-85
[51] Computational Inelasticity, Springer Verlag, 1997
[52] Consistent tangent operators for rate-independent plasticity, Comput. Methods Appl. Mech. Eng., Volume 48 (1985), pp. 101-118
[53] Nonlinear Finite Element Methods, Springer, 2008
[54] A direct method to predict cyclic steady states of elastoplastic structures (D. Weichert; G. Maier, eds.), Inelastic Analysis of Structures Under Variable Loads, Kluwer Academic Publishers, 2000, pp. 213-232
[55] A simplified method to predict the steady cyclic stress state of creeping structures, J. Appl. Mech., Volume 69 (2002), pp. 149-153
[56] Separated representation of incremental elastoplastic simulations, Key Eng. Mater., Volume 651–653 (2015), pp. 1285-1293
[57] Single crystal modelling for structural calculations. Part 2: finite element implementation, J. Eng. Mater. Technol., Volume 113 (1991), pp. 171-182
[58] Micromechanical study of the loading path effect in high cycle fatigue, Int. J. Fatigue, Volume 59 (2014), pp. 64-75
[59] Single crystal modelling for structural calculations. Part 2: finite element implementation, Int. J. Plast., Volume 17 (2001), pp. 537-563
[60] Etude théorique et expérimentale du comportement élastoplastique des monocristaux métalliques se déformant par glissement: modélisation pour un chargement complexe quasi statique, Université Paris-Nord (Paris-13), 1984 (PhD thesis)
[61] Numerical investigations of the free surface effect in three-dimensional polycrystalline aggregates, Comput. Mater. Sci., Volume 70 (2013), pp. 150-162
[62] Recent advances in sparse direct solvers, San Francisco, CA, USA (2013)
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