Mixed discrete least squares meshless method for solving the linear and non-linear propagation problems

Abstract

A Mixed formulation of Discrete Least Squares Meshless (MDLSM) as a truly meshfree method is presented in this paper for solving both linear and non-linear propagation problems. In DLSM method, the irreducible formulation was deployed which needs to calculate the costly second derivatives of the MLS shape functions. In the proposed MDLSM method, the complex and costly second derivatives of shape functions are not required. Furthermore, using the mixed formulation, both unknown parameters and their gradients are simultaneously obtained circumventing the need for post-processing procedure performed in irreducible formulation to calculate the gradients. Therefore, the accuracy of gradients of unknown parameters is increased. In MDLSM method, the set of simultaneous algebraic equations are built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional defined as the sum of squared residuals of the differential equation and its boundary condition. The proposed method automatically leads to symmetric and positive-definite system of equations and, therefore, is not subject to the Ladyzenskaja-Babuska-Brezzi (LBB) condition. The proposed MDLSM method is validated and verified by a set of benchmark problems. The results indicate the ability of proposed method to efficiently and effectively solve the linear and non-linear propagation problems.