# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 2 Meshless Methods  
The majority of meshless methods follow a generic procedure, well discribed in the literature [1; 2]. After the problem geometry and the natural and essential boundary conditions are described and established, the problem domain and the respective boundary are discretized by using a nodal set, which can follow a regular or irregular distribution. Similar to mesh-dependent methods combined with weak formulations and applied to elasticity problems, the accuracy of the solution provided by meshless methods depends on the nodal density of the discretization. Thus, a higher density of nodes generally returns more accurate results. Moreover, in specific problems wherein particular geometrical or loading details can have a considerable effect on the stress distribution (material discontinuities or concentrated loads), a higher nodal density in those locations is required to obtain better results. In meshless methods, the nodal distribution does not form a mesh because there is no preestablished connectivity between the nodes. After the nodal discretization of the problem domain, the background integration points are defined. The complete set of integration points is known as integration mesh. Background integration meshes can be obtained by using background integration cells (fitted to the problem domain, as elements in the FEM) or by using nodal integration schemes (as the ones used in NEM or NNRPIM). This last procedure uses the mathematical concepts of natural neighbours, Voronoi diagrams and Delaunay tessellation to construct the integration points uniquely based on the nodal distribution, and thus producing truly meshless formulations. After discretizing the problem domain with nodes and integration points, it is possible to establish the nodal connectivity. In FEM, as input, it is necessary to declare the nodes forming each element. Thus, integration points inside each element interact with the nodes forming that element, and elements interact with other elements possessing common nodes. In meshless methods, because there is no preestablished declared relation between the nodes, it is necessary to enforce such connectivity. Therefore, it is necessary to apply the concept of influence domain. Each integration point \(\mathbf{x}_{I}\) will search for nodes inside a certain radius. The nodes inside such radius will form the influence domain of \(\mathbf{x}_{I}\). Due to its simplicity, several meshless formulations apply such concepts (RPIM, EFGM, MLPG, RKPM, etc.). There are several variants to this concept [1], some authors use fixed radial searches and other use variable radial searches. Fixing a radius will produce uneven influence domains. Integration points near a physical boundary will possess a lower number of nodes inside their influence domains when compared with the influence domains of the nodes inside the problem domain. Alternatively, fixing the number of nodes inside each influence domain (variable radius searches), will produce influence domains of the same size for all integration points. Alternatively, meshless formulations using the natural neighbour concept, such as NEM or NNRPIM, already possess the nodal connectivity established due to the Voronoi diagram [31] of the nodal discretization. Each integration point \(\mathbf{x}_{I}\) has an associated node \(\mathbf{x}_{i}\), and each node possesses a set of natural neighbours: \(\mathbf{V}_{i}\). Thus, the influence domain of \(\mathbf{x}_{I}\) is the set of natural neighbours of \(\mathbf{x}_{i}\): \(\mathbf{V}_{i}\), and \(\mathbf{x}_{i}\) itself. Afterward, the shape functions are constructed by using approximation or interpolation functions according to the technique adopted.