Prediction of transmission, reflection and absorption coefficients of periodic structures using a hybrid wave based – finite element unit cell method. (English) Zbl 1380.74110
Summary: This paper presents a hybrid wave based method - finite element unit cell method to predict the absorption, reflection and transmission properties of arbitrary, two-dimensional periodic structures. The planar periodic structure, represented by its unit cell combined with Bloch-Floquet periodicity boundary conditions, is modelled within the finite element method, allowing to represent complex geometries and to include any type of physics. The planar periodic structure is coupled to semi-infinite acoustic domains above and/or below, in which the dynamic pressure field is modelled with the wave based method, applying a wave function set that fulfills the Helmholtz equation and satisfies the Sommerfeld radiation condition and the Bloch-Floquet periodicity conditions inherently. The dynamic fields described within both frameworks are coupled using a direct coupling strategy, accounting for the mutual dynamic interactions via a weighted residual formulation. The method explicitly accounts for the interaction between the unit cell and the surrounding acoustic domain, also accounting for higher order periodic waves. The convergence of the method is analysed and its applicability is shown for a variety of problems, proving it to be a useful tool combining the strengths of two methods.
MSC:
| 74S05 | Finite element methods applied to problems in solid mechanics |
| 74E05 | Inhomogeneity in solid mechanics |
| 74J99 | Waves in solid mechanics |
Keywords:
periodic structures; absorption; transmission; wave-based method; finite element method; Bloch-FloquetReferences:
| [1] | Taub, A. I.; Krajewski, P. E.; Luo, A. A.; Owens, J. N., The evolution of technology for materials processing over the last 50 years: the automotive example, J. Miner. Met. Mater. Soc., 59, 2, 48-57 (2007) |
| [2] | Goines, L.; Hagler, L., Noise pollution: a modern plague, South. Med. J., 100, 287-294 (2007) |
| [3] | Sgard, F. C.; Olny, X.; Atalla, N.; Castel, F., On the use of perforations to improve the sound absorption of porous materials, Appl. Acoust., 66, 625-651 (2005) |
| [4] | Groby, J.-P.; Brouard, B.; Dazel, O.; Nennig, B.; Kelders, L., Enhancing rigid frame porous layer absorption with three-dimensional periodic irregularities, J. Acoust. Soc. Am., 133, 821-831 (2013) |
| [5] | Groby, J.-P.; Lagarrigue, C.; Brouard, B.; Dazel, O.; Tournat, V., Using simple shape three-dimensional rigid inclusions to enhance porous layer absorption, J. Acoust. Soc. Am., 136, 1139-1148 (2014) |
| [6] | Weisser, T.; Groby, J.-P.; Dazel, O.; Gaultier, F.; Deckers, E.; Futatsugi, S.; Monteiro, L., Acoustic behavior of a rigidly backed poroelastic layer with periodic resonant inclusions by a multiple scattering approach, J. Acoust. Soc. Am., 139, 617-629 (2016) |
| [7] | Boutin, C.; Bécot, F.-X., Theory and experiments on poro-acoustics with inner resonators, Wave Motion, 54, 76-99 (2015) |
| [8] | Lagarrigue, C.; Groby, J.-P.; Tournat, V.; Dazel, O.; Umnova, O., Absorption of sound by porous layers with embedded periodic arrays of resonant inclusions, J. Acoust. Soc. Am., 134, 4670-4680 (2013) |
| [9] | Groby, J.-P.; Lagarrigue, C.; Brouard, B.; Dazel, O.; Tournat, V.; Nennig, B., Enhancing the absorption properties of acoustic porous plates by periodically embedding Helmholtz resonators, J. Acoust. Soc. Am., 137, 273-280 (2015) |
| [10] | 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., 102, 49-54 (2016) |
| [11] | Sheng, P.; Zhang, X. X.; Liu, Z.; Chan, C. T., Locally resonant sonic materials, Physica B, Condens. Matter, 338, 1, 201-205 (2003) |
| [12] | Zhou, X.; Liu, X.; Hu, G., Elastic metamaterials with local resonances: an overview, Theor. Appl. Mech. Lett., 2, 4, Article 041001 pp. (2012) |
| [13] | Hussein, M. I.; Leamy, M. J.; Ruzzene, M., Dynamics of phononic materials and structures: historical origins, recent progress, and future outlook, Appl. Mech. Rev., 66, 4, Article 040802 pp. (2014) |
| [14] | Claeys, C.; Vergote, K.; Sas, P.; Desmet, W., On the potential of tuned resonators to obtain low-frequency vibrational stop bands in periodic panels, J. Sound Vib., 332, 6, 1418-1436 (2013) |
| [15] | Goffaux, C.; Sánchez-Dehesa, J.; Yeyati, A.; Lambin, P.; Khelif, A.; Vasseur, J.; Djafari-Rouhani, B., Evidence of Fano-like interference phenomena in locally resonant materials, Phys. Rev. Lett., 88, Article 225502 pp. (2002) |
| [16] | Claeys, C.; Deckers, E.; Pluymers, B.; Desmet, W., A lightweight vibro-acoustic metamaterial demonstrator: numerical and experimental investigation, Mech. Syst. Signal Process., 70-71, 853-880 (2016) |
| [17] | Xiao, Y.; Wen, J.; Wen, X., Sound transmission loss of metamaterial-based thin plates with multiple subwavelength arrays of attached resonators, J. Sound Vib., 331, 25, 5408-5423 (2012) |
| [18] | Zhang, H.; Wen, J.; Xiao, Y.; Wang, G.; Wen, X., Sound transmission loss of metamaterial thin plates with periodic subwavelength arrays of shunted piezoelectric patches, J. Sound Vib., 343 (2015) |
| [19] | Song, Y.; Feng, L.; Wen, J.; Yu, D.; Wen, X., Reduction of the sound transmission of a periodic sandwich plate using the stop band concept, Compos. Struct., 128, 428-436 (2015) |
| [20] | Yang, Z.; Dai, H. M.; Chan, N. H.; Ma, G. C.; Sheng, P., Acoustic metamaterial panels for sound attenuation in the 50-1000 Hz regime, Appl. Phys. Lett., 96, Article 041906 pp. (2010) |
| [21] | Mei, J.; Ma, G.; Yang, M.; Yang, Z.; Wen, W.; Sheng, P., Dark acoustic metamaterials as super absorbers for low-frequency sound, Nat. Commun., 3, 756 (2012) |
| [22] | Hu, X.; Chan, C.; Zi, J., Two-dimensional sonic crystals with Helmholtz resonators, Phys. Rev. E, 71, Article 055601 pp. (2005) |
| [23] | Elford, D.; Chalmers, L.; Kusmartsev, F.; Swallowe, G., Matryoshka locally resonant sonic crystal, J. Acoust. Soc. Am., 130, 2746-2755 (2011) |
| [24] | Miniaci, M.; Krushynska, A.; Movchan, A. B.; Bosia, F.; Pugno, N. M., Spider web-inspired acoustic metamaterials, Appl. Phys. Lett., 109, Article 071905 pp. (2016) |
| [25] | Li, Y.; Liang, B.; Ming, Z.; Ye Zou, X.; Zhao, X., Reflected wavefront manipulation based on ultrathin planar acoustic metasurfaces, Sci. Rep., 3, Article 02546 pp. (2013) |
| [26] | Ma, G.; Yang, M.; Xiao, S.; Sheng, P., Acoustic metasurface with hybrid resonances, Nat. Mater., 13, 873-878 (2015) |
| [27] | Schwan, L.; Umnova, O.; Boutin, C., Sound absorption and reflection from a resonant metasurface: homogenisation model with experimental validation, Wave Motion, 72, 154-172 (2017) |
| [28] | Ren, S. W.; Meng, H.; Xin, F. X.; Lu, T. J., Ultrathin multi-slit metamaterial as excellent sound absorber: influence of micro-structure, J. Appl. Phys., 119, Article 014901 pp. (2016) |
| [29] | Yang, J.; Lee, J. S.; Kim, Y. Y., Metaporous layer to overcome the thickness constraint for broadband sound absorption, J. Appl. Phys., 117, Article 174903 pp. (2015) |
| [30] | Ruiz, H.; Claeys, C.; Deckers, E.; Desmet, W., Numerical and experimental study of the effect of microslits on the normal absorption of structural metamaterials, Mech. Syst. Signal Process., 70, 904-918 (2016) |
| [31] | Groby, J.-P.; Huang, W.; Lardeau, A.; Aurégan, Y., The use of slow sound to design simple sound absorbing materials, J. Appl. Phys., 117, Article 124903 pp. (2015) |
| [32] | Groby, J.-P.; Pommier, R.; Aurégan, Y., Use of slow sound to design perfect and broadband passive sound absorbing materials, J. Acoust. Soc. Am., 139, Article 16601671 pp. (2016) |
| [33] | 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., 109, Article 121902 pp. (2016) |
| [34] | Yang, J.; Lee, J. S.; Kim, Y. Y., Multiple slow waves in metaporous layers for broadband sound absorption, J. Phys. D, Appl. Phys., 50, Article 015301 pp. (2017) |
| [35] | Setaki, F.; Tenpierik, M.; Turin, M.; van Timmeren, A., Acoustic absorbers by additive manufacturing, Build. Environ., 72, 188-200 (2014) |
| [36] | Bloch, F., Über die Quantenmechanik der Elektronen in Kristallgittern, Z. Phys. A, Hadrons Nucl., 52, 555-600 (1929) · JFM 54.0990.01 |
| [37] | Brillouin, L., Wave Propagation in Periodic Structures (1946), McGraw-Hill Book Company · Zbl 0063.00607 |
| [38] | Claeys, C.; Sas, P.; Desmet, W., On the acoustic radiation efficiency of local resonance based stop band materials, J. Sound Vib., 333, 3203-3213 (2014) |
| [39] | Mead, D. J., Plates with regular stiffening in acoustic media: vibration and radiation, J. Acoust. Soc. Am., 88, 1, 391-401 (1990) |
| [40] | Li, P.; Yao, S.; Zhou, X., Effective medium theory of thin-plate acoustic metamaterials, J. Acoust. Soc. Am., 135, 1844-1852 (2014) |
| [41] | Oudich, M.; Zhou, X.; Badreddine Assouar, M., General analytical approach for sound transmission loss analysis through a thick metamaterial plate, J. Appl. Phys., 116, 19, Article 193509 pp. (2014) |
| [42] | Brouard, B.; Lafarge, D.; Allard, J. F., A general method of modeling sound propagation in layered media, J. Sound Vib., 183, 129-142 (1995) · Zbl 0973.74579 |
| [43] | Parrinello, A.; Ghiringhelli, G. L., Transfer matrix representation for periodic planar media, J. Sound Vib., 371, 196-209 (2016) |
| [44] | Deckers, E.; Claeys, C.; Atak, O.; Groby, J.-P.; Dazel, O.; Desmet, W., A wave based method to predict the absorption, reflection and transmission coefficient of two-dimensional rigid frame porous structures with periodic inclusions, J. Comput. Phys., 312, 115-138 (2016) · Zbl 1351.76273 |
| [45] | Groby, J.-P.; Wirgin, A.; Ogam, E., Acoustic response of a periodic distribution of macroscopic inclusions within a rigid frame porous plate, Waves Random Complex Media, 18, 409-433 (2008) · Zbl 1271.76308 |
| [46] | Cook, R. R.; Malkus, D. S.; Pelsha, M. E.; Witt, R. J., Concepts and Applications of Finite Element Analysis (2002), Wiley: Wiley Madison |
| [47] | Givoli, D., High-order local non-reflecting boundary conditions: a review, Wave Motion, 39, 319-326 (2004) · Zbl 1163.74356 |
| [48] | Bettess, P., Infinite Elements (1992), Penshaw Press |
| [49] | Berenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 185-200 (1994) · Zbl 0814.65129 |
| [50] | Bécache, E.; Joly, P.; Tsogka, C., Fictitious domains, mixed finite elements and perfectly matched layers for 2D elastic wave propagation, J. Comput. Acoust., 9, 3, 1175-1203 (2001) · Zbl 1360.74156 |
| [51] | Bermúdez, A.; Hervella-Nieto, L.; Prieto, A.; Rodríguez, R., Perfectly matched layers for time-harmonic second order elliptic problems, Arch. Comput. Methods Eng., 17, 77-107 (2010) · Zbl 1359.76217 |
| [52] | Assouar, B.; Oudich, M.; Zhou, X., Acoustic metamaterials for sound mitigation, C. R. Phys., 17 (2016) |
| [53] | Johnson, S. G., Notes on perfectly matched layers |
| [54] | Desmet, W., A Wave Based Prediction Technique for Coupled Vibro-Acoustic Analysis (1998), KU Leuven, division PMA, PhD thesis 98D12 |
| [55] | Deckers, E.; Atak, O.; Coox, L.; D’Amico, R.; Devriendt, H.; Jonckheere, S.; Koo, K.; Pluymers, B.; Vandepitte, D.; Desmet, W., The Wave Based Method: an overview of 15 years of research, Wave Motion, 51, 550-565 (2014) · Zbl 1456.35006 |
| [56] | Trefftz, E., Ein Gegenstück zum Ritzschen Verfahren, (Proceedings of the 2nd International Congress on Applied Mechanics. Proceedings of the 2nd International Congress on Applied Mechanics, Zurich, Switzerland (1926)), 131-137 · JFM 52.0483.02 |
| [57] | Collet, M.; Ouisse, M.; Ruzzene, M.; Ichchou, M. N., Floquet-Bloch decomposition for the computation of dispersion of two-dimensional periodic, damped mechanical systems, Int. J. Solids Struct., 48, 2837-2848 (2011) |
| [58] | Mace, B.; Manconi, E., Modelling wave propagation in two-dimensional structures using finite element analysis, J. Sound Vib., 318, 3169-3180 (2008) |
| [59] | Landau, L. D.; Lifshitz, E. M., Theory of Elasticity (1975), Pergamon Press |
| [60] | Reissner, E., The effect of transverse shear deformation on the bending of elastic plates, J. Appl. Mech., 12, 69-77 (1945) · Zbl 0063.06470 |
| [61] | Mindlin, R. D., Influence of rotary inertia and shear on flexural motions of isotropic, elastic plates, J. Appl. Mech., 18, 31-38 (1951) · Zbl 0044.40101 |
| [62] | Biot, M. A., The theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. II. Higher frequency range, J. Acoust. Soc. Am., 28, 168-191 (1956) |
| [63] | Atalla, N.; Panneton, R.; Debergue, P., Enhanced weak integral formulation for the mixed \((u_{\_}, p_{\_})\) poroelastic equations, J. Acoust. Soc. Am., 109, 3065-3068 (2001) |
| [64] | Jonckheere, S., Wave Based and Hybrid Methodologies for Vibro-Acoustic Simulation with Complex Damping Treatments (2015), KU Leuven, PhD thesis |
| [65] | Zienkiewicz, O. C.; Taylor, R. L., The Finite Element Method, vol. 3 (1977), McGraw-Hill: McGraw-Hill London · Zbl 0435.73072 |
| [66] | Colton, D.; Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory (1998), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0893.35138 |
| [67] | Van Genechten, B.; Atak, O.; Bergen, B.; Deckers, E.; Jonckheere, S.; Lee, J. S.; Maressa, A.; Vergote, K.; Pluymers, B.; Vandepitte, D.; Desmet, W., An efficient Wave Based Method for solving Helmholtz problems in three-dimensional bounded domains, Eng. Anal. Bound. Elem., 36, 63-75 (2012) · Zbl 1259.76057 |
| [68] | Huybrechs, Daan, On the benefits of ill-conditioning in Trefftz-type methods and other non-polynomial discretisation of wave problems, (Innovations in Wave Modelling 2017 (July 2017)) |
| [69] | Van Genechten, B.; Vergote, K.; Vandepitte, D.; Desmet, W., A Multi-Level Wave Based numerical modelling framework for the steady-state dynamic analysis of bounded Helmholtz problems with multiple inclusions, Comput. Methods Appl. Mech. Eng., 199, 1881-1905 (2010) · Zbl 1231.76222 |
| [70] | van Hal, B.; Desmet, W.; Vandepitte, D.; Sas, P., Hybrid finite element - wave based method for acoustic problems, Comput. Assist. Mech. Eng. Sci., 11, 375-390 (2003) · Zbl 1360.76157 |
| [71] | Schur, I., Über die Potenzreihen die im Innern des Einheitkreises beschränkt sind, I, J. Reine Angew. Math., 147, 205-232 (1917) · JFM 46.0475.01 |
| [72] | Schur, I., Über die Potenzreihen die im Innern des Einheitkreises beschränkt sind, II, J. Reine Angew. Math., 148, 122-145 (1918) · JFM 46.0475.01 |
| [73] | MATLAB, Solve systems of linear equations \(A x = B\) for \(x (``2017)\), Online |
| [74] | Allard, J. F.; Atalla, N., Propagation of Sound in Porous Media: Modeling Sound Absorbing Materials (2009), John Wiley & Sons: John Wiley & Sons West Sussex, United Kingdom |
| [75] | Bouillard, P.; Ihlenburg, R., Error estimation and adaptivity for the finite element method in acoustics: 2D and 3D applications, Comput. Methods Appl. Mech. Eng., 176, 147-163 (1999) · Zbl 0954.76040 |
| [76] | Hörlin, N.-E.; Nordström, M.; Göransson, P., A 3-D hierarchical FE formulation of Biot’s equations for elasto-acoustic modelling of porous media, J. Sound Vib., 245, 633-652 (2001) |
| [77] | Groby, J.-P.; Duclos, A.; Dazel, O.; Boeckx, L.; Kelders, L., Enhancing absorption coefficient of a backed rigid frame porous layer by embedding circular periodic inclusions, J. Acoust. Soc. Am., 130, 3771-3780 (2011) |
| [78] | Van Genechten, B.; Vandepitte, D.; Desmet, W., A direct hybrid finite element - Wave based modelling technique for efficient coupled vibro-acoustic analysis, Comput. Methods Appl. Mech. Eng., 200, 742-764 (2011) · Zbl 1225.74110 |
| [79] | Jonckheere, S.; Deckers, E.; Van Genechten, B.; Vandepitte, D.; Desmet, W., A direct hybrid Finite Element - Wave Based Method for steady-state analysis of acoustic cavities with poro-elastic damping layers using the coupled Helmholtz-Biot equations, Comput. Methods Appl. Mech. Eng., 263, 144-157 (2013) · Zbl 1286.76133 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
