Introduction to Meshfree and Particle Methods

This chapter presents the history of the development of meshfree methods, starting from smoothed particle hydrodynamics and the generalized finite difference methods to the diffuse element method and the general meshfree approach for solving PDEs. The construction of the approximation in terms of nodal points without element or any adjoining connectivity defines a meshfree method, which is the primary distinction from the finite element method (FEM). The shape functions can also be constructed with arbitrary smoothness, and the orders of continuity and completeness can be made independent in most meshfree methods, which is opposed to FEM and isogeometric analysis. Furthermore, the arbitrary smoothness in the meshfree approximation allows the solution of PDEs by using the standard weak form and the strong form directly. These unique properties make meshfree methods more effective in solving problems with severe deformations without the difficulties of element distortion and entanglement, and adaptive refinement can be carried out straightforwardly compared to FEM. Demonstrations of meshfree modeling of various challenging engineering problems are presented, and the wide variety of meshfree methods that have been proposed over the years are discussed.