A Mixed Discrete Least Square Meshless (MDLSM) method is proposed for the solution of planar elasticity problems. In this approach, the differential equations governing the planar elasticity problems are written in terms of the stresses and displacements which are approximated independently using the same shape functions. Since the resulting governing equations are of the first order, both the displacement and stress boundary conditions are of the Dirichlet-type which is easily incorporated via a penalty method. Because least squares based algorithm of MDLSM method, the proposed method does not need to be satisfied by the LBB condition. The performance of the proposed method is tested on a benchmark example from theory of elasticity namely the problem of infinite plate with a circular hole and the results are presented and compared with those of the analytical solution and the solutions obtained using the irreducible DLSM formulation. The results indicate that the proposed MDLSM method is more accurate than the DLSM method. The results show that the numerical solutions of the MDLSM method can be obtained with lower computational cost and with higher accuracy. Also its performance is marginally affected by the irregularity of the nodal distribution.@en

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.@en

In this paper, an adaptive refinement procedure is proposed to be used with Discrete Least Squares Meshless (DLSM) method for accurate solution of planar elasticity problems. DLSM method is a newly introduced meshless method based on the least squares concept. The method is based on the minimization of a least squares functional defined as the weighted summation of the squared residual of the governing differential equation and its boundary conditions at nodal points used to discretize the domain and its boundaries. A Moving Least Square (MLS) method is used to construct the shape function making the approach a fully least squares based approach. An error estimate and adaptive refinement strategy is proposed in this paper to increase the efficiency of the DLSM method. For this, a residual based error estimator is introduced and used to discover the region of higher errors. The proposed error estimator has the advantages of being available at the end of each analysis contributing to the efficiency of the proposed method. An enrichment method is then used by adding more nodes to the area of higher errors as indicated by the error estimator. A Voronoi diagram is used to locate the position of the nodes to be added to the current nodal configuration. Efficiency and effectiveness of the proposed procedure is examined by adaptively solving two benchmark problems. The results show the ability of the proposed strategy for accurate simulation of elasticity problems.@en