An Implicit Incompressible SPH Method for Free Surface Flow Problems

A consistent implicit Incompressible Smoothed Particle Hydrodynamics (ISPH) method based on projection approach is proposed for solving violent free surface flow problems. In this way، two consistent discretization schemes are employed for first and second spatial derivatives. In this study، it is shown that in explicit ISPH solvers، the field variables and the positions of particles in the process of numerical differentiation are estimated at two different time steps. So، the incompressibility is not completely satisfied. In the present approach، an iteration loop is implemented، in each time-step. Thus، at the end of each time-step both velocity and the positions used in divergence estimation are at the new time-level. The proposed ISPH method is validated in free surface flow problems involving 2-D dam break benchmarks in which both wet and dry beds are considered. Among the advantages of the present implicit method is being more accurate and stable than the explicit one، despite use of lower number of particles and greater time-step sizes. Also، it provides significant improvement in free surface simulations and pressure distribution results.