A Comparative Study on Two Mixed Least Squares Meshless Models with Improved SPH, MPS and CPM Methods to Solve Elasticity Problems

Abstract

This paper investigates the accuracy of several meshless methods to solve elasticity problems. The methods include the well-known smoothed particle hydrodynamics (SPH) (called model 1 in this study), and Voronoi-based SPH (model 2), moving particle semi-implicit (MPS) (model 3), and Voronoi-based MPS (model 4) methods and three different proposed least squares models based on Taylor series expansion (TSE). The accuracy of both the SPH and MPS methods is improved by employing the Voronoi diagram as an alternative approach to estimate the computational node volume parameter. One of the least squares methods (model 5) uses TSE truncated to the second-order to solve the standard quadratic differential equations of elasticity problems, considering displacements as unknown variables. The two last methods employ the first-order (model 6) and second-order (model 7) TSE to approximate the function in the mixed formulation, where the governing equations can be written as a system of the first-order differential equations with unknown variables of stresses and displacements. The mixed formulation improves the prediction accuracy of unknown parameters, especially stress, by eliminating the second derivative calculations. The results indicate that the least squares methods, particularly model 5, can achieve higher accuracy and computational efficiency than SPH and MPS methods, especially in stress calculations. Noteworthy, the second-order mixed model exhibits considerable superiority over the first-order model while requiring approximately the same computational effort.