Accuracy analysis of improved SPH methods for solution of quadratic PDEs representing potential flows and elasticity problems

This paper compares the accuracy of four different higher-order smoothed particles hydrodynamics (SPH) models with the standard SPH model for the solution of potential flows and planar elasticity problems described by quadratic PDEs. Two of these models were previously proposed for the solution of elliptic partial differential equations (PDEs) Shobeyri and Ardakani (Iran J Sci Technol Trans Civ Eng 41:345-350, 2017; Iran J Sci Technol Trans Civil Eng 43(4):791-805, 2019). In this study, these schemes including the Voronoi-based approach and the hybrid model improved by Taylor series expansion (TSE) and moving least squares (MLS) method in the context of SPH are developed for elasticity problems with more complicated PDEs. One of the newly proposed models uses a higher-order TSE and the second one employs the mixed formulation where the costly calculation of second-order derivatives is eliminated. The results show that the model with higher-order TSE gets the smallest errors while the performance of the hybrid and mixed models is almost the same and the Voronoi-based model shows less improvement. Each of the improved models can be more appropriate in the fields of fluid or solid mechanics based on its inherent properties discussed in this study.