(A) study on the least-squares meshfree method in the analysis of elastic-plastic deformation

Abstract

The least-squares meshfree methods for the linear elasticity, rigid-plasticity and elasto-plasticity are presented. The methods are based on the proposed first-order least-squares formulations and the moving least-squares approximation. The main benefit of the proposed methods is the full achievement of meshfree strategy for the numerical analysis of the problems in solid mechanics. To be a truly meshfree method, the approximation, the domain integration of variational formulation, the treatment of incompressible locking and the remodeling could be performed without any structure of mesh. The proposed methods satisfy these demands. Despite the recent development of the meshfree approximations such as the moving least-squares or reproducing kernel approximations, their applications to the Galerkin formulation require accurate integration for which element-like cells are often employed. Recently, it has been shown that the least-squares formulation is robust to integration errors. Thus a simple or cell-free integration scheme can be effectively used. For this purpose, the support integration scheme, where the integration points are distributed within nodal supports, is presented in the present work. First, the least-squares meshfree method is applied to the linear elasticity. For this, two first-order least-squares formulations, the conventional and compatibility-imposed formulations, are presented. Both formulations achieve the solution accuracy comparable to that of Galerkin formulation. The compatibility-imposed formulation shows the optimal rate of convergence for both primal and dual variables. It is also shown that the least-squares meshfree methods work well with cell-free integration schemes. Another merit of the least-squares method is its uniform convergence behavior in the incompressible condition with equal-order shape functions for both primal and dual variables. The mixed Galerkin method requires lower-order approximation for dual variables, and ...