mls shape functions

The smoothed particle hydrodynamics (SPH) and moving particle semi-implicit (MPS) are well-known and efficient mesh-less numerical methods widely used to investigate a wide range of complicated practical engineering problems. Recently, two modified Laplacian models [1, 2] have been proposed by using different efficient mathematical techniques, and the analogy between SPH and MPS methods. These two models exhibit significantly superior precision in comparison with several existing modified schemes [3-9] but still suffer from lower accuracy near calculation domain boundaries as they work with the conventional weight or interpolation functions. In this paper, the models were reformulated and further improved by replacing the weight functions with well-known moving least squares (MLS) shape functions without requiring dummy calculation nodes beyond boundaries. The proposed Laplacian models in this study could achieve very accurate results compared with the existing models [1, 2] for the solution of four different two-dimensional Poisson equations on irregular node distributions.