Abstract. A Mixed formulation of Discrete Least Squares Meshless (MDLSM) as atruly mesh-free method is presented in this paper for solving both linear and non-linear propagation problems. In DLSM method, the irreducible formulation is 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 is built by minimizing a least squares functional with respect to the nodal parameters. The least squares functional is de ned as the sum of squared residuals of the di erential equation and its boundary condition. The proposed method automatically leads to symmetric and positive-de nite system of equations and, therefore, is not subject to the Ladyzenskaja-Babuska-Brezzi (LBB) condition. The proposed MDLSM method is validated and veri ed by a set of benchmark problems. The results indicate the ability of the proposed method to e ciently and e ectively solve the linear and non-linear propagation problems.

In this paper, a collocated Mixed Discrete Least Squares Meshless (MDLSM) method is proposed and used to attain an e cient solution to engineering problems. Background mesh is not required in the MDLSM method; hence, the method is a truly meshless method. Nodal points are used in the MDLSM methods to construct the shape functions, while collocated points are used to form the least squares functional. In the original MDLSM method, the locations of the nodal points and collocated points are the same. In the proposed Collocated Mixed Discrete Least Squares Meshless (CMDLSM) method, a set of additional collocated points is introduced. It is expected that the accuracy of results may improve by using the additional collocated points. It is noted that the size of coe cient matrix is not increased in the proposed CMDLSM method compared with the MDLSM method. Therefore, the required computational e ort for solving the linear algebraic system of equations is same as that in MDLSM method. A set of benchmark numerical examples, cited in the literature, is used to evaluate the performance of the proposed method. The results indicate that the accuracy of solutions is improved by using additional collocated points in the proposed CMDLSM method.

Abstract. One of the main di culties in the development of meshless methods using the Moving Least-Squares approximation, such as Mixed Discrete Least-Squares Meshless (MDLSM) method, is the imposition of the essential boundary conditions. In this paper, the RPIM shape function, which satis es the properties of the Kronecker delta condition, is employed in the Mixed Discrete Least-Squares Meshless (MDLSM) method for solving the elasticity problems. Accordingly, two new MDLSM formulations are proposed in this article, namely RPIM-based MDLSM and coupled MLS-RPIM MDLSM formulation. The essential boundary conditions can be imposed directly on the both presented methods. The proposed methods are used for the solution of three benchmark elasticity problems, and the results are presented and compared with the available analytical solutions and those of MLS-based MDLSM formulations. In addition, in each example, di erent types of nodal distributions, uniform, and re ned con gurations are considered to test the performance of the presented methods. The numerical tests indicate higher accuracy of the suggested approaches in comparison with the MLS-based MDLSM method.