Implicit-splitting physics-driven particle relaxation for enhancement of Eulerian SPH

Physics-driven particle relaxation, driven by either constant background pressure or the kernel gradient correction (KGC) matrix, has been proposed to generate isotropic and body-fitted particle distributions for complex geometries while ensuring zero-order consistency in smoothed particle hydrodynamics (SPH). However, this relaxation process often encounters challenges, such as a low decay rate of residuals and difficulties in achieving convergence to minor zero-order consistency errors, particularly for three-dimensional complex geometries and scenarios with a small ratio of smoothing length $h$ to particle spacing $\Delta x$. This limitation hinders the development of Eulerian SPH (ESPH), as its accuracy depends on the initial particle configurations, while the smoothing length determines the computational cost. In this work, we introduce an implicit-splitting approach to improve the relaxation process, aiming to obtain isotropic particle distributions with negligible zero-order consistency errors at reduced $h/\Delta x$ values. This approach provides the initial particle distribution for ESPH, enhancing its computational efficiency by reducing the number of neighboring particles. Extensive relaxation examples demonstrate that the proposed method significantly reduces the relaxation residual and achieves substantially smaller zero-order consistency errors. Subsequently, different incompressible numerical examples based on relaxed particles have been validated using Eulerian SPH. A value of $h/\Delta x$ smaller than 1.0 was selected to reduce neighboring particles, and the reverse kernel gradient correction (RKGC) was adopted to ensure first-order consistency. The numerical results consistently show good accuracy and smoother contours than those obtained from the finite volume method (FVM) implemented in an SPH framework based on unstructured meshes.