Boundary conditions for SPH through energy conservation

TL;DR

This work tackles the challenging problem of imposing wall boundary conditions in SPH while preserving energy, a key ingredient for unconditional stability. It introduces an energy-conserving wall boundary condition within a Boundary Integrals framework, deriving operators that ensure power balance and correct wall energy exchange for both fixed and moving walls. The method is tested on several canonical and complex scenarios (normal impact, adiabatic piston, dam break, spacecraft water landing), demonstrating stable, energy-consistent behavior and improved wall penetrability control over existing BI formulations. While the approach achieves strong stability and energy conservation, it carries low-order consistency and leaves room for higher-order boundary operators and tensile-instability mitigation. Overall, the proposed formulation advances practical SPH wall treatments by achieving unconditional stability without requiring artificial dissipation, with clear paths for future refinement.

Abstract

Dealing with boundary conditions in Smoothed Particle Hydrodynamics (SPH) poses significant difficulties, indeed being one of the SPHERIC Grand Challenges. In particular, wall boundary conditions have been pivotal in SPH model development since it evolved from astrophysics to more generic fluid dynamics simulations. Despite considerable attention from researchers and numerous publications dedicated to formulating and assessing wall boundary conditions, few of them have addressed the crucial aspect of energy conservation. This work introduces a novel boundary condition designed with energy conservation as a primary consideration, effectively extending the unconditional stability of SPH to problems involving wall boundary conditions. The result is formulated within the framework of the Boundary Integrals technique. The proposal is tested on a number of cases: normal impact against a wall, adiabatic oscillations of a piston, dam break, and the water landing of a spacecraft.

Figures (25)

• Figure 1: Schematic view of the fluid jet impact initial condition.
• Figure 2: Pressure field snapshot at $t = T / 4$. Left: Normal fluid jet impact against a solid wall. Right: Normal fluid jet impact cercospita_2023_energy.
• Figure 3: Energy profile. Left: Normal fluid jet impact against a solid wall. Right: Normal fluid jet impact cercospita_2023_energy. Red: kinetic energy, blue: compression energy, black: total energy.
• Figure 4: Schematic view of the adiabatic expansion.
• Figure 5: Simulation snapshot at $t / T = 0.2$. The color bar represents pressure, in units of $\rho_0 c_0^2$Ma$^2$.