A new variational finite element method is developed for nonlinear free surface gravity water waves. This method also handles waves generated by a wave maker. Its formulation stems from Miles' variational principle for water waves together with a space-time finite element discretization that is continuous in space and discontinuous in time. The key features of this formulation are: (i) a discrete variational approach that gives rise to conservation of discrete energy and phase space and preservation of variational structure; and (ii) a space-time approach that guarantees satisfaction of the geometric conservation law which is crucial in handling the deforming flow domain due to the wave maker and free surface motion.
The numerical discretization is a combination of a second order finite element discretization in space and a second order symplectic Stormer-Verlet discretization in time. The resulting numerical scheme is verified against nonlinear analytical solutions and discrete energy conservation is demonstrated for long time simulations. We also validated the scheme with experimental data of waves generated in a wave basin of the Maritime Research Institute Netherlands.
|Publisher||University of Twente, Department of Applied Mathematics|
- Finite Element Method
- Variational formulation
- Symplectic time integration
- Deforming grids
- Nonlinear water waves
- Galerkin approximation