Bloch wave homogenization of linear elasticity system

Sista Sivaji Ganesh; Muthusamy Vanninathan

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 4, page 542-573
  • ISSN: 1292-8119

Abstract

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In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome these difficulties. The existence of a directionally regular Bloch spectrum is proved and is used in the homogenization. As a consequence an interesting relation between homogenization process and wave propagation in the homogenized medium is obtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequence it is proved that higher modes apart from the first three do not contribute to the homogenization process.

How to cite

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Ganesh, Sista Sivaji, and Vanninathan, Muthusamy. "Bloch wave homogenization of linear elasticity system." ESAIM: Control, Optimisation and Calculus of Variations 11.4 (2010): 542-573. <http://eudml.org/doc/90777>.

@article{Ganesh2010,
abstract = { In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome these difficulties. The existence of a directionally regular Bloch spectrum is proved and is used in the homogenization. As a consequence an interesting relation between homogenization process and wave propagation in the homogenized medium is obtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequence it is proved that higher modes apart from the first three do not contribute to the homogenization process. },
author = {Ganesh, Sista Sivaji, Vanninathan, Muthusamy},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Bloch waves; homogenization; linear elasticity.; directionally regular Bloch spectrum; spectral gap},
language = {eng},
month = {3},
number = {4},
pages = {542-573},
publisher = {EDP Sciences},
title = {Bloch wave homogenization of linear elasticity system},
url = {http://eudml.org/doc/90777},
volume = {11},
year = {2010},
}

TY - JOUR
AU - Ganesh, Sista Sivaji
AU - Vanninathan, Muthusamy
TI - Bloch wave homogenization of linear elasticity system
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 4
SP - 542
EP - 573
AB - In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of the lower Bloch spectrum. For the three dimensional linear elasticity system, the first eigenvalue is degenerate of multiplicity three and hence existence of such a regular Bloch spectrum is not guaranteed. The aim here is to develop all necessary spectral tools to overcome these difficulties. The existence of a directionally regular Bloch spectrum is proved and is used in the homogenization. As a consequence an interesting relation between homogenization process and wave propagation in the homogenized medium is obtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequence it is proved that higher modes apart from the first three do not contribute to the homogenization process.
LA - eng
KW - Bloch waves; homogenization; linear elasticity.; directionally regular Bloch spectrum; spectral gap
UR - http://eudml.org/doc/90777
ER -

References

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  1. G. Allaire, Homogenization and two scale convergence. SIAM J. Math. Anal.23 (1992) 1482–1518.  
  2. G. Allaire and C. Conca, Bloch wave homogenization for a spectral problem in fluid-solid structures. Arch. Rational Mech. Anal.135 (1996) 197–257.  
  3. G. Allaire and C. Conca, Boundary layers in the homogenization of a spectral problem in fluid-solid structures. SIAM J. Math. Anal.29 (1997) 343–379.  
  4. A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic analysis for periodic structures. North Holland, Amsterdam (1978).  
  5. C. Conca, S. Natesan and M. Vanninathan, Numerical solution of elliptic partial differential equations by Bloch waves method, XVII CEDYA: Congress on differential equations and applications/VII CMA: Congress on applied mathematics, Dep. Mat. Appl., Univ. Salamanca, Salamanca (2001) 63–83.  
  6. C. Conca, R. Orive and M. Vanninathan, Bloch approximation in homogenization and applications. SIAM J. Math. Anal.33 (2002) 1166–1198.  
  7. C. Conca, J. Planchard and M. Vanninathan, Fluids and periodic structures. John Wiley & Sons, New York, and Masson, Paris (1995).  
  8. C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math.57 (1997) 1639–1659.  
  9. C. Conca and M. Vanninathan, Fourier approach to homogenization. ESAIM: COCV8 (2002) 489–511.  
  10. A.P. Cracknell and K.C. Wong, The Fermi surface. Clarendon press, Oxford (1973).  
  11. G. Dal maso, An introduction to Γ-convergence. Birkhäuser, Boston (1993).  
  12. P. Gérard, Microlocal defect measures. Commun. PDE16 (1991) 1761–1794.  
  13. P. Gérard, P.A. Markowich, N.J. Mauser and F. Poupaud, Homogenization limits and Wigner transforms. Comm. Pure Appl. Math.50 (1997) 323–379.  
  14. V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential operators and Integral functionals. Berlin, Springer-Verlag (1994).  
  15. T. Kato, Perturbation theory for linear operators. 2nd edition, Springer-Verlag, Berlin (1980).  
  16. F. Murat and L. Tartar, H-Convergence, Topics in the Mathematical Modeling of Composite Materials, A. Charkaev and R. Kohn Eds. PNLDE 31, Birkhäuser, Boston (1997).  
  17. G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608–623.  
  18. O.A. Oleinik, A.S. Shamaev and G.A. Yosifian, Mathematical problems in elasticity and homogenization. North Holland, Amsterdam (1992).  
  19. F. Rellich, Perturbation theory of eigenvalue problems. Gordon and Breach science publishers, New York (1969).  
  20. M. Roseau, Vibrations in Mechanical systems: Analytical methods and applications. Springer-Verlag, Berlin (1987).  
  21. W. Rudin, Functional analysis. 2nd edition, Mc-Graw Hill, New York (1991).  
  22. J. Sínchez-Hubert and E. Sínchez-Palencia, Vibration and coupling of continuous systems: asymptotic methods. Springer-Verlag, Berlin (1989).  
  23. E. Sínchez-Palencia, Non-homogeneous media and vibration theory. Lect. Notes Phys. 127 (1980).  
  24. F. Santosa and W.W. Symes, A dispersive effective medium for wave propagation in periodic composites. SIAM J. Appl. Math.51 (1991) 984–1005.  
  25. S. Sivaji Ganesh and M. Vanninathan, Bloch wave homogenization of scalar elliptic operators. Asymptotic Analysis39 (2004) 15–44.  
  26. L. Tartar, H-measures, a new approach for studying homogenization, oscillations and concentration effects in partial differential equations. Proc. Roy. Soc. Edin. Sect. A115 (1990) 193–230.  
  27. N. Turbé, Applications of Bloch decomposition to periodic elastic and viscoelastic media. Math. Meth. Appl. Sci.4 (1982) 433–449.  

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