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International Journal for Multiscale Computational Engineering
Fator do impacto: 1.016 FI de cinco anos: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN Imprimir: 1543-1649
ISSN On-line: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.2019030576
pages 563-582


Y. Kamoun
Department of Aerospace Engineering, Technion−Israel Institute of Technology Haifa 32000, Israel
Dan Givoli
Department of Aerospace Engineering, Technion−Israel Institute of Technology, Haifa 32000, Israel; Faculty of Civil Engineering & Geosciences, Technical University of Delft, 2600 GA Delft, The Netherlands


Finding the sound pressure level (SPL) distribution near the ground due to aircraft noise is an important problem in environmental engineering. Since the human hearing range is very wide, ranging from 20 Hz to 20 kHz, the determination of the SPL distribution for a given source spectrum is a difficult multiscale problem, and requires the repeated solution, for many different wave numbers, of the Helmholtz equation in the upper half space, while imposing a given impedance boundary condition on the ground. Previously, a simple computational scheme, based on the use of fictitious sources, was proposed for the efficient solution of such problems, for aflat ground with a given constant impedance. In the present study, this scheme is improved and extended in several ways. First, the ground impedance is allowed to vary with location, representing a varying type of ground (soil, water, asphalt, etc.). Second, a mechanism for verification of the method and for error estimation is developed, whereas previously only the boundary condition residual was evaluated. Third, the use of the appropriate Green's function, associated with a ringlike source, is made precise. Two simplifying assumptions which are maintained are that the ground is flat and that its impedance function is axially symmetric. Numerical experiments are used to demonstrate the performance of the scheme.


  1. Baharav, Z. and Leviatan, Y., Scattering Analysis using Fictitious Wavelet Array Sources, J. Electromagn. Waves Appl, vol. 10, pp. 1683-1697,1996.

  2. Bai, M.R., Hsu, H., and Wen, J.C., Spatial Sound Field Synthesis and Upmixing based on the Equivalent Source Method, J Acoust. Soc. Am., vol. 135, pp. 269-282,2014.

  3. Barton, G., Elements of Green's Functions and Propagation, Oxford, U.K.: Clarendon Press, 1989.

  4. Bender, C.M. and Orszag, S.A., Advanced Mathematical Methods for Scientists and Engineers, Berlin: Springer, 1999.

  5. Bi, C.X. and Bolton, J.S., An Equivalent Source Technique for Recovering the Free Sound Field in a Noisy Environment, J. Acoust. Soc. Am., vol. 131, pp. 1260-1270,2012.

  6. Boag, A. and Leviatan, Y., Analsys of 3-Dimensional Acoustic Scattering from Doubly Periodic Structures Using a Source Model, J. Acoust. Soc. Am, vol. 91, pp. 572-580,1992.

  7. Briggs, W.L., A Multigrid Tutorial, Philadelphia, PA: SIAM, 2000.

  8. Bruno, O.R. and Maas, M., Shifted Equivalent Sources and FFT Acceleration for Periodic Scattering Problems, Including Wood Anomalies, J. Comput. Phys., vol. 378, pp. 548-572,2019.

  9. Chien, C.F. and Soroka, W.W., Sound Propagation along an Impedance Plane, J. Sound Vib., vol. 43, pp. 9-20,1975.

  10. Chien, C.F. and Soroka, W.W., A Note on the Calculation of Sound Propagation along an Impedance Surface, J. Sound Vib, vol. 69, pp. 340-343,1980.

  11. de la Rubia, V. and Mrozowski, M.A., Compact Basis for Reliable Fast Frequency Sweep via the Reduced-Basis Method, IEEE Trans. Microwave Theory Tech., vol. 66, pp. 4367-4382,2018.

  12. Delany, M.E. andBazley, E.N., Acoustical Properties of Fibrous Absorbent Materials, Appl. Acoust., vol. 3, pp. 105-116,1970.

  13. Duran, M., Muga, I., and Nedelec, J.C., The Helmholtz Equation with Impedance in a Half-Space, C. R. Math, vol. 341, pp. 561-566,2005.

  14. Embleton, T. and Daigle, G., Atmospheric Propagation, in Aeroacoustics of Flight Vehicles: Theory and Practice, H.H. Hubbard, Ed., Washington, DC: NASA, vol. 2, Chap. 12, pp. 58-71,1995.

  15. Fernandez-Grande, E., Xenaki, A., and Gerstoft, P.A., Sparse Equivalent Source Method for Near-Field Acoustic Holography, J. Acoust. Soc. Am, vol. 141, pp. 532-542,2017.

  16. Fidell, S., Mestre, V., Schomer, P., Horonjeff, R., and Reid, T., A Systematic Rationale for Defining the Significance of Aircraft Noise Impacts, J. Acoust. Soc. Am., vol. 136, pp. 1129-1138,2014.

  17. Fikioris, G. and Tsitsas, N.L., On Convergence and Inherent Oscillations within Computational Methods Employing Fictitious Sources, Comput. Math. Appl., vol. 69, pp. 636-649,2015.

  18. Genesca, M., Directional Monitoring Terminal for Aircraft Noise, J. Sound Vib., vol. 374, pp. 77-91,2016.

  19. Givoli, D., Non-Reflecting Boundary Conditions: A Review, J. Comput. Phys., vol. 94, pp. 1-29, 1991.

  20. Givoli, D., High-Order Local Non-Reflecting Boundary Conditions: A Review, Wave Motion, vol. 39, pp. 319-326,2004.

  21. Gounot, Y J.R. and Musafir, R.E., Simulation of Scattered Fields: Some Guidelines for the Equivalent Source Method, J. Sound Vib, vol. 330, pp. 3698-3709,2011.

  22. Gur, I. and Givoli, D.A., Fictitious Source Method for a Multifrequency Acoustic Source over Ground with Given Impedance, Int. J. Multiscale Comput. Eng, vol. 6, pp. 533-548,2008.

  23. Habault, D., Sound Propagation above an Inhomogeneous Plane-Boundary Integral-Equation Methods, J. Sound Vib., vol. 100, pp. 55-67,1985.

  24. Hagstrom, T., Radiation Boundary Conditions for the Numerical Simulation of Waves, Acta Numerica, vol. 8, pp. 47-106,1999.

  25. Hetmaniuk, U., Tezaur, R., and Farhat, C., An Adaptive Scheme for a Class of Interpolatory Model Reduction Methods for Frequency Response Problems, Int. J. Numer. Methods Eng., vol. 93, pp. 1109-1124,2013.

  26. Ihlenburg, F. and Babuska, I., Dispersion Analysis and Error Estimation of Galerkin Finite Element Methods for the Helmholtz Equation, Int. J. Numer. Methods Eng., vol. 38, pp. 3745-3774,1995.

  27. Kamoun, Y., Fictitious Source Method for the Analysis of Noise Level from a Distant Source on Ground with Variable Impedance, MSc, Department of Aerospace Engineering Technion, Haifa, Israel, 2016.

  28. Karageorghis, A., Johansson, B.T., and Lesnic, D., The Method of Fundamental Solutions for the Identification of a Sound-Soft Obstacle in Inverse Acoustic Scattering, Appl. Numer. Math, vol. 62, pp. 1767-1780,2012.

  29. Keener, J.P., Principles of Applied Mathematics, Redwood City, CA: Addison-Wesley, 1988.

  30. Lee, S., Review: The Use of Equivalent Source Method in Computational Acoustics, J. Comput. Acoust., vol. 25, pp. 1630001:1-11,2017.

  31. Lenzi, M.S., Lefteriu, S, Beriot, H., and Desmet, W., A Fast Frequency Sweep Approach Using Pade Approximations for Solving Helmholtz Finite Element Models, J. Sound Vib., vol. 332, pp. 1897-1917,2013.

  32. Leviatan, Y, Baharav, Z., andHeyman, E., Analysis of Electromagnetic Scattering Using Arrays of Fictitious Sources, IEEE Trans. AntennasPropag., vol. 43, pp. 1091-1098,1995.

  33. Leviatan, Y., Erez, E., and Beran, M.J., A Source-Model Technique for Analysis of Flexural Wave Scattering in a Heterogeneous Thin Plate, J. Mech. Appl. Math, vol. 45, pp. 499-514,1992.

  34. Li, W.L., Wu, T.W., and Seybert, A.F., A Half-Space Boundary-Element Method for Acoustic Problems with a Reflecting Plane of Arbitrary Impedance, J. Sound Vib., vol. 171, pp. 173-184,1994.

  35. Liang, T.F., Wang, J.P., Xiao, J.Y., and Wen, L.H., Coupled BE-FE based Vibroacoustic Modal Analysis and Frequency Sweep Using a Generalized Resolvent Sampling Method, Comput. Methods Appl. Mech. Eng., vol. 345, pp. 518-538,2019.

  36. Ochmann, M., The Complex Equivalent Source Method for Sound Propagation over an Impedance Plane, J. Acoust. Soc. Am., vol. 116, pp. 3304-3311,2004.

  37. Ochmann, M. and Brick, H., Acoustical Radiation and Scattering above an Impedance Plane, in Computational Acoustics of Noise Propagation in Fluids, S. Marburg and B. Nolte, Eds., Berlin: Springer, Chap. 17, pp. 459-494,2008.

  38. Perrey-Debain, E., Trevelyan, J., and Bettess, P., On Wave Boundary Elements for Radiation and Scattering Problems with Piece-wise Constant Impedance, IEEE Trans. Antennas Propag., vol. 53, pp. 876-879,2005.

  39. Pierce, A.D., Acoustics, New York: Acoustical Society of America, 1989.

  40. Polacsek, C., Desquesnes, G., and Reboul, G., An Equivalent-Source Model for Simulating Noise Generation in Turbofan Engines, J. Sound Vib, vol. 323, pp. 697-717,2009.

  41. Reitbort, R., Gur, I., and Givoli, D., Computational Methods for Analyzing Aircraft Noise above Ground with General Topography and Impedance, J. Comput. Acoust, vol. 20, pp. 1240001:1-18,2012.

  42. Rickley, E.J. and Fleming, G.G., Computing the Absorption of Sound by the Atmosphere and Its Applicability to Aircraft Noise Certification, John A. Volpe National Transportation Systems Center, Cambridge, MA, Rep. No. DTS-34-FA853-LR2,1998.

  43. Rumpler, R., Goransson, P., and Deue, J.F., A Finite Element Approach Combining a Reduced-Order System, Pade Approximants, and an Adaptive Frequency Windowing for Fast Multifrequency Solution of Poro-Acoustic Problems, Int. J. Numer. Methods Eng., vol. 97, pp. 759-784,2014.

  44. Sorensen, M., Aircraft Noise Exposure and Hypertension, Occup., Environ. Med., vol. 74, pp. 85-86,2017.

  45. Swift, S.H., Blaisdell, G.A., and Lyrintzis, A.S., An Efficient Time-Domain Equivalent Source Method for Acoustic Scattering, Int. J. Aeroacoust., vol. 14, pp. 133-160,2015.

  46. Torjesen, I., Long-Term Aircraft Noise is Linked to Incidence of High Blood Pressure, Br. Med. J, vol. 357, pp. j2872:1-9,2017.

  47. Trefethen, L.N. andBau,D.,III, Numerical Linear Algebra, Philadelphia: SIAM, 1997.

  48. Wixom, A.S. and McDaniel, J.G., Fast Frequency Sweeps with Many Forcing Vectors through Adaptive Interpolatory Model Order Reduction, Int. J. Numer. Methods Eng., vol. 100, pp. 442-457,2014.

  49. Wu, H., Jiang, W., and Zhang, H., A Mapping Relationship based Near-Field Acoustic Holography with Spherical Fundamental Solutions for Helmholtz Equation, J. Sound Vib, vol. 373, pp. 66-88,2016.

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