Theoretical Approach to Element Free Galerkin Method and Its Mathematical Implementation

Numerical methods such as FVM, FDM, FVM, and BVM are eminent for solving the physical problems in engineering and science. Mentioned numerical methods are based on the predefined topological map, generally called “mesh,” Meshes are required to establish the relations between nodes, which becomes vital for the creation of shape functions. The problems with mesh-based methods are (i) They require the qualitative mesh, which is a somewhat tedious, time-consuming & messy task (ii) Meshing & re-meshing for a sizeable computational domain is time consuming, tedious, and costly task also requires the skills (iii) In complex computational domains, the mesh-based method fails in terms of accuracy (iv) Glass hour and shear locking phenomena generally found in the traditional finite element method. In the last two and a half decades, many engineers and mathematicians have proposed a new class of numerical methods known as meshfree methods. Meshfree methods are independent of mesh and approximate the governing PDE based on the set of nodes only. This chapter seeds light on an eminent meshfree method called EFGM. Chapter deals with the introduction and background of meshfree methods, the EFGM method, and its mathematical formulations. The chapter also comprises two elastostatic numerical problems, the 1D problem of a bar with body forces and 2D Timoshenko cantilever beam with traction at the tip, numerical results have been evaluated & compared with exact results. The convergence of both 1D and 2D problems have been discussed. This work built a sound foundation on EFGM and will act as a stepping stone for novices in the field of meshfree methods. Keywords: Advanced numerical approach, Mesh-free methods, Element free Galerkin method