A Numerical Comparison of Different Approximation Techniques in Meshfree Methods

There is a great interest in applying numerical methods for solution of partial differential equations of various types of engineering problems. Numerical solution of partial differential equation consist of two main parts: approximating the solution of equation (or unknown function) then determining the unknown function by embedding approximated solution in the governing equation in the weak or strong form and applying boundary conditions to determine the unknown function. Accuracy and efficiency of numerical method is strictly dependent to accuracy and efficiency of these two parts. During last decade, various methods have been evolved rapidly and efficient methods such as EFG, SPH, PIM, MFS and RBF were successfully used for different engineering problems. In this paper we compare accuracy and efficiency of different techniques used in meshfree methods which proposed and used by researchers for approximating the solution of equation. Results show that there is a great difference between accuracy and time efficiency of these meshfree methods. Some numerical example in elasticity problems were studied in this paper. Stress and displacement approximation of a cracked media also studied in this paper to reveal the efficiency of mesh free methods in approximating singular functions and its derivatives