A new Lagrangian–Eulerian incompressible SPH method for simulating free surface flows

In this study, a novel and efficient Lagrangian–Eulerian approach is introduced in the context of incompressible smoothed particle hydrodynamics (ISPH) method. SPH is a well-known mesh-less approach in which function approximation is done using integral interpolants. The method due to its fundamental properties has been extensively applied in Lagrangian form. In spite of significant advantages of Lagrangian form of numerical methods such as simple and straightforward method of tracking free surfaces, higher computational costs and lower accuracy because of continuous movement of fluid particles and their corresponding configurations are the main shortcomings of these approaches. The present arbitrary Lagrangian–Eulerian (ALE) approach is developed based on dividing the computational domain into three separate Lagrangian, Eulerian and transition regions. Eulerian SPH formulations are used to discretize the fluid flow governing equations in the Eulerian region which is chosen far from the free surface for many successive time steps. Then, the Lagrangian formulations are employed for this zone for a few time steps (one or two). A reverse pattern is chosen for the transition zone while the pure Lagrangian formulations are used for all time steps for the Lagrangian region. This procedure is repeated until the desired simulation time. This algorithm prevents particle disorder in the transition zone. It should be noted that the modified incompressible SPH G. Shobeyri (JBSMSE 41:421 2019) which works well with highly irregular particle distributions is implemented for the proposed ALE approach. The present method is validated for different free surface flows.