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August  2017, 10(4): 815-835. doi: 10.3934/dcdss.2017041

Effective acoustic properties of a meta-material consisting of small Helmholtz resonators

Agnes Lamacz and  Ben Schweizer ,

Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87,44227 Dortmund, Germany

* Corresponding author: Ben Schweizer

Received  March 2016 Revised  October 2016 Published  April 2017

We investigate the acoustic properties of meta-materials that are inspired by sound-absorbing structures. We show that it is possible to construct meta-materials with frequency-dependent effective properties, with large and/or negative permittivities. Mathematically, we investigate solutions $u^\varepsilon : \Omega_\varepsilon \to \mathbb{R}$ to a Helmholtz equation in the limit $\varepsilon \to 0$ with the help of two-scale convergence. The domain $\Omega_\varepsilon $ is obtained by removing from an open set $\Omega\subset \mathbb{R}^n$ in a periodic fashion a large number (order $\varepsilon ^{-n}$) of small resonators (order $\varepsilon $). The special properties of the meta-material are obtained through sub-scale structures in the perforations.

Keywords: Helmholtz equation, homogenization, resonance, perforated domain, frequency dependent effective properties.
Mathematics Subject Classification: 78M40, 35P25, 35J05.
Citation: Agnes Lamacz, Ben Schweizer. Effective acoustic properties of a meta-material consisting of small Helmholtz resonators. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 815-835. doi: 10.3934/dcdss.2017041
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[2]

G. Allaire and M. Briane, Multiscale convergence and reiterated homogenisation, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 297-342.  doi: 10.1017/S0308210500022757.  Google Scholar

[3]

G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures Appl., 77 (1998), 153-208.  doi: 10.1016/S0021-7824(98)80068-8.  Google Scholar

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M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 407-436.   Google Scholar

[5]

M. Bellieud and I. Gruais, Homogenization of an elastic material reinforced by very stiff or heavy fibers. Non-local effects. Memory effects, J. Math. Pures Appl., 84 (2005), 55-96.  doi: 10.1016/j.matpur.2004.02.003.  Google Scholar

[6]

G. Bouchitté and M. Bellieud, Homogenization of a soft elastic material reinforced by fibers, Asymptot. Anal., 32 (2002), 153-183.   Google Scholar

[7]

G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris, 347 (2009), 571-576.   Google Scholar

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G. Bouchitté and D. Felbacq, Low frequency scattering by a set of parallel metallic rods, In Mathematical and numerical aspects of wave propagation (Santiago de Compostela, 2000), pages 226-230. SIAM, Philadelphia, PA, 2000. Google Scholar

[9]

G. Bouchitté and D. Felbacq, Homogenization near resonances and artificial magnetism from dielectrics, C. R. Math. Acad. Sci. Paris, 339 (2004), 377-382.   Google Scholar

[10]

G. Bouchitté and D. Felbacq, Homogenization of a wire photonic crystal: the case of small volume fraction, SIAM J. Appl. Math., 66 (2006), 2061-2084.   Google Scholar

[11]

G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations in a split ring geometry, Multiscale Model. Simul., 8 (2010), 717-750.   Google Scholar

[12]

G. Bouchitté and B. Schweizer, Plasmonic waves allow perfect transmission through subwavelength metallic gratings, Netw. Heterog. Media, 8 (2013), 857-878.   Google Scholar

[13]

Y. Chen and R. Lipton, Tunable double negative band structure from non-magnetic coated rods, New Journal of Physics, 12 (2010), 083010.  doi: 10.1088/1367-2630/12/8/083010.  Google Scholar

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K. D. Cherednichenko, V. P. Smyshlyaev and V. V. Zhikov, Non-local homogenized limits for composite media with highly anisotropic periodic fibres, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 87-114.  doi: 10.1017/S0308210500004455.  Google Scholar

[15]

V. Chiadò Piat and M. Codegone, Scattering problems in a domain with small holes, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 97 (2003), 447-454.   Google Scholar

[16]

D. Cioranescu and F. Murat, A strange term coming from nowhere, In Topics in the mathematical modelling of composite materials, volume 31 of Progr. Nonlinear Differential Equations Appl. , pages 45-93. Birkhäuser Boston, Boston, MA, 1997. doi: 10.1007/978-1-4612-2032-9_4.  Google Scholar

[17]

D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607.  doi: 10.1016/0022-247X(79)90211-7.  Google Scholar

[18]

D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, volume 136 of Applied Mathematical Sciences, Springer-Verlag, New York, 1999. Google Scholar

[19]

C. Dörlemann, M. Heida and B. Schweizer, Transmission conditions for the Helmholtz equation in perforated domains, Vietnam J. Math. , 2016. doi: 10.1007/s10013-016-0222-y.  Google Scholar

[20]

S. Guenneau, F. Zolla and A. Nicolet, Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability, Waves Random Complex Media, 17 (2007), 653-697.  doi: 10.1080/17455030701607013.  Google Scholar

[21]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian[G. A. Iosifyan]. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[22]

R. Kohn and S. Shipman, Magnetism and homogenization of micro-resonators, Multiscale Modeling & Simulation, 7 (2007), 62-92.   Google Scholar

[23]

R. V. Kohn, J. Lu, B. Schweizer and M. I. Weinstein, A variational perspective on cloaking by anomalous localized resonance, Comm. Math. Phys., 328 (2014), 1-27.  doi: 10.1007/s00220-014-1943-y.  Google Scholar

[24]

A. Lamacz and B. Schweizer, Effective Maxwell equations in a geometry with flat rings of arbitrary shape, SIAM J. Math. Anal., 45 (2013), 1460-1494.  doi: 10.1137/120874321.  Google Scholar

[25]

A. Lamacz and B. Schweizer, A negative index meta-material for Maxwell's equations, SIAM J. Math. Anal., 48 (2016), 4155-4174.  doi: 10.1137/16M1064246.  Google Scholar

[26]

V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations, volume 46 of Progress in Mathematical Physics, Birkhäuser Boston, Inc. , Boston, MA, 2006. Translated from the 2005 Russian original by M. Goncharenko and D. Shepelsky. Google Scholar

[27]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.  Google Scholar

[28]

S. O'Brien and J. Pendry, Magnetic activity at infrared frequencies in structured metallic photonic crystals, J. Phys. Condens. Mat., 14 (2002), 6383-6394.   Google Scholar

[29]

J. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), p3966.  doi: 10.1103/PhysRevLett.85.3966.  Google Scholar

[30]

E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, volume 127 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1980. Google Scholar

[31]

B. Schweizer, The low-frequency spectrum of small Helmholtz resonators, Proc. A., 0339 (2014), 20140339.  doi: 10.1098/rspa.2014.0339.  Google Scholar

[32]

B. Schweizer, Resonance meets homogenization -Construction of meta-materials with astonishing properties, Jahresberichte der DMV, 2016. doi: 10.1365/s13291-016-0153-2.  Google Scholar

[33]

V. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ, Soviet Physics Uspekhi, 10 (1968), 509-514.   Google Scholar

[34]

V. V. Zhikov, Two-scale convergence and spectral problems of homogenization, Tr. Semin. im. Petrovskogo I. G., 22 (2002), 105-120.   Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[2]

G. Allaire and M. Briane, Multiscale convergence and reiterated homogenisation, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 297-342.  doi: 10.1017/S0308210500022757.  Google Scholar

[3]

G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis, J. Math. Pures Appl., 77 (1998), 153-208.  doi: 10.1016/S0021-7824(98)80068-8.  Google Scholar

[4]

M. Bellieud and G. Bouchitté, Homogenization of elliptic problems in a fiber reinforced structure. Nonlocal effects, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 407-436.   Google Scholar

[5]

M. Bellieud and I. Gruais, Homogenization of an elastic material reinforced by very stiff or heavy fibers. Non-local effects. Memory effects, J. Math. Pures Appl., 84 (2005), 55-96.  doi: 10.1016/j.matpur.2004.02.003.  Google Scholar

[6]

G. Bouchitté and M. Bellieud, Homogenization of a soft elastic material reinforced by fibers, Asymptot. Anal., 32 (2002), 153-183.   Google Scholar

[7]

G. Bouchitté, C. Bourel and D. Felbacq, Homogenization of the 3D Maxwell system near resonances and artificial magnetism, C. R. Math. Acad. Sci. Paris, 347 (2009), 571-576.   Google Scholar

[8]

G. Bouchitté and D. Felbacq, Low frequency scattering by a set of parallel metallic rods, In Mathematical and numerical aspects of wave propagation (Santiago de Compostela, 2000), pages 226-230. SIAM, Philadelphia, PA, 2000. Google Scholar

[9]

G. Bouchitté and D. Felbacq, Homogenization near resonances and artificial magnetism from dielectrics, C. R. Math. Acad. Sci. Paris, 339 (2004), 377-382.   Google Scholar

[10]

G. Bouchitté and D. Felbacq, Homogenization of a wire photonic crystal: the case of small volume fraction, SIAM J. Appl. Math., 66 (2006), 2061-2084.   Google Scholar

[11]

G. Bouchitté and B. Schweizer, Homogenization of Maxwell's equations in a split ring geometry, Multiscale Model. Simul., 8 (2010), 717-750.   Google Scholar

[12]

G. Bouchitté and B. Schweizer, Plasmonic waves allow perfect transmission through subwavelength metallic gratings, Netw. Heterog. Media, 8 (2013), 857-878.   Google Scholar

[13]

Y. Chen and R. Lipton, Tunable double negative band structure from non-magnetic coated rods, New Journal of Physics, 12 (2010), 083010.  doi: 10.1088/1367-2630/12/8/083010.  Google Scholar

[14]

K. D. Cherednichenko, V. P. Smyshlyaev and V. V. Zhikov, Non-local homogenized limits for composite media with highly anisotropic periodic fibres, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 87-114.  doi: 10.1017/S0308210500004455.  Google Scholar

[15]

V. Chiadò Piat and M. Codegone, Scattering problems in a domain with small holes, RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat., 97 (2003), 447-454.   Google Scholar

[16]

D. Cioranescu and F. Murat, A strange term coming from nowhere, In Topics in the mathematical modelling of composite materials, volume 31 of Progr. Nonlinear Differential Equations Appl. , pages 45-93. Birkhäuser Boston, Boston, MA, 1997. doi: 10.1007/978-1-4612-2032-9_4.  Google Scholar

[17]

D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes, J. Math. Anal. Appl., 71 (1979), 590-607.  doi: 10.1016/0022-247X(79)90211-7.  Google Scholar

[18]

D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, volume 136 of Applied Mathematical Sciences, Springer-Verlag, New York, 1999. Google Scholar

[19]

C. Dörlemann, M. Heida and B. Schweizer, Transmission conditions for the Helmholtz equation in perforated domains, Vietnam J. Math. , 2016. doi: 10.1007/s10013-016-0222-y.  Google Scholar

[20]

S. Guenneau, F. Zolla and A. Nicolet, Homogenization of 3D finite photonic crystals with heterogeneous permittivity and permeability, Waves Random Complex Media, 17 (2007), 653-697.  doi: 10.1080/17455030701607013.  Google Scholar

[21]

V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian[G. A. Iosifyan]. doi: 10.1007/978-3-642-84659-5.  Google Scholar

[22]

R. Kohn and S. Shipman, Magnetism and homogenization of micro-resonators, Multiscale Modeling & Simulation, 7 (2007), 62-92.   Google Scholar

[23]

R. V. Kohn, J. Lu, B. Schweizer and M. I. Weinstein, A variational perspective on cloaking by anomalous localized resonance, Comm. Math. Phys., 328 (2014), 1-27.  doi: 10.1007/s00220-014-1943-y.  Google Scholar

[24]

A. Lamacz and B. Schweizer, Effective Maxwell equations in a geometry with flat rings of arbitrary shape, SIAM J. Math. Anal., 45 (2013), 1460-1494.  doi: 10.1137/120874321.  Google Scholar

[25]

A. Lamacz and B. Schweizer, A negative index meta-material for Maxwell's equations, SIAM J. Math. Anal., 48 (2016), 4155-4174.  doi: 10.1137/16M1064246.  Google Scholar

[26]

V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations, volume 46 of Progress in Mathematical Physics, Birkhäuser Boston, Inc. , Boston, MA, 2006. Translated from the 2005 Russian original by M. Goncharenko and D. Shepelsky. Google Scholar

[27]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.  Google Scholar

[28]

S. O'Brien and J. Pendry, Magnetic activity at infrared frequencies in structured metallic photonic crystals, J. Phys. Condens. Mat., 14 (2002), 6383-6394.   Google Scholar

[29]

J. Pendry, Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), p3966.  doi: 10.1103/PhysRevLett.85.3966.  Google Scholar

[30]

E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, volume 127 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1980. Google Scholar

[31]

B. Schweizer, The low-frequency spectrum of small Helmholtz resonators, Proc. A., 0339 (2014), 20140339.  doi: 10.1098/rspa.2014.0339.  Google Scholar

[32]

B. Schweizer, Resonance meets homogenization -Construction of meta-materials with astonishing properties, Jahresberichte der DMV, 2016. doi: 10.1365/s13291-016-0153-2.  Google Scholar

[33]

V. Veselago, The electrodynamics of substances with simultaneously negative values of ε and μ, Soviet Physics Uspekhi, 10 (1968), 509-514.   Google Scholar

[34]

V. V. Zhikov, Two-scale convergence and spectral problems of homogenization, Tr. Semin. im. Petrovskogo I. G., 22 (2002), 105-120.   Google Scholar

Figure 1.  Sketch of the scattering problem. Left: The sub-region $D\subset \Omega$ contains the small Helmholtz resonators, given by $\Sigma_\varepsilon \subset D$. The number of resonators in the region $D$ is of order $\varepsilon ^{-n}$. We are interested in the effective properties of the meta-material in $D$. Right: The microscopic geometry with the single resonator $R_Y$. The channel width inside $Y$ is of the order $\varepsilon ^p$
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Figure 2.  Sketch of the geometry around the end-point of the channel
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