Design of acoustic metamaterials through nonlinear programming

The dispersive wave propagation in a periodic metamaterial with tetrachiral topology and inertial local resonators is investigated. The Floquet-Bloch spectrum of the metamaterial is compared with that of the tetrachiral beam lattice material without resonators. The resonators can be designed to open and shift frequency band gaps, that is, spectrum intervals in which harmonic waves do not propagate. Therefore, an optimal passive control of the frequency band structure can be pursued in the metamaterial. To this aim, a suitable constrained nonlinear optimization problem on a compact set of admissible geometrical and mechanical parameters is stated. According to functional requirements, the particular set of parameters which determines the largest low-frequency band gap between a pair of consecutive branches of the Floquet-Bloch spectrum is obtained. The optimization problem is successfully solved by means of a version of the method of moving asymptotes, combined with a quasi-Monte Carlo multi-start technique. Subjects: Materials Science (cond-mat.mtrl-sci) Cite as: arXiv:1603.07717 [cond-mat.mtrl-sci] (or arXiv:1603.07717v2 [cond-mat.mtrl-sci] for this version

Metamaterial filter design via surrogate optimization

Recently, an increasing research effort has been dedicated to analyse transmission and dispersion properties of periodic metamaterials containing resonators, and to optimize the amplitude of selected acoustic band gaps between consecutive dispersion curves in the Floquet-Bloch spectrum. Potential novel applications of this research are in the design of passive mechanical filters/diodes. The present work proposes a way to interpolate the objective functions in such band gap optimization problems, using Radial Basis Functions. The study is motivated by the high computational effort often needed for an exact evaluation of the original objective functions, when using iterative optimization algorithms. By replacing such functions with surrogate objective functions, well-performing suboptimal solutions can be obtained with a small computational effort. Numerical results demonstrate the feasibility of the approach

Towards a cyber physical system for personalised and automatic OSA treatment

Obstructive sleep apnea (OSA) is a breathing disorder that takes place in the course of the sleep and is produced by a complete or a partial obstruction of the upper airway that manifests itself as frequent breathing stops and starts during the sleep. The real-time evaluation of whether or not a patient is undergoing OSA episode is a very important task in medicine in many scenarios, as for example for making instantaneous pressure adjustments that should take place when Automatic Positive Airway Pressure (APAP) devices are used during the treatment of OSA. In this paper the design of a possible Cyber Physical System (CPS) suited to real-time monitoring of OSA is described, and its software architecture and possible hardware sensing components are detailed. It should be emphasized here that this paper does not deal with a full CPS, rather with a software part of it under a set of assumptions on the environment. The paper also reports some preliminary experiments about the cognitive and learning capabilities of the designed CPS involving its use on a publicly available sleep apnea database

Multi-objective optimal design of mechanical metafilters based on principal component analysis

In this paper, an advanced computational method is proposed, whose aim is to obtain an approximately optimal design of a particular class of acoustic metamaterials, by means of a novel combination of multiobjective optimization and dimensionality reduction. Metamaterials are modeled as beam lattices with internal local resonators coupled with the microstructure through a viscoelastic phase. The dynamics is governed by a set of integro-differential equations, that are transformed into the Z-Laplace space in order to derive an eigenproblem whose solution provides the dispersion relation of the free in-plane propagating Bloch waves. A multi-objective optimization problem is stated, whose aim is to achieve the largest multiplicative trade-off between the bandwidth of the first stop band and the one of the successive pass band in the metamaterial frequency spectrum. Motivated by the multi-dimensionality of the design parameters space, the goal above is achieved by integrating numerical optimization with machine learning. Specifically, the problem is solved by combining a sequential linear programming algorithm with principal component analysis, exploited as a data dimensionality reduction technique and applied to a properly sampled field of gradient directions, with the aim to perform an optimized sensitivity analysis. This represents an original way of applying principal component analysis in connection with multi-objective optimization. Successful performances of the proposed optimization method and its computational savings are demonstrated

Design of acoustic metamaterials made of Helmholtz resonators for perfect absorption by using the complex frequency plane

[Otros] Dans cette revue, nous présentons des résultats sur l'absorption acoustique parfaite sub-longueur d'onde faisant appel à des métamatériaux acoustiques avec des résonateurs Helmholtz pour différentes configurations. L'absorption parfaite à basse fréquence nécessite une augmentation du nombre d'états aux basses fréquences ainsi que de trouver les bonnes conditions pour une adaptation d'impédance avec le milieu environnant. Si en outre, on souhaite réduire les dimensions géométriques des structures proposées pour des questions pratiques, on peut utiliser des résonateurs locaux judicieusement conçus afin d'attendre une absorption parfaite sub-longueur d'onde. Les résonateurs de Helmholtz se sont révélés de bons candidats en raison de leur accordabilité aisée de la géométrie, donc de la fréquence de résonance, de la fuite d'énergie et des pertes intrinsèques. Lorsqu'ils sont branchés à un guide d'ondes ou à un milieu environnant, ils se comportent comme des systèmes ouverts, avec pertes et résonances caractérisés par leur fuite d'énergie et leurs pertes intrinsèques. L'équilibre entre ces deux aspects représente la condition de couplage critique et donne lieu à un maximum d'absorption d'énergie. Le mécanisme de couplage critique est ici représenté dans le plan de fréquence complexe afin d'interpréter la condition d'adaptation d'impédance. Dans cette revue, nous discutons en détail la possibilité d'obtenir une absorption parfaite par ces conditions de couplage critiques dans différents systèmes tels que la réflexion (à un port), la transmission (à deux ports) ou les systèmes à trois ports.[EN] In this review, we present the results on sub-wavelength perfect acoustic absorption using acoustic metamaterials made of Helmholtz resonators with different setups. Low frequency perfect absorption requires to increase the number of states at low frequencies and finding the good conditions for impedance matching with the background medium. If, in addition, one wishes to reduce the geometric dimensions of the proposed structures for practical issues, one can use properly designed local resonators and achieve subwavelength perfect absorption. Helmholtz resonators have been shown good candidates due to their easy tunability of the geometry, so of the resonance frequency, the energy leakage and the intrinsic losses. When plugged to a waveguide or a surrounding medium they behave as open, lossy and resonant systems characterized by their energy leakage and intrinsic losses. The balance between these two represents the critical coupling condition and gives rise to maximum energy absorption. The critical coupling mechanism is represented here in the complex frequency plane in order to interpret the impedance matching condition. In this review we discuss in detail the possibility to obtain perfect absorption by these critical coupling conditions in different systems such as reflection (one-port), transmission (two-ports) or three-ports systems.The authors gratefully acknowledge the ANR-RGC METARoom (ANR-18-CE08-0021) project and the project HYPERMETA funded under the program Étoiles Montantes of the Région Pays de la Loire. NJ acknowledges financial support from the Spanish Ministry of Science, Innovation and Universities (MICINN) through grant ¿Juan de la Cierva-Incorporación¿ (IJC2018-037897- I). This article is based upon work from COST Action DENORMS CA15125, supported by COST (European Cooperation in Science and Technology).Romero-García, V.; Jimenez, N.; Theocharis, G.; Achilleos, V.; Merkel, A.; Richoux, O.; Tournat, V.... (2020). Design of acoustic metamaterials made of Helmholtz resonators for perfect absorption by using the complex frequency plane. Comptes Rendus Physique. 21(7-8):713-749. https://doi.org/10.5802/crphys.32S713749217-8[1] Law, M.; Greene, L. E.; Johnson, J. C.; Saykally, R.; Yang, P. Nanowire dye-sensitized solar cells, Nat. Mater., Volume 4 (2005) no. 6, pp. 455-459[2] Derode, A.; Roux, P.; Fink, M. Robust acoustic time reversal with high-order multiple scattering, Phys. Rev. Lett., Volume 75 (1995) no. 23, pp. 4206-4209[3] Chong, Y.; Ge, L.; Cao, H.; Stone, A. D. Coherent perfect absorbers: time-reversed lasers, Phys. Rev. Lett., Volume 105 (2010) no. 5, 053901[4] Mei, J.; Ma, G.; Yang, M.; Yang, Z.; Wen, W.; Sheng, P. Dark acoustic metamaterials as super absorbers for low-frequency sound, Nat. Commun., Volume 3 (2012), 756[5] Engheta, N.; Ziolkowski, R. W. Metamaterials: Physics and Engineering Explorations, John Wiley & Sons, 2006[6] Romero-García, V.; Hladky-Hennion, A. Fundamentals and Applications of Acoustic Metamaterials: From Seismic to Radio Frequency, ISTE Willey, 2019[7] Craster, R.; Guenneau, S. Acoustic Metamaterials, Springer Series in Materials Science, vol. 166, Spinger, 2013[8] Yu, X.; Zhou, J.; Liang, H.; Jiang, Z.; Wu, L. Mechanical metamaterials associated with stiffness, rigidity and compressibility: A brief review, Prog. Mater. Sci., Volume 94 (2018), pp. 114-173[9] Mu, D.; Shu, H.; Zhao, L.; An, S. A review of research on seismic metamaterials, Adv. Eng. Mater., Volume 22 (2020) no. 4, 1901148[10] Liu, Z.; Zhang, X.; Mao, Y.; Zhu, Y. Y.; Yang, Z.; Chan, C. T.; Sheng, P. Locally resonant sonic materials, Science, Volume 289 (2000) no. 5485, pp. 1734-1736[11] Fang, N.; Xi, D.; Xu, J.; Ambati, M.; Srituravanich, W.; Sun, C.; Zhang, X. Ultrasonic metamaterials with negative modulus, Nat. Mater., Volume 5 (2006) no. 6, pp. 452-456[12] Acoustic Metamaterials and Phononic Crystals (Deymier, P., ed.), Springer-Verlag, Berlin, Heidelberg, 2013[13] Ma, G.; Sheng, P. Acoustic metamaterials: From local resonances to broad horizons, Sci. Adv., Volume 2 (2016) no. 2, e1501595[14] Yang, M.; Sheng, P. Sound absorption structures: From porous media to acoustic metamaterials, Annu. Rev. Mater. Res., Volume 47 (2017), pp. 83-114[15] Bobrovnitskii, Y. I.; Tomilina, T. Sound absorption and metamaterials: A review, Acoust. Phys., Volume 64 (2018) no. 5, pp. 519-526[16] Lagarrigue, C.; Groby, J.-P.; Tournat, V.; Dazel, O.; Umnova, O. Absorption of sound by porous layers with embedded periodic array of resonant inclusions, J. Acoust. Soc. Am., Volume 134 (2013), pp. 4670-4680[17] Groby, J.-P.; Nennig, B.; Lagarrigue, C.; Brouard, B.; Dazel, O.; Tournat, V. Enhancing the absorption properties of acoustic porous plates by periodically embedding Helmholtz resonators, J. Acoust. Soc. Am., Volume 137 (2015) no. 1, pp. 273-280[18] Lagarrigue, C.; Groby, J.-P.; Dazel, O.; Tournat, V. Design of metaporous supercells by genetic algorithm for absorption optimization on a wide frequency band, Appl. Acoust., Volume 102 (2016), pp. 49-54[19] Cai, X.; Guo, Q.; Hu, G.; Yang, J. Ultrathin low-frequency sound absorbing panels based on coplanar spiral tubes or coplanar Helmholtz resonators, Appl. Phys. Lett., Volume 105 (2014) no. 12, 121901[20] Li, Y.; Assouar, B. M. Acoustic metasurface-based perfect absorber with deep subwavelength thickness, Appl. Phys. Lett., Volume 108 (2016) no. 6, 063502[21] Wang, Y.; Zhao, H.; Yang, H.; Zhong, J.; Wen, J. A space-coiled acoustic metamaterial with tunable low-frequency sound absorption, Europhys. Lett., Volume 120 (2018) no. 5, 54001[22] Shen, C.; Cummer, S. A. Harnessing multiple internal reflections to design highly absorptive acoustic metasurfaces, Phys. Rev. Appl., Volume 9 (2018) no. 5, 054009[23] Yang, Z.; Mei, J.; Yang, M.; Chan, N.; Sheng, P. Membrane-type acoustic metamaterial with negative dynamic mass, Phys. Rev. Lett., Volume 101 (2008) no. 20, 204301[24] Guo, X.; Gusev, V.; Bertoldi, K.; Tournat, V. Manipulating acoustic wave reflexion by a nonlinear elastic metasurface, J. Appl. Phys., Volume 123 (2018) no. 12, 124901[25] Guo, X.; Gusev, V.; Tournat, V.; Deng, B.; Bertoldi, K. Frequency-doubling effect in acoustic reflection by a nonlinear, architected rotating-square metasurface, Phys. Rev. E, Volume 99 (2019), 052209[26] Bradley, C. E. Acoustic bloch wave propagation in a periodic waveguide Tech. rep., Technical Report of Applied Research Laboratories, Report No. ARL-TR-91-19 (July), The University of Texas at Austin (1991)[27] Sugimoto, N.; Horioka, T. Dispersion characteristics of sound waves in a tunnel with an array of Helmholtz resonators, J. Acoust. Soc. Am., Volume 97 (1995) no. 3, pp. 1446-1459[28] Romero-García, V.; Theocharis, G.; Richoux, O.; Merkel, A.; Tournat, V.; Pagneux, V. Perfect and broadband acoustic absorption by critically coupled sub-wavelength resonators, Sci. Rep., Volume 6 (2016), 19519[29] Ma, G.; Yang, M.; Xiao, S.; Yang, Z.; Sheng, P. Acoustic metasurface with hybrid resonances, Nat. Mater., Volume 13 (2014) no. 9, pp. 873-878[30] Yang, M.; Meng, C.; Fu, C.; Li, Y.; Yang, Z.; Sheng, P. Subwavelength total acoustic absorption with degenerate resonators, Appl. Phys. Lett., Volume 107 (2015) no. 10, 104104[31] Aurégan, Y. Ultra-thin low frequency perfect sound absorber with high ratio of active area, Appl. Phys. Lett., Volume 113 (2018), 201904[32] Groby, J.-P.; Huang, W.; Lardeau, A.; Aurégan, Y. The use of slow waves to design simple sound absorbing materials, J. Appl. Phys., Volume 117 (2015) no. 12, 124903[33] Leroy, V.; Strybulevych, A.; Lanoy, M.; Lemoult, F.; Tourin, A.; Page, J. H. Superabsorption of acoustic waves with bubble metascreens, Phys. Rev. B, Volume 91 (2015), 020301[34] Fernández-Marín, A. A.; Jiménez, N.; Groby, J.-P.; Sánchez-Dehesa, J.; Romero-García, V. Aerogel-based metasurfaces for perfect acoustic energy absorption, Appl. Phys. Lett., Volume 115 (2019), 061901[35] Long, H.; Cheng, Y.; Tao, J.; Liu, X. Perfect absorption of low-frequency sound waves by critically coupled subwavelength resonant system, Appl. Phys. Lett., Volume 110 (2017) no. 2, 023502[36] Romero-García, V.; Theocharis, G.; Richoux, O.; Pagneux, V. Use of complex frequency plane to design broadband and sub-wavelength absorbers, J. Acoust. Soc. Am., Volume 139 (2016) no. 6, pp. 3395-3403[37] Jiménez, N.; Huang, W.; Romero-García, V.; Pagneux, V.; Groby, J.-P. Ultra-thin metamaterial for perfect and quasi-omnidirectional sound absorption, Appl. Phys. Lett., Volume 109 (2016) no. 12, 121902[38] Jiménez, N.; Romero-García, V.; Pagneux, V.; Groby, J.-P. Quasiperfect absorption by subwavelength acoustic panels in transmission using accumulation of resonances due to slow sound, Phys. Rev. B, Volume 95 (2017), 014205[39] Jiménez, N.; Romero-García, V.; Pagneux, V.; Groby, J.-P. Rainbow-trapping absorbers: Broadband, perfect and asymmetric sound absorption by subwavelength panels for transmission problems, Sci. Rep., Volume 7 (2017) no. 1, 13595[40] Jiménez, N.; Cox, T. J.; Romero-García, V.; Groby, J.-P. Metadiffusers: Deep-subwavelength sound diffusers, Sci. Rep., Volume 7 (2017) no. 1, 5389[41] Jiménez, N.; Romero-García, V.; Groby, J.-P. Perfect absorption of sound by rigidly-backed high-porous materials, Acta Acust. United Acust., Volume 104 (2018) no. 3, pp. 396-409[42] Merkel, A.; Theocharis, G.; Richoux, O.; Romero-García, V.; Pagneux, V. Control of acoustic absorption in one-dimensional scattering by resonant scatterers, Appl. Phys. Lett., Volume 107 (2015) no. 24, 244102[43] Achilleos, V.; Richoux, O.; Theocharis, G. Coherent perfect absorption induced by the nonlinearity of a Helmholtz resonator, J. Acoust. Soc. Am., Volume 140 (2016), EL94[44] Long, H.; Cheng, Y.; Liu, X. Asymmetric absorber with multiband and broadband for low-frequency sound, Appl. Phys. Lett., Volume 111 (2017) no. 14, 143502[45] Merkel, A.; Romero-García, V.; Groby, J.-P.; Li, J.; Christensen, J. Unidirectional zero sonic reflection in passive pt-symmetric willis media, Phys. Rev. B, Volume 98 (2018), 201102(R)[46] Monsalve, E.; Maurel, A.; Petitjeans, P.; Pagneux, V. Perfect absorption of water waves by linearor nonlinear critical coupling, Appl. Phys. Lett., Volume 114 (2018), 013901[47] Schwan, L.; Umnova, O.; Boutin, C. Sound absorption and reflection from a resonant metasurface: Homogenisation model with experimental validation, Wave Motion, Volume 72 (2017), pp. 154-172[48] Huang, S.; Fang, X.; Wang, X.; Assouar, B.; Cheng, Q.; Li, Y. Acoustic perfect absorbers via spiral metasurfaces with embedded apertures, Appl. Phys. Lett., Volume 113 (2018) no. 23, 233501[49] Elayouch, A.; Addouche, M.; Khelif, A. Extensive tailorability of sound absorption using acoustic metamaterials, J. Appl. Phys., Volume 124 (2018) no. 15, 155103[50] Maurel, A.; Mercier, J.-F.; Pham, K.; Marigo, J.-J.; Ourir, A. Enhanced resonance of sparse arrays of Helmholtz resonators—application to perfect absorption, J. Acoust. Soc. Am., Volume 145 (2019) no. 4, pp. 2552-2560[51] Romero-García, V.; Jiménez, N.; Groby, J.-P.; Merkel, A.; Tournat, V.; Theocharis, G.; Richoux, O.; Pagneux, V. Perfect absorption in mirror-symmetric acoustic metascreens, Phys. Rev. Appl. (2020), 054055[52] Bliokh, K. Y.; Bliokh, Y. P.; Freilikher, V.; Savel’ev, S.; Nori, F. Colloquium: Unusual resonators: Plasmonics, metamaterials, and random media, Rev. Mod. Phys., Volume 80 (2008) no. 4, pp. 1201-1213[53] Richoux, O.; Achilleos, V.; Theocharis, G.; Brouzos, I. Subwavelength interferometric control of absorption in three-port acoustic network, Sci. Rep., Volume 8 (2018) no. 1, 12328[54] Piper, J. R.; Liu, V.; Fan, S. Total absorption by degenrate critical coupling, Appl. Phys. Lett., Volume 104 (2014), 251110[55] Chong, Y.; Ge, L.; Cao, H.; Stone, A. Coherent perfect absorbers: Time-reversed lasers, Phys. Rev. Lett., Volume 105 (2010), 053901[56] Wei, P.; Croënne, C.; Chu, S. T.; Li, J. Symmetrical and anti-symmetrical coherent prefect absorption for acoustic waves, Appl. Phys. Lett., Volume 104 (2014), 121902[57] Groby, J.-P.; Pommier, R.; Aurégan, Y. Use of slow sound to design perfect and broadband passive sound absorbing materials, J. Acoust. Soc. Am., Volume 139 (2016) no. 4, pp. 1660-1671[58] Luk, T.; Campione, S.; Kim, I.; Feng, S.; Jun, Y.; Liu, S.; Wright, J.; Brener, I.; Catrysse, P.; Fan, S.; Sinclair, M. Directional perfect absorption using deep subwavelength lowpermittivity films, Phys. Rev. B, Volume 90 (2014), 085411[59] Pagneux, V. Trapped modes and edge resonances in acoustics and elasticity, Dynamic Localization Phenomena in Elasticity, Acoustics and Electromagnetism (CISM International Centre for Mechanical Sciences, vol. 547), Springer, Vienna, 2013, pp. 181-223[60] Stinson, M. R. The propagation of plane sound waves in narrow and wide circular tubes, and generalization to uniform tubes of arbitrary cross-sectional shape, J. Acoust. Soc. Am., Volume 89 (1991) no. 2, pp. 550-558[61] Theocharis, G.; Richoux, O.; Romero-Garcìa, V.; Merkel, A.; Tournat, V. Limits of slow sound and transparency in lossy locally resonant periodic structures, New J. Phys., Volume 16 (2014), 093017[62] Kergomard, J.; Garcia, A. Simple discontinuities in acoustic waveguides at low frequencies: critical analysis and formulae, J. Sound Vib., Volume 114 (1987) no. 3, pp. 465-479[63] Dubos, V.; Kergomard, J.; Khettabi, A.; Dalmont, J.-P.; Keefe, D.; Nederveen, C. Theory of sound propagation in a duct with a branched tube using modal decomposition, Acta Acust. United Acust., Volume 85 (1999) no. 2, pp. 153-169[64] Mechel, F. P. Formulas of Acoustics, Springer Science & Business Media, Springer-Verlag, Heidelberg, 2008[65] Born, M.; Wolf, E. Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Ligth, Cambridge University Press, UK, 1999[66] Zwikker, C.; Kosten, C. Sound Absorbing Materials, Elsevier Publishing Company, Amsterdam, 1949[67] Xu, Q.; Sandhu, S.; Povinelli, M. L.; Shakya, J.; Fan, S.; Lipson, M. Experimental realization of an on-chip all-optical analogue to electromagnetically induced transparency, Phys. Rev. Lett., Volume 96 (2006), 123901[68] Yang, X.; Yu, M.; Kwong, D.-L.; Wong, C. W. All-optical analog to electromagnetically induced transparency in multiple coupled photonic crystal cavities, Phys. Rev. Lett., Volume 102 (2009), 173902[69] Boudouti, E. H. E.; Mrabti, T.; Al-Wahsh, H.; Djafari-Rouhani, B.; Akjouj, A.; Dobrzynski, L. Transmission gaps and Fano resonances in an acoustic waveguide: analytical model, J. Phys.: Condens. Matter, Volume 20 (2008), 255212[70] Santillan, A.; Bozhevolnyi, S. I. Acoustic transparency and slow sound using detuned acoustic resonators, Phys. Rev. B, Volume 84 (2011), 064394[71] Mouadili, A.; Boudouti, E. H. E.; Soltani, A.; Talbi, A.; Djafari-Rouhani, B.; Akjouj, A.; Haddadi, K. Electromagnetically induced absorption in detuned stub waveguides: a simple analytical and experimental model, J. Phys.: Condens. Matter, Volume 26 (2014), 505901[72] Xu, Y.; Li, Y.; Lee, R. K.; Yariv, A. Scattering-theory analysis of waveguide–resonator coupling, Phys. Rev. E, Volume 62 (2000), pp. 7389-7404[73] Powell, M. J. A fast algorithm for nonlinearly constrained optimization calculations, Numerical Analysis, Springer, 1978, pp. 144-157[74] Cox, T. J.; D’Antonio, P. Acoustic Absorbers and Diffusers: Theory, Design and Application, CRC Press, 2016[75] Groby, J.-P.; Ogam, E.; De Ryck, L.; Sebaa, N.; Lauriks, W. Analytical method for the ultrasonic characterization of homogeneous rigid porous materials from transmitted and reflected coefficients, J. Acoust. Soc. Am., Volume 127 (2010) no. 2, pp. 764-772[76] Tsakmakidis, K. L.; Boardman, A. D.; Hess, O. Trapped rainbow storage of light in metamaterials, Nature, Volume 450 (2007) no. 7168, pp. 397-401[77] Zhu, J.; Chen, Y.; Zhu, X.; Garcia-Vidal, F. J.; Yin, X.; Zhang, W.; Zhang, X. Acoustic rainbow trapping, Sci. Rep., Volume 3 (2013), 1728[78] Romero-Garcia, V.; Picó, R.; Cebrecos, A.; Sanchez-Morcillo, V.; Staliunas, K. Enhancement of sound in chirped sonic crystals, Appl. Phys. Lett., Volume 102 (2013) no. 9, 091906[79] Ni, X.; Wu, Y.; Chen, Z.-G.; Zheng, L.-Y.; Xu, Y.-L.; Nayar, P.; Liu, X.-P.; Lu, M.-H.; Chen, Y.-F. Acoustic rainbow trapping by coiling up space, Sci. Rep., Volume 4 (2014), 7038[80] Colombi, A.; Colquitt, D.; Roux, P.; Guenneau, S.; Craster, R. V. A seismic metamaterial: The resonant metawedge, Sci. Rep., Volume 6 (2016), 27717[81] Jan, A. U.; Porter, R. Transmission and absorption in a waveguide with a metamaterial cavity, J. Acoust. Soc. Am., Volume 144 (2018) no. 6, pp. 3172-318