Extended parametric meshless Galerkin method

The parametric meshless Galerkin method (PMGM), which has recently emerged in the field of meshless methods, enhances the efficiency of meshless methods through the use of meshless shape functions which are mapped from a parametric space to the physical space. This property helps to save the running-time needed to calculate the meshless shape functions, which is a time consuming and complicated task. However, the meshless shape functions need to be regenerated in the vicinity of the boundaries as well as some other locations after mapping. In this paper, a new extension of the PMGM, called extended parametric meshless Galerkin method (X-PMGM) is presented. In this method, it is not necessary to regenerate the meshless shape functions near the boundaries, and instead, the approximation is enriched using discontinuous fields through a partition of unity method. Therefore the amount of shape functions regeneration is decreased which improves the efficiency of the method. Numerical examples show that very accurate results and computational time savings of about 80% are achieved comparing with the element-free Galerkin method results.