Computational Mechanics

, Volume 57, Issue 5, pp 717–732 | Cite as

Numerical modeling of undersea acoustics using a partition of unity method with plane waves enrichment

Original Paper


A new 2D numerical model to predict the underwater acoustic propagation is obtained by exploring the potential of the Partition of Unity Method (PUM) enriched with plane waves. The aim of the work is to obtain sound pressure level distributions when multiple operational noise sources are present, in order to assess the acoustic impact over the marine fauna. The model takes advantage of the suitability of the PUM for solving the Helmholtz equation, especially for the practical case of large domains and medium frequencies. The seawater acoustic absorption and the acoustic reflectance of the sea surface and sea bottom are explicitly considered, and perfectly matched layers (PML) are placed at the lateral artificial boundaries to avoid spurious reflexions. The model includes semi-analytical integration rules which are adapted to highly oscillatory integrands with the aim of reducing the computational cost of the integration step. In addition, we develop a novel strategy to mitigate the ill-conditioning of the elemental and global system matrices. Specifically, we compute a low-rank approximation of the local space of solutions, which in turn reduces the number of degrees of freedom, the CPU time and the memory footprint. Numerical examples are presented to illustrate the capabilities of the model and to assess its accuracy.


Underwater acoustic propagation Helmholtz equation Partition of unity method Plane waves Complex wavenumber Low-rank approximation Singular value decomposition 



This work is partially supported by KIC InnoEnergy and European Institute of Innovation and Technology (EIT) through project Offshore Test Station (OTS; 03_2011_LH03 Industry Energy Efficiency).


  1. 1.
    Nedwell JR, Langworthy J, Howell D (2003) Assessment of sub-sea acoustic noise and vibration from offshore wind turbines and its impact on marine wildlife; initial measurements of underwater noise during construction of offshore windfarms, and comparison with background noise, Subacoustech Report ref: 544R0424, published by COWRIEGoogle Scholar
  2. 2.
    Ainslie MA, de Jong CAF, Dol HS, Blaquière G, Marasini C (2009) Assessment of natural and anthropogenic sound sources and acoustic propagation in the North Sea. TNO report TNO-DV, p. C085Google Scholar
  3. 3.
    Thomsen F, Lüdemann K, Kafemann R, Piper W (2006) Effects of offshore wind farm noise on marine mammals and fish. Biola, Hamburg, Germany on behalf of COWRIE Ltd, p. 62Google Scholar
  4. 4.
    Thomsen F (2009) Assessment of the environmental impact of underwater noise. OSPAR Commission. Biodiversity Series, vol. 436Google Scholar
  5. 5.
    Southall BL, Bowles AE, Ellison WT, Finneran JJ, Gentry RL, Greene CL Jr, Kastak D, Ketten DR, Miller JH, Nachtigall PE, Richardson WJ, Thomas JA, Tyack PL (2007) Marine mammal noise exposure criteria: initial scientific recommendations. Aquat Mamm 33(4):411–521CrossRefGoogle Scholar
  6. 6.
    Marine Mammal Commission (1972) ‘The Marine Mammal Protection Act of 1972Google Scholar
  7. 7.
    OSPAR Commission (1992) Convention for the Protection of the Marine Environment of the North-East Atlantic. SintraGoogle Scholar
  8. 8.
    European Commission (2008) Directive 2008/56/EC establishing a framework for community action in the field of marine environmental policy (Marine Strategy Framework Directive). European Parliament and CouncilGoogle Scholar
  9. 9.
    Dekeling RPA, Tasker ML, Ainslie MA, Andersson M, Andr M, Castellote M, Borsani JF, Dalen J, Folegot T, Leaper R, Liebschner A, Pajala J, Robinson SP, Sigray P, Sutton G, Thomsen F, Van der Graaf AJ, Werner S, Wittekind D, Young JV (2013) Monitoring guidance for underwater noise in European seas. Second Report of the technical subgroup on underwater noise (TSG Noise)Google Scholar
  10. 10.
    Hazelwood RA, Connelly J (2005) Estimation of underwater noise—a simplified method. Int J Soc for Underw Technol 26(3):51–57Google Scholar
  11. 11.
    Tappert FD (1977) The parabolic approximation method, vol. 70. Lecture Notes in physics, Courant Institute of Mathematical Sciences, New York University: Springer, chap  5, pp 224–287Google Scholar
  12. 12.
    Collins MD (1993) A split-step Padé solution for the parabolic equation method. J Acoust Soc Am 93:1736–1742CrossRefGoogle Scholar
  13. 13.
    Babuška I, Sauter SA (1997) Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J Numer Anal 34:2392–2423MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Ihlenburg F (1998) Finite element analysis of acoustic scattering, vol. 132 of applied mathematical sciences, 1st edn. Springer-Verlag, New YorkCrossRefMATHGoogle Scholar
  15. 15.
    Deraemaeker A, Babuška I, Bouillard P (1999) Dispersion and pollution of the FEM solution for the Helmholtz equation in one, two and three dimensions. Int J Numer Methods Engrg 46:471–499CrossRefMATHGoogle Scholar
  16. 16.
    Melenk JM (1995) On Generalized Finite Element Methods. PhD thesis, The University of MarylandGoogle Scholar
  17. 17.
    Melenk JM, Babuška I (1996) The partition of unity finite element method: basic theory and applications. Comput Methods Appl Mech Eng 139:289–314MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Babuška I, Melenk JM (1997) The partition of unity method. Int J Numer Methods Engrg 40:727–758MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Herrera I, Sabina FJ (1978) Connectivity as an alternative to boundary integral equations: construction of bases. Proc Natl Acad Sci USA 75:2059–2063MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Perrey-Debain E, Laghrouche O, Bettess P, Trevelyan J (2004) Plane-wave basis finite elements and boundary elements for three-dimensional wave scattering. Phil Trans R Soc Lond A 362:561–577MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Wang D, Tezaur R, Toivanen J, Farhat C (2012) Overview of the discontinuous enrichment method, the ultra-weak variational formulation, and the partition of unity method for acoustic scattering in the medium frequency regime and performance comparisons. Int J Numer Methods Eng 89:403–417MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Cessenat O, Després B (1998) Application of an ultra weak variational formulation of elliptic PDEs to the two-dimensional Helmholtz problem. SIAM J Numer Anal 35:255–299MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Huttunen T, Monk P, Kaipio JP (2002) Computational aspects of the ultra-weak variational formulation. J Comput Phys 182:27–46MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Farhat C, Harari I, Franca LP (2001) The discontinuous enrichment method. Comput Methods Appl Mech Eng 190:6455–6479MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Farhat C, Harari I, Hetmaniuk U (2003) A discontinuous Galerkin method with Lagrange multipliers for the solution of Helmholtz problems in the mid-frequency regime. Comput Methods Appl Mech Eng 132:1389–1419MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Babuška I, Ihlenburg F, Paik ET, Sauter SA (1995) A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput Methods Appl Mech Eng 128:325–359MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Mayer P, Mandel J (1997) The finite ray element method for the Helmholtz equation of scattering: first numerical experiments. University of Colorado at Denver, Center for Computational Mathematics, DenverGoogle Scholar
  28. 28.
    Laghrouche O, Bettess P (2000) Short wave modelling using special finite elements. J Comput Acoust 8(1):189–210MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ortiz P, Sanchez E (2001) An improved partition of unity finite element model for diffraction problems. Int J Numer Methods Eng 50:2727–2740CrossRefMATHGoogle Scholar
  30. 30.
    Laghrouche O, Bettess P, Astley RJ (2002) Modelling of short wave diffraction problems using approximating systems of plane waves. Int J Numer Methods Eng 54:1501–1533CrossRefMATHGoogle Scholar
  31. 31.
    Laghrouche O, Bettess P, Perrey-Debain E, Trevelyan J (2003) Plane wave basis finite-elements for wave scattering in three dimensions. Commun Numer Meth Eng 19:715–723CrossRefMATHGoogle Scholar
  32. 32.
    Ortiz P (2004) Finite elements using a plane-wave basis for scatering of surface water waves. Phil Trans R Soc Lond A 362:525–540MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Laghrouche O, Bettess P, Perrey-Debain E, Trevelyan J (2005) Wave interpolation finite elements for Helmholtz problems with jumps in the wave speed. Comput Methods Appl Mech Eng 194:367–381MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    De Bel E, Villon P, Bouillard P (2005) Forced vibrations in the medium frequency range solved by a partition of unity method with local information. Int J Numer Methods Eng 62:1105–1126CrossRefMATHGoogle Scholar
  35. 35.
    Strouboulis T, Hidajat R (2006) Partition of unity method for Helmholtz equation: q-convergence for plane waves and wave-band local bases. Appl Math 51(2):181–204MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Bettess P, Shirron J, Laghrouche O, Peseux B, Sugimoto R, Trevelyan J (2003) A numerical integration scheme for special finite elements for the Helmholtz equation. Int J Numer Methods Eng 56:531–552CrossRefMATHGoogle Scholar
  37. 37.
    Sugimoto R, Bettess P, Trevelyan J (2003) A numerical integration scheme for special quadrilateral finite elements for the Helmholtz equation. Commun Numer Meth Eng 19:233–245MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Kuperman WA, Lynch JF (2004) Shallow-water acoustics. Phys today 57:55–61CrossRefGoogle Scholar
  39. 39.
    European Wind Energy Association (EWEA) and others (2013) Deep water—the next step for offshore wind energy. Brussels: A report by the European Wind Energy Association 2013Google Scholar
  40. 40.
    Ainslie MA, McColm JG (1998) A simplified formula for viscous and chemical absorption in sea water. J Acoust Soc Am 103:1671–1672CrossRefGoogle Scholar
  41. 41.
    Wong GSK, Zhu S (1995) Speed of sound in seawater as a function of salinity, temperature, and pressure. J Acoust Soc Am 97:1732–1736CrossRefGoogle Scholar
  42. 42.
    Chen C-T, Millero FJ (1977) Speed of sound in seawater at high pressures. J Acoust Soc Am 62:1129–1135CrossRefGoogle Scholar
  43. 43.
    Leroy CC, Parthiot F (1998) Depth-pressure relationships in the oceans and seas. J Acoust Soc Am 103:1346–1352CrossRefGoogle Scholar
  44. 44.
    Isaacson M, Qu S (1990) Waves in a harbour with partially reflecting boundaries. Coast Eng 14:193–214CrossRefGoogle Scholar
  45. 45.
    Berkhoff JCW (1976) Mathematical models for simple harmonic linear water waves. Wave diffraction and refraction. PhD thesis, TU Delft, Delft, NederlandsGoogle Scholar
  46. 46.
    Urick RJ (1983) Principles of underwater sound, 3rd edn. Peninsula Publishing, Los AltosGoogle Scholar
  47. 47.
    Burdic WS (1983) Underwater acoustic system analysis. Prentice-Hall signal processing series, Prentice-Hall Inc., LondonGoogle Scholar
  48. 48.
    Rappaport CM (1995) Perfectly matched absorbing boundary conditions based on anisotropic lossy mapping of space. IEEE Microw Guided Wave Lett 5:90–92MathSciNetCrossRefGoogle Scholar
  49. 49.
    Bérenger J-P (1994) A perfectly matched layer for the absorption of electromagnetic waves. J Comput Phys 114:185–200MathSciNetCrossRefMATHGoogle Scholar
  50. 50.
    Johnson SG (2007) Notes on Perfectly Matched Layers (PMLs). Notes for MIT courses 18.369 and 18.336Google Scholar
  51. 51.
    Kinsler LE, Frey AR, Coppens AB, Sanders JV (2000) Fundamentals of acoustics, 4th edn. Wiley, New YorkGoogle Scholar
  52. 52.
    Strouboulis T, Babuška I, Hidajat R (2006) The generalized finite element method for Helmholtz equation: theory, computation, and open problems. Comput Methods Appl Mech Eng 195:4711–4731MathSciNetCrossRefMATHGoogle Scholar
  53. 53.
    Strouboulis T, Hidajat R, Babuška I (2008) The generalized finite element method for Helmholtz equation. Part II: effect of choice of handbook functions, error due to absorbing boundary conditions and its assessment. Comput Methods Appl Mech Eng 197:364–380CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Raúl Hospital-Bravo
    • 1
  • Josep Sarrate
    • 1
  • Pedro Díez
    • 1
  1. 1.Laboratori de Càlcul Numèric (LaCàN), Departament de Matemàtica Aplicada IIIUniversitat Politècnica de CatalunyaBarcelonaSpain

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