Numerical investigation of the element-free Galerkin method (EFGM) with appropriate temporal discretization techniques for transient wave propagation problems

In the earlier paper, it is known that the solution accuracy usually can not be monotonically increased as the temporal discretization interval decreases when the standard finite element method (FEM) with the conventional temporal discretization approach is exploited for elastodynamics, hence the time integration step should be carefully determined for sufficiently fine solutions. The present work aims to investigate the behaviors of the classical element-free Galerkin method (EFGM), which is a typical meshless approach, with the Bathe temporal discretization scheme for elastodynamics. The main insights are that we explicitly show the total numerical dispersion errors in the computed numerical results actually consists of two different parts corresponding to the spatial and temporal discretization, respectively; both of them are responsible for solution accuracy. By performing the dispersion analysis, how the solution accuracy is affected by temporal and spatial discretization is shown, it is seen that we can improve the solution accuracy monotonically as the temporal step size decreases as long as the spatial dispersion error is sufficiently small and the related mathematical proofs are also given. From several supporting numerical examples, we can clearly see that the EFGM with the Bathe time integration scheme can basically provide monotonically convergent solutions as long as the reasonable node arrangement pattern and sufficiently large shape function supports are employed, namely the monotonic convergence property with respect to the temporal discretization interval can be achieved. This attractive and important feature makes the EFGM more competitive than the FE approach for elastodynamics.