Abstract
Inversely designing and optimizing topological structures of phononic crystals that dominate extraordinary wave characteristics has become a research hotspot. In this study, a joint framework combining a data-driven deep learning model with the semi-analytical two-dimensional periodic spectral finite element method is applied to achieve the inverse design and optimization of topology. A convolutional neural network and a generative adversarial network are trained for inverse design. Meanwhile, the semi-analytical periodic approach is utilized to analyze the wave characteristics of two-dimensional phononic crystals. Through the proactive tuning of the band structures, the topologies of phononic crystals are inversely on-demand designed and optimized for the anticipated partial bandgap or complete bandgap, respectively, which revealed the unidirectional wave transmission or vibration isolation characteristics within the desired frequency segment. The unidirectional wave propagation and vibration isolation performance are validated through numerical simulations. This work holds the potential to benefit the design, optimization, and application of metamaterials.
Similar content being viewed by others
References
Kushwaha, M.S., Halevi, P., Dobrzynski, L., Djafari-Rouhani, B.: Acoustic band structure of periodic elastic composites. Phys. Rev. Lett. 71, 2022 (1993)
Wang, Y.F., Wang, Y.Z., Wu, B., Chen, W., Wang, Y.S.: Tunable and active phononic crystals and metamaterials. Appl. Mech. Rev. 72, 040801 (2020)
Zhang, X., Liu, Z.: Negative refraction of acoustic waves in two-dimensional phononic crystals. Appl. Phys. Lett. 85, 341–343 (2004)
Lee, M.K., Ma, P.S., Lee, I.K., Kim, H.W., Kim, Y.Y.: Negative refraction experiments with guided shear-horizontal waves in thin phononic crystal plates. Appl. Phys. Lett. 98, 011909 (2011)
Li, X.F., Ni, X., Feng, L., Lu, M.H., He, C., Chen, Y.F.: Tunable unidirectional sound propagation through a sonic-crystal-based acoustic diode. Phys. Rev. Lett. 106, 084301 (2011)
Li, Y., Shen, C., Xie, Y., Li, J., Wang, W., Cummer, S.A., Jing, Y.: Tunable asymmetric transmission via lossy acoustic metasurfaces. Phys. Rev. Lett. 119, 035501 (2017)
Ma, N.F., Han, Q., Han, S.H., Li, C.L.: Hierarchical re-entrant honeycomb metamaterial for energy absorption and vibration insulation. Int. J. Mech. Sci. 250, 108307 (2023)
Jiang, T.J., Han, Q., Li, C.L.: Design and bandgap optimization of multi-scale composite origami-inspired metamaterials. Int. J. Mech. Sci. 248, 108233 (2023)
Jiang, T.J., Han, Q., Li, C.L.: Topologically tunable local-resonant origami metamaterials for wave transmission and impact mitigation. J. Sound Vib. 548, 117548 (2023)
Xie, B., Wang, H.X., Zhang, X., Zhan, P., Jiang, J.H., Lu, M., Chen, Y.: Higher-order band topology. Nat. Rev. Phys. 3, 520–532 (2021)
Wang, P., Lu, L., Bertoldi, K.: Topological phononic crystals with one-way elastic edge waves. Phys. Rev. Lett. 115, 104302 (2015)
Huang, H., Chen, J., Huo, S.: Simultaneous topological Bragg and locally resonant edge modes of shear horizontal guided wave in one-dimensional structure. J. Phys. D Appl. Phys. 50, 275102 (2017)
Kulpe, J.A., Sabra, K.G., Leamy, M.J.: Bloch-wave expansion technique for predicting wave reflection and transmission in two-dimensional phononic crystals. J. Acoust. Soc. Am. 135, 1808–1819 (2014)
Palermo, A., Marzani, A.: Extended Bloch mode synthesis: ultrafast method for the computation of complex band structures in phononic media. Int. J. Solids Struct. 100, 29–40 (2016)
Huang, G.L., Sun, C.T.: Band gaps in a multiresonator acoustic metamaterial. J. Vib. Acoust. 132 (2010)
Yu, D., Liu, Y., Wang, G., Zhao, H., Qiu, J.: Flexural vibration band gaps in Timoshenko beams with locally resonant structures. J. Appl. Phys. 100, 124901 (2006)
Zhou, W., Lim, C.W.: Topological edge modeling and localization of protected interface modes in 1D phononic crystals for longitudinal and bending elastic waves. Int. J. Mech. Sci. 159, 359–372 (2019)
Li, Z.N., Yuan, B., Wang, Y.Z., Shui, G.S., Zhang, C., Wang, Y.S.: Diode behavior and nonreciprocal transmission in nonlinear elastic wave metamaterial. Mech. Mater. 133, 85–101 (2019)
Li, Z.N., Wang, Y.Z., Wang, Y.S.: Electro-mechanical coupling diode of elastic wave in nonlinear piezoelectric metamaterials. J. Acoust. Soc. Am. 150, 891–905 (2021)
Dal Poggetto, V.F., Serpa, A.L.: Elastic wave band gaps in a three-dimensional periodic metamaterial using the plane wave expansion method. Int. J. Mech. Sci. 184, 105841 (2020)
Dal Poggetto, V.F., Serpa, A.L.: Flexural wave band gaps in a ternary periodic metamaterial plate using the plane wave expansion method. J. Sound Vib. 495, 115909 (2021)
Dal Poggetto, V.F., de Franca Arruda, J.R.: Widening wave band gaps of periodic plates via shape optimization using spatial Fourier coefficients. Mech. Syst. Signal Pr. 147, 107098 (2021)
Liu, X.N., Hu, G.K., Sun, C.T., Huang, G.L.: Wave propagation characterization and design of two-dimensional elastic chiral metacomposite. J. Sound Vib. 330, 2536–2553 (2011)
Veres, I.A., Berer, T., Matsuda, O.: Complex band structures of two dimensional phononic crystals: analysis by the finite element method. J. Appl. Phys. 114, 083519 (2013)
Palermo, A., Marzani, A.: A reduced Bloch operator finite element method for fast calculation of elastic complex band structures. Int. J. Solids Struct. 191, 601–613 (2020)
Wu, Z., Li, F.M., Zhang, C.: Band-gap analysis of a novel lattice with a hierarchical periodicity using the spectral element method. J. Sound Vib. 421, 246–260 (2018)
Wu, Z., Li, F.M., Zhang, C.: Vibration band-gap properties of three-dimensional Kagome lattices using the spectral element method. J. Sound Vib. 341, 162–173 (2015)
Wu, Z., Li, F.M.: Spectral element method and its application in analysing the vibration band gap properties of two-dimensional square lattices. J. Vib. Control 22, 710–721 (2016)
Li, C.L., Jiang, T.J., Liu, S., Han, Q.: Dispersion and band gaps of elastic guided waves in the multi-scale periodic composite plates. Aerosp. Sci. Technol. 124, 107513 (2022)
Han, S.H., Han, Q., Jiang, T.J., Li, C.L.: Complex dispersion relations and evanescent waves in periodic magneto-electro curved phononic crystal plates. Appl. Math. Model. 119, 373–390 (2023)
Oudich, M., Gerard, N.J.R.K., Deng, Y., Jing, Y.: Tailoring Structure-Borne Sound through Bandgap Engineering in phononic crystals and metamaterials: a comprehensive review. Adv. Funct. Mater. 33, 2206309 (2023)
Xie, L.X., Xia, B.Z., Liu, J., Huang, G.L., Lei, J.R.: An improved fast plane wave expansion method for topology optimization of phononic crystals. Int. J. Mech. Sci. 120, 171–181 (2017)
Sharma, K.A., Kosta, M., Shmuel, G., Amir, O.: Gradient-based topology optimization of soft dielectrics as tunable phononic crystals. Compos. Struct. 280, 114846 (2022)
Dalklint, A., Wallin, M., Bertoldi, K., Tortorelli, D.: Tunable phononic bandgap materials designed via topology optimization. J. Mech. Phys. Solids 163, 104849 (2022)
Li, W., Meng, F., Chen, Y., Li, Y.F., Huang, X.: Topology optimization of photonic and phononic crystals and metamaterials: a review. Adv. Theor. Simul. 2, 1900017 (2019)
Chen, Y., Meng, F., Huang, X.: Creating acoustic topological insulators through topology optimization. Mech. Syst. Signal Pr. 146, 107054 (2021)
Chen, Y., Meng, F., Zhu, J., Huang, X.: Inverse design of second-order photonic topological insulators in C3-symmetric lattices. Appl. Math. Model. 102, 194–206 (2022)
Dong, H.W., Zhao, S.D., Zhu, R., Wang, Y.S., Cheng, L., Zhang, C.: Customizing acoustic dirac cones and topological insulators in square lattices by topology optimization. J. Sound Vib. 493, 115687 (2021)
Dong, H.W., Su, X.X., Wang, Y.S., Zhang, C.: Topological optimization of two-dimensional phononic crystals based on the finite element method and genetic algorithm. Struct. Multidiscip. O 50, 593–604 (2014)
Dong, H.W., Zhao, S.D., Miao, X.B., Shen, C., Zhang, X., Zhao, Z., Zhang, C., Wang, Y.S., Cheng, L.: Customized broadband pentamode metamaterials by topology optimization. J. Mech. Phys. Solids 152, 104407 (2021)
Sanchez-Lengeling, B., Aspuru-Guzik, A.: Inverse molecular design using machine learning: generative models for matter engineering. Science 361, 360–365 (2018)
Killoran, N., Lee, L.J., Delong, A., Duvenaud, D., Frey, B.J.: Generating and Designing DNA with Deep Generative Models. arXiv preprint. arXiv:1712.06148(2017)
Jin, Y., He, L., Wen, Z., Mortazavi, B., Guo, H., Torrent, D., Djafari-Rouhani, B., Rabczuk, T., Zhuang, X.Y., Li, Y.: Intelligent on-demand design of phononic metamaterials. Nanophotonics 11(3), 439–460 (2022)
Kennedy, J., Lim, C.W.: Machine learning and deep learning in phononic crystals and metamaterials a review. Mater. Today Commun. 104606 (2022)
Jiang, W., Zhu, Y., Yin, G., Lu, H., Xie, L., Yin, M.: Dispersion relation prediction and structure inverse design of elastic metamaterials via deep learning. Mater. Today Phys. 22, 100616 (2022)
Liu, C.X., Yu, G.L.: Inverse design of locally resonant metabarrier by deep learning with a rule-based topology dataset. Comput. Method Appl. M. 394, 114925 (2022)
Maghami, A., Hosseini, S.M.: IAutomated design of phononic crystals under thermoelastic wave propagation through deep reinforcement learning. Eng. Struct. 263, 114385 (2022)
He, L., Wen, Z., Jin, Y., Torrent, D., Zhuang, X., Rabczuk, T.: Inverse design of topological metaplates for flexural waves with machine learning. Mater. Design. 199, 109390 (2021)
Ma, W., Cheng, F., Xu, Y., Wen, Q., Liu, Y.: Probabilistic representation and inverse design of metamaterials based on a deep generative model with semi-supervised learning strategy. Adv. Mater. 31, 1901111 (2019)
Han, S.H., Han, Q., Li, C.L.: Deep-learning-based inverse design of phononic crystals for anticipated wave attenuation. J. Appl. Phys. 132, 154901 (2022)
Chen, C.T., Gu, G.X.: Generative deep neural networks for inverse materials design using backpropagation and active learning. Adv. Sci. 7, 1902607 (2020)
Ahmed, W.W., Farhat, M., Zhang, X., Wu, Y.: Deterministic and probabilistic deep learning models for inverse design of broadband acoustic cloak. Phys. Rev. Res. 3, 013142 (2021)
Challapalli, A., Patel, D., Li, G.: Inverse machine learning framework for optimizing lightweight metamaterials. Mater. Design. 208, 109937 (2021)
Li, X., Ning, S., Liu, Z., Yan, Z., Luo, C., Zhuang, Z.: Designing phononic crystal with anticipated band gap through a deep learning based data-driven method. Comput. Method Appl. M 361, 112737 (2020)
Schattschneider, D.: The plane symmetry groups: their recognition and notation. Am. Math. Mon. 85, 439–450 (1978)
Goodfellow, I., Pouget-Abadie, J., Mirza, M., Xu, B., Warde-Farley, D., Ozair, S., Courville, A., Bengio, Y.: Generative adversarial networks. Commun. ACM 63, 139–144 (2020)
Isola, P., Zhu, J. Y., Zhou, T., Efros, A. A.: Image-to-image translation with conditional adversarial networks. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 1125–1134 (2017)
Amari, S.: Backpropagation and stochastic gradient descent method. Neurocomputing 5, 185–196 (1993)
Kingma, D.P., Ba, J.: Adam: A Method for Stochastic Optimization. arXiv preprint. arXiv:1412.6980 (2014)
Pathak, D., Krahenbuhl, P., Donahue, J., Darrell, T., Efros, A. A.: Context encoders: feature learning by inpainting. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. pp. 2536–2544 (2016)
Acknowledgements
The authors wish to acknowledge the support from National Natural Science Foundation of China (11972160), Guangdong Basic and Applied Basic Research Foundation(2022A1515010143), Young Talent Support Project of Guangzhou Association for Science and Technology (QT2023013) and Science and Technology Program of Guangzhou (2023A04J1302).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Han, S., Han, Q., Jiang, T. et al. Inverse design of phononic crystals for anticipated wave propagation by integrating deep learning and semi-analytical approach. Acta Mech 234, 4879–4897 (2023). https://doi.org/10.1007/s00707-023-03634-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-023-03634-y