Acoustic fields with impedance boundary conditions
have high engineering applications, such as noise control and evaluation of
sound insulation materials, and can be approximated by three-dimensional
Helmholtz boundary value problems. Finite difference method is widely applied
to solving these problems due to its ease of use. However, when the wave number
is large, the pollution effects are still a major difficulty in obtaining accurate
numerical solutions. We develop a fast algorithm for solving three-dimensional
Helmholtz boundary problems with large wave numbers. The boundary of
computational domain is discrete based on high-order compact difference scheme.
Using the properties of the tensor product and the discrete Fourier sine
transform method, the original problem is solved by splitting it into
independent small tridiagonal subsystems. Numerical examples with impedance
boundary conditions are used to verify the feasibility and accuracy of the proposed algorithm. Results demonstrate that the
algorithm has a fourth-order convergence in ？and？-norms, and costs less CPU calculation time and
random access memory.
References
[1]
Kuo, C.W., Veltin, J. and McLaughlin, D.K. (2014) Acoustic Measurements of Models of Military Style Supersonic Nozzle Jets. Chinese Journal of Aeronautics, Chinese Journal of Aeronautics, 27, 23-33. https://doi.org/10.1016/j.cja.2013.12.006
[2]
Altmann, J. (2004) Acoustic and Seismic Signals of Heavy Military Vehicles for Co-Operative Verification. Journal of Sound and Vibration, 273, 713-740. https://doi.org/10.1016/j.jsv.2003.05.002
[3]
Sarvazyan, A.P., Urban, M.W. and Greenleaf (2013) Acoustic Waves in Medical Imaging and Diagnostics. Ultrasound in Medicine & Biology, 39, 1133-1146. https://doi.org/10.1016/j.ultrasmedbio.2013.02.006
[4]
Hughes-Riley, T. and Dias, T. (2018) Developing an Acoustic Sensing Yarn for Health Surveillance in a Military Setting. Sensors, 18, 1590. https://doi.org/10.3390/s18051590
[5]
Marshall, A. and Boussakta, S. (2007) Signal Analysis of Medical Acoustic Sounds with Applications to Chest Medicine. Journal of the Franklin Institute, 344, 230-242. https://doi.org/10.1016/j.jfranklin.2006.08.003
[6]
Behnia, A., Chai, H.K. and Shiotani, T. (2014) Advanced Structural Health Monitoring of Concrete Structures with the Aid of Acoustic Emission. Construction and Building Materials, 65, 282-302. https://doi.org/10.1016/j.conbuildmat.2014.04.103
[7]
He, C., Ni, X., et al. (2016) Acoustic Topological Insulator and Robust One-Way Sound Transport. Nature Physics, 12, 1124-1129. https://doi.org/10.1038/nphys3867
[8]
Chiatti, G., Chiavola, O. and Palmieri, F. (2017) A Vibration and Acoustic Characteristics of a City-Car Engine Fueled with Biodiesel Blends. Journal of Physical Society of Japan, 85, 664-670. https://doi.org/10.1016/j.apenergy.2016.10.119
[9]
Schelkunoff, S.A. (1938) The Impedance Concept and Its Application to Problems of Reflection, Refraction, Shielding and Power Absorption. Bell System Technical Journal, 17, 17-48. https://doi.org/10.1002/j.1538-7305.1938.tb00774.x
[10]
Shchukin, A.N. (1940) Propagation of Radio Waves. Svyazizdat, Moscow.
[11]
Leontovich, M.A. (1944) Methods of Solution for Problems of Electromagnetic Waves Propagation along the Earth Surface. Bulletin of the Academy of Sciences of the USSR. Physical Series, 8, 1622.
[12]
Leontovich, M. (1948) Approximate Boundary Conditions for Electromagnetic Field on the Surface of Conducting Bodies. In: Vvedenskii, B.A., Ed., Investigations on Propagation of Radio Waves, AN SSSR, Moscow, 5-10.
[13]
Alpert, Y.L. (1940) To the Problem of Electromagnetic Waves Propagation in Tubes. Journal of Technical Physics, 10, 1358-1364.
[14]
Zhang, X., Hu, X., Gong, H., Zhang, J., Lv, Z. and Hong, W. (2020) Experimental Study on the Impact Sound Insulation of Cross Laminated Timber and Timber-Concrete Composite Floors. Applied Acoustics, 161, Article ID: 107173. https://doi.org/10.1016/j.apacoust.2019.107173
[15]
Calleri, C., Astolfi, A., et al. (2019) Characterization of the Sound Insulation Properties of a Two-Layers Lightweight Concrete Innovative Façade, Applied Acoustics, 145, 267-277. https://doi.org/10.1016/j.apacoust.2018.10.003
[16]
Tsirigoti, D., Giarma, C. and Tsikaloudaki, K. (2020) Indoor Acoustic Comfort Provided by an Innovative Preconstructed Wall Module: Sound Insulation Performance Analysis. Sustainability, 12, 8666. https://doi.org/10.3390/su12208666
[17]
Kuo, S.M. and Tsai, J. (1994) Residual Noise Shaping Technique for Active Noise Control Systems. The Journal of the Acoustical Society of America, 95, 1665-1668. https://doi.org/10.1121/1.408555
[18]
Bodson, M., Jensen, J. and Douglas, S. (2001) Active Noise Control for Periodic Disturbances. IEEE Transactions on Control Systems Technology, 9, 200-205. https://doi.org/10.1109/87.896760
[19]
Sagartzazu, X., Hervella-Nieto, L. and Pagalday, J.M. (2008) Review in Sound Absorbing Materials. Archives of Computational Methods in Engineering, 15, 311-342. https://doi.org/10.1007/s11831-008-9022-1
[20]
Xia, B. and Yu, D. (2012) Modified Sub-Interval Perturbation Finite Element Method for 2D Acoustic Field Prediction with Large Uncertain-But-Bounded Parameters. Journal of Sound and Vibration, 311, 3774-3790. https://doi.org/10.1016/j.jsv.2012.03.024
[21]
Xia, B. and Yu, D. (2012) Interval Analysis of Acoustic Field with Uncertain-But-Bounded Parameters. Computers & Structures, 112-113, 235-244. https://doi.org/10.1016/j.compstruc.2012.08.010
[22]
Babuška, I.M. and Sauter, S.A. (1997) Is the Pollution Effect of the FEM Avoidable for the Helmholtz Equation Considering High Wave Numbers? SIAM Journal on Numerical Analysis, 34, 2392-2423. https://doi.org/10.1137/S0036142994269186
[23]
Lorca, J.P.L., Beams, N., Beecroft, D. and Gillman, A. (2023) An Iterative Solver for the HPS Discretization Applied to Three Dimensional Helmholtz Problems.
[24]
Galkowski, J., Lafontaine, D., Spence, E.A. and Wunsch, J. (2022) The hp-FEM Applied to the Helmholtz Equation with PML Truncation Does Not Suffer from the Pollution Effect.
[25]
Spence, E.A. (2022) A Simple Proof That the hp-FEM Does Not Suffer from the Pollution Effect for the Constant-Coefficient Full-Space Helmholtz Equation. Advances in Computational Mathematics, 49, Article No. 27. https://doi.org/10.1007/s10444-023-10025-3
[26]
Kumar, N. and Dubey, R.K. (2019) Development of a New Sixth Order Accurate Compact Scheme for Two and Three Dimensional Helmholtz Equation.
[27]
Feng, Q., Han, B. and Michelle, M. (2023) Sixth Order Compact Finite Difference Method for 2D Helmholtz Equations with Singular Sources and Reduced Pollution Effect.
[28]
An, S., Gu, G. and Zhao, M. (2018) A Fast High Order Algorithm for 3D Helmholtz Equation with Dirichlet Boundary. Applied and Computational Mathematics, 7, 180-187. https://doi.org/10.11648/j.acm.20180704.11
[29]
Zhu, N. and Zhao, M. (2018) A Fast Fourth-Order Method for 3D Helmholtz Equation with Neumann Boundary. American Journal of Computational Mathematics, 8, 222-232. https://doi.org/10.4236/ajcm.2018.83018
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