Meshless methods (MMs) have become interesting and promising methods in solving partial differential equations, because of their flexibility in practical applications when compared with the standard finite-element method (e.g. crack propagation, large deformations, and so on). Implementation of these methods requires a good understanding of the influence of some specific selectable parameters. In this article, those parameters are analysed for one of the most popular MMs, the element-free Galerkin method, considering both accuracy and computational cost. Thus, the dependence of the solutions on grid irregularity, order of the polynomial basis, type of weight function, and the support size is investigated, and conclusions are drawn with respect to recommended or ‘optimal’ values for one-dimensional bar problems with applied axial loads.

In the recent decades, meshless methods (MMs), like the element-free Galerkin method (EFGM), have been widely studied and interesting results have been reached when solving partial differential equations. However, such solutions show a problem around boundary conditions, where the accuracy is not adequately achieved. This is caused by the use of moving least squares or residual kernel particle method methods to obtain the shape functions needed in MM, since such methods are good enough in the inner of the integration domains, but not so accurate in boundaries. This way, Bernstein curves, which are a partition of unity themselves, can solve this problem with the same accuracy in the inner area of the domain and at their boundaries.

Since 1994, two main meshless methods have been developed and widely used: these are the element free Galerkin method and the meshless local Petrov-Galerkin method. Both methods solve partial differential equations by posing a numerical approximation to the solution using the moving least squares technique. Using Bernstein polynomials as the shape functions of Galerkin weak form-based methods improves the numerical approximation achieved at boundaries without losing accuracy inside the domain.

Continuing with the analysis performed for the one-dimensional axially loaded bar problem, a beam in bending is analysed to understand the influence of the characteristic parameters that have any influence in the solution of this problem using the element-free Galerkin method (EFGM), one of the most popular meshless methods. Both accuracy and time cost are considered as the evaluation functions to perform such an analysis. Both functions provide a reasonable idea to consider EFGM as an adequate method to solve the problem considered in this article. As in a one-dimensional axially loaded bar problem, the parameters to be considered will be those that affect the solution: number of nodes in which the domain is modelled, the nodes scatter, the order of the polynomial base to generate shape functions, the order of the quadrature to solve integrals, and the support radius. Besides, as in a one-dimensional axially loaded problem, some cases with different loading and stiffness conditions are considered. However, in this analysis a generalized moving least squares method is used to create shape functions instead of the moving least squares.

The moving least squares (MLS)-based element-free Galerkin method (EFGM) has been widely studied, yielding valuable results and becoming a robust mesh-free technique for structural analysis. Due to the non-interpolating nature of MLS shape functions, the EFGM procedure leads to stiffness and mass matrices based on nodal parameters instead of true nodal displacements. The interpolating moving least squares (IMLS) methods with singular weight functions or the weighted nodal least squares (WNLS) methods were proposed in order to arrange the formulation in terms of actual displacements at the nodes. In this work, a later transformation matrix to retrieve the classic stiffness and mass matrices is discussed and numerically tested in problems with discrete prescribed displacements at boundaries, for which a straightforward finite element method (FEM)-type imposition of boundary conditions is possible. Mathematics Subject Classification