Abstract
Smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods are representative meshfree particle methods used to compute Lagrangian mechanics. The approximations of differential operators in the SPH and MPS methods have several similarities, but the theoretical discussion of the difference between them is limited. This study mathematically describes the difference via a comprehensive derivation of the first- and second-order derivative operators for each method. The comprehensive derivation indicates that the SPH and MPS operators are consistent with the pressure Poisson equation and moving least-squares approximation, respectively. The variation in consistency can explain the difference in the schemes of the incompressible flow problem. Additionally, the comprehensive derivation of the MPS operators can result in novel second-order and anisotropic operators. This study strengthens the theoretical understanding of the SPH and MPS methods and facilitates the selection of the appropriate method by users. Furthermore, the proposed MPS operators contribute to developing methods with adaptive or multiscale particle distributions.
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