# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 1 Introduction
LRBFCM uses radial basis functions to construct trial functions, and it is highly dependent on the shape parameters of the corresponding radial basis function. The accuracy and efficiency of LRBFCM strongly depend on the values of such shape parameters, which must be analysed and optimised for distinct problems [14; 15]. The literature offers several variants of collocation methods, such as the Trefftz collocation method [16; 17]. Approximation methods, such as EFGM, RKPM, and MLPG, produce highly accurate solutions. However, their shape functions lack the Kronecker delta property, hindering the imposition of the essential and natural boundary conditions, which can only be enforced with exact technique, such Lagrange multipliers [1]. Therefore, efforts were made to develop interpolating meshless techniques. The natural element method (NEM), was one of the first to be developed [18]. Using the natural neighbour and Voronoi diagram concepts, the nodal connectivity, the background integration mesh and the shape functions (and corresponding spatial derivatives), are constructed automatically, producing a true meshless method approach with interpolating properties. In 2001, [PERSON] and his research team developed the point interpolator method (PIM) [19], in which the shape functions are constructed by using a polynomial basis. Although simple and easy to apply, the process is not capable of producing shape functions when nodes are perfectly aligned (most common in regular nodal distributions). Therefore, a radial basis function was added to the formulation, allowing for the development of one of the most popular interpolating meshless methods: the radial point interpolation method (RPIM) [20]. The use of the radial-based functions eliminated the singularities occurring from the nodal alignment. Later, [PERSON] and coworkers included in the formulation the natural neighbour concept and developed the natural neighbour radial point interpolation method (NNRPIM) [1; 21]. With the NNRPIM, both the nodal connectivity and the node-dependent integration background mesh are constructed resorting to the Voronoi tessellation and to the Delaunay triangulation. This procedure guarantees that the shape functions developed have the Kronecker delta property and a formulation that is truly meshless. The literature shows that both RPIM and NNRPIM are highly accurate and present high convergence rates [1], being solid alternatives for structural analysis. It is possible to find in the literature several works successfully combining meshless methods with elastoplastic formulations. In 1999, [PERSON] and [PERSON] [22] extended the EFGM to elastoplasticity considering a 3D formulation with small strains, analysing a thick-walled pressure vessel with internal pressure and an extension of a strip with a circular. Compared with reference solutions [23], the method was capable of producing results very close to the ones obtained in the FEM. In 2006, [PERSON] and [PERSON] [24; 25; 26] used the EFGM in the analysis of two-dimensional problems, plates, and laminates in the elastoplastic regime. Various modified Newton-Raphson algorithms were used to obtain the nonlinear solution. A Cook membrane and a sandwich material, under distributed load, were studied. The obtained results were in accordance with FEM solutions. Several modified Newton-Raphson algorithm versions were used to obtain the nonlinear solution and their performance and computational cost was also analysed. The results showed that the EFGM is a solid alternative to such nonlinear problems, being capable of obtaining similar results to the FEM analysis. More recently, in 2014, [PERSON] et al. [27], developed a novel interpolating EFGM (IEFGM) for two-dimensional elastoplasticity. By using the interpolating moving least-squares method (IMLS) to construct the shape functions, the Kronecker delta property is satisfied, and the essential boundary conditions can be applied directly. The method has some advantages, such as simpler formulae and the direct application of the boundary conditions.