Numerical study of PPE source term errors in the incompressible SPH models

In a recent work by Gui et al. [13], an incompressible SPH model was presented that employs a mixed pressure Poisson equation (PPE) source term combining both the density-invariant and velocity divergence-free formulations. The present work intends to apply the model to a wider range of fluid impact situations in order to quantify the numerical errors associated with different formulations of the PPE source term in incompressible SPH (ISPH) models. The good agreement achieved between the model predictions and the documented data is taken as a further demonstration that the mixed source term formulation can accurately predict the fluid impact pressures and forces, both in the magnitude and in the spatial and temporal patterns. Furthermore, an in-depth numerical analysis using either the pure density-invariant or velocity divergence-free formulation has revealed that the pure density-invariant formulation can lead to relatively large divergence errors while the velocity divergence-free formulation may cause relatively large density errors. As compared with these two approaches, the mixed source term formulation performs much better having the minimum total errors in all test cases. Although some recent studies found that the weakly compressible SPH models perform somewhat better than the incompressible SPH models in certain fluid impact problems, we have shown that this could be largely caused by the particular formulation of PPE source term in the previous ISPH models and a better formulation of the source term can significantly improve the accuracy of ISPH models.