Authors

E. Gagarina, V.R. Ambati, J.J.W. van der Vegt, O. Bokhove

Abstract

A new variational finite element method is developed for nonlinear free surface gravity water waves using the potential flow approximation. This method also handles waves generated by a wave maker. Its formulation stems from Miles’ variational principle for water waves together with a finite element discretization that is continuous in space and discontinuous in time. One novel feature of this variational finite element approach is that the free surface evolution is variationally dependent on the mesh deformation vis-à-vis the mesh deformation being geometrically dependent on free surface evolution. Another key feature is the use of a variational (dis)continuous Galerkin finite element discretization in time. Moreover, in the absence of a wave maker, it is shown to be equivalent to the second order symplectic Störmer–Verlet time stepping scheme for the free-surface degrees of freedom. These key features add to the stability of the numerical method. Finally, the resulting numerical scheme is verified against nonlinear analytical solutions with long time simulations and validated against experimental measurements of driven wave solutions in a wave basin of the Maritime Research Institute Netherlands.

Keywords

Nonlinear water waves; Finite element Galerkin method; Variational formulation; Symplectic time integration; Deforming grids

Citation

  • Journal: Journal of Computational Physics
  • Year: 2014
  • Volume: 275
  • Issue:
  • Pages: 459–483
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.jcp.2014.06.035

BibTeX

@article{Gagarina_2014,
  title={{Variational space–time (dis)continuous Galerkin method for nonlinear free surface water waves}},
  volume={275},
  ISSN={0021-9991},
  DOI={10.1016/j.jcp.2014.06.035},
  journal={Journal of Computational Physics},
  publisher={Elsevier BV},
  author={Gagarina, E. and Ambati, V.R. and van der Vegt, J.J.W. and Bokhove, O.},
  year={2014},
  pages={459--483}
}

Download the bib file

References

  • Ambati, (2008)
  • Arnold, D. N., Brezzi, F., Cockburn, B. & Marini, L. D. Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems. SIAM Journal on Numerical Analysis vol. 39 1749–1779 (2002) – 10.1137/s0036142901384162
  • Beale, J. T. A convergent boundary integral method for three-dimensional water waves. Mathematics of Computation vol. 70 977–1030 (2000) – 10.1090/s0025-5718-00-01218-7
  • Bettess, P. Operation counts for boundary integral and finite element methods. International Journal for Numerical Methods in Engineering vol. 17 306–308 (1981) – 10.1002/nme.1620170214
  • Broeze, J., van Daalen, E. F. G. & Zandbergen, P. J. A three-dimensional panel method for nonlinear free surface waves on vector computers. Computational Mechanics vol. 13 12–28 (1993) – 10.1007/bf00350699
  • Cotter, C. & Bokhove, O. Variational water-wave model with accurate dispersion and vertical vorticity. Journal of Engineering Mathematics vol. 67 33–54 (2009) – 10.1007/s10665-009-9346-3
  • Dal Maso, Definition and weak stability of nonconservative products. J. Math. Pures Appl. (1995)
  • Fenton, J. D. & Rienecker, M. M. A Fourier method for solving nonlinear water-wave problems: application to solitary-wave interactions. Journal of Fluid Mechanics vol. 118 411 (1982) – 10.1017/s0022112082001141
  • Fochesato, C. & Dias, F. A fast method for nonlinear three-dimensional free-surface waves. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences vol. 462 2715–2735 (2006) – 10.1098/rspa.2006.1706
  • Fochesato, C., Grilli, S. & Dias, F. Numerical modeling of extreme rogue waves generated by directional energy focusing. Wave Motion vol. 44 395–416 (2007) – 10.1016/j.wavemoti.2007.01.003
  • Gagarina, A Hamiltonian Boussinesq model with horizontally sheared currents. (2012)
  • Gagarina, E., van der Vegt, J. & Bokhove, O. Horizontal circulation and jumps in Hamiltonian wave models. Nonlinear Processes in Geophysics vol. 20 483–500 (2013) – 10.5194/npg-20-483-2013
  • Gagarina, (2014)
  • Guerber, E., Benoit, M., Grilli, S. T. & Buvat, C. A fully nonlinear implicit model for wave interactions with submerged structures in forced or free motion. Engineering Analysis with Boundary Elements vol. 36 1151–1163 (2012) – 10.1016/j.enganabound.2012.02.005
  • Hairer, (2006)
  • Hennig, Experimental variation of focussing wave groups for the investigation of their predictability. (2009)
  • Hou, Convergence of a boundary integral method for 3-D water waves. SIAM J. Numer. Anal. (2002)
  • Johnson, (1997)
  • Kim, J. W. & Bai, K. J. A finite element method for two-dimensional water-wave problems. International Journal for Numerical Methods in Fluids vol. 30 105–122 (1999) – 10.1002/(sici)1097-0363(19990515)30:1<105::aid-fld822>3.0.co;2-f
  • Kim, J. W., Bai, K. J., Ertekin, R. C. & Webster, W. C. A Strongly-Nonlinear Model for Water Waves in Water of Variable Depth—The Irrotational Green-Naghdi Model. Journal of Offshore Mechanics and Arctic Engineering vol. 125 25–32 (2003) – 10.1115/1.1537722
  • KLOPMAN, G., VAN GROESEN, B. & DINGEMANS, M. W. A variational approach to Boussinesq modelling of fully nonlinear water waves. Journal of Fluid Mechanics vol. 657 36–63 (2010) – 10.1017/s0022112010001345
  • Kristina, W., Bokhove, O. & van Groesen, E. Effective coastal boundary conditions for tsunami wave run-up over sloping bathymetry. Nonlinear Processes in Geophysics vol. 21 987–1005 (2014) – 10.5194/npg-21-987-2014
  • Latifah, A. L. & van Groesen, E. Coherence and predictability of extreme events in irregular waves. Nonlinear Processes in Geophysics vol. 19 199–213 (2012) – 10.5194/npg-19-199-2012
  • Lesoinne, M. & Farhat, C. Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Computer Methods in Applied Mechanics and Engineering vol. 134 71–90 (1996) – 10.1016/0045-7825(96)01028-6
  • The deformation of steep surface waves on water - I. A numerical method of computation. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences vol. 350 1–26 (1976) – 10.1098/rspa.1976.0092
  • Luke, J. C. A variational principle for a fluid with a free surface. Journal of Fluid Mechanics vol. 27 395–397 (1967) – 10.1017/s0022112067000412
  • 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. International Journal for Numerical Methods in Fluids vol. 36 265–285 (2001) – 10.1002/fld.131
  • 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. International Journal for Numerical Methods in Fluids vol. 36 287–308 (2001) – 10.1002/fld.133
  • Ma, Q. W. & Yan, S. Quasi ALE finite element method for nonlinear water waves. Journal of Computational Physics vol. 212 52–72 (2006) – 10.1016/j.jcp.2005.06.014
  • Miles, J. W. On Hamilton’s principle for surface waves. Journal of Fluid Mechanics vol. 83 153–158 (1977) – 10.1017/s0022112077001104
  • Rhebergen, (2010)
  • Rhebergen, S., Bokhove, O. & van der Vegt, J. J. W. Discontinuous Galerkin finite element methods for hyperbolic nonconservative partial differential equations. Journal of Computational Physics vol. 227 1887–1922 (2008) – 10.1016/j.jcp.2007.10.007
  • Romate, J. E. & Zandbergen, P. J. Boundary integral equation formulations for free-surface flow problems in two and three dimensions. Computational Mechanics vol. 4 276–282 (1989) – 10.1007/bf00301385
  • Satish, Efficient management of parallelism in object oriented numerical software libraries. (1997)
  • Satish,
  • Satish, (2004)
  • Tsai, W. & Yue, D. K. P. Computation of Nonlinear Free-Surface Flows. Annual Review of Fluid Mechanics vol. 28 249–278 (1996) – 10.1146/annurev.fl.28.010196.001341
  • van der Vegt, J. J. W. & Tomar, S. K. Discontinuous Galerkin Method for Linear Free-Surface Gravity Waves. Journal of Scientific Computing vols 22–23 531–567 (2005) – 10.1007/s10915-004-4149-1
  • van der Vegt, J. J. W. & Xu, Y. Space–time discontinuous Galerkin method for nonlinear water waves. Journal of Computational Physics vol. 224 17–39 (2007) – 10.1016/j.jcp.2006.11.031
  • Vinje, T. & Brevig, P. Numerical simulation of breaking waves. Advances in Water Resources vol. 4 77–82 (1981) – 10.1016/0309-1708(81)90027-0
  • Volpert, The spaces BV and quasilinear equations. Mat. Sb. (N.S.) (1967)
  • Wang, C. Z. & Wu, G. X. Interactions between fully nonlinear water waves and cylinder arrays in a wave tank. Ocean Engineering vol. 37 400–417 (2010) – 10.1016/j.oceaneng.2009.12.006
  • Wang, C. & Wu, G. A brief summary of finite element method applications to nonlinear wave-structure interactions. Journal of Marine Science and Application vol. 10 127–138 (2011) – 10.1007/s11804-011-1052-7
  • Westhuis, (2001)
  • Wu, G. X. & Hu, Z. Z. Simulation of nonlinear interactions between waves and floating bodies through a finite-element-based numerical tank. Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences vol. 460 2797–2817 (2004) – 10.1098/rspa.2004.1302