Skip to main content
SpringerLink
Log in

A coupled weak-form meshfree method for underwater noise prediction

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

A meshfree weak-form method based on combining a radial point interpolation method (RPIM) and modified Dirichlet-to-Neumann (MDtN) boundary condition is proposed for use in analyzing underwater acoustic radiation. To apply a meshfree weak-form method to exterior acoustic radiation prediction, an unbounded problem domain is truncated by an artificial boundary to yield a finite computational domain. To improve the interpolation accuracy, RPIM is used to form an acoustic shape function without use of a mesh or connectivity of nodes to implement field variable interpolations. An MDtN boundary condition is imposed on the artificial boundary to guarantee a unique solution. The factors affecting the performance of the devised method are investigated, and numerical examples are used to test its performance. Simulations indicate that the method can produce more accurate results and converge faster and more efficiently, and is less sensitive to the acoustic wavenumber than the finite element scheme. Therefore, this proposed method is competitive at predicting underwater acoustic radiation.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20
Fig. 21
Fig. 22
Fig. 23
Fig. 24
Fig. 25
Fig. 26
Fig. 27

Similar content being viewed by others

References

  1. Ciskowski RD, Brebbia CA (2005) Boundary element methods in acoustics. Kluwer Academic Publishers Group, The Netherlands

    MATH  Google Scholar 

  2. Kirkup S (2007) The boundary element method in acoustics. Integrated Sound Software

  3. Liu XJ, Wu HJ, Jiang WK (2017) Hybrid approximation hierarchical boundary element methods for acoustic problems. J Comput Acoust 25(3):1750013

    MathSciNet  Google Scholar 

  4. Marburg S (2016) The burton and miller method: Unlocking another mystery of its coupling parameter. J Comput Acoust 24(1):1550016

    MathSciNet  MATH  Google Scholar 

  5. Marburg S (2018) A pollution effect in the boundary element method for acoustic problems. J Theor Comp Acout 26:1850018

    MathSciNet  Google Scholar 

  6. Wu YH, Dong CY, Yang HS, Sun FL (2021) Isogeometric symmetric FE-BE coupling method for acoustic-structural interaction. Appl Math Comput 393:125758

    MathSciNet  MATH  Google Scholar 

  7. Takahashi T, Yamamoto T, Shimba Y, Isakari H, Matsumoto T (2019) A framework of shape optimisation based on the isogeometric boundary element method toward designing thin-silicon photovoltaic devices. Eng Comput 35:423–449

    Google Scholar 

  8. Gong JY, An JY, Ma L, Xu HT (2017) Numerical quadrature for singular and near-singular integrals of boundary element method and its applications in large-scale acoustic problems. Chin J Acoust 36:289–301

    Google Scholar 

  9. Kirkup S (2019) The boundary element method in acoustics: a survey. Appl Sci 9(8):1642

    Google Scholar 

  10. Ihlenburg F (1998) Finite element analysis of acoustic scattering. Springer, New York

    MATH  Google Scholar 

  11. Kaltenbacher M (2018) Computational acoustics. Springer, Berlin

    MATH  Google Scholar 

  12. Servan-Camas B, Gutierrez-Romero JE, Garcia-Espinosa J (2018) A time-domain second-order FEM model for the wave diffraction-radiation problem. Validation with a semisubmersible platform. Mar Struct 58:278–300

    Google Scholar 

  13. He ZH, Chen ZH, Jiang YB, Cao XF, Zhao T (2020) Effects of the standoff distance on hull structure damage subjected to near-field underwater explosion. Mar Struct 74:102839

    Google Scholar 

  14. Thompson LL (2006) A review of finite-element methods for time-harmonic acoustics. J Acoust Soc Am 119(3):1315–1330

    Google Scholar 

  15. Zienkiewicz O, Bando K, Bettess P, Chiam T (1985) Mapped infinite elements for exterior wave problems. Int J Numer Meth Eng 21:1229–1251

    MathSciNet  MATH  Google Scholar 

  16. Hu X, Cui XY, Zhang QY, Wang G, Li GY (2017) The stable node-based smoothed finite element method for analyzing acoustic radiation problems. Eng Anal Bound Elem 80:142–151

    MathSciNet  MATH  Google Scholar 

  17. Xu XY, Zhang GY, Zhou B, Wang HY, Tang Q (2019) Analysis of acoustic radiation problems using the cell-based smoothed radial point interpolation method with Dirichlet-to-Neumann boundary condition. Eng Anal Bound Elem 108:447–458

    MathSciNet  MATH  Google Scholar 

  18. Dai TF, Jin X, Yang HZ, Lin TR, Gu YT (2020) Smoothed finite element methods for predicting the mid to high frequency acoustic response in the cylinder-head chamber of a diesel engine. Int J Comp Meth 17(9):1950060

    MathSciNet  MATH  Google Scholar 

  19. Liu GR, Zhang GY (2013) Smoothed point interpolation methods: G space theory and weakened weak forms. World Scientific, Singapore

    MATH  Google Scholar 

  20. Chai YB, Li W, Gong ZX, Li TY (2016) Hybrid smoothed finite element method for two-dimensional under water acoustic scattering problems. Ocean Eng 116(1):129–141

    Google Scholar 

  21. Liu GR, Zhang GY, Zong Z, Li M (2013) Meshfree cell-based smoothed alpha radial point interpolation method (CS-\(\alpha\) RPIM) for solid mechanics problems. Int J Comp Meth 10(4):1350020

    MATH  Google Scholar 

  22. Li W, Chai YB, Lei M, Li TY (2017) Numerical investigation of the edge-based gradient smoothing technique for exterior Helmholtz equation in two dimensions. Comput Struct 182(1):149–164

    Google Scholar 

  23. Chai YB, Li W, Li TY, Gong ZX, You XY (2016) Analysis of underwater acoustic scattering problems using stable node-based smoothed finite element method. Eng Anal Bound Elem 72:27–41

    MathSciNet  MATH  Google Scholar 

  24. Chai YB, Li W, Liu GR, Gong ZX, Li TY (2017) A superconvergent alpha finite element method (S\(\alpha\)FEM) for static and free vibration analysis of shell structures. Comput Struct 179(15):27–47

    Google Scholar 

  25. Keller J, Givoli D (1989) Exact non-reflecting boundary conditions. J Comput Phys 82(1):172–192

    MathSciNet  MATH  Google Scholar 

  26. Givoli D, Patlashenko I (1998) Optimal local non-reflecting boundary conditions. Appl Numer Math 27:3670384

    MathSciNet  MATH  Google Scholar 

  27. Grote M, Keller J (1995) On nonreflecting boundary conditions. J Comput Phys 122:231–243

    MathSciNet  MATH  Google Scholar 

  28. Hsiao G, Nigam N, Pasciak J, Xu L (2011) Error analysis of the DtN-FEM for the scattering problem in acoustics via fourier analysis. J Comput Appl Math 235:4949–4965

    MathSciNet  MATH  Google Scholar 

  29. Geng HR, Yin T, Xu LW (2017) A priori error estimates of the DtN-FEM for the transmission problem in acoustics. J Comput Appl Math 313:1–17

    MathSciNet  MATH  Google Scholar 

  30. Oberai AA, Malhotra M, Pinsky PM (1998) On the implementation of the DtN radiation condition for iterative solution of the Helmholtz equation. Appl Numer Math 27:443–464

    MathSciNet  MATH  Google Scholar 

  31. Li PJ, Yuan XK (2020) Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures. Comput Method Appl M 360:112722

    MathSciNet  MATH  Google Scholar 

  32. Koyama D (2009) Error estimates of the finite element method for the exterior Helmholtz problem with a modified DtN boundary condition. J Comput Appl Math 232:109–121

    MathSciNet  MATH  Google Scholar 

  33. Grote MJ, Kirsch C (2004) Dirichlet-to-Neumann boundary conditions for multiple scattering problems. J Comput Phys 201:630–650

    MathSciNet  MATH  Google Scholar 

  34. Ofir Y, Givoli D (2015) DtN-based mixed-dimensional coupling using a Boundary Stress Recovery technique. Comput Method Appl M 287:31–53

    MathSciNet  MATH  Google Scholar 

  35. Liu GR, Gu YT (2005) An introduction to meshfree methods and their programming. Springer Verlag, The Netherlands

    Google Scholar 

  36. Abbasbandy S, Ghehsareh HR, Hashim I (2012) Numerical analysis of a mathematical model for capillary formation in tumor angiogenesis using a meshfree method based on the radial basis function. Eng Anal Bound Elem 36(12):1811–1818

    MathSciNet  MATH  Google Scholar 

  37. Dehghan M, Shokri A (2008) A numerical method for solution of the two dimensional sine-Gordon equation using the radial basis functions. Math Comput Simul 79(3):700–715

    MathSciNet  MATH  Google Scholar 

  38. Belytschko T, Lu YY, Gu L (1994) Element-free Galerkin methods. Int J Numer Methods Eng 37(2):229–256

    MathSciNet  MATH  Google Scholar 

  39. Zhang Z, Hao SY, Liew KM, Cheng YM (2013) The improved element-free Galerkin method for two-dimensional elastodynamics problems. Eng Anal Bound Elem 37(12):1576–1584

    MathSciNet  MATH  Google Scholar 

  40. Liu GR, Wu YL, Ding H (2004) Meshfree weak-strong (MWS) form method and its application to incompressible flow problems. Int J Numer Meth Fl 46(10):1025–1047

    MathSciNet  MATH  Google Scholar 

  41. Gu YT, Liu GR (2005) A meshfree weak-strong (MWS) form method for time dependent problems. Compt Mech 35(2):134–145

    MATH  Google Scholar 

  42. Tian X, Lin J (2020) A novel radial basis function method for 3D linear and nonlinear advection diffusion reaction equations with variable coefficients. Eng Comput https://doi.org/10.1007/s00366-020-01161-1

  43. Dehghan M, Shafieeabyaneh N (2021) Local radial basis function-finite-difference method to simulate some models in the nonlinear wave phenomena: regularized long-wave and extended Fisher-Kolmogorov equations. Eng Comput 37:1159–1179

    Google Scholar 

  44. Hashemi MS (2020) Numerical study of the one-dimensional coupled nonlinear sine-Gordon equations by a novel geometric meshless method. Eng Comput https://doi.org/10.1007/s00366-020-01001-2

  45. Oruç \(\ddot{\rm O}\) (2021) An efficient meshfree method based on Pascal polynomials and multiple-scale approach for numerical solution of 2-D and 3-D second order elliptic interface problems. J Comput Phys 428:110070

    MathSciNet  MATH  Google Scholar 

  46. Oruç \(\ddot{\rm O}\) (2019) Numerical solution to the deflection of thin plates using the two-dimensional Berger equation with a meshless method based on multiple-scale Pascal polynomials. Appl Math Model 74:441–456

    MathSciNet  MATH  Google Scholar 

  47. Oruç \(\ddot{\rm O}\) (2020) A local hybrid kernel meshless method for numerical solutions of two-dimensional fractional cable equation in neuronal dynamics. Numer Meth Part D E 36(6):1699–1717

    MathSciNet  Google Scholar 

  48. Oruç \(\ddot{\rm O}\) (2022) A strong-form local meshless approach based on radial basis function-finite difference (RBF-FD) method for solving multi-dimensional coupled damped Schrödinger system appearing in Bose-Einstein condensates. Commun Nonlinear Sci 104:106042

    MATH  Google Scholar 

  49. Liu GR, Gu YT (2001) A point interpolation method for two-dimensional solids. Int J Numer Meth Eng 50(4):937–951

    MATH  Google Scholar 

  50. Liu GR, Gu YT (2003) A matrix triangularization algorithm for the polynomial point interpolation method. Comput Method Appl M 192(19):2269–2295

    MATH  Google Scholar 

  51. Liu GR, Gu YT (2001) A local radial point interpolation method (LR-PIM) for free vibration analyses of 2-D solids. J Sound Vib 246(1):29–46

    Google Scholar 

  52. Wang JG, Liu GR (2002) A point interpolation meshless method based on radial basis functions. Int J Numer Meth Eng 54(11):1623–1648

    MATH  Google Scholar 

  53. Oruç \(\ddot{\rm O}\) (2020) Two meshless methods based on local radial basis function and barycentric rational interpolation for solving 2D viscoelastic wave equation. Comput Math Appl 79:3272–3288

    MathSciNet  MATH  Google Scholar 

  54. Oruç \(\ddot{\rm O}\) (2021) A local radial basis function-finite difference (RBF-FD) method for solving 1D and 2D coupled Schrödinger-Boussinesq (SBq) equations. Eng Anal Bound Elem 129:55–66

    MathSciNet  MATH  Google Scholar 

  55. Liu GR, Dai KY, Lim KM, Gu YT (2003) A radial point interpolation method for simulation of two-dimensional piezoelectric structures. Smart Mater Struct 12(2):171–180

    Google Scholar 

  56. Liu L, Liu GR, Tan VBC (2002) Element free method for static and free vibration analysis of spatial thin shell structures. Comput Method Appl M 191(51–52):5923–5942

    MATH  Google Scholar 

  57. Wang JG, Liu GR, Lin P (2002) Numerical analysis of Biot’s consolidation process by radial point interpolation method. Int J Solids Struct 39(6):1557–1573

    MATH  Google Scholar 

  58. Atluri S, Zhu T (1998) A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput Mech 22:117–127

    MathSciNet  MATH  Google Scholar 

  59. Liu GR, Gu YT (2001) A local point interpolation method for stress analysis of two-dimensional solids. Struct Eng Mech 11:221–236

    Google Scholar 

  60. Liu GR, Yan L, Wang JG, Gu YT (2002) Point interpolation method based on local residual formulation using radial basis functions. Struct Eng Mech 14:713–732

    Google Scholar 

  61. Oliveira T, Vélez W, Santana W, Araújo T, Mendonça F, Portela A (2019) A local mesh free method for linear elasticity and fracture mechanics. Eng Anal Bound Elem 101:221–242

    MathSciNet  MATH  Google Scholar 

  62. Santana E, Oliveira T, Vélez W, Araújo A, Martins F, Portela A (2020) A local mesh free numerical method with automatic parameter optimization. Eng Anal Bound Elem 113:55–71

    MathSciNet  MATH  Google Scholar 

  63. Liu GR, Gu YT (2004) Boundary meshfree methods based on the boundary point interpolation methods. Eng Anal Bound Elem 28(5):475–487

    MATH  Google Scholar 

  64. Araújo A, Martins F, Vélez W, Portela A (2021) Automatic mesh-free boundary analysis: multi-objective optimization. Eng Anal Bound Elem 125:264–279

    MathSciNet  MATH  Google Scholar 

  65. Wenterodt C, Estorff OV (2011) Optimized meshfree methods for acoustics. Comput Method Appl M 200(25):2223–2236

    MathSciNet  MATH  Google Scholar 

  66. Zhang GY, Chen ZC, Sui ZX, Tao DS (2019) A cell-based smoothed radial point interpolation method with virtual nodes for three-dimensional mid-frequency acoustic problems. Int J Numer Methods Eng 119(6):548–566

    MathSciNet  Google Scholar 

  67. Wu SW, Xiang Y, Yao JC (2018) A meshfree radial point interpolation coupled with infinite acoustic wave envelope element method for computing acoustic fields. Acta Acust United Ac 104:64–78

    Google Scholar 

  68. Qi LB, Wu YS, Zou MS, Yu Y (2019) Propeller-shaft-hull coupled vibration and its impact on acoustic radiation utilizing sono-elasticity theory. Ocean Eng 171:391–398

    Google Scholar 

  69. Qi LB, Zou MS, Liu SX, Yu Y (2019) Use of impedance mismatch in the control of coupled acoustic radiation of the submarine induced by propeller-shaft system. Mar Struct 65:249–258

    Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 51909016), the Natural Science Foundation of Chongqing, China (Grant No. cstc2020jcyj-msxmX0070), the Science and Technology Research Program of Chongqing Municipal Education Commission (Grant No. KJZD-K202100702 and KJQN201900705) and the Key Laboratory of Marine Power Engineering & Technology (Wuhan University of Technology), Ministry of Transport (Grant No. KLMPET2019-04).

Author information

Author notes

  1. Shaowei Wu, Yang Xiang and Guangnian Li contributed equally to this work.

Authors and Affiliations

  1. School of Shipping and Naval Architecture, Chongqing Jiaotong University, Chongqing, 400074, China

    Shaowei Wu

  2. School of Energy and Power Engineering, Wuhan University of Technology, Wuhan, 430063, Hubei, China

    Yang Xiang

  3. Key Laboratory of Mine Power Engineering & Technology (Ministry of Communications), Wuhan University of Technology, Wuhan, 430063, Hubei, China

    Shaowei Wu & Yang Xiang

  4. Faculty of Maritime and Transportation, Ningbo University, Ningbo, 315611, Zhejiang, China

    Guangnian Li

Authors
  1. Shaowei Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Xiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Guangnian Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaowei Wu.

Ethics declarations

Conflict of interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wu, S., Xiang, Y. & Li, G. A coupled weak-form meshfree method for underwater noise prediction. Engineering with Computers 38, 5091–5109 (2022). https://doi.org/10.1007/s00366-021-01593-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-021-01593-3

Keywords

Search

Navigation