A meshfree method: meshfree weak?strong (MWS) form method, for 2-D solids

Abstract

A novel meshfree weak-strong (MWS) form method is proposed based on a combined formulation of both the strong-form and the local weak-form. In the MWS method, the problem domain and its boundary is represented by a set of distributed points or nodes. The strong form or the collocation method is used for all nodes whose local quadrature domains do not intersect with natural (Neumann) boundaries. Therefore, no numerical integration is required for these nodes. The local weak-form, which needs the local numerical integration, is only used for nodes on or near the natural boundaries. The locally supported radial point interpolation method and the moving least squares approximation are used to construct the meshfree shape functions. The final system matrix will be sparse and banded for computational efficiency. Numerical examples of two-dimensional solids are presented to demonstrate the efficiency, stability, accuracy and convergence of the proposed meshfree method.

A meshfree weak-strong (MWS) form method for time dependent problems

“…1, the problem domain and boundaries are represented by properly scattered field nodes. The key idea of the MWS method (Liu and Gu, 2003) is that in establishing the discrete system equations, both the strong form and the local weak form are used for the same problem, but for different field nodes. In Fig.…”

“…Hence, the combination of the strong-form and the local weak-form has been used to propose a novel meshfree method, the meshfree weak-strong (MWS) form method (Liu and Gu, 2003), and the MWS method has been used for two-dimensional elasto-statics. In this paper, a MWS formulation is developed for time dependent problems.…”

Equilibrium on line method (ELM) for imposition of Neumann boundary conditions in the finite point method (FPM)

“…By considering the fact that in meshless strong-form methods, instability and computational error is mainly induced by the Neumann boundary conditions, Liu [22] proposed meshfree weak-strong (MWS) form method. In this method, the strong-form or collocation method is used for all nodes whose local quadrature domains do not intersect with Neumann boundaries, while the local weak form is only used for nodes on or near the natural boundaries.…”

Modified equilibrium on line method for imposition of Neumann boundary conditions in meshless collocation methods

“…By substituting Equations (6b), (7b), (15) into Equation (14) and rewriting the derivatives with respect to the global coordinate system (x, y), final formulation can be obtained as follows:…”