Skip to main content
SpringerLink
Log in

A Mixed Eulerian–Lagrangian Spectral Element Method for Nonlinear Wave Interaction with Fixed Structures

  • Original Article
  • Published:
Water Waves Aims and scope Submit manuscript

Abstract

We present a high-order nodal spectral element method for the two-dimensional simulation of nonlinear water waves. The model is based on the mixed Eulerian–Lagrangian (MEL) method. Wave interaction with fixed truncated structures is handled using unstructured meshes consisting of high-order iso-parametric quadrilateral/triangular elements to represent the body surfaces as well as the free surface elevation. A numerical eigenvalue analysis highlights that using a thin top layer of quadrilateral elements circumvents the general instability problem associated with the use of asymmetric mesh topology. We demonstrate how to obtain a robust MEL scheme for highly nonlinear waves using an efficient combination of (i) global \(L^2\) projection without quadrature errors, (ii) mild modal filtering and (iii) a combination of local and global re-meshing techniques. Numerical experiments for strongly nonlinear waves are presented. The experiments demonstrate that the spectral element model provides excellent accuracy in prediction of nonlinear and dispersive wave propagation. The model is also shown to accurately capture the interaction between solitary waves and fixed submerged and surface-piercing bodies. The wave motion and the wave-induced loads compare well to experimental and computational results from the literature.

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

Similar content being viewed by others

References

  1. Aristodemo, F., Tripepi, G., Davide Meringolo, D., Veltri, P.: Solitary wave-induced forces on horizontal circular cylinders: laboratory experiments and SPH simulations. Coast. Eng. 129, 17–35 (2017)

    Google Scholar 

  2. Beck, R., Reed, A.: Modern computational methods for ships in a seaway. Trans. Soc. Nav. Archit. Mar. Eng. 109, 1–52 (2001)

    Google Scholar 

  3. Bjøntegaard, T., Rønquist, E.M.: Accurate interface-tracking for arbitrary Lagrangian–Eulerian schemes. J. Comput. Phys. 228(12), 4379–4399 (2009)

    MathSciNet  MATH  Google Scholar 

  4. Cai, X., Langtangen, H.P., Nielsen, B.F., Tveito, A.: A finite element method for fully nonlinear water waves. J. Comput. Phys. 143, 544–568 (1998)

    MathSciNet  MATH  Google Scholar 

  5. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods—fundamentals in single domains. Springer, New York (2006)

    MATH  Google Scholar 

  6. Cavaleri, L., Alves, J.-H.G.M., Ardhuin, F., Babanin, A., Banner, M., Belibassakis, K., Benoit, M., Donelan, M., Groeneweg, J., Herbers, T.H.C., Hwang, P., Janssen, P.A.E.M., Janssen, T., Lavrenov, I.V., Magne, R., Monbaliu, J., Onorato, M., Polnikov, V., Resio, D., Rogers, W.E., Sheremet, A., McKee Smith, J., Tolman, H.L., van Vledder, G., Wolf, J., Young, I.: Wave modelling—the state of the art. Prog. Oceanogr. 75(4), 603–674 (2007)

    Google Scholar 

  7. Chern, M .J., Borthwick, A .G .L., Eatock Taylor, R.: A pseudospectral \(\sigma \)-transformation model of 2-D nonlinear waves. J. Fluids Struct. 13(5), 607–630 (1999)

    Google Scholar 

  8. Chian, C., Ertekin, R.C.: Diffraction of solitary waves by submerged horizontal cylinders. Wave Motion 15(2), 121–142 (1992)

    MathSciNet  MATH  Google Scholar 

  9. Christiansen, T., Engsig-Karup, A.P., Bingham, H.B.: Efficient pseudo-spectral model for nonlinear water waves. In Proceedings of the 27th International Workshop on Water Waves and Floating Bodies (IWWWFB) (2012)

  10. Clement, A., Mas, S.: Hydrodynamic forces induced by a solitary wave on a submerged circular cylinder. Proc. Int. Offshore Polar Eng. Conf. 1, 339–347 (1995)

    Google Scholar 

  11. Cooker, M.J., Peregrine, D.H., Vidal, C., Dold, J.W.: The interaction between a solitary wave and a submerged cylinder. J. Fluid Mech. 215(1), 1–22 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Cooker, M.J., Weidman, P.D., Bale, D.S.: Reflection of a high-amplitude solitary wave at a vertical wall. Oceanogr. Lit. Rev. 7, 1279 (1998)

    MATH  Google Scholar 

  13. Dommermuth, D.G., Yue, D.K.P.: A high-order spectral method for the study of nonlinear gravity waves. J. Fluid Mech. 184, 267–288 (1987)

    MATH  Google Scholar 

  14. Donea, J., Huerta, A., Ponthot, J.P., Rodriguez-Ferran, A.: Arbitrary Lagrangian–Eulerian methods, encyclopedia of computational mechanics, Chapter 14, vol. 1. Wiley, Hoboken (2004)

    Google Scholar 

  15. Ducrozet, G., Bingham, H.B., Engsig-Karup, A.P., Ferrant, P.: High-order finite difference solution for 3D nonlinear wave–structure interaction. J. Hydrodyn. Ser. B 22(5), 225–230 (2010)

    Google Scholar 

  16. Dutykh, D., Clamond, D.: Efficient computation of steady solitary gravity waves. Wave Motion 51(1), 86–99 (2014)

    MathSciNet  MATH  Google Scholar 

  17. Engsig-Karup, A.P.: Analysis of efficient preconditioned defect correction methods for nonlinear water waves. Int. J. Numer. Method. Fluids 74(10), 749–773 (2014)

    MathSciNet  Google Scholar 

  18. Engsig-Karup, A.P., Bingham, H.B., Lindberg, O.: An efficient flexible-order model for 3D nonlinear water waves. J. Comput. Phys. 228, 2100–2118 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Engsig-Karup, A.P., Eskilsson, C., Bigoni, D.: A stabilised nodal spectral element method for fully nonlinear water waves. J. Comput. Phys. 318(6), 1–21 (2016)

    MathSciNet  MATH  Google Scholar 

  20. Engsig-Karup, A.P., Eskilsson, C., Bigoni, D.: Unstructured spectral element model for dispersive and nonlinear wave propagation. In Proceedings 26th International Ocean and Polar Engineering Conference (ISOPE), Greece (2016)

  21. Engsig-Karup, A.P., Glimberg, S.L., Nielsen, A.S., Lindberg, O.: Fast hydrodynamics on heterogenous many-core hardware. In: Raphäel, C. (ed.) Designing Scientific Applications on GPUs, Lecture Notes in Computational Science and Engineering. CRC Press, Cambridge (2013)

    Google Scholar 

  22. Engsig-Karup, A.P., Hesthaven, J.S., Bingham, H.B., Madsen, P.A.: Nodal DG-FEM solutions of high-order Boussinesq-type equations. J. Eng. Math. 56, 351–370 (2006)

    MathSciNet  MATH  Google Scholar 

  23. Engsig-Karup, A.P., Madsen, M.G., Glimberg, S.L.: A massively parallel GPU-accelerated model for analysis of fully nonlinear free surface waves. Int. J. Numer. Method. Fluids 70(1), 20–36 (2011)

    MathSciNet  MATH  Google Scholar 

  24. Ferrant, P.: Simulation of strongly nonlinear wave generation and wave-body interactions using a 3-D MEL model. In: Proceedings of the 21st ONR Symposium on Naval Hydrodynamics, Trondheim, Norway, pp. 93–109 (1996)

  25. Fischer, P., Mullen, J.: Filter-based stabilization of spectral element methods. Comptes Rendus De L Academie Des Sciences Serie I-mathematique 332, 265–270,02 (2001)

    MathSciNet  MATH  Google Scholar 

  26. Gagarina, E., Ambati, V.R., van der Vegt, J.J.W., Bokhove, O.: Variational space-time (dis)continuous Galerkin method for nonlinear free surface water waves. J. Comput. Phys. 275, 459–483 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Gordon, W.J., Hall, C.A.: Construction of curvilinear coordinate systems and their applications to mesh generation. Int. J. Numer. Method. Eng. 7, 461–477 (1973)

    MATH  Google Scholar 

  28. Gouin, M., Ducrozet, G., Ferrant, P.: Propagation of 3D nonlinear waves over an elliptical mound with a High-Order Spectral method. Eur. J. Mech. B Fluids 63, 9–24 (2017)

    MathSciNet  MATH  Google Scholar 

  29. Greaves, D .M., Wu, G .X., Borthwick, A .G .L., Eatock Taylor, R.: A moving boundary finite element method for fully nonlinear wave simulations. J. Ship Res. 41(3), 181–194 (1997)

    Google Scholar 

  30. Harris, J.C., Dombre, E., Benoit, M., Grilli, S.T.: A comparison of methods in fully nonlinear boundary element numerical wave tank development. In: 14émes Journées de l’Hydrodynamique (2014)

  31. Herrmann, L.R.: Laplacian-isoparametric grid generation scheme. J. Mar. Sci. Appl. 102(5), 749–756 (1976)

    Google Scholar 

  32. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2008)

    MATH  Google Scholar 

  33. Kim, C.H., Clément, A.H., Tanizawa, K.: Recent research and development of numerical wave tanks—a review. Int. J. Offshore Polar Eng. 9(4), 241–256 (1999)

    Google Scholar 

  34. Klettner, C., Eames, I.: Momentum and energy of a solitary wave interacting with a submerged semi-circular cylinder. J. Fluid Mech. 708, 576–595 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, 199–215 (1972)

    MathSciNet  Google Scholar 

  36. Laskowski, W., Bingham, H.B., Engsig-Karup, A.P.: Modelling of wave-structure interaction for cylindrical structures using a spectral element multigrid method. In Proceedings of the 34th International Workshop on Water Waves and Floating Bodies (2019)

  37. Lin, P.: A multiple-layer \(\sigma \)-coordinate model for simulation of wave-structure interaction. Comput. Fluids 35(2), 147–167 (2006)

    MATH  Google Scholar 

  38. Longuet-Higgins, M .S., Cokelet, E .D.: The deformation of steep surface waves on water. I. A numerical method of computation. Proc. R. Soc. Lond. Ser. A 350(1660), 1–26 (1976)

    MathSciNet  MATH  Google Scholar 

  39. Luke, J.C.: A variational principle for a fluid with a free surface. J. Fluid Mech. 27(2), 395–397 (1967)

    MathSciNet  MATH  Google Scholar 

  40. Ma, Q.W. (ed.): Advances in numerical simulation of nonlinear water waves, vol. 11. World Scientific Publishing Co. Pte. Ltd., Singapore (2010)

    MATH  Google Scholar 

  41. Ma, Q .W., Wu, G .X., Eatock Taylor, R.: Finite element simulation of fully non-linear interaction between vertical cylinders and steep waves. Part 1: methodology and numerical procedure. Int. J. Numer. Method. Fluids 36(3), 265–285 (2001)

    MATH  Google Scholar 

  42. Ma, Q .W., Wu, G .X., Eatock Taylor, R.: Finite element simulations of fully non-linear interaction between vertical cylinders and steep waves. Part 2: numerical results and validation. Int. J. Numer. Method Fluids 36(3), 287–308 (2001)

    Google Scholar 

  43. Ma, Q.W., Yan, S.: Quasi ALE finite element method for nonlinear water waves. J. Comput. Phys. 212(1), 52–72 (2006)

    MathSciNet  MATH  Google Scholar 

  44. Ma, Q.W., Yan, S.: QALE-FEM for numerical modelling of non-linear interaction between 3d moored floating bodies and steep waves. Int. J. Numer. Method Eng. 78(6), 713–756 (2009)

    MATH  Google Scholar 

  45. Madsen, P.A., Bingham, H.B., Liu, H.: A new boussinesq method for fully nonlinear waves from shallow to deep water. J. Fluid Mech. 462, 1–30 (2002)

    MathSciNet  MATH  Google Scholar 

  46. Maxworthy, T.: Experiments on the collision between solitary waves. J. Fluid Mech. 76(1), 177–185 (1976)

    Google Scholar 

  47. Olson, L.: Algebraic multigrid preconditioning of high-order spectral elements for elliptic problems on a simplicial mesh. SIAM J. Sci. Comput. 29, 2189–2209, 01 (2007)

    MathSciNet  MATH  Google Scholar 

  48. Patera, A.T.: A Spectral element method for fluid dynamics: Laminar flow in a channel expansion. J. Comput. Phys. 54, 468–488 (1984)

    MATH  Google Scholar 

  49. Rienecker, M.M., Fenton, J.D.: A Fourier approximation method for steady water waves. J. Fluid Mech. 104, 119–137 (1981)

    MATH  Google Scholar 

  50. Robertson, I., Sherwin, S.J.: Free-surface flow simulation using hp/spectral elements. J. Comput. Phys. 155, 26–53 (1999)

    MathSciNet  MATH  Google Scholar 

  51. Rønquist, E .M., Patera, A .T.: Spectral element multigrid. i. Formulation and numerical results. J. Sci. Comput. 2(4), 389–406 (1987)

    MathSciNet  MATH  Google Scholar 

  52. Shao, Y.-L., Faltinsen, O.M.: A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics. J. Comput. Phys. 274, 312–332 (2014)

    MathSciNet  MATH  Google Scholar 

  53. Sibley, P., Coates, L.E., Arumugam, K.: Solitary wave forces on horizontal cylinders. Appl. Ocean Res. 4(2), 113–117 (1982)

    Google Scholar 

  54. Tanizawa, K.: A nonlinear simulation method of 3-D body motions in waves (1st report). J. Soc. Nav. Archit. Jpn. 178, 179–191 (1995)

    Google Scholar 

  55. Tsai, W.-T., Yue, D.K.P.: Computation of nonlinear free-surface flows. In Annual review of fluid mechanics, Vol. 28, pp. 249–278. Annual Reviews, Palo Alto, CA (1996)

    MathSciNet  Google Scholar 

  56. Turnbull, M .S., Borthwick, A .G .L., Eatock Taylor, R.: Numerical wave tank based on a \(\sigma \)-transformed finite element inviscid flow solver. Int. J. Numer. Method Fluids 42(6), 641–663 (2003)

    MATH  Google Scholar 

  57. van der Vegt, J.J.W., Tomar, S.K.: Discontinuous galerkin method for linear free-surface gravity waves. J. Sci. Comput. 22(1), 531–567 (2005)

    MathSciNet  MATH  Google Scholar 

  58. Wang, C.-Z., Wu, G.-X.: A brief summary of finite element method applications to nonlinear wave–structure interactions. J. Mar. Sci. Appl. 10, 127–138 (2011)

    Google Scholar 

  59. Wang, L., Tang, H., Wu, Y.: Study on interaction between a solitary wave and a submerged semi-circular cylinder using acceleration potential. In Proceedings of The Annual International Offshore and Polar Engineering Conference (ISOPE) (2013)

  60. Warburton, T.: An explicit construction for interpolation nodes on the simplex. J. Eng. Math. 56(3), 247–262 (2006)

    MathSciNet  MATH  Google Scholar 

  61. West, B.J., Brueckner, K.A., Janda, R.S., Milder, D.M., Milton, R.L.: A new numerical method for surface hydrodynamics. J. Geophys. Res. Oceans 92(C11), 11803–11824 (1987)

    Google Scholar 

  62. Westhuis, J.-H.: The numerical simulation of nonlinear waves in a hydrodynamic model test basin. PhD thesis, Department of Mathematics, University of Twente, The Netherlands (2001)

  63. Wu, G .X., Eatock Taylor, R.: Finite element analysis of two-dimensional non-linear transient water waves. Appl. Ocean Res. 16(6), 363–372 (1994)

    Google Scholar 

  64. Wu, G .X., Eatock Taylor, R.: Time stepping solutions of the two-dimensional nonlinear wave radiation problem. Ocean Eng. 22(8), 785–798 (1995)

    Google Scholar 

  65. Yan, H., Liu, Y.: An efficient high-order boundary element method for nonlinear wave–wave and wave–body interactions. J. Comput. Phys. 230(2), 402–424 (2011)

    MathSciNet  MATH  Google Scholar 

  66. Yeung, E.W.: Numerical methods in free-surface flows. In: Annual Review of Fluid Mechanics, Vol. 14, pp. 395–442. Annual Reviews, Palo Alto, Calif. (1982)

    MathSciNet  Google Scholar 

  67. Zakharov, V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 190–194 (1968)

    Google Scholar 

  68. Zhu, Q., Liu, Y., Yue, D.K.P.: Three-dimensional instability of standing waves. J. Fluid Mech. 496, 213–242 (2003)

    MathSciNet  MATH  Google Scholar 

  69. Zienkewitch, O.C.: The finite element method in engineering science, 2nd edn. McGraw-Hill, New York (1971)

    Google Scholar 

  70. Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P.: Chapter 6—free surface and buoyancy driven flows. In: Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P. (eds.) The Finite Element Method for Fluid Dynamics, 7th edn, pp. 195–224. Butterworth-Heinemann, Oxford (2014)

    MATH  Google Scholar 

Download references

Acknowledgements

This work contributed to the activities in the research project Multi-fidelity Decision making tools for Wave Energy Systems (MIDWEST) that is supported by the OCEAN-ERANET program. The DTU Computing Center (DCC) supported the work with access to computing resources. Claes Eskilsson was partially supported by the Swedish Energy Agency through Grant no. 41125-1.

Author information

Authors and Affiliations

  1. Department of Applied Mathematics and Computer Science, Center for Energy Resources Engineering, Technical University of Denmark, Lyngby, Denmark

    Allan P. Engsig-Karup & Carlos Monteserin

  2. Maritime Research, Research Institutes of Sweden, Gothenburg, Sweden

    Claes Eskilsson

  3. Department of Civil Engineering, Aalborg University, Aalborg, Denmark

    Claes Eskilsson

Authors
  1. Allan P. Engsig-Karup
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Carlos Monteserin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Claes Eskilsson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Carlos Monteserin.

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

Engsig-Karup, A.P., Monteserin, C. & Eskilsson, C. A Mixed Eulerian–Lagrangian Spectral Element Method for Nonlinear Wave Interaction with Fixed Structures. Water Waves 1, 315–342 (2019). https://doi.org/10.1007/s42286-019-00018-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s42286-019-00018-5

Keywords

Search

Navigation