Abstract
A topology optimization method for coupled vibro-acoustic problem based on hybrid finite element-wave based method is proposed. The hybrid finite element-wave based method is used to model the coupled free layer damping structure-acoustics system, and a gradient-based topology optimization formulation is developed by solid isotropic microstructures with penalization method, to obtain optimized layout of free layer damping with the objective of minimizing sound pressure in the acoustic cavity and with constraint defined as the amount of free layer damping. The adjoint variable method for the coupled vibro-acoustic model based on the hybrid finite element-wave based method is also derived as a part of the sensitivity analysis. Numerical example shows that the optimized distribution reduce sound pressure effectively and the calculation time of proposed method is less than the conventional method for topology optimization base on finite element method. Experimental results verify the accuracy of the hybrid finite element-wave based method and the effect of optimized damping material layout is also validated by experiment.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Akl W, El-Sabbagh A, Al-Mitani K, Baz A (2009) Topology optimization of a plate coupled with acoustic cavity. Int J Solids Struct 46(10):2060–2074. https://doi.org/10.1016/j.ijsolstr.2008.05.034
Assaf S, Guerich M, Cuvelier P (2011) Vibration and damping analysis of plates with partially covered damping layers. Acta Acust United Acust 97(4):553–568. https://doi.org/10.3813/aaa.918436
Bendsøe MP, Kikuchi N (1988) Generating optimal topologies in structural design using a homogenization method. Comput Methods Appl Mech Eng 71(2):197–224. https://doi.org/10.1016/0045-7825(88)90086-2
Bendsøe MP, Sigmund O (2003) Topology optimization: theory, methods and applications. Springer, New York
Bouillard P, Ihlenburg F (1999) Error estimation and adaptivity for the finite element method in acoustics: 2d and 3d applications. Comput Methods Appl Mech Eng 176(1):147–163. https://doi.org/10.1016/S0045-7825(98)00334-X
Bourdin B (2001) Filters in topology optimization. Int J Numer Methods Eng 50(9):2143–2158
Chen N, Yu DJ, Xia BZ, Liu J, Ma ZD (2017) Microstructural topology optimization of structural-acoustic coupled systems for minimizing sound pressure level. Struct Multidisc Optim 56(6):1259–1270. https://doi.org/10.1007/s00158-017-1718-0
Christiansen RE (2016) Topology optimization for wave propagation problems with experimental validation. Thesis, Technical University of Denmark
Creixell-Mediante E, Jensen JS, Naets F, Brunskog J, Larsen M (2018) Adaptive parametric model order reduction technique for optimization of vibro-acoustic models: application to hearing aid design. J Sound Vib 424:208–223. https://doi.org/10.1016/j.jsv.2018.03.013
Desmet W (1998) A wave based prediction technique for coupled vibro-acoustic analysis. Thesis, KU Leuven
Dilgen CB, Dilgen SB, Aage N, Jensen JS (2019) Topology optimization of acoustic mechanical interaction problems: a comparative review. Struct Multidisc Optim 60:779–801. https://doi.org/10.1007/s00158-019-02236-4
El-Sabbagh A, Baz A (2014) Topology optimization of unconstrained damping treatments for plates. Eng Optim 46(9):1153–1168. https://doi.org/10.1080/0305215x.2013.832235
Gao R, Zhang Y, Kennedy D (2019) Topology optimization of sound absorbing layer for the mid-frequency vibration of vibro-acoustic systems. Struct Multidisc Optim 59(5):1733–1746
Genechten BV, Atak O, Bergen B, Deckers E, Jonckheere S, Lee JS, Maressa A, Vergote K, Pluymers B, Vandepitte D, Desmet W (2012) An efficient wave based method for solving Helmholtz problems in three-dimensional bounded domains. Eng Anal Bound Elem 75:63–75
Goo S, Wang S, Kook J, Koo K, Hyun J (2017) Topology optimization of bounded acoustic problems using the hybrid finite element-wave based method. Comput Methods Appl Mech Eng 313:834–856. https://doi.org/10.1016/j.cma.2016.10.027
Goo S, Kook J, Wang S (2020) Topology optimization of vibroacoustic problems using the hybrid finite element-wave based method. Comput Methods Appl Mech Eng 364:112932. https://doi.org/10.1016/j.cma.2020.112932
Hal Bv (2004) Automation and performance optimization of the wave based method for interior structural-acoustic problems. Thesis, KU Leuven
Isakari H, Kondo T, Takahashi T, Matsumoto T (2017) A level-set-based topology optimisation for acoustic-elastic coupled problems with a fast BEM-FEM solver. Comput Methods Appl Mech Eng 315:501–521. https://doi.org/10.1016/j.cma.2016.11.006
Jensen JS (2018) A simple method for coupled acoustic-mechanical analysis with application to gradient-based topology optimization. Struct Multidisc Optim 59:1567–1580. https://doi.org/10.1007/s00158-018-2147-4
Jensen JS, Sigmund O (2006) Topology optimization of wave-propagation problems. In: Bendsøe MP, Olhoff N, Sigmund O (eds) IUTAM Symposium on Topological Design Optimization of Structures, Machines and Materials. Solid Mechanics and Its Applications, vol 137. Springer, Dordrecht. https://doi.org/10.1007/1-4020-4752-5_38
Jonckheere S, Deckers E, Van Genechten B, Vandepitte D, Desmet W (2013) A direct hybrid finite element-wave based method for the 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
Kang Z, Zhang XP, Jiang SG, Cheng GD (2012) On topology optimization of damping layer in shell structures under harmonic excitations. Struct Multidisc Optim 46(1):51–67. https://doi.org/10.1007/s00158-011-0746-4
Kim SY, Mechefske CK, Kim IY (2013) Optimal damping layout in a shell structure using topology optimization. J Sound Vib 332(12):2873–2883. https://doi.org/10.1016/j.jsv.2013.01.029
Koo K, Pluymers B, Desmet W, Wang S (2011) Vibro-acoustic design sensitivity analysis using the wave-based method. J Sound Vib 330(17):4340–4351
Kook J, Jensen JS, Wang S (2013) Acoustical topology optimization of Zwicker’s loudness with Pade approximation. Comput Methods Appl Mech Eng 255:40–46. https://doi.org/10.1016/j.cma.2012.10.022
Li W, He Y, Xu Z, Zhang Z (2019) A reduced passive constrained layer damping finite element model based on the modified improved reduced system method. J Sandw Struct Mater 21(2):758–783. https://doi.org/10.1177/1099636217699022
Marburg S (2002) Six boundary elements per wavelength: is that enough? J Comput Acoust 10(01):25–51
Parthasarathy G, Reddy CVR, Ganesan N (1985) Partial coverage of rectangular-plates by unconstrained layer damping treatments. J Sound Vib 102(2):203–216. https://doi.org/10.1016/S0022-460x(85)80053-5
Parthasarthy G, Ganesan N, Reddy CVR (1986) Study of unconstrained layer damping treatments applied to rectangular-plates having central cutouts. Comput Struct 23(3):433–443. https://doi.org/10.1016/0045-7949(86)90233-6
Pluymers B (2006) Wave based modelling methods for steady-state vibro-acoustics. Thesis, KU Leuven
Shang L, Zhao G, Zhai J (2017) Topology optimization for coupled acoustic-structural systems under random excitation. Struct Multidisc Optim 56(4):809–822. https://doi.org/10.1007/s00158-017-1687-3
Sigmund O, Jensen J (2003) Systematic design of phononic band-gap materials and structures by topology optimization. Philos Trans R Soc Lond Ser A 361(1806):1001–1019
Sondergaard MB, Pedersen CBW (2014) Applied topology optimization of vibro-acoustic hearing instrument models. J Sound Vib 333(3):683–692. https://doi.org/10.1016/j.jsv.2013.09.029
Van Genechten B, Vandepitte D, Desmet W (2011) A direct hybrid finite element-wave based modelling technique for efficient coupled vibro-acoustic analysis. Comput Methods Appl Mech Eng 200(5–8):742–764. https://doi.org/10.1016/j.cma.2010.09.017
Wang F, Lazarov BS, Sigmund O (2010) On projection methods, convergence and robust formulations in topology optimization. Struct Multidisc Optim 43(6):767–784. https://doi.org/10.1007/s00158-010-0602-y
Xu W, Zhang Z, Yu L, Xu ZM (2017) Topology optimization for noise reduction of structures with free damping. J Vib Shock (in Chin) 36(11):192–198
Yoon GH (2010) Structural topology optimization for frequency response problem using model reduction schemes. Comput Methods Appl Mech Eng 199(25–28):1744–1763. https://doi.org/10.1016/j.cma.2010.02.002
Zhang D, Wang S, Zheng L (2018) A comparative study on acoustic optimization and analysis of CLD/plate in a cavity using ESO and GA. Shock Vib 2018:1–16. https://doi.org/10.1155/2018/7146580
Zhao W, Zheng C, Liu C, Chen H (2017) Minimization of sound radiation in fully coupled structural-acoustic systems using FEM-BEM based topology optimization. Struct Multidisc Optim 58(1):115–128. https://doi.org/10.1007/s00158-017-1881-3
Zheng L, Zhang DD, Liu CF, Li YN, Xiang SH, Ye L, Fang GQ (2016) Topology optimization of a constrained layer damping plate coupled with an acoustical cavity. Int J Acoust Vib 21(4):394–405. https://doi.org/10.20855/ijav.2016.21.4434
Zheng W, Lei Y, Li S, Huang Q (2013) Topology optimization of passive constrained layer damping with partial coverage on plate. Shock Vib 20(2):199–211. https://doi.org/10.3233/SAV-2012-00738
Zheng WG, Yang TL, Huang QB, He Z (2016) Topology optimization of PCLD on plates for minimizing sound radiation at low frequency resonance. Struct Multidisc Optim 53(6):1231–1242. https://doi.org/10.1007/s00158-015-1371-4
Acknowledgements
The authors acknowledge the support from the National Key Laboratory of Science and Technology on Helicopter Transmission (Nanjing University of Aeronautics and Astronautics, Grant No. HTL-0-19G01)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Replication of results
All the datasets in this paper are generated using MATLAB codes. The source codes to perform topology optimization for the numerical examples in Sect. 4 can be obtained in the supplementary material.
Additional information
Responsible Editor: Qing Li
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Wu, B., Fan, W., Xu, Z. et al. Topology optimization of damping material layout in coupled vibro-acoustic system using hybrid finite element-wave based method. Struct Multidisc Optim 64, 3819–3834 (2021). https://doi.org/10.1007/s00158-021-03063-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00158-021-03063-2