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

Abstract

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.

Numerical Analysis of Crack-Tip Fields Using a Meshless Method in Linear Elastic Materials

“…The penalty value should be large enough to keep the coefficient matrix symmetric. Although a certain algorithm was proposed to estimate the penalty coefficients [32], the appropriate value for this coefficient is usually determined by trial and error. In this study, these coefficients are assumed to be α 1 = 10 8 and α 2 = 10 4 .…”

“…Af-shar and Arzani [23] proposed a novel LSM-derived method called the discrete leastsquares meshless (DLSM) method to provide a solution to Poisson's equation. This method has been successfully adopted to solve elliptic PDEs [24], as well as fluid mechanics [25][26][27][28] and elasticity [29][30][31][32][33][34] problems, alone or in conjunction with other numerical methods. In this approach, the solution domain is discretized by nodes employed for constructing the shape functions using the MLS interpolant.…”