On the accuracy of SPH formulations with boundary integral terms

Abstract

Handling boundaries in smoothed particle hydrodynamics (SPH) is believed to be one of the most critical problems in studies and applications of SPH. In the last two decades, excellent progress has been made in exploring the evaluation of the boundary integral terms directly to improve SPH approximation on the boundary. However, because of the fundamental difficulties, formulations in this regard are difficult to be made consistent when using SPH particle approximations, which significantly affect the solution accuracy. In this work, an intensive investigation is made in carefully analyzing the order of accuracy of the SPH approximation when boundary integral terms are included. Both kernel consistent and particle consistent boundary integral formulations of SPH are examined in great detail. Notably, a novel particle consistent boundary integral formulation for first-order derivatives is derived. An approximation of this new formulation, which is weakly consistent, is also discussed. The theoretical analysis and numerical experiments show that the SPH with particle consistent boundary integral formulation outperforms significantly in terms of accuracy compared to the correction factors used in traditional SPH implementations.