Derivation of Higher-Order Terms in FFT-Based Numerical Homogenization

  • Felix DietrichEmail author
  • Dennis Merkert
  • Bernd Simeon
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 126)


In this paper, we first introduce the reader to the Basic Scheme of Moulinec and Suquet in the setting of quasi-static linear elasticity, which takes advantage of the fast Fourier transform on homogenized microstructures to accelerate otherwise time-consuming computations. By means of an asymptotic expansion, a hierarchy of linear problems is then derived, whose solutions are looked at in detail. It is highlighted how these generalized homogenization problems depend on each other. We extend the Basic Scheme to fit this new problem class and give some numerical results for the first two problem orders.



The collaboration with H. Andrä, M. Kabel and M. Schneider, Fraunhofer ITWM Kaiserslautern, is gratefully acknowledged.


  1. 1.
    C. Boutin, Microstructural effects in elastic composites. Int. J. Solids Struct. 33, 1023–1051 (1996)CrossRefGoogle Scholar
  2. 2.
    S. Brisard, L. Dormieux, FFT-based methods for the mechanics of composites: A general variational framework. Comput. Mater. Sci. 49, 663–671 (2010)CrossRefGoogle Scholar
  3. 3.
    P. Eisenlohr et al., A spectral method solution to crystal elasto-viscoplasticity at finite strains. Int. J. Plast. 46, 37–53 (2013)CrossRefGoogle Scholar
  4. 4.
    Z. Hashin, The elastic moduli of heterogeneous materials. J. Appl. Mech. 29, 143–150 (1962)MathSciNetCrossRefGoogle Scholar
  5. 5.
    M. Kabel, H. Andrä, Fast numerical computation of precise bounds of effective elastic moduli, Berichte des Fraunhofer ITWM, 224 (2013)Google Scholar
  6. 6.
    M. Kabel, D. Merkert, M. Schneider, Use of composite voxels in FFT-based homogenization. Comput. Methods Appl. Mech. Eng. 294, 168–188 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    V. Monchiet, G. Bonnet, A polarization-based FFT iterative scheme for computing the effective properties of elastic composites with arbitrary contrast. Int. J. Numer. Methods Eng. 89, 1419–1436 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    H. Moulinec, P. Suquet, A fast numerical method for computing the linear and nonlinear properties of composites. Comptes rendus de l’Académie des Sci. Série II, Mécanique, physique, chimie, astronomie 318, 1417–1423 (1994)Google Scholar
  9. 9.
    J. Spahn et al., A multiscale approach for modeling progressive damage of composite materials using fast Fourier transforms. Comput. Methods Appl. Mech. Eng. 268, 871–883 (2014)MathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Suquet, N. Lahellec, Elasto-plasticity of heterogeneous materials at different scales, Procedia IUTAM 10, 247–262 (2014)CrossRefGoogle Scholar
  11. 11.
    T. Tran, V. Monchiet, G. Bonnet, A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media. Int. J. Solids Struct. 49, 783–792 (2012)CrossRefGoogle Scholar
  12. 12.
    J. Vondřejc, J. Zeman, I. Marek, An FFT-based Galerkin method for homogenization of periodic media. Comput. Math. Appl. 68, 156–173 (2014)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Technische Universität KaiserslauternKaiserslauternGermany

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