Investigating the efficiency of modified element-free Galerkin method in solving static problems and optimization

The finite element method has been used comprehensively in traditional and academic works. The common finite element method is a powerful method in solving boundary value problems that transforms strong differential form equations into weak form equations using domain discretization. Even though the finite element method has sufficient accuracy in displacements, but calculating the stress field by FEM has low accuracy. This paper uses the modified element-free Galerkin method to solve some numerical elastostatics problems. At first, a one-dimensional elastic bar is considered, which is subjected to a volumetric force with linear changes along the length of the bar. A comparison between the original element-free Galerkin method, the modified element-free Galerkin method and the exact solution has been made to check the accuracy, efficiency and the required time cost. The presented study indicates that these mentioned methods have the same accuracy, but the modified EFG method can be very time-consuming compared to others, mainly when a large number of degrees of freedom is used with a large size of the support domain. The numerical solution of the modified and original element-free Galerkin methods is compared with Timoshenko’s analytical responses for the bending of an elastic beam. This comparison exhibits that modified and original methods have excellent agreements with the analytical ones in calculating displacement values. Despite the same accuracy in estimating the displacements, the calculation of the stress field indicates that the modified method is less accurate than the original method. It is shown that by increasing the number of degrees of freedom, the accuracy of the modified method for estimating the stress field improves. Increased degrees of freedom are used for introducing the domain of the beam. In this study accuracy of the stress solution in the modified EFG method is improved. However, the modified EFG method is yet more time-consuming than others. According to the results, the modified element-free Galerkin method can be nominated as a powerful mesh-free method based on moving least squares that has shape functions with interpolation properties. Having interpolator shape functions in this method makes it possible to combine it with other numerical methods and apply boundary conditions with less computational cost. The results exhibit that the displacement calculation error in the presented method was at most 5% compared to the analytical solution method. Also, the maximum error rate in the presented method for stress estimation was equal to 15%.