Quasi ALE finite element method for nonlinear water waves

Abstract

This paper presents a newly developed quasi arbitrary Lagrangian–Eulerian finite element method (QALE-FEM) for simulating water waves based on fully nonlinear potential theory. The main difference of this method from the conventional finite element method developed by one of authors of this paper and others (see e.g. [Q.W. Ma, G.X. Wu, R. Eatock Taylor, Finite element simulation of fully non-linear interaction between vertical cylinders and steep waves. Part 1: Methodology and numerical procedure and Part 2: Numerical results and validation, Int. J. Numer. Methods Fluids, 36 (2001) 265–308.] and [G.X. Wu, Z.Z. Hu, Simulation of nonlinear interactions between waves and floating bodies through a finite-element-based numerical tank, Proc. R. Soc. A 460 (2004) 2050, 3037–3058.]) is that the complex mesh is generated only once at the beginning and is moved at all other time steps in order to conform to the motion of the free surface and structures. This feature allows one to use an unstructured mesh with any degree of complexity without the need of regenerating it every time step, which is generally inevitable and very costly. Due to this feature, the QALE-FEM has high potential in enhancing computational efficiency when applied to problems associated with the complex interaction between large steep waves and structures since the use of an unstructured mesh in such a case is likely to be necessary. To achieve overall high efficiency, the numerical techniques involved in the QALE-FEM are developed, including the method to move interior nodes, technique to re-distribute the nodes on the free surface, scheme to calculate velocities and so on. The model is validated by water waves generated by a wavemaker in a tank and the interaction between water waves and periodic bars on the bed of tank. Satisfactory agreement is achieved with analytical solutions, experimental data and numerical results from other methods.