# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 1 Introduction  
Computational mechanics is the discipline concerned with the use of computational numerical methods to study phenomena governed by the principles of solid and fluid mechanics. It is used every day to help predict, develop, and improve not only structural and fluid applications in the industrial sense but also several other science areas, such as medicine, biology, chemistry, etc. [1]. With the constant evolution of engineering, the challenges for computational mechanics are numerous. For example, when analysing the simulation of manufacturing processes, such as extrusion or moulding, it is necessary to deal with large displacements of mesh, viscoplastic effects and contact conditions, whereas in simulations of failure processes, it is required to model the fracture initiation and its propagation path, leading to the development of demanding remeshing algorithms and fracture-damage criteria. Problems dealing with large mesh distortions or localised remeshing requirements are not efficiently solved by mesh-dependent discretization techniques, such as the finite element method (FEM) [2]. Meshless methods are an alternative to classic mesh-dependent discretization methods. In meshless methods, the problem domain is discretized by a random set of nodes, without any preestablished connectivity, rather than a structured element mesh. To define the nodal connectivity, meshless methods use the influence-domain concept, allowing one to establish automatically the nodal connectivity in a later stage of the preprocessing phase [3, 4]. Meshless methods and their origin can be traced to the smooth particle hydrodynamics method (SPH) [5]. It was applied to polytropic stellar models and used statistical techniques to recover analytical expressions for the physical variables from a known distribution of fluid elements, starting with a nonaxisymmetric distribution of approximately 80 particlesin three dimensions. This method was applied to solid mechanics by [PERSON] and [PERSON] [6] in 1991, in which an elastic-perfectly-plastic constitutive model was formulated. It was found that excellent features of the SPH, such as robustness, simplicity, and ease of adding new physics, could be transferred to the analysis of solid mechanics problems [7]. SPH is a particle method, in which the set of nodes (the particles) do not form a solid continuum in the same sense as in FEM, being more suited for free surface fluid analyses. Thus, other meshless technique started to be developed, such as the diffuse element method (DEM) [8], the first mature meshless method using a weak formulation and shape functions constructed by using the moving least-squares (MLS) approximants approach. Then, [PERSON] and coworkers improved the DEM and proposed in 1994 the element free Galerkin method (EFGM) [9], one of the most popular meshless methods. At the same period, other approximation meshless techniques were developed, such as the reproducing kernel particle method (RKPM) [10] and the meshless local Petrov-Galerkin method (MLPG) [11], a truly meshless method (in the sense that it does not require a background nodal independent integration mesh) based on a local symmetric weak form (LSWF). MLPG does not need an element mesh for the interpolation of the solution variables. Integrals are solved over a regular-shaped domain and its boundaries. The generalized finite difference method (GFDM) is another popular approximation method [12; 13]. Because it can be applied to solve problems discretized by using only an irregular nodal distribution, GFDM is also considered truly meshless. GFDM uses the Taylor series expansions and the MLS approximation to derive explicit expressions for the required partial derivatives of unknown variables. Collocation meshless methods are another class of meshless methods available in the literature, such as the local radial basis function collocation method (LRBFCM) [14; 15], which uses the strong-form formulation to establish the system of equations and obtain the variable field.