Meshfree Methods

Meshfree methods have undergone substantial development and have received much attention in the last two decades. This new family of numerical methods is designed to inherit the main advantages of the finite element method such as compact supports of shape functions and good approximation properties while, at the same time, overcome the main disadvantages of the finite element method caused by the mesh dependence. The meshfree methods share a common feature that no mesh is needed and shape functions are constructed from sets of points, thus eliminating the need for time consuming mesh generation. The most significant advantage of meshfree methods is the flexibility in customizing approximation functions for desired regularity and for capturing essential physics and features of the particular problems of interest. Adaptivity formulation and multiple-scale solution strategies also can be implemented with relative ease. It has become clear that the meshfree methods provide considerable advantages over the conventional finite element methods in solving problems involving moving discontinuities, evolving material interfaces, multiple-scale phenomena, large material distortion and structural deformation, and fracture and damage processes. This Chapter gives an overview of many classes of meshfree methods, with more detailed discussions on Smoothed Particle Hydrodynamics (SPH), the Reproducing Kernel Particle Method (RKPM), Peridynamics (PD), the Material Point Method (MPM), as well as their applications in various challenging engineering problems.22 Some contents in Sections “Introduction”, “Approximations Based on Least-Squares Methods”, “Kernel Estimate”, “Reproducing Kernel Approximation”, “Discrete Reproducing Kernel Approximation”, “Partition of Unity Methods”, and “Conclusions and Outlook” are republished by permission from American Society of Civil Engineers: Journal of Engineering Mechanics, Meshfree Methods: Progress Made after 20 Years, Chen et al. (2017b). Some contents in Sections “Introduction”, “Implicit Gradients”, “Solving PDEs by the Galerkin Method”, “Large Deformation Analysis by Lagrangian Reproducing Kernel Approximation and Discretization”, “Large Deformation Analysis by Semi-Lagrangian Reproducing Kernel Approximation and Discretization”, “RKPM Smooth Contact Algorithm”, “Domain Integration in Galerkin Meshfree Methods”, “Stabilization of Nodal Integration”, “Application of Reproducing Kernel Particle Method” are republished by permission from John Wiley and Sons, Encyclopedia of Computational Mechanics Second Edition, Reproducing Kernel Particle Method for Solving Partial Differential Equations, Chen et al. (2017b). Content in Section “Stability of SPH” is republished by permission from John Wiley and Sons, Meshfree and Particle Methods, Belytschko et al. (2024).