Stabilization of Loosely Coupled Fluid Rigid-Body for Incompressible Smoothed Particle Hydrodynamics
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Violent flows in fluid-solid interactions need computational domains that can adapt to extreme deformations, making mesh-less methods well-suited for such problems. As a mesh-less and Lagrangian method, the Smoothed Particle Hydrodynamics (SPH) is suitable to handle large body movements while intrinsically enforcing free-surface boundary conditions. However, SPH is computationally expensive compared to mesh-based methods. Incompressible SPH (ISPH) addresses this limitation by significantly improving the computational efficiency. Despite these advantages, ISPH coupled with loosely integrated solid body solvers exhibits instabilities under specific configurations: low solid-to-fluid density ratios or thin bodies. These instabilities intensify with decreasing time steps, rendering simulations unconditionally unstable. This class of instability is already known in classical numerical methods and is related to the violation of the incompressibility constraint. While strongly coupled or implicit schemes could resolve these instabilities, they negate the ISPH computational advantages. We propose a stabilization procedure that maintains a loose coupling while ensuring stability. Following the approach proposed by Burman and Fernández and Degroote et al., a stabilized explicit ISPH-rigid body coupling scheme is developed in the present work. A rigorous stability analysis for coupled ISPH is performed. Precisely, the non-linear stability is proven in simplified configurations and errors introduced by the stabilizing term are quantified. Thus, the error (in discrete norm) associated with the added term is proven to scale as the Chorin-Temam temporal discretization error.
