Using a modified MPS gradient model to improve accuracy of SPH method for Poisson equations

In this study, a more accurate and efficient Laplacian model with the promising feature of kernel gradient-free is formulated. For this purpose, a hybrid approach in the context of the smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) methods is used. A newly developed gradient model (Shobeyri in Iran J Sci Technol Trans Civil Eng, 2022. https://doi.org/10.1007/s40996-022-01013-6) which is derived by combining two MPS gradient models with the promising performance in free surface flows is applied in this paper (Chen et al. in Int J Numer Meth Fluids 80(6):358–374, 2016; Wang et al. in Int J Numer Meth Fluids 85(2):69–89, 2017). The proposed MPS gradient model is used in the standard Laplacian formulation (Shao and Lo in Adv Water Resour 26(7):787–800, 2003) to derive the improved Laplacian model. A comprehensive accuracy analysis for the solution of four 2-D Poisson equations subjected to both Dirichlet and Neumann boundary conditions on irregular calculation node distributions shows that this model can yield smaller errors compared with several SPH Laplacian models. Despite more complex formulations and additional terms, the proposed Laplacian model requires lower CPU usage times for solving the test problems which indicates noticeable computational efficiency. The introduced model can be effectively applied for simulation of different engineering applications such as fluid and solid mechanics problems.