Skip to main content
SpringerLink
Log in

A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures

  • Published:
SeMA Journal Aims and scope Submit manuscript

Abstract

In this paper we make a comparison between the two-scale asymptotic expansion method for periodic homogenization and the so-called Bloch wave method. It is well-known that the homogenized tensor coincides with the Hessian matrix of the first Bloch eigenvalue when the Bloch parameter vanishes. In the context of the two-scale asymptotic expansion method, there is the notion of high order homogenized equation (Bakhvalov and Panasenko in Homogenization: averaging processes in periodic media. Kluwer, Dordrecht, 1989) where the homogenized equation can be improved by adding small additional higher order differential terms. The next non-zero high order term is a fourth-order term, accounting for dispersion effects (see e.g. Santosa and Symes in SIAM J Appl Math 51:984–1005, 1991; Lamacz in Math Models Methods Appl Sci 21(9):1871–1899, 2011; Dohnal et al. Multiscale Model Simul 12(2):488–513, 2014). Surprisingly, this homogenized fourth-order tensor is not equal to the fourth-order tensor arising in the Taylor expansion of the first Bloch eigenvalue, which is often called Burnett tensor. Here, we establish an exact relation between the homogenized fourth-order tensor and the Burnett fourth-order tensor. It was proved in Conca et al. (J Math Phys 47(3):11, 2006) that the Burnett fourth-order tensor has a sign. For the special case of a simple laminate we prove that the homogenized fourth-order tensor may change sign. In the elliptic case we explain the difference between the homogenized and Burnett fourth-order tensors by a difference in the source term which features an additional corrector term. Finally, for the wave equation, the two fourth-order tensors coincide again, so dispersion is unambiguously defined, and only the source terms differ as in the elliptic case.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abdulle, A., Grote, M., Stohrer, Ch.: Finite element heterogeneous multiscale method for the wave equation: long-time effects. Multiscale Model. Simul. 12, 1230–1257 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abdulle, A., Pouchon, T.N.: A priori error analysis of the finite element heterogeneous multiscale method for the wave equation in heterogeneous media over long time, preprint (2015)

  3. Abdulle, A., Pouchon, T.N.: Effective models for the multidimensional wave equation in heterogeneous media over long time, preprint (2016)

  4. Allaire, G.: Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, vol. 146, p. 456. Springer-Verlag, New-York (2002)

  5. Allaire, G., Palombaro, M., Rauch, J.: Diffractive geometric optics for bloch wave packets. Arch. Ration. Mech. Anal. 202, 373–426 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bakhvalov, N., Panasenko, G.: Homogenization: Averaging Processes in Periodic Media. Kluwer, Dordrecht (1989)

    Book  MATH  Google Scholar 

  7. Bensoussan, A., Lions, J.-L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures, corrected reprint of the 1978 original, p. 398. AMS Chelsea Publishing, Providence (2011)

  8. Brahim-Otsmane, S., Francfort, G., Murat, F.: Correctors for the homogenization of the wave and heat equations. J. Math. Pures Appl. 9(71), 197–231 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Christov, C.I., Maugin, G.A., Velarde, M.G.: Well-posed Boussinesq paradigm with purely spatial higher-order derivatives. Phys. Rev. E 54, 3621–3638 (1996)

    Article  Google Scholar 

  10. Conca, C., Vanninathan, M.: Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57, 1639–1659 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  11. Conca, C., Orive, R., Vanninathan, M.: Bloch approximation in homogenization and applications. SIAM J. Math. Anal. 33(5), 1166–1198 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Conca, C., Orive, R., Vanninathan, M.: On Burnett coefficients in periodic media. J. Math. Phys. 47(3), 11 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Conca, C., San Martín, J., Smaranda, L., Vanninathan, M.: On Burnett coefficients in periodic media in low contrast regime. J. Math. Phys. 49(5), 23 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  14. Conca, C., San Martín, J., Smaranda, L., Vanninathan, M.: Optimal bounds on dispersion coefficient in one-dimensional periodic media. Math. Models Methods Appl. Sci. 19(9), 1743–1764 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Conca, C., San Martín, J., Smaranda, L., Vanninathan, M.: Burnett coefficients and laminates. Appl. Anal. 91(6), 1155–1176 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dohnal, T., Lamacz, A., Schweizer, B.: Bloch-wave homogenization on large time scales and dispersive effective wave equations. Multiscale Model. Simul. 12(2), 488–513 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fish, J., Chen, W., Nagai, G.: Non-local dispersive model for wave propagation in heterogeneous media: multi-dimensional case. Int. J. Numer. Methods Eng. 54(3), 347–363 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ganesh, S.S., Vanninathan, M.: Bloch wave homogenization of linear elasticity system. ESAIM Control Optim. Calc. Var. 11(4), 542–573 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lamacz, A.: Dispersive effective models for waves in heterogeneous media. Math. Models Methods Appl. Sci. 21(9), 1871–1899 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lombardo, M., Askes, H.: Elastic wave dispersion in microstructured membranes. Proc. R. Soc. A 466, 1789–1807 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Moskow, S., Vogelius, M.: First order corrections to the homogenized eigenvalues of a periodic composite medium. A convergence proof. Proc. R. Soc. Edinb. 127, 1263–1295 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. Academic Press, New York (1978)

    MATH  Google Scholar 

  23. Sanchez-Palencia, E.: Non-homogeneous media and vibration theory. Lecture Notes in Physics, vol. 127, pp. 398. Springer-Verlag, Berlin, New York (1980)

  24. Santosa, F., Symes, W.: A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math. 51, 984–1005 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  25. Smyshlyaev, V.P., Cherednichenko, K.D.: On rigorous derivation of strain gradient effects in the overall behaviour of periodic heterogeneous media. The J. R. Willis 60th anniversary volume. J. Mech. Phys. Solids 48(6–7), 1325–1357 (2000)

  26. Tartar, L.: The general theory of homogenization: a personalized introduction, Lecture Notes of the Unione Matematica Italiana, vol. 7, p. 470. Springer - Verlag, Berlin (2009)

  27. Wilcox, C.: Theory of Bloch waves. J. Anal. Math. 33, 146–167 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This research was partially supported by the Indo-French Centre for Applied Mathematics (IFCAM). G. A. is a member of the DEFI project at INRIA Saclay Ile-de-France.

Author information

Authors and Affiliations

  1. CMAP, École Polytechnique, CNRS UMR 7641, Université Paris-Saclay, 91128, Palaiseau, France

    G. Allaire

  2. INSA de Rennes, Institut de Recherche Mathématique de Rennes, 35000, Rennes, France

    M. Briane

  3. TIFR-CAM, Bangalore, India

    M. Vanninathan

Authors
  1. G. Allaire
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Briane
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Vanninathan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Allaire.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allaire, G., Briane, M. & Vanninathan, M. A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures. SeMA 73, 237–259 (2016). https://doi.org/10.1007/s40324-016-0067-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40324-016-0067-z

Keywords

Mathematics Subject Classification

Search

Navigation