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The reduced model multiscale method (R3M) for the nonlinear homogenization of hyperelastic media at finite strains. (English) Zbl 1163.74048
Summary: This paper presents a new multi-scale method for the homogenization analysis of hyperelastic solids undergoing finite strains. The key contribution is to use an incremental nonlinear homogenization technique in tandem with a model reduction method, in order to alleviate the complexity of multiscale procedures, which usually involve a large number of nonlinear nested problems to be solved. The problem associated with the representative volume element (RVE) is solved via a model reduction method (proper orthogonal decomposition). The reduced basis is obtained through pre-computations on the RVE. The technique, coined as reduced model multiscale method (R3M), allows reducing significantly the computation times, as no large matrix needs to be inverted, and as the convergence of both macro and micro problems is enhanced. Furthermore, the R3M drastically reduces the size of the data base describing the history of the micro problems. In order to validate the technique in the context of porous elastomers at finite strains, a comparison between a full and a reduced multiscale analysis is performed through numerical examples, involving different micro and macro structures, as well as different nonlinear models (Neo-Hookean, Mooney-Rivlin). It is shown that the R3M gives good agreement with the full simulations, at lower computational and data storage requirements.

MSC:
74S05 Finite element methods applied to problems in solid mechanics
74B20 Nonlinear elasticity
74Q05 Homogenization in equilibrium problems of solid mechanics
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[1] M.F.A. Azeez, A.F. Vakakis, Numerical and experimental analysis of the nonlinear dynamics due to impacts of a continuous overhung rotor, in: Proceedings of DETC’97, ASME Design Engineering Technical Conferences, Sacramento, CA, 1997. · Zbl 1342.74002
[2] M.F.A. Azeez, A.F. Vakakis, Proper Orthogonal decomposition (POD) of a class of vibroimpact oscillations, Internal Report, Department of Mechanical and Industrial Engineering, University of Illinois at Urbana Champain, USA, 1998. · Zbl 0910.34053
[3] Chaterjee, A., An introduction to the proper orthogonal decomposition, Curr. sci., 78, 808-817, (2000)
[4] deBotton, G.; Hariton, I.; Socolsky, E.A., Neo-Hookean fibber reinforced composites in finite elasticity, J. mech. phys. solids, 54, 3, 533-559, (2006) · Zbl 1120.74317
[5] Dür, A., On the optimality of the discrete karhunen – loève expansion, SIAM J. contr. optim., 36, 6, 1937-1939, (1998) · Zbl 0914.93074
[6] Dvorak, G.; Bahei-El-Din, Y.A.; Wafa, A.M., The modeling of inelastic composite materials with the transformation field analysis, Modell. simul. mater. sci. eng., 2, 571-586, (1994) · Zbl 0835.73038
[7] Feeny, B.F.; Kappagantu, R., On the physical interpretation of proper orthogonal modes in vibrations, J. sound vibr., 219, 189-192, (1998)
[8] Feyel, F.; Chaboche, J.-L., FE^{2} multiscale approach for modelling the elastoviscoplastic behaviour of long fibber sic/ti composite materials, Comput. methods appl. mech. eng., 183, 309-330, (2000) · Zbl 0993.74062
[9] 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. eng., 192, 3233-3244, (2003) · Zbl 1054.74727
[10] Feng, W.; Christensen, R.M., Nonlinear deformation of elastomeric foams, Int. J. non-linear mech., 17, 335-367, (1982)
[11] Fish, J.; Shek, K.; Pandheeradi, M.; Shepard, M.S., Computational plasticity for composite structures based on mathematical homogenization: theory and practice, Comput. methods appl. mech. eng., 148, 53-73, (1997) · Zbl 0924.73145
[12] Gent, A.N.; Thomas, A.G., The deformation of foamed elastic materials, J. appl. polym. sci., 1, 107-113, (1959)
[13] Golub, G.H.; Van Loan, C.F., Matrix computations, (1983), North Oxford Academic Oxford · Zbl 0559.65011
[14] Ghosh, S.; Lee, K.; Moorthy, S., Two scale analysis of heterogeneous elastic – plastic materials with asymptotic homogenization and Voronoi cell finite element model, Comput. methods appl. mech. eng., 132, 63-116, (1996) · Zbl 0892.73061
[15] Ghosh, S.; Lee, K.; Raghavan, P., A multi-level computational model for multi-scale damage analysis in composite and porous materials, Int. J. solids struct., 38, 2335-2385, (2001) · Zbl 1015.74058
[16] Guedes, J.M.; Kikuchi, N., Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods, Comput. methods appl. mech. eng., 83, 143-198, (1990) · Zbl 0737.73008
[17] Harman, H., Modern factor analysis, (1960), University of Chicago Press · Zbl 0095.13403
[18] Hashin, Z., Large isotropic elastic deformation of composites and porous media, Int. J. solids struct., 21, 711-720, (1985) · Zbl 0575.73051
[19] Hill, R., On constitutive macro-variables for heterogeneous solids at finite stains, Proc. R. soc. London ser. A, 326, 131-147, (1972) · Zbl 0229.73004
[20] Hill, R.; Rice, J.R., Elastic potentials and the structure of inelastic constitutive laws, SIAM J. appl. math., 25, 448-461, (1973) · Zbl 0275.73028
[21] Hotelling, H., Analysis of complex statistical variables in principal components, J. exp. psy., 24, 417, (1953)
[22] Holmes, P.; Lumley, J.L.; Berkooz, G., Turbulence, coherent structures, dynamical systems and symmetry, (1996), Cambridge University Press Cambridge · Zbl 0890.76001
[23] Holzapfel, G.A., Nonlinear solid mechanics, A continuum approach for engineering, (2001), Wiley New York
[24] Humphrey, J.D., Cardiovascular solid mechanics: cells, tissues and organs, (2002), Springer New York
[25] Joyner, M.L., Comparison of two techniques for implementing the proper orthogonal decomposition method in damage detection problems, Math. comput. modelling, 40, 553-571, (2004) · Zbl 1112.65114
[26] Karhunen, K., Zur spektraltheorie stochastischer prozesse, Ann. acad. sci. fennicae, 37, (1946) · Zbl 0063.03144
[27] Kousnetzova, V.; Brekelmans, W.A.M.; Baaijens, F.T.P., An approach to micro – macro modelling of heterogeneous materials, Comput. mech., 27, 37-48, (2001) · Zbl 1005.74018
[28] Kousnetzova, V.G.; Geers, M.G.D.; Brekelmans, W.A.M., Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy, Comput. methods appl. mech. eng., 193, 5525-5550, (2004) · Zbl 1112.74469
[29] Ladevèze, P.; Loiseau, O.; Dureisseix, D., A micro – macro and parallel computational strategy for highly heterogeneous structures, Int. J. numer. methods eng., 52, 121-138, (2001)
[30] Lall, S.; Krysl, P.; Marsden, J.E., Structure-preserving model reduction for mechanical systems, Physica D, 184, 304-318, (2003) · Zbl 1041.70011
[31] Liang, Y.C.; Lee, H.P.; Lim, S.P.; Lin, W.Z.; Lee, K.H., Proper orthogonal decomposition and its applications – part I: theory, J. sound vibr., 252, 3, 527-544, (2002) · Zbl 1237.65040
[32] T. Lieu, M. Lesoinne, Parameters adaptation of reduced order models for three-dimensional flutter analysis, AIAA Papers 2004-0888.
[33] Lieu, T.; Farhat, C.; Lesoinne, M., Reduced-order fluid/structure modelling of a complete aircraft configuration, Comput. methods appl. mech. eng., 195, 41-43, 5730-5742, (2006) · Zbl 1124.76042
[34] Loève, M.M., Probability theory, (1955), Van Nostrand NJ · Zbl 0108.14202
[35] Lopez-Pamies, O.; Castañeda, Ponte, Second-order estimated for the macroscopic response and loss of ellipticity in porous rubbers at large deformations, J. elasticity, 76, 247-287, (2005) · Zbl 1086.74032
[36] E.N. Lorenz, Empirical orthogonal eigenfunctions and statistical weather prediction, Technical Report, MIT, Department of Meteorology, Statistical Forecasting Project, 1956.
[37] Lumley, J.L., The structure of inhomogeneous turbulent flows, (), 166-178
[38] Michel, J.C.; Moulinec, H.; Suquet, P., Effective properties of composite materials with periodic microstructure: a computational approach, Comput. methods appl. mech. eng., 172, 109-143, (1999) · Zbl 0964.74054
[39] Michel, J.C.; Suquet, P., Nonuniform transformation fields analysis, Int. J. solids struct., 40, 6937-6955, (2003) · Zbl 1057.74031
[40] Michel, J.C.; Suquet, P., Computational analysis of nonlinear composite structures using the nonuniform transformation fields analysis, Comput. methods appl. mech. eng., 193, 48-51, 5477-5502, (2004) · Zbl 1112.74471
[41] Miehe, C.; Schröder; Schotte, J., Computational homogenization analysis in finite plasticity. simulation of texture development in polycrystalline materials, Comput. methods appl. mech. eng., 171, 387-418, (1999) · Zbl 0982.74068
[42] Miehe, C.; Koch, A., Computational micro-to-macro transition of discretized microstructures undergoing small strains, Arch. appl. mech., 72, 300-317, (2002) · Zbl 1032.74010
[43] Miehe, C., Computational micro-to-macro transitions for discretized micro-structures of heterogeneous materials at finite strains based on the minimization of averaged incremental energy, Comput. methods appl. mech. eng., 192, 559-591, (2003) · Zbl 1091.74530
[44] Miehe, C.; Schotte, J.; Lambrecht, M., Homogenization of inelastic solid materials at finite strains based on incremental minimization principles. application to the texture analysis of polycrystals, J. mech. phys. solids, 50, 2123-2167, (2002) · Zbl 1151.74403
[45] Müller, S., Homogenization of nonconvex integral functionals and cellular elastic materials, Arch. rational mech. anal., 99, 189-212, (1987) · Zbl 0629.73009
[46] Nemat-Nasser, S.; Hori, M., Micromechanics: overall properties of heterogeneous materials, () · Zbl 0924.73006
[47] A. Newman, P.S. Krishnaprasad, Nonlinear model reduction for RTCVD, in: Proceedings of the 32nd Conference in Information Sciences and Systems, Princeton, NJ, 1998.
[48] Ogden, R.W., On the overall moduli of non-linear elastic composites materials, J. mech. phys. solids, 22, 541-553, (1974) · Zbl 0293.73003
[49] Ogden, R.W., Extremum principles in non-linear elastic and their application to composites - I. theory, Int. J. solids struct., 14, 265-282, (1978) · Zbl 0384.73022
[50] Pearson, K., On lines and planes of closest fit to systems of points in space, Philos. mag., 6, 559, (1901) · JFM 32.0246.07
[51] Ponte Castañeda, P., The overall constitutive behavior of nonlinearly elastic composites, Proc. R. soc. London ser. A, 422, 147-171, (1989) · Zbl 0673.73005
[52] Ponte Castañeda, P.; Suquet, P., Nonlinear composites, Adv. appl. mech., 34, 171-302, (1998) · Zbl 0889.73049
[53] Ponte Castañeda, P., Second-order homogenization estimates for nonlinear composites incorporating field fluctuations - I. theory, J. mech. phys. solids, 50, 737-757, (2002) · Zbl 1116.74412
[54] Schmidt, E., Zur theorie der linearen und nichtlinearen integralgleichungen. I teil: etwicklung willkürlicher funktion nach systemen vorgeschriebener, Math. ann., 63, 433-476, (1907) · JFM 38.0377.02
[55] R. Schmidt, M. Glauser, Improvements in low dimensional tools for flow-structure interaction problems: using global POD, AIAA Papers 2004-0889.
[56] Sheng, N.; Boyce, M.C.; Parks, D.M.; Rutledge, G.C.; Abes, J.I.; Cohen, R.E., Multiscale micromechanical modelling of polymer/Clay nanocomposites and their effective Clay particle, Polymer, 45, 487-506, (2004)
[57] Smit, R.J.M.; Brekelmans; Meijer, H.E.H., Prediction of the mechnanical behaviour of non-linear heterogeneous systems by multi-level finite element modeling, Comput. methods appl. mech. eng., 155, 181-192, (1998) · Zbl 0967.74069
[58] Suquet, P.M., Local and global aspects in the mathematical theory of plasticity, (), 279-310
[59] Suquet, P.M., Elements of homogenization for inleastic solid mechanics, (), 193-278
[60] J. Tatyor, Dynamics of large scale structures in turbulent shear layers, Ph.D. Thesis, Clarkson University, 2001.
[61] Terada, K.; Kikuchi, N., Nonlinear homogenization method for practical applications, (), 279-310
[62] Terada, K.; Kikuchi, A class of general algorithms for multi-scale analysis of heterogeneous media, Comput. methods appl. mech. eng., 190, 5427-5464, (2001) · Zbl 1001.74095
[63] Willis, J.R., Variational and related methods for the overall properties of composites, (), 1-78 · Zbl 0476.73053
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