Cheng, Gengdong; Li, Xikui; Nie, Yinghao; Li, Hengyang FEM-cluster based reduction method for efficient numerical prediction of effective properties of heterogeneous material in nonlinear range. (English) Zbl 1440.74383 Comput. Methods Appl. Mech. Eng. 348, 157-184 (2019). Summary: A novel FEM-Cluster based reduced order method or FEM-Cluster based Analysis method (FCA) which enables efficient numerical prediction of effective properties of heterogeneous material in nonlinear range is proposed. The cluster concept initially presented in the work by WK Liu et al. is introduced and extended to derive a full FEM multi-scale formulation of the Representative Unit Cell (RUC) to circumvent the heavy burden due to huge computational efforts required for a direct numerical simulation (DNS) of the high-fidelity RUC. The proposed FEM-Cluster based reduced order method is formulated in a consistent framework of finite element method. The offline clustering process with construction of the cluster-interaction matrix derived under the assumption of the linear elasticity is carried out by the devised FE procedure of RUC. The online elasto-plastic process is performed by the incremental non-linear FE analysis using the constant cluster-interaction matrix, which plays a role in the present work conceptually similar to the initial elastic modular matrix used in the “initial stiffness method” for the traditional incremental elasto-plastic analysis. Accurate and efficient numerical prediction of effective properties of heterogeneous material in nonlinear range are developed in a consistent way. The performances of the proposed reduced order model and its numerical implementation are studied and demonstrated. Several numerical examples show its efficiency and applicability. Cited in 2 ReviewsCited in 18 Documents MSC: 74S05 Finite element methods applied to problems in solid mechanics 74Q20 Bounds on effective properties in solid mechanics Keywords:nonlinear effective properties’ prediction; FEM-cluster based reduced order method; cluster-interaction matrix; elasto-plastic analysis PDFBibTeX XMLCite \textit{G. Cheng} et al., Comput. Methods Appl. Mech. Eng. 348, 157--184 (2019; Zbl 1440.74383) Full Text: DOI References: [1] Zohdi, T. I.; Wriggers, P., An introduction to computational micromechanics, Lect. Notes Appl. Comput. Mech., 20, 2176-2185 (2008) [2] Heimbs, S.; Mehrens, T.; Middendorf, P.; Maier, M.; Schumacher, A., Numerical determination of the nonlinear effective mechanical properties of folded core structures for aircraft sandwich panels, (European Ls-Dyna Users Conference (2007)) [3] Eshelby, J. D., The determination of the elastic field of an ellipsoidal inclusion, and related problems, Proc. Roy. Soc. Lond., 241, 376-396 (1957) · Zbl 0079.39606 [4] Hill, R., The elastic behaviour of a crystalline aggregate, Proc. Phys. Soc., 65, 349-354 (1952) [5] Kanit, T.; Forest, S.; Galliet, I.; Mounoury, V.; Jeulin, D., Determination of the size of the representative volume element for random composites: statistical and numerical approach, Int. J. Solids Struct., 40, 3647-3679 (2003) · Zbl 1038.74605 [6] Hill, R., A self-consistent mechanics of composite materials, J. Mech. Phys. Solids, 13, 213-222 (1965) [7] Christensen, R. M.; Lo, K. H., Solutions for effective shear properties in three phase sphere and cylinder models, J. Mech. Phys. Solids, 27, 315-330 (1979) · Zbl 0419.73007 [8] Benveniste, Y., A new approach to the application of Mori-Tanaka’s theory in composite materials, Mech. Mater., 6, 147-157 (1987) [9] Mori, T.; Tanaka, K., Average stress in matrix and average elastic energy of materials with misfitting inclusions, Acta Metall., 21, 571-574 (1973) [10] Hashin, Z.; Shtrikman, S., A variational approach to the theory of the elastic behaviour of polycrystals, J. Mech. Phys. Solids, 10, 343-352 (1962) · Zbl 0119.40105 [11] Bakhvalov, N.; Panasenko, G., Homogenisation: Averaging Processes in Periodic Media (1989), Springer Netherlands · Zbl 0692.73012 [12] Bensoussan, A.; Lions, J. L.; Papanicolaou, G., Asymptotic analysis for periodic structures, Encyclopedia Math. Appl., 20, 307-309 (1991) [13] Oleinik, O. A.; Shamaev, A. S.; Yosifian, G. A., Mathematical Problems in Elasticity and Homogenization (1992), North-Holland: North-Holland Amsterdam · Zbl 0768.73003 [14] Feyel, F., A multilevel finite element method \((FE^2\) ) to describe the response of highly non-linear structures using generalized continua, Comput. Methods Appl. Mech. Engrg., 192, 3233-3244 (2003) · Zbl 1054.74727 [15] Feyel, F.; Chaboche, J. L., \(FE^2\) multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials, Comput. Methods Appl. Mech. Engrg., 183, 309-330 (2000) · Zbl 0993.74062 [16] Terada, K.; Hirayama, N.; Yamamoto, K.; Kato, J.; Kyoya, T.; Matsubara, S.; Arakawa, Y.; Ueno, Y.; Miyanaga, N., Applicability of micro-macro decoupling scheme to two-scale analysis of fiber-reinforced plastics, Adv. Compos. Mater., 23, 421-450 (2014) [17] Terada, K.; Kato, J.; Hirayama, N.; Inugai, T.; Yamamoto, K., A method of two-scale analysis with micro-macro decoupling scheme: application to hyperelastic composite materials, Comput. Mech., 52, 1199-1219 (2013) · Zbl 1388.74026 [18] Cai, Y. W.; Xu, L.; Cheng, G. D., Novel numerical implementation of asymptotic homogenization method for periodic plate structures, Int. J. Solids Struct., 51, 284-292 (2014) [19] Cheng, G. D.; Cai, Y. W.; Xu, L., Novel implementation of homogenization method to predict effective properties of periodic materials, Acta Mech. Sin., 29, 550-556 (2013) · Zbl 1346.74158 [20] Yi, S. N.; Xu, L.; Cheng, G. D.; Cai, Y. W., FEM formulation of homogenization method for effective properties of periodic heterogeneous beam and size effect of basic cell in thickness direction, Comput. Struct., 156, 1-11 (2015) [21] Zhao, J.; Li, H. Y.; Cheng, G. D.; Cai, Y. W., On predicting the effective elastic properties of polymer nanocomposites by novel numerical implementation of asymptotic homogenization method, Comput. Struct., 135, 297-305 (2016) [22] Li, Y.; Abbès, F.; Hoang, M. P.; Abbès, B.; Guo, Y., Analytical homogenization for in-plane shear, torsion and transverse shear of honeycomb core with skin and thickness effects, Comput. Struct., 140, 453-462 (2016) [23] Cheng, G. D.; Xu, L., Two-scale topology design optimization of stiffened or porous plate subject to out-of-plane buckling constraint, Struct. Multidiscip. Optim., 54, 1-14 (2016) [24] Yi, S. N.; Cheng, G. D.; Xu, L., Stiffness design of heterogeneous periodic beam by topology optimization with integration of commercial software, Comput. Struct., 172, 71-80 (2016) [25] Trovalusci, P.; Bellis, M. L.D.; Ostoja-Starzewski, M., A Statistically-Based Homogenization Approach for Particle Random Composites as Micropolar Continua (2016), Springer International Publishing [26] Reccia, E.; Bellis, M. L.D.; Trovalusci, P.; Masiani, R., Sensitivity to material contrast in homogenization of random particle composites as micropolar continua, Composites B, 136, 39-45 (2018) [27] Kubair, D. V.; Ghosh, S., Statistics informed boundary conditions for statistically equivalent representative volume elements of clustered composite microstructures, Mech. Adv. Mater. Struct., 1-9 (2017) [28] MacKay, C., D. J. Information Theory, Inference, and Learning Algorithms (2003), Cambridge University Press · Zbl 1055.94001 [29] Noor, A. K.; Kamel, H. A.; Fulton, R. E., Substructuring techniques—status and projections, Comput. Struct., 8, 621-632 (1978) · Zbl 0377.73034 [30] Guyan, R. J., Reduction of stiffness and mass matrices, AIAA J., 3 (1965), 380-380 [31] Warburton, G. B., The Dynamical Behaviour of Structures, 643 (1964) [32] Leung, Y. T., An accurate method of dynamic substructuring with simplified computation, Int. J. Numer. Methods Eng., 14, 1241-1256 (1979) · Zbl 0423.73051 [33] Kuran, B.; Özgüven, H. N., A modal superposition method for non-linear structures, J. Sound Vib., 189, 315-339 (1996) [34] Su, T. J.; Craig Jr, R. R., Model reduction and control of flexible structures using krylov vectors, J. Guid. Control Dyn., 14, 260-267 (1991) [35] Berkooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, 539-575 (1993) [36] Yvonnet, J.; He, Q. C., The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains, J. Comput. Phys., 223, 341-368 (2007) · Zbl 1163.74048 [37] Dvorak, G. J., Transformation field analysis of inelastic composite materials, Proc. Roy. Soc. A Math. Phys. Eng. Sci., 437, 311-327 (1992) · Zbl 0748.73007 [38] Michel, J. C.; Suquet, P., Nonuniform transformation field analysis, Int. J. Solids Struct., 40, 6937-6955 (2003) · Zbl 1057.74031 [39] Roussette, S.; Michel, J. C.; Suquet, P., Nonuniform transformation field analysis of elastic – viscoplastic composites, Compos. Sci. Technol., 69, 22-27 (2009) [40] Karhunen, K., Zur spektraltheorie stochastischer prozesse, Ann. Acad. Sci. Fenn. (1946) · Zbl 0063.03144 [41] Loève, M., Probability Theory. Foundations. Random Sequences (1955), D. Van Nostrand Company: D. Van Nostrand Company New York · Zbl 0066.10903 [42] Jolliffe, I.; Jolliffe, N. A., (Principal Component Analysis. Principal Component Analysis, Series: Springer Series in Statistics (2002)) · Zbl 1011.62064 [43] Goury, O.; Amsallem, D.; Bordas, S. P.A.; Liu, W. K.; Kerfriden, P., Automatised selection of load paths to construct reduced-order models in computational damage micromechanics: from dissipation-driven random selection to Bayesian optimization, Comput. Mech., 58, 213-234 (2016) · Zbl 1398.74053 [44] Néron, D.; Boucard, P. A.; Relun, N., Time-space PGD for the rapid solution of 3D nonlinear parametrized problems in the many-query context, Int. J. Numer. Methods Eng., 103, 275-292 (2015) · Zbl 1352.74075 [45] Ladevèze, P., Nonlinear Computational Structural Mechanics: New Approaches and non-Incremental Methods of Calculation (1999), Springer · Zbl 0912.73003 [46] Ammar, A.; Mokdad, B.; Chinesta, F.; Keunings, R., 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., 144, 98-121 (2007) · Zbl 1196.76047 [47] Ladevèze, P.; Passieux, J.-C.; Néron, D., The LATIN multiscale computational method and the proper generalized decomposition, Comput. Methods Appl. Mech. Engrg., 199, 1287-1296 (2010) · Zbl 1227.74111 [48] Chinesta, F.; Ammar, A.; Leygue, A.; Keunings, R., An overview of the proper generalized decomposition with applications in computational rheology, J. Non-Newton. Fluid Mech., 166, 578-592 (2011) · Zbl 1359.76219 [49] Ghnatios, C.; Masson, F.; Huerta, A.; Leygue, A.; Cueto, E.; Chinesta, F., Proper generalized decomposition based dynamic data-driven control of thermal processes, Comput. Methods Appl. Mech. Engrg., 213-216, 29-41 (2012) [50] Liu, Z.; Bessa, M. A.; Liu, W. K., Self-consistent clustering analysis: An efficient multi-scale scheme for inelastic heterogeneous materials, Comput. Methods Appl. Mech. Engrg., 306, 319-341 (2016) · Zbl 1436.74070 [51] Tang, S. Q.; Zhang, L.; Liu, W. K., From virtual clustering analysis to self-consistent clustering analysis: a mathematical study, Comput. Mech., 1-18 (2018) [52] Yan, J.; Cheng, G. D.; Liu, S.; Liu, L., Comparison of prediction on effective elastic property and shape optimization of truss material with periodic microstructure, Int. J. Mech. Sci., 48, 400-413 (2006) · Zbl 1192.74312 [53] Macqueen, J., Some methods for classification and analysis of multivariate observations, (Proc. of Berkeley Symposium on Mathematical Statistics and Probability (1965)), 281-297 [54] Lloyd, S., Least squares quantization in PCM, IEEE Trans. Inform. Theory, 28, 129-137 (1982) · Zbl 0504.94015 [55] Hill, R., Elastic properties of reinforced solids: Some theoretical principles, J. Mech. Phys. Solids, 11, 357-372 (1963) · Zbl 0114.15804 [56] Hill, R., The essential structure of constitutive laws for metal composites and polycrystals, J. Mech. Phys. Solids, 15, 79-95 (1967) [57] Mandel, J., Plasticité Classique et Viscoplasticité (1972), Springer-Verlag: Springer-Verlag Vienna-New York · Zbl 0285.73018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.