×

zbMATH — the first resource for mathematics

Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials. (English) Zbl 1297.74020
Summary: This paper is devoted to a computational stochastic multiscale analysis of nonlinear structures made up of heterogeneous hyperelastic materials. At the microscale level, the nonlinear constitutive equation of the material is characterized by a stochastic potential for which a polynomial chaos representation is used. The geometry of the microstructure is random and characterized by a high number of random parameters. The method is based on a deterministic non-concurrent multiscale approach devoted to micro-macro nonlinear mechanics which leads us to characterize the nonlinear constitutive equation with an explicit continuous form of the strain energy density function with respect to the large scale Cauchy Green strain states. To overcome the curse of dimensionality, due to the high number of involved random variables, the problem is transformed into another one consisting in identifying the potential on a polynomial chaos expansion. Several strategies, based on novel algorithms dedicated to high stochastic dimension, are used and adapted for the class of multi-modal random variables which may characterize the potential. Numerical examples, at both small and large scales, allow analyzing the efficiency of the approach through comparisons with classical methods.

MSC:
74B20 Nonlinear elasticity
74A40 Random materials and composite materials
74Q99 Homogenization, determination of effective properties in solid mechanics
PDF BibTeX Cite
Full Text: DOI
References:
[1] Aboudi, J., Finite strain micromechanical modeling of multiphase composites, Int. J. Multiscale Comput. Engrg., 6, 5, 411-434, (2008)
[2] Arnst, M.; Ghanem, R.; Soize, C., Identification of Bayesian posteriors for coefficients of chaos expansion, J. Comput. Phys., 229, 9, 3134-3154, (2010) · Zbl 1184.62034
[3] Babuska, I.; Tempone, R.; Zouraris, G. E., Solving elliptic boundary value problems with uncertain coefficients by the finite element method: the stochastic formulation, Comput. Methods Appl. Mech. Engrg., 194, 12-16, 1251-1294, (2005) · Zbl 1087.65004
[4] Botev, Z. I.; Grotowski, J. F.; Kroese, D. P., Kernel density estimation via diffusion, Ann. Stat., 38, 5, 2916-2957, (2010) · Zbl 1200.62029
[5] Bowman, A. W.; Azzalini, A., Applied smoothing techniques for data analysis, (1997), Oxford University Press · Zbl 0889.62027
[6] Clément, A.; Soize, C.; Yvonnet, J., Computational nonlinear stochastic homogenization using a non-concurrent multiscale approach for hyperelastic heterogeneous microstructures analysis, Int. J. Numer. Methods Engrg., 91, 8, 799-824, (2012)
[7] deBotton, G.; Shmuel, G., A new variational estimate for the effective response of hyperelastic composites, J. Mech. Phys. Solids, 58, 4, 466-483, (2010) · Zbl 1244.74112
[8] Desceliers, C.; Ghanem, R.; Soize, C., Maximum likelihood estimation of stochastic chaos representations from experimental data, Int. J. Numer. Methods Engrg., 66, 6, 978-1001, (2006) · Zbl 1110.74826
[9] Desceliers, C.; Soize, C.; Ghanem, R., Identification of chaos representations of elastic properties of random media using experimental vibration tests, Comput. Mech., 39, 6, 831-838, (2007) · Zbl 1161.74016
[10] Doostan, A.; Owhadi, H., A non-adapted sparse approximation of PDEs with stochastic inputs, J. Comput. Phys., 230, 3015-3034, (2011) · Zbl 1218.65008
[11] Feyel, F.; Chaboche, J. L., FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fiber sic/ti composite materials, Comput. Methods Appl. Mech. Engrg., 183, 309-330, (2000) · Zbl 0993.74062
[12] Ganapathysubramanian, B.; Zabaras, N., Sparse grid collocation schemes for stochastic natural convection problems, J. Comput. Phys., 225, 652-685, (2007) · Zbl 1343.76059
[13] Ghanem, R.; Spanos, P. D., Stochastic finite elements: A spectral approach, (1991), Springer-Verlag New York · Zbl 0722.73080
[14] Ghanem, R.; Kruger, R. M., Numerical solution of spectral stochastic finite element system, Comput. Methods Appl. Mech. Engrg., 129, 289-303, (1996) · Zbl 0861.73071
[15] Ghanem, R., Ingredients for a general purpose stochastic finite elements implementation, Comput. Methods Appl. Mech. Engrg., 168, 19-34, (1999) · Zbl 0943.65008
[16] Ghanem, R.; Doostan, A., On the construction and analysis of stochastic models; characterization and propagation of the errors associated with limited data, J. Comput. Phys., 217, 63-81, (2006) · Zbl 1102.65004
[17] Guilleminot, J.; Soize, C.; Kondo, D., Mesoscale probabilistic model for the elasticity tensor of fiber reinforced composites: experimental identification and numerical aspects, Mech. Mater., 41, 1309-1322, (2009)
[18] Hiriyur, B.; Waisman, H.; Deodatis, G., Uncertainty quantification in homogenization of heterogeneous microstructures modeled by XFEM, Int. J. Numer. Methods Engrg., 86, 257-278, (2011) · Zbl 1242.74125
[19] Holzapfel, G., Nonlinear solid mechanics: A continuum approach for engineering, (2000), John Wiley and Sons Chichester · Zbl 0980.74001
[20] Knio, O. M.; Le Maitre, O. P., Uncertainty propagation in CFD using polynomial chaos decomposition, Fluid Dyn. Res., 38, 9, 616-640, (2006) · Zbl 1178.76297
[21] Le-Maitre, O. P.; Knio, O. M., Spectral methods for uncertainty quantification with applications to computational fluid dynamics, (2010), Springer Heidelberg · Zbl 1193.76003
[22] Le-Maitre, O. P.; Knio, O. M.; Najm, H. J.; Ghanem, R. G., Uncertainty propagation using Wiener-Haar expansions; J. Comput. Phys., 197, 28-57, (2004) · Zbl 1052.65114
[23] Lopez-Pamies, O.; Ponte Castañeda, P., Second-order estimates for the macroscopic response and loss of ellipticity in porous rubbers at large deformations, J. Elast., 76, 247-287, (2004) · Zbl 1086.74032
[24] Nezamabadi, S.; Yvonnet, J.; Zahrouni, H.; Pottier-Ferry, M., A multilevel computational strategy for handling microscopic and macroscopic instabilities, Comput. Methods Appl. Mech. Engrg., 198, 2099-2110, (2009) · Zbl 1227.74026
[25] Nouy, A.; Le Maître, O. P., Generalized spectral decomposition for stochastic nonlinear problems, J. Comput. Phys., 228, 1, 202-235, (2009) · Zbl 1157.65009
[26] Nouy, A., Identifications of multi-modal random variables through mixtures of polynomial chaos expansions, C. R. Méc., 338, 12, 698-703, (2010) · Zbl 1221.62162
[27] Nouy, A.; Chevreuil, M.; Safatly, E., Fictitious domain method and separated representations for the solution of boundary value problems on uncertain parameterized domains, Comput. Methods Appl. Mech. Engrg., 200, 3066-3082, (2011) · Zbl 1230.65135
[28] Nouy, A.; Clément, A., An extended stochastic finite element method for the numerical simulation of random multi-phased materials, Int. J. Numer. Methods Engrg., 83, 1312-1344, (2010) · Zbl 1202.74182
[29] Ogden, R. W.; Saccomandi, G.; Sgura, I., Fitting hyperelastic models to experimental data, Comput. Mech., 34, 484-502, (2004) · Zbl 1109.74320
[30] Ponte-Castañeda, P.; Tiberio, E., Second-order homogenization method in finite elasticity and applications to black-filled elastomers, J. Mech. Phys. Solids, 48, 6-7, 1389-1411, (2000) · Zbl 0984.74070
[31] Schueller, G. I., On the treatment of uncertainties in structural mechanics and analysis, Comput. Struct., 85, 5, 235-243, (2007)
[32] Serfling, R. J., Approximation theorems of mathematical statistics, (1980), John Wiley & Sons · Zbl 0456.60027
[33] Smit, R.; Brekelmans, W.; Meijer, H., Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling, Comput. Methods Appl. Mech. Engrg., 155, 181-192, (1998) · Zbl 0967.74069
[34] Soize, C., Generalized probabilistic approach of uncertainties in computational dynamics using random matrices and polynomial chaos decompositions, Int. J. Numer. Methods Engrg., 81, 8, 939-970, (2010) · Zbl 1183.74381
[35] Soize, C., Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data, Comput. Methods Appl. Mech. Engrg., 199, 33-36, 2150-2164, (2010) · Zbl 1231.74501
[36] Soize, C.; Desceliers, C., Computational aspects for constructing realizations of polynomial chaos in high dimension, SIAM J. Sci. Comput., 32, 5, 2820-2831, (2010) · Zbl 1225.60118
[37] Soize, C., A computational inverse method for identification of non-gaussian random fields using the Bayesian approach in very high dimension, Comput. Methods Appl. Mech. Engrg., 200, 3083-3099, (2011) · Zbl 1230.74241
[38] Spall, J. C., Introduction to stochastic search and optimization, (2003), John Wiley and Sons Hoboken, New Jersey · Zbl 1088.90002
[39] Stefanou, G.; Nouy, A.; Clément, A., Identification of random shapes from images through polynomial chaos expansion of random level-set functions, Int. J. Numer. Methods Engrg., 79, 127-155, (2009) · Zbl 1171.74449
[40] Takano, N.; Zako, M.; Ohnishi, Y., Macro-micro uncoupled homogenization procedure for microscopic nonlinear behavior analysis of composites, Mater. Sci. Res. Int., 2, 2, 81-86, (1996)
[41] Temizer, I.; Wriggers, P., An adaptive method for homogenization in orthotropic nonlinear elasticity, Comput. Methods Appl. Mech. Engrg., 35-36, 3409-3423, (2007) · Zbl 1173.74378
[42] Terada, K.; Kikuchi, N., A class of general algorithms for multi-scale analysis of heterogeneous media, Comput. Methods Appl. Mech. Engrg., 190, 5427-5464, (2001) · Zbl 1001.74095
[43] Tootkaboni, M.; Graham-Brady, L., A multi-scale spectral stochastic method for homogenization of multi-phase periodic composites with random material properties, Int. J. Numer. Methods Engrg., 83, 59-90, (2010) · Zbl 1193.74166
[44] Torquato, S., Random heterogeneous materials: microstructures and macroscopic properties, (2002), Springer New York · Zbl 0988.74001
[45] Yvonnet, J.; He, Q. C., The reduced model multiscale method (R3M) for the nonlinear homogenization of hyperelastic media at finite strains, J. Comput. Phys., 223, 341-368, (2007) · Zbl 1163.74048
[46] Yvonnet, J.; Gonzalez, D.; He, Q. C., Numerically explicit potentials for the homogenization of nonlinear elastic heterogeneous materials, Comput. Methods Appl. Mech. Engrg., 198, 2723-2737, (2009) · Zbl 1228.74067
[47] Yvonnet, J.; He, Q. C., A non-concurrent multiscale method for computing the response of nonlinear elastic heterogeneous structures, Eur. J. Comput. Mech., 19, 105-116, (2010) · Zbl 1426.74050
[48] J. Yvonnet, E. Monteiro, Q.C. He, Computational homogenization of hyperelastic heterogeneous structures: a non-concurrent approach, International Journal for Multiscale Computational Engineering (2012), accepted for publication.
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.