Improvements to the meshless generalized finite difference method

Abstract

This study was dedicated to develop an improved version of meshless generalized finite difference method (GFDM) which benefits from various aspects: Present method, uses stars (group of nodes around a central node) with only six nodes for calculating derivatives; it is not dependent on weighting functions and it is not based on moving least square (MLS) methods. Developing the method, the approximation schemes were suggested for estimating first and second partial derivatives along with the Laplacian term. Moreover, three sources for singularity of coefficient matrix were found and the remedies were suggested to avoid them. All these aspects made the method as a stable and consistent one. Implementing various boundary conditions was also explained where a clever technique was proposed to implement Neumann boundary condition in its exact form. The order of accuracy of the method was obtained theoretically and confirmed by the numerical tests. The applicability of the method was approved through its excellent results obtained for the heat conduction problem in two geometries. Clearly, the present method has improved efficiency while its general applications are the same as other GFDMs.