Meshless method – Review on recent developments

Abstract

The present article outlines the historical development of Meshless method. The high computational cost associated with conventional numerical methods and their lack of accuracy, particularly in the area of astrophysics is the motivation and an initiation towards the invention of latest meshless methods. The review starts with Smoothed particle hydrodynamics meshless method which was invented by Gingold, Monghan in 1977 for astrophysics applications. Furthermore, shortcomings of the SPH method, it’s solution, and recent development has been reviewed for different applications. After that advent features of Element free Galerkin method with mean least square method and Lagrangian multiplier have been critically reviewed for different cases. Later, the radial basis function method is explained along with its extended version to solve a partial differential equation. Many types of RBF methods have been compared. Also, several consistency and accuracy problems are reviewed for Reproducing Kernel Particle Method. Recent advancement like generalization of RKPM method, hybrid of RKPM with Lagrangian multiplier has been briefed for a different approach. Additionally, feasibility, flexibility and several applications of Meshless local Petrov Galerkin have been reviewed. Lastly recent development of MLPG with Laplace transform to solve PDE have been studied. Afterwards, a different attribute of point collocation method for inherent multiresolution capability and error estimation has been reviewed for different applications. Briefly, the article provides a condensed overview of several meshless methods, its recent advancement and their applications.