Node moving adaptive refinement strategy for planar elasticity problems using discrete least squares meshless method

In this paper, an error indicator and adaptive refinement procedure in conjunction with the Discrete Least Squares Meshless (DLSM) method is presented for the effective and efficient analysis of planar elasticity problems. The DLSM method is a truly meshless method, which has been used for the solution of different problems ranging from solid to fluid mechanics problems. The method is based on the minimization of the least squares functional with respect to the nodal parameters. The least squares functional is formed as the weighted summation of the residual of the differential equation and its boundary condition. The DLSM method enjoys from providing a natural error indicator defined as the value of the functional at nodal points, which can be efficiently used to identify zones of larger numerical errors. The proposed error indicator has the additional advantage of easy computation since most of its components are already available from the main DLSM computation for analysis. Since both the approximation method and discretization method are meshless, repositioning of the nodes can be easily exploited for refining the numerical solution without any geometrical difficulties usually encountered in mesh based methods. Here, a node moving strategy based on spring analogy is proposed to displace the nodal points to the areas indicated by the higher values of the error indicator. A Voronoi diagram is used to identify neighboring nodes that should be connected to each other using springs. The efficiency and effectiveness of proposed adaptive refinement technique is tested on some benchmark examples with available analytical solutions and the results are presented. The results show that the proposed adaptive refinement technique is quiet effective for the accurate and efficient solution of elasticity problems.