Collocated Mixed Discrete Least Squares Meshless (CMDLSM) method for solving quadratic partial differential equations

Abstract

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.

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

“…Mixed formulation also provides the possibility of simultaneously calculating both unknown parameters and their gradients without any post-processing procedure that is essential in the DLSM method for computing the gradients. Since the post-processing procedure involves less accurate derivatives of shape functions, the gradients are accurately computed in the MDLSM method compared to the DLSM method [30,35,36]. The MDLSM method is based on minimizing a least squares functional with respect to the nodal parameters.…”

“…This property is more useful to solve the Navier-Stokes equations [37] and will be investigated in future studies. MDLSM method was successfully employed to solve equilibrium problems such as linear elasticity problem [30,36] and linear quadratic di erential equations [35]. The obtained results indicated high accuracy and e ciency of MDLSM method compared to the DLSM method.…”