Computational Mechanics

, Volume 57, Issue 1, pp 123–147 | Cite as

Modified SFEM for computational homogenization of heterogeneous materials with microstructural geometric uncertainties

  • Dmytro Pivovarov
  • Paul SteinmannEmail author
Original Paper


In the current work we examine the application of the stochastic finite element method (SFEM) to the modeling of representative volume elements for heterogeneous materials. Uncertainties in the geometry of the microstructure result in the random nature of the solution fields thus requiring use of the stochastic version of the finite element method. For considering large differences in the material properties of matrix and inclusions a standard SFEM approach proves not stable and results in high numerical errors compared to a brute-force Monte-Carlo evaluation. Therefore in order to stabilize the SFEM we propose an alternative Gauss integration rule as resulting from a truncation of the probability density function for the random variable. In addition we propose new basis functions substituting the common polynomial chaos expansion, resulting in higher accuracy for the standard deviation in the homogenized stress at the macro scale.


Stochastic FEM Computational homogenization Random geometry 



The support of this work by the ERC Advanced Grant MOCOPOLY and Deutsche Forschungs-Gemeinschaft (DFG) through the Cluster of Excellence Engineering of Advanced Materials is gratefully acknowledged. The authors also thank Mr. Bastian Walter for preparation of elastomer samples and providing corresponding SEM images.


  1. 1.
    Alsayednoor J, Harrison P, Guo Z (2013) Large strain compressive response of 2-d periodic representative volume element for random foam microstructures. Mech Mater 66:7–20. doi: 10.1016/j.mechmat.2013.06.006 CrossRefGoogle Scholar
  2. 2.
    Andrianov I, Danishevsky V, Tokarzewski S (2000) Quasifractional approximants in the theory of composite materials. Acta Appl Math 61(1–3):29–35. doi: 10.1023/A:1006455311626 MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andrianov I, Danishevs’kyy VV, Weichert D (2008) Simple estimation on effective transport properties of a random composite material with cylindrical fibres. Z angew Math Phys 59(5):889–903. doi: 10.1007/s00033-007-6146-3 MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Andrianov IV, Danishevs’kyy VV, Kholod EG (2012) Homogenization of viscoelastic composites with fibres of diamond-shaped cross-section. Acta Mech 223(5):1093–1100. doi: 10.1007/s00707-011-0608-6 MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Andrianov IV, Danishevs’kyy VV, Weichert D (2002) Asymptotic determination of effective elastic properties of composite materials with fibrous square-shaped inclusions. Eur J Mech A/Solids 21(6):1019–1036. doi: 10.1016/S0997-7538(02)01250-0 CrossRefzbMATHGoogle Scholar
  6. 6.
    Castaneda PP, Galipeau E (2011) Homogenization-based constitutive models for magnetorheological elastomers at finite strain. J Mech Phys Solids 59(2):194–215. doi: 10.1016/j.jmps.2010.11.004 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chatzigeorgiou G, Javili A, Steinmann P (2013) Unified magnetomechanical homogenization framework with application to magnetorheological elastomers. Math Mech Solids 2012:193–211. doi: 10.1177/1081286512458109 MathSciNetGoogle Scholar
  8. 8.
    Cottereau R (2013) A stochastic-deterministic coupling method for multiscale problems. Application to numerical homogenization of random materials. Procedia IUTAM 6(0):35–43. doi: 10.1016/j.piutam.2013.01.004. IUTAM symposium on multiscale problems in stochastic mechanics
  9. 9.
    Cottereau R, Clouteau D, Ben Dhia H (2011) Localized modeling of uncertainty in the arlequin framework. In: Belyaev AK, Langley RS (eds) IUTAM symposium on the vibration analysis of structures with uncertainties, IUTAM Bookseries, vol. 27. Springer Netherlands, pp 457–468Google Scholar
  10. 10.
    Dimas LS, Giesa T, Buehler MJ (2014) Coupled continuum and discrete analysis of random heterogeneous materials: elasticity and fracture. J Mech Phys Solids 63:481–490. doi: 10.1016/j.jmps.2013.07.006 MathSciNetCrossRefGoogle Scholar
  11. 11.
    Ernst O, Powell C, Silvester D, Ullmann E (2009) Efficient solvers for a linear stochastic galerkin mixed formulation of diffusion problems with random data. SIAM J Sci Comput 31(2):1424–1447. doi: 10.1137/070705817 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ernst OG, Mugler A, Starkloff HJ, Ullmann E (2012) On the convergence of generalized polynomial chaos expansions. ESAIM. Math Model Numer Anal 46:317–339. doi: 10.1051/m2an/2011045 MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ernst OG, Ullmann E (2010) Stochastic galerkin matrices. SIAM J Matrix Anal Appl 31(4):1848–1872. doi: 10.1137/080742282 MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fries TP, Belytschko T (2010) The extended/generalized finite element method: an overview of the method and its applications. Int J Numer Methods Eng 84(3):253–304. doi: 10.1002/nme.2914 MathSciNetzbMATHGoogle Scholar
  15. 15.
    Galipeau E, Castaneda PP (2012) The effect of particle shape and distribution on the macroscopic behavior of magnetoelastic composites. Int J Solids Struct 49(1):1–17. doi: 10.1016/j.ijsolstr.2011.08.014 CrossRefGoogle Scholar
  16. 16.
    Galipeau E, Castaneda PP (2013) A finite-strain constitutive model for magnetorheological elastomers: magnetic torques and fiber rotations. J Mech Phys Solids 61(4):1065–1090. doi: 10.1016/j.jmps.2012.11.007 MathSciNetCrossRefGoogle Scholar
  17. 17.
    Galipeau E, Rudykh S, deBotton G, Castaneda PP (2014) Magnetoactive elastomers with periodic and random microstructures. Int J Solids Struct 51(8):3012–3024. doi: 10.1016/j.ijsolstr.2014.04.013 CrossRefGoogle Scholar
  18. 18.
    Ghanem RG, Spanos PD (2003) Stochastic finite elements: a spectral approach. Dover Publications, inc., New YorkGoogle Scholar
  19. 19.
    Hadigol M, Doostan A, Matthies HG, Niekamp R (2014) Partitioned treatment of uncertainty in coupled domain problems: a separated representation approach. Comput Methods Appl Mech Eng 274:103–124. doi: 10.1016/j.cma.2014.02.004 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Javili A, Chatzigeorgiou G, Steinmann P (2013) Computational homogenization in magneto-mechanics. Int J Solids Struct 50(25—-26):4197–4216. doi: 10.1016/j.ijsolstr.2013.08.024 CrossRefGoogle Scholar
  21. 21.
    Khoromskij B, Litvinenko A, Matthies H (2009) Application of hierarchical matrices for computing the karhunen-loeve expansion. Computing 84(1–2):49–67. doi: 10.1007/s00607-008-0018-3 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kucerova A, Sykora J, Rosic B, Matthies HG (2012) Acceleration of uncertainty updating in the description of transport processes in heterogeneous materials. J Comput Appl Math 236(18):4862–4872. doi: 10.1016/ FEMTEC 2011: 3rd International Conference on Computational Methods in Engineering and Science, May 9–13, 2011
  23. 23.
    Leclerc W, Karamian-Surville P, Vivet A (2013) An efficient stochastic and double-scale model to evaluate the effective elastic properties of 2d overlapping random fibre composites. Comput Mater Sci 69:481–493. doi: 10.1016/j.commatsci.2012.10.036 CrossRefGoogle Scholar
  24. 24.
    Legrain G, Cartraud P, Perreard I, Moes N (2011) An x-fem and level set computational approach for image-based modelling: application to homogenization. Int J Numer Methods Eng 86(7):915–934. doi: 10.1002/nme.3085 CrossRefzbMATHGoogle Scholar
  25. 25.
    Ma J, Sahraee S, Wriggers P, De Lorenzis L (2015) Stochastic multiscale homogenization analysis of heterogeneous materials under finite deformations with full uncertainty in the microstructure. Comput Mech 55(5):819–835. doi: 10.1007/s00466-015-1136-3 MathSciNetCrossRefGoogle Scholar
  26. 26.
    Ma J, Zhang J, Li L, Wriggers P, Sahraee S (2014) Random homogenization analysis for heterogeneous materials with full randomness and correlation in microstructure based on finite element method and monte-carlo method. Comput Mech 54(6):1395–1414. doi: 10.1007/s00466-014-1065-6 MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Moes N, Cloirec M, Cartraud P, Remacle JF (2003) A computational approach to handle complex microstructure geometries. Comput Methods Appl Mech Eng 192(2830):3163–3177. doi: 10.1016/S0045-7825(03)00346-3 CrossRefzbMATHGoogle Scholar
  28. 28.
    Moes N, Dolbow J, Belytschko T (1999) A finite element method for crack growth without remeshing. Int J Numer Methods Eng 46(1):131–150. doi: 10.1002/(SICI)1097-0207(19990910)46 CrossRefzbMATHGoogle Scholar
  29. 29.
    Nouy A, Clement A, Schoefs F, Moes N (2008) An extended stochastic finite element method for solving stochastic partial differential equations on random domains. Comput Methods Appl Mech Eng 197(51—-52):4663–4682. doi: 10.1016/j.cma.2008.06.010 MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Pajonk O, Rosic BV, Matthies HG (2013) Sampling-free linear bayesian updating of model state and parameters using a square root approach. Comput Geosci 55(0):70–83. doi: 10.1016/j.cageo.2012.05.017 CrossRefGoogle Scholar
  31. 31.
    Papoulis A, Pillai SU (2001) Probability, random variables and stochastic processes. McGraw-Hill Education, New YorkGoogle Scholar
  32. 32.
    Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1992) Numerical recipes in FORTRAN: the art of scientific computing, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
  33. 33.
    Rosic B, Matthies H (2008) Computational approaches to inelastic media with uncertain parameters. J Serbian Soc Comput Mech 2(1):28–43Google Scholar
  34. 34.
    Rosic B, Matthies H, Zivkovic M (2011) Uncertainty quantification of inifinitesimal elastoplasticity. Sci Tech Rev 61(2):3–9Google Scholar
  35. 35.
    Rosic B, Matthies HG (2011) Plasticity described by uncertain parameters: a variational inequality approach. In: Proceedings of XI international conference on computational plasticity, fundamentals and applications (COMPLAS), pp 385–395Google Scholar
  36. 36.
    Rosic BV (2012) Variational formulations and functional approximation algorithms in stochastic plasticity of materials. Ph.D. thesis, Faculty of Engineering, Kragujevac. doi: 10.2298/KG20121116ROSIC
  37. 37.
    Sakata S, Ashida F (2011) Hierarchical stochastic homogenization analysis of a particle reinforced composite material considering non-uniform distribution of microscopic random quantities. Comput Mech 48(5):529–540. doi: 10.1007/s00466-011-0604-7 MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Sakata S, Ashida F, Enya K (2012) A microscopic failure probability analysis of a unidirectional fiber reinforced composite material via a multiscale stochastic stress analysis for a microscopic random variation of an elastic property. Comput Mater Sci 62(0):35–46. doi: 10.1016/j.commatsci.2012.05.008 CrossRefGoogle Scholar
  39. 39.
    Sakata S, Ashida F, Kojima T (2008) Stochastic homogenization analysis on elastic properties of fiber reinforced composites using the equivalent inclusion method and perturbation method. Int J Solids Struct 45(2526):6553–6565. doi: 10.1016/j.ijsolstr.2008.08.017 CrossRefzbMATHGoogle Scholar
  40. 40.
    Sakata S, Ashida F, Zako M (2008) Kriging-based approximate stochastic homogenization analysis for composite materials. Comput Methods Appl Mech Eng 197(2124):1953–1964. doi: 10.1016/j.cma.2007.12.011 CrossRefzbMATHGoogle Scholar
  41. 41.
    Shynk JJ (2012) Probability, random variables, and random processes: theory and signal processing applications. Wiley-Interscience, HobokenGoogle Scholar
  42. 42.
    Spieler C, Kaestner M, Goldmann J, Brummund J, Ulbricht V (2013) Xfem modeling and homogenization of magnetoactive composites. Acta Mech 224(11):2453–2469. doi: 10.1007/s00707-013-0948-5 MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Ullmann E, Elman HC, Ernst OG (2012) Efficient iterative solvers for stochastic galerkin discretizations of log-transformed random diffusion problems. SIAM J Sci Comput 34(2):659–682. doi: 10.1137/110836675 MathSciNetCrossRefGoogle Scholar
  44. 44.
    Zaccardi C, Chamoin L, Cottereau R, Ben Dhia H (2013) Error estimation and model adaptation for a stochastic-deterministic coupling method based on the arlequin framework. Int J Numer Meth Eng 96(2):87–109. doi: 10.1002/nme.4540 MathSciNetCrossRefGoogle Scholar
  45. 45.
    Zohdi T, Feucht M, Gross D, Wriggers P (1998) A description of macroscopic damage through microstructural relaxation. Int J Numer Methods Eng 43(3):493–506. doi: 10.1002/(SICI)1097-0207(19981015)43 CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Chair of Applied MechanicsUniversity of Erlangen-NurembergErlangenGermany
  2. 2.Chair of Applied MechanicsUniversity of Erlangen-NurembergErlangenGermany

Personalised recommendations