Coupled acoustic-elastic finite element modeling of underwater sound propagation

Abstract

Underwater sound propagation modeling is fundamental to sonar design and marine exploration. The parabolic equation (PE) method has been the dominant tool, but carries inherent limitations: the one-way approximation neglects backscattered energy, acoustic-only implementations ignore shear-wave conversion at the fluid-solid interface, and range-marching discretization approximates irregular bathymetry as a staircase boundary. These limitations become significant at low frequencies (below 200 Hz), where acoustic wavelengths reach 7.5–30 m and a substantial fraction of energy penetrates into the elastic seabed, exciting both compressional and shear waves. This study develops a two-dimensional frequency-domain finite element model coupling the Helmholtz equation in the water column with the Navier equation in the elastic seabed, with pressure and normal displacement continuity enforced at the fluid-solid interface. Irregular bathymetry is represented by a terrain-following curvilinear mesh via transfinite interpolation, eliminating staircase errors. A perfectly matched layer truncates open boundaries. Validation against RAMGEO for a Pekeris waveguide yields a mean absolute error of 1.8 dB (mean bias −0.17 dB). In a lossless comparison, mean |ΔTL| between acoustic-only and coupled models is approximately 6 dB, nearly independent of frequency across 50–200 Hz. Under realistic attenuation, mean |ΔTL| ranges from 0.8 dB (soft sediment) to 5.1 dB (hard bottom), with local maxima exceeding 10 dB at interference nulls. Staircase meshes yield displacement errors of 53–116% and pressure errors of 23–24% versus the curvilinear solution. The proposed model provides a rigorous forward engine for accurate transmission loss prediction in complex shallow-water environments.

Keywords

Finite element method / Acoustic-elastic coupling / Transmission loss / Underwater sound propagation / Curvilinear mesh

References

[1] Jensen FB, Kuperman WA, Porter MB, Schmidt H. Computational Ocean Acoustics . 2nd ed. New York, NY: Springer; 2011. doi: 10.1007/978—1—4419—8678—8

[2] Hovem JM. Marine Acoustics: The Physics of Sound in Underwater Environments . Los Altos, CA: Peninsula Publishing; 2010.

[3] Etter PC. Underwater Acoustic Modeling and Simulation . 5th ed. Boca Raton, FL: CRC Press; 2018. doi: 10.1201/9781315166346

[4] Collins MD. A split—step Padé solution for the parabolic equation method. J Acoust Soc Am . 1993; 93(4): 1736-1742. doi: 10.1121/1.406739

[5] Porter MB, Reiss EL. A numerical method for ocean—acoustic normal modes. J Acoust Soc Am . 1984; 76(1): 244-252. doi: 10.1121/1.391101

[6] Westwood EK, Tindle CT, Chapman NR. A normal mode model for acousto—elastic ocean environments. J Acoust Soc Am. 1996; 100(6): 3631-3645. doi: 10.1121/1.417226

[7] Collins MD. A higher—order parabolic equation for wave propagation in an ocean overlying an elastic bottom. J Acoust Soc Am. 1989; 86(4): 1459-1464. doi: 10.1121/1.398706

[8] Collins MD, Westwood EK. A higher—order energy—conserving parabolic equation for range—dependent ocean depth, sound speed, and density. J Acoust Soc Am. 1991; 89(3): 1068-1075. doi: 10.1121/1.400526

[9] Schmidt H, Jensen FB. A full wave solution for propagation in multilayered viscoelastic media with application to Gaussian beam reflection at fluid—solid interfaces. J Acoust Soc Am. 1985; 77(3): 813-825. doi: 10.1121/1.392050

[10] Hamilton EL. Geoacoustic modeling of the sea floor. J Acoust Soc Am. 1980; 68(5): 1313-1340. doi: 10.1121/1.385100

[11] Virieux J. P—SV wave propagation in heterogeneous media: Velocity—stress finite—difference method. Geophysics. 1986; 51(4): 889-901. doi: 10.1190/1.1442147

[12] Levander AR. Fourth—order finite—difference P—SV seismograms. Geophysics. 1988; 53(11): 1425-1436. doi: 10.1190/1.1442422

[13] Pratt RG. Seismic waveform inversion in the frequency domain, Part 1: Theory and verification in a physical scale model. Geophysics. 1999; 64(3): 888-901. doi: 10.1190/1.1444597

[14] Kozitskiy S. Coupled—mode parabolic equations for the modeling of sound propagation in a shallow—water waveguide with weak elastic bottom. J Mar Sci Eng . 2022; 10(10): 1355. doi: 10.3390/jmse10101355

[15] Dosso SE, Dettmer J. Bayesian matched—field geoacoustic inversion. Inverse Probl. 2011; 27(5): 055009. doi: 10.1088/0266—5611/27/5/055009

[16] Gerstoft P. Inversion of seismoacoustic data using genetic algorithms and a posteriori probability distributions. J Acoust Soc Am. 1994; 95(2): 770-782. doi: 10.1121/1.408387

[17] Sonnemann T, Dettmer J, Holland CW, Dosso SE. Meso—scale seabed quantification with geoacoustic inversion. Commun Eng . 2024; 3: 60. doi: 10.1038/s44172—024—00204—5

[18] Liu W, Zhang L, Wang W, et al. A three—dimensional finite difference model for ocean acoustic propagation and benchmarking for topographic effects. J Acoust Soc Am. 2021; 150(2): 1140-1156. doi: 10.1121/10.0005853

[19] Li Q, Wu G, Jia Z, Duan P. Full—waveform inversion in acoustic—elastic coupled media with irregular seafloor based on the generalized finite—difference method. Geophysics . 2023; 88(2): T83-T100. doi: 10.1190/geo2022—0408.1

[20] Gordon WJ, Hall CA. Construction of curvilinear co—ordinate systems and applications to mesh generation. Int J Numer Methods Eng. 1973; 7(4): 461-477. doi: 10.1002/nme.1620070405

[21] Thompson JF, Warsi ZUA, Mastin CW. Numerical Grid Generation: Foundations and Applications . Amsterdam, Netherlands: North—Holland; 1985.

[22] Ihlenburg F. Finite Element Analysis of Acoustic Scattering . New York, NY: Springer; 1998. doi: 10.1007/b98828

[23] Zienkiewicz OC, Taylor RL. The Finite Element Method, Vol. 1: The Basis . 5th ed. Oxford, UK: Butterworth—Heinemann; 2000.

[24] Atalla N, Sgard F. Finite Element and Boundary Methods in Structural Acoustics and Vibration . Boca Raton, FL: CRC Press; 2015. doi: 10.1201/b18366

[25] Komatitsch D, Tromp J. Introduction to the spectral element method for three—dimensional seismic wave propagation. Geophys J Int. 1999; 139(3): 806-822. doi: 10.1046/j.1365—246x.1999.00967.x

[26] Bottero A, Cristini P, Komatitsch D, Asch M. An axisymmetric time—domain spectral—element method for full—wave simulations: Application to ocean acoustics. J Acoust Soc Am. 2016; 140(5): 3520-3530. doi: 10.1121/1.4965964

[27] Cristini P, Komatitsch D. Some illustrative examples of the use of a spectral—element method in ocean acoustics. J Acoust Soc Am. 2012; 131(3): EL229-EL235. doi: 10.1121/1.3682459

[28] Isakson MJ, Chotiros NP. Finite element modeling of reverberation and transmission loss in shallow water waveguides with rough boundaries. J Acoust Soc Am. 2011; 129(3): 1273-1279. doi: 10.1121/1.3531810

[29] Vendhan CP, Diwan GC, Bhattacharyya SK. Finite—element modeling of depth and range dependent acoustic propagation in oceanic waveguides. J Acoust Soc Am. 2010; 127(6): 3319-3326. doi: 10.1121/1.3392440

[30] Zhou YQ, Luo WY. A finite element model for underwater sound propagation in 2—D environment. J Mar Sci Eng. 2021; 9(9): 956. doi: 10.3390/jmse9090956

[31] Gui Q, Zhang GY, Chai YB, Li W. A finite element method with cover functions for underwater acoustic propagation problems. Ocean Eng. 2022; 243: 110174. doi: 10.1016/j.oceaneng.2021.110174

[32] He T, Wang B, Mo S, Fang E. Predicting range—dependent underwater sound propagation from structural sources in shallow water using coupled finite element/equivalent source computations. Ocean Eng. 2023; 272: 113904. doi: 10.1016/j.oceaneng.2023.113904

[33] Wang Y, Tu H, Xu G, Gao D. A review of the application of spectral methods in computational ocean acoustics. Phys Fluids . 2023; 35(12): 121301. doi: 10.1063/5.0176116

[34] Zhang Y, Tu H, Wang Y, Xu G, Gao D. A normal mode model based on the spectral element method for simulating horizontally layered acoustic waveguides. J Mar Sci Eng. 2024; 12(9): 1499. doi: 10.3390/jmse12091499

[35] Xu W, Fu Z, Xu W, Xue MA, Rashed YF, Zheng J. FEM—PIKFNN for underwater acoustic propagation induced by structural vibrations in different ocean environments. Comput Math Appl. 2024; 166: 46-54. doi: 10.1016/j.camwa.2024.09.007

[36] Tu H, Wang Y, Yang C, Liu W, Wang X. A Chebyshev—Tau spectral method for coupled modes of underwater sound propagation in range—dependent ocean environments. Phys Fluids. 2023; 35(3): 037113. doi: 10.1063/5.0138012

[37] Bérenger JP. A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys. 1994; 114(2): 185-200. doi: 10.1006/jcph.1994.1159

[38] Collino F, Tsogka C. Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics. 2001; 66(1): 294-307. doi: 10.1190/1.1444908

[39] Komatitsch D, Martin R. An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation. Geophysics. 2007; 72(5): SM155-SM167. doi: 10.1190/1.2757586

[40] Pled F, Desceliers C. Review and recent developments on the perfectly matched layer (PML) method for the numerical modeling and simulation of elastic wave propagation in unbounded domains. Arch Comput Methods Eng. 2022; 29: 471-518. doi: 10.1007/s11831—021—09581—y

[41] Liseikin VD. Grid Generation Methods . 2nd ed. Dordrecht, Netherlands: Springer; 2010. doi: 10.1007/978—90—481—2912—6

[42] Schenk O, Gärtner K. Solving unsymmetric sparse systems of linear equations with PARDISO. Future Gener Comput Syst . 2004; 20(3): 475-487. doi: 10.1016/j.future.2003.07.011

[43] Chew WC, Weedon WH. A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates. Microw Opt Technol Lett. 1994; 7(13): 599-604. doi: 10.1002/mop.4650071304

[44] Käser M, Dumbser M. An arbitrary high—order discontinuous Galerkin method for elastic waves on unstructured meshes I: The two—dimensional isotropic case with external source terms. Geophys J Int. 2006; 166(2): 855-877. doi: 10.1111/j.1365—246X.2006.03051.x

[45] Francois RE, Garrison GR. Sound absorption based on ocean measurements. Part II: Boric acid contribution and equation for total absorption. J Acoust Soc Am. 1982; 72(6): 1879-1890. doi: 10.1121/1.388673

[46] Blanch JO, Robertsson JOA, Symes WW. Modeling of a constant Q: Methodology and algorithm for an efficient and optimally inexpensive viscoelastic technique. Geophysics. 1995; 60(1): 176-184. doi: 10.1190/1.1443744

[47] Jung Y, Lee K. Observation of the relationship between ocean bathymetry and acoustic bearing—time record patterns acquired during a reverberation experiment in the southwestern continental margin of the Ulleung Basin, Korea. J Mar Sci Eng. 2021; 9(11): 1259. doi: 10.3390/jmse9111259