Velocity calculation methods in finite element based mel formulation

The simulation of nonlinear waves can be carried out by using the conventional methods like Finite Element Method (FEM), Boundary Element Method (BEM) based on Mixed Eulerian and Lagrangian (MEL) formulation. The simulation based on FEM has the advantages of extending the code easily to viscous flow and to three-dimensional (3D) tank with complex geometry. While adopting FEM, the derivatives are usually found from differentiating the shape function, which is the direct differentiation of the velocity potential. The approximation of velocity field thus obtained is inferior than the approximation of the velocity potential. In time-dependent problems, this play an important role. Thus, researchers have been focusing on obtaining the derivatives through different methods such as Global Projection, Local Finite Difference (FD), mapped FD, least square method or by using cubic spline approximation. The present chapter shows a detailed review of these methods for calculating the derivatives including the advantages and disadvantages in the context of simulation of nonlinear free surface waves using structured/unstructured FEM.