Meshfree Methods in Computational Mechanics—State of the Art

The finite element method (FEM) has widespread use in solving problems in computational mechanics and applied sciences. However, researchers continue to develop and implement new numerical methods to solve problems that involve complex geometry, material non-linearity, and fracture mechanisms, including crack formation and propagation with moving and discontinuous boundaries. Meshfree methods have seen a significant increase in their application to solve partial differential equations (PDE). These methods involve modelling and solving procedures that depends on a cloud of nodes or points for geometry representation and discretization. In the field of computational fracture mechanics, the meshfree nature of meshfree methods has gained considerable attention for modelling two-dimensional and three-dimensional crack growth. While FEM uses interpolation methods to formulate shape functions, meshfree approaches use approximation methods. This review aims to examine the developments, utility, limitations, and potential for refinements of meshfree techniques.