The Least-Squares Meshfree Finite Element Method

Most meshfree methods are based on the Galerkin principle and employ the moving least-squares approximation for construction of shape functions [1]. Although there is no need for an explicit mesh in the construction of moving least-squares shape functions, a separate background mesh is required to integrate the weak form, so they are not truly meshfree methods. Due to the non-interpolative character of the moving least-squares approximation, the enforcement of essential boundary conditions in the Galerkin formulation is quite awkward. Moreover, the moving least-squares approximation is more expensive computationally than the finite element interpolation. In this research, we have developed a least-squares meshfree particle finite element method which combines the features of the least-squares finite element method [2] and the meshfree particle method [3]. The least-squares finite element method which is based on minimization of the L2 norm of the residuals of a first-order system of differential equations, is a simple, efficient and robust technique, and can solve almost any kind of partial differential equation with the same mathematical/computational formulation. Since the least-squares method doesn’t make use of the integration by parts for converting domain integration into boundary integration, and the meshfree particle method employs the usual finite element interpolations based on particles, all troubles that plague the Garlerkin-based meshfree methods disappear. Numerical examples show that the proposed method possesses superconvergence for elasticity and thin plate bending problems.