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Bloch wave homogenisation of quasiperiodic media

Part of: Qualitative theory Partial differential equations General theory of linear operators Elliptic equations and systems

Published online by Cambridge University Press:  05 October 2020

SISTA SIVAJI GANESH  and
VIVEK TEWARY Open the ORCID record for VIVEK TEWARY [Opens in a new window]
SISTA SIVAJI GANESH
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India emails: [email protected]; [email protected]
VIVEK TEWARY
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India emails: [email protected]; [email protected]
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Abstract

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

Keywords

Homogenisationalmost periodicityBloch eigenvaluesquasiperiodicity

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure
Secondary: 35J15: Second-order elliptic equations 47A55: Perturbation theory 34C27: Almost and pseudo-almost periodic solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 1 , February 2022 , pp. 58 - 78
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Allaire, G. & Piatnitski, A. (2005) Homogenization of the Schrödinger equation and effective mass theorems. Comm. Math. Phys. 258(1), 1–22.CrossRefGoogle Scholar
Allais, M. (1983) Sur la distribution normale des valeurs à des instants régulièrement espacés d’une somme de sinusodes. C. R. Acad. Sci. Paris Sér. I Math. 296(19), 829–832.Google Scholar
Bakhvalov, N. & Panasenko, G. (1989) Homogenisation: Averaging Processes in Periodic Media, Vol. 36, Kluwer Academic Publishers Group, Dordrecht.CrossRefGoogle Scholar
Baumgärtel, H. (1985) Analytic Perturbation Theory for Matrices and Operators , Operator Theory: Advances and Applications, Vol. 15, Birkhäuser Verlag, Basel.Google Scholar
Bellissard, J. & Testard, D. (1981) Almost periodic Hamiltonians: an algebraic approach. Technical Report CNRS-CPT–81-P-1311, Centre National de la Recherche Scientifique.Google Scholar
Benoit, A. & Gloria, A. (2019) Long-time homogenization and asymptotic ballistic transport of classical waves. Ann. Sci. Éc. Norm. Supér. (4) 52(3), 703–759.CrossRefGoogle Scholar
Bensoussan, A., Lions, J.-L. & Papanicolaou, G. (2011) Asymptotic Analysis for Periodic Structures, AMS Chelsea Publishing, Providence, RI.Google Scholar
Birman, M. & Suslina, T. A. (2004) Second order periodic differential operators. Threshold properties and homogenization. St. Petersburg Math. J. 15(5), 639–714.CrossRefGoogle Scholar
Blanc, X., Le Bris, C. & Lions, P.-L. (2015) Local profiles for elliptic problems at different scales: defects in, and interfaces between periodic structures. Comm. Part. Differ. Equ. 40(12), 2173–2236.CrossRefGoogle Scholar
Blot, J. & Pennequin, D. (2001) Spaces of quasi-periodic functions and oscillations in differential equations. Acta Appl. Math. 65(1–3), 83–113.CrossRefGoogle Scholar
Bourgeat, A. & Piatnitski, A. (2004) Approximations of effective coefficients in stochastic homogenization. Ann. Inst. H. Poincaré Probab. Statist. 40(2), 153–165.CrossRefGoogle Scholar
Braides, A., Chiadò Piat, V. & Defranceschi, A. (1992) Homogenization of almost periodic monotone operators. Ann. Inst. H. Poincaré Anal. Non Linéaire 9(4), 399–432.CrossRefGoogle Scholar
Cluni, F. & Gusella, V. (2018) Estimation of residuals for the homogenized solution of quasi-periodic media. Probabilistic Eng. Mech. 54, 110–117.CrossRefGoogle Scholar
Conca, C., Natesan, S. & Vanninathan, M. (2006) Numerical experiments with the Bloch-Floquet approach in homogenization. Int. J. Numer. Methods Eng. 65(9), 1444–1471.CrossRefGoogle Scholar
Conca, C., Orive, R. & Vanninathan, M. (2002) Bloch approximation in homogenization and applications. SIAM J. Math. Anal. 33(5), 1166–1198.CrossRefGoogle Scholar
Conca, C. & Vanninathan, M. (1997) Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57(6), 1639–1659.CrossRefGoogle Scholar
Craster, R. V., Guenneau, S. R. L., Hutridurga, H. R. & Pavliotis, G. A. (2018) Cloaking via mapping for the heat equation. Multiscale Model. Simul. 16(3), 1146–1174.CrossRefGoogle Scholar
Damanik, D., Fillman, J. & Gorodetski, A. (2019) Multidimensional Almost-Periodic Schrödinger operators with Cantor spectrum. Annales Henri Poincaré 20(4), 1393–1402.CrossRefGoogle Scholar
Dubois, J.-M. (2012) Properties-and applications of quasicrystals and complex metallic alloys. Chem. Soc. Rev. 41(20), 6760–6777.CrossRefGoogle ScholarPubMed
Gloria, A. & Habibi, Z. (2016) Reduction in the resonance error in numerical homogenization II: correctors and extrapolation. Found. Comput. Math. 16(1), 217–296.CrossRefGoogle Scholar
Hofstadter, D. R. (1976) Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B 14, 2239–2249.CrossRefGoogle Scholar
Jikov, V. V., Kozlov, S. M. & Oleĭnik, O. A. (1994) Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Kato, T. (1995) Perturbation Theory for Linear Operators , Classics in Mathematics, Springer-Verlag, Berlin.Google Scholar
Kozlov, S. M. (1978) Averaging of differential operators with almost periodic rapidly oscillating coefficients. Mat. Sb. (N.S.) 107(149)(2), 199–217, 317.Google Scholar
Ladyzhenskaya, O. A. & Ural’tseva, N. N. (1968) Linear and Quasilinear Elliptic Equations, Academic Press, New York-London.Google Scholar
Levi, L., Rechtsman, M., Freedman, B., Schwartz, T., Manela, O. & Segev, M. (2011) Disorder-enhanced transport in photonic quasicrystals. Science 332(6037), 1541–1544.CrossRefGoogle ScholarPubMed
Man, W., Megens, M., Steinhardt, P. J. & Chaikin, P. M. (2005) Experimental measurement of the photonic properties of icosahedral quasicrystals. Nature 436(7053), 993–996.CrossRefGoogle ScholarPubMed
Martínez, A. J., Porter, M. A. & Kevrekidis, P. G. (2018) Quasiperiodic granular chains and Hofstadter butterflies. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 376(2127), 20170139.CrossRefGoogle ScholarPubMed
Oleinik, O. A. & Zhikov, V. V. (1982) On the homogenization of elliptic operators with almost-periodic coefficients. Rendiconti del Seminario Matematico e Fisico di Milano 52(1), 149–166.CrossRefGoogle Scholar
Olenik, O. A. & Radkevič, E. V. (1973) Second Order Equations with Nonnegative Characteristic Form, Plenum Press, New York-London.CrossRefGoogle Scholar
Pal, R. K., Rosa, M. I. N. & Ruzzene, M. (2019) Topological bands and localized vibration modes in quasiperiodic beams. New J. Phys. 21(9), 093017.CrossRefGoogle Scholar
Rechtsman, M. C., Jeong, H.-C., Chaikin, P. M., Torquato, S. & Steinhardt, P. J. (2008) Optimized structures for photonic quasicrystals. Phys. Rev. Lett. 101(7), 073902.CrossRefGoogle ScholarPubMed
Reed, M. & Simon, B. (1978) Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York-London.Google Scholar
Rodriguez, A. W., McCauley, A. P., Avniel, Y. & Johnson, S. G. (2008) Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem. Phys. Rev. B 77(10), 104201.CrossRefGoogle Scholar
Sánchez-Palencia, E. (1980) Nonhomogeneous Media and Vibration Theory , Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York.Google Scholar
Shechtman, D., Blech, I., Gratias, D. & Cahn, J. W. (1984) Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953.CrossRefGoogle Scholar
Shen, Z. (2015) Convergence rates and Hölder estimates in almost-periodic homogenization of elliptic systems. Anal. PDE 8(7), 1565–1601.CrossRefGoogle Scholar
Shubin, M. A. (1978) Almost periodic functions and partial differential operators. Russian Math. Surv. 33(2), 1.CrossRefGoogle Scholar
Sivaji Ganesh, S. & Tewary, V. (2019) Bloch approach to almost periodic homogenization and approximations of effective coefficients. https://arxiv.org/abs/1908.07977.Google Scholar
Sivaji Ganesh, S. & Vanninathan, M. (2004) Bloch wave homogenization of scalar elliptic operators. Asymptot. Anal. 39(1), 15–44.Google Scholar
Sivaji Ganesh, S. & Vanninathan, M. (2005) Bloch wave homogenization of linear elasticity system. ESAIM Control Optim. Calc. Var. 11(4), 542–573.CrossRefGoogle Scholar
Wang, Y. & Sigmund, O. (2020) Quasiperiodic mechanical metamaterials with extreme isotropic stiffness. Extreme Mech. Lett. 34, 100596.CrossRefGoogle Scholar
Wellander, N., Guenneau, S. & Cherkaev, E. (2018) Two-scale cut-and-projection convergence; homogenization of quasiperiodic structures. Math. Methods Appl. Sci. 41(3), 1101–1106.CrossRefGoogle Scholar
Wellander, N., Guenneau, S. & Cherkaev, E. (2019) Homogenization of quasiperiodic structures and two-scale cut-and-projection convergence. https://arXiv.org/abs/1911.03560.Google Scholar
Yu Xi, X. & Hong Sun, X. (2019) The investigation of photonic band gap of 2D multi-fold photonic quasicrystals. J. Phys. Conf. Ser. 1237, 032083.CrossRefGoogle Scholar
Zhu, W., Fang, Z., Li, D., Sun, Y., Li, Y., Jing, Y. & Chen, H. (2018) Simultaneous observation of a topological edge state and exceptional point in an open and non-Hermitian acoustic system. Phys. Rev. Lett. 121(12), 124501.CrossRefGoogle Scholar