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Original Articles

A new boundary condition for homogenization of high-contrast random heterogeneous materials

, &
Pages 2012-2027 | Received 10 Nov 2014, Accepted 06 Aug 2015, Published online: 23 Sep 2015

References

  • G. Allaire and K. El Ganaoui, Homogenization of a conductive and radiative heat transfer problem, Multiscale Model. Sim. 7 (2009), pp. 1148–1170. doi: 10.1137/080714737
  • A. Bensoussan, J.L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North–Holland Publishing Company, Amsterdam, 1978.
  • I.I. Bogdanov, V.V. Mourzenko, J.F. Thovert, and P.M. Adler, Effective permeability of fractured porous media with power-law distribution of fracture sizes, Phys. Rev. E. 76 (2007), p. 15. doi: 10.1103/PhysRevE.76.036309
  • A. Bourgeat and A. Piatnitski, Approximations of effective coefficients in stochastic homogenization, Ann. I. H. Poincare-Pr. 40 (2004), pp. 153–165. doi: 10.1016/j.anihpb.2003.07.003
  • D. Braess, Finite elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, Cambridge, 2001.
  • V.M. Calo, Y. Efendiev, and J. Galvis, Asymptotic expansions for high-contrast elliptic equations, Math. Mod. Meth. Appl. S. 24 (2014), pp. 465–494. doi: 10.1142/S0218202513500565
  • C.M. Chen, Introduction to Scientific Computing, Science Press, Beijing, 2007.
  • H.X. Chen, X.J. Xu, and W.Y. Zheng, Local multilevel methods for second-order elliptic problems with highly discontinuous coefficients, J. Comput. Math. 30 (2012), pp. 223–248. doi: 10.4208/jcm.1109-m3401
  • L.J. Durlofsky, Numerical-calculation of equivalent grid block permeability tensors for heterogeneous porous-media, Water Resour. Res. 27 (1991), pp. 699–708. doi: 10.1029/91WR00107
  • W. E, B. Engquist, X.T. Li, W.Q. Ren, and E. Vanden-Eijnden, Heterogeneous multiscale methods: A review, Commun. Comput. Phys. 2 (2007), pp. 367–450.
  • Y. Efendiev, J. Galvis, R. Lazarov, and J. Willems, Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms, Esaim-Math. Model. Num. 46 (2012), pp. 1175–1199. doi: 10.1051/m2an/2011073
  • R. Ewing, O. Iliev, R. Lazarov, I. Rybak, and J. Willems, A simplified method for upscaling composite materials with high contrast of the conductivity, Siam J. Sci. Comput. 31 (2009), pp. 2568–2586. doi: 10.1137/080731906
  • M.G.D. Geers, V.G. Kouznetsova, and W.A.M. Brekelmans, Multi-scale computational homogenization: Trends and challenges, J. Comput. Appl. Math. 234 (2010), pp. 2175–2182. doi: 10.1016/j.cam.2009.08.077
  • A. Gloria, Reduction of the resonance error – part 1: Approximation of homogenized coefficients, Math. Mod. Meth. Appl. S. 21 (2011), pp. 1601–1630. doi: 10.1142/S0218202511005507
  • G.H. Golup and C.F. Van Loan, Matrix Computations, Johns hopkins studies in the mathematical sciences, Johns Hopkins University Press, Baltimore and Landon, 1996.
  • F. Greco, L. Leonetti, and P. Lonetti, A two-scale failure analysis of composite materials in presence of fiber/matrix crack initiation and propagation, Compos. Struct. 95 (2013), pp. 582–597. doi: 10.1016/j.compstruct.2012.08.035
  • J.Z. Huang, L.Q. Cao, and S. Yang, A molecular dynamics-continuum coupled model for heat transfer in composite materials, Multiscale Model. Sim. 10 (2012), pp. 1292–1316. doi: 10.1137/120864696
  • T. Kanit, S. Forest, I. Galliet, V. Mounoury, and D. Jeulin, Determination of the size of the representative volume element for random composites: Statistical and numerical approach, Int. J. Solids Struct. 40 (2003), pp. 3647–3679. doi: 10.1016/S0020-7683(03)00143-4
  • G.C. Papanicolaou and S.R.S. Varadhan, Boundary Value Problems with Rapidly Oscillating Random Coefficients, in Random Fields, vol. 27, North-Holland Publishing Company, Amsterdam, 1981, pp. 835–873.
  • M. Ptashnyk, Two-scale convergence for locally periodic microstructures and homogenization of plywood structures, Multiscale Model. Sim. 11 (2013), pp. 92–117. doi: 10.1137/120862338
  • L.Z. Qin and X.J. Xu, On a parallel robin-type nonoverlapping domain decomposition method, Siam J. Numer. Anal. 44 (2006), pp. 2539–2558. doi: 10.1137/05063790X
  • K. Sab, On the homogenization and the simulation of random materials, Eur. J. Mech. A-Solid. 11 (1992), pp. 585–607.
  • K. Sab and B. Nedjar, Periodization of random media and representative volume element size for linear composites, CR Mecanique 333 (2005), pp. 187–195. doi: 10.1016/j.crme.2004.10.003
  • K. Terada, M. Hori, T. Kyoya, and N. Kikuchi, Simulation of the multi-scale convergence in computational homogenization approaches, Int. J. Solids Struct. 37 (2000), pp. 2285–2311. doi: 10.1016/S0020-7683(98)00341-2
  • K.I. Tserpes and A. Chanteli, Parametric numerical evaluation of the effective elastic properties of carbon nanotube-reinforced polymers, Compos. Struct. 99 (2013), pp. 366–374. doi: 10.1016/j.compstruct.2012.12.004
  • S.S. Vel and A.J. Goupee, Multiscale thermoelastic analysis of random heterogeneous materials part i: Microstructure characterization and homogenization of material properties, Comp. Mater. Sci. 48 (2010), pp. 22–38. doi: 10.1016/j.commatsci.2009.11.015
  • L.H. Wang and X.J. Xu, The Mathematical Basis of Finite Element Method, Science Press, Beijing, 2004.
  • X.H. Wu, Y. Efendiev, and T.Y. Hou, Analysis of upscaling absolute permeability, Discrete Cont. Dyn-B. 2 (2002), pp. 185–204. doi: 10.3934/dcdsb.2002.2.185
  • Y.T. Wu, Y.F. Nie, and Z.H. Yang, Prediction of effective properties for random heterogeneous materials with extrapolation, Arch. Appl. Mech. 84 (2014), pp. 247–261. doi: 10.1007/s00419-013-0797-7
  • Q.S. Yang and W. Becker, A comparative investigation of different homogenization methods for prediction of the macroscopic properties of composites, Cmes-Comp. Model. Eng. 6 (2004), pp. 319–332.
  • Z.Q. Yang, J.Z. Cui, Y.F. Nie, and Q. Ma, The second-order two-scale method for heat transfer performances of periodic porous materials with interior surface radiation, Cmes-Comp. Model. Eng. 88 (2012), pp. 419–442.
  • X.Y. Yue and W.N. E, The local microscale problem in the multiscale modeling of strongly heterogeneous media: Effects of boundary conditions and cell size, J. Comput. Phys. 222 (2007), pp. 556–572. doi: 10.1016/j.jcp.2006.07.034
  • V.V. Zhikov, S.M. Kozlov, and O.A. Oleǐnik, Homogenization of Differential Operators and Integral Functionals, Springer–Verlag, Berlin, 1994.
  • T.I. Zohdi and P. Wriggers, An Introduction to Computational Micromechanics, Lecture Notes in Applied and Computational Mechanics, Springer–Verlag, Berlin, Heidelberg, 2005.

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