Smoothed Particle Hydrodynamics (SPH): towards implicit time integration

This thesis investigates the smoothed particle hydrodynamics (SPH) method, which is a numerical mesh-less method particularly suited for the simulation of free surface flows. The formulation is based on a total Lagrangian description of a system of first-order conservation laws, expressed in terms of the linear momentum and the Jacobian of the deformation. In this framework, the evaluation of spatial integrals is performed with respect to the initial undeformed configuration, resulting in an efficient approach that completely circumvents the need for continuous particle neighbor searching. To ensure stability and consistency at the SPH discretisation level, a characteristic-based Riemann solver is employed, designed to preserve both the accuracy and the conservation properties of the overall algorithm. This work presents a novel implicit formulation of the Smoothed Particle Hydrodynamics (SPH) method. The chosen time integration scheme is the second-order Crank–Nicolson method, while the nonlinearities arising from the implicit formulation are handled through a Newton–Raphson iterative procedure. Unlike traditional explicit schemes, the proposed method removes the restrictions on the time step size imposed by the Courant–Friedrichs- Lewy (CFL) condition, enabling simulations over longer time scales and expanding the range of applications. The new formulation is implemented and validated through a set of benchmark tests.