Technische Universität Dortmund, Fakultät für Mathematik, Vogelpothsweg 87,44227 Dortmund, Germany |
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.
| [1] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [6] |
G. Bouchitté and M. Bellieud,
Homogenization of a soft elastic material reinforced by fibers, Asymptot. Anal., 32 (2002), 153-183.
|
| [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.
|
| [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. |
| [9] |
G. Bouchitté and D. Felbacq,
Homogenization near resonances and artificial magnetism from dielectrics, C. R. Math. Acad. Sci. Paris, 339 (2004), 377-382.
|
| [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.
|
| [11] |
G. Bouchitté and B. Schweizer,
Homogenization of Maxwell's equations in a split ring geometry, Multiscale Model. Simul., 8 (2010), 717-750.
|
| [12] |
G. Bouchitté and B. Schweizer,
Plasmonic waves allow perfect transmission through subwavelength metallic gratings, Netw. Heterog. Media, 8 (2013), 857-878.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [18] |
D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, volume 136 of Applied Mathematical Sciences, Springer-Verlag, New York, 1999. |
| [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. |
| [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. |
| [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. |
| [22] |
R. Kohn and S. Shipman,
Magnetism and homogenization of micro-resonators, Multiscale Modeling & Simulation, 7 (2007), 62-92.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [29] |
J. Pendry,
Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), p3966.
doi: 10.1103/PhysRevLett.85.3966. |
| [30] |
E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, volume 127 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1980. |
| [31] |
B. Schweizer,
The low-frequency spectrum of small Helmholtz resonators, Proc. A., 0339 (2014), 20140339.
doi: 10.1098/rspa.2014.0339. |
| [32] |
B. Schweizer, Resonance meets homogenization -Construction of meta-materials with astonishing properties, Jahresberichte der DMV, 2016.
doi: 10.1365/s13291-016-0153-2. |
| [33] |
V. Veselago,
The electrodynamics of substances with simultaneously negative values of ε and μ, Soviet Physics Uspekhi, 10 (1968), 509-514.
|
| [34] |
V. V. Zhikov,
Two-scale convergence and spectral problems of homogenization, Tr. Semin. im. Petrovskogo I. G., 22 (2002), 105-120.
|
show all references
| [1] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
| [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. |
| [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. |
| [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.
|
| [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. |
| [6] |
G. Bouchitté and M. Bellieud,
Homogenization of a soft elastic material reinforced by fibers, Asymptot. Anal., 32 (2002), 153-183.
|
| [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.
|
| [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. |
| [9] |
G. Bouchitté and D. Felbacq,
Homogenization near resonances and artificial magnetism from dielectrics, C. R. Math. Acad. Sci. Paris, 339 (2004), 377-382.
|
| [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.
|
| [11] |
G. Bouchitté and B. Schweizer,
Homogenization of Maxwell's equations in a split ring geometry, Multiscale Model. Simul., 8 (2010), 717-750.
|
| [12] |
G. Bouchitté and B. Schweizer,
Plasmonic waves allow perfect transmission through subwavelength metallic gratings, Netw. Heterog. Media, 8 (2013), 857-878.
|
| [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. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [18] |
D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, volume 136 of Applied Mathematical Sciences, Springer-Verlag, New York, 1999. |
| [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. |
| [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. |
| [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. |
| [22] |
R. Kohn and S. Shipman,
Magnetism and homogenization of micro-resonators, Multiscale Modeling & Simulation, 7 (2007), 62-92.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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.
|
| [29] |
J. Pendry,
Negative refraction makes a perfect lens, Phys. Rev. Lett., 85 (2000), p3966.
doi: 10.1103/PhysRevLett.85.3966. |
| [30] |
E. Sánchez-Palencia, Nonhomogeneous Media and Vibration Theory, volume 127 of Lecture Notes in Physics, Springer-Verlag, Berlin, 1980. |
| [31] |
B. Schweizer,
The low-frequency spectrum of small Helmholtz resonators, Proc. A., 0339 (2014), 20140339.
doi: 10.1098/rspa.2014.0339. |
| [32] |
B. Schweizer, Resonance meets homogenization -Construction of meta-materials with astonishing properties, Jahresberichte der DMV, 2016.
doi: 10.1365/s13291-016-0153-2. |
| [33] |
V. Veselago,
The electrodynamics of substances with simultaneously negative values of ε and μ, Soviet Physics Uspekhi, 10 (1968), 509-514.
|
| [34] |
V. V. Zhikov,
Two-scale convergence and spectral problems of homogenization, Tr. Semin. im. Petrovskogo I. G., 22 (2002), 105-120.
|
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