# Elastoplastic Analysis of Plates with Radial Point Interpolation Meshless Methods
## 5 Conclusions  
In this work, the radial point interpolation method (RPIM) was combined with an elastoplastic formulation to analyse 3D plates. Original nonlinear RPIM and FEM codes were written in order to have a more detailed comprehension of the differences and similarities between both numerical methods (at the formulation and results levels). This process allowed us to directly compare performances (global computational times and efficiency) and the results accuracy.  
Concerning the nonlinear solutions algorithms tested, it was found that using the KT-All algorithm is more efficient. Although it recalculates the stiffness matrix in every iteration, it required many fewer iterations than KT0 or KT1 when a large part of the material starts to yield. Thus, the overall computation time is less than the one required by KT0 or KT1 algorithms to achieve the same solution.  
Regarding the stress distribution along the thickness of the plate, the results show that generally FEM and RPIM produce similar normal stresses distributions. However, concerning shear stresses, the results are not as close. In opposition to FEM, RPIM is capable of producing the expected parabolic shear stress distribution along the thickness, demonstrating an higher accuracy. The stress distributions shown were acquired at the nodes. Therefore, such results are an extrapolation of the stress values on the integration points. This explains why shear stress obtained with FEM and RPIM is not null at the top and bottom layers of the plate (as it should be).  
The presented stress distribution maps allowed us to visualise the expected rupture lines of both benchmarks. In the simply supported plate, the diagonal rupture line can identified, as well as in the clamped plate the boundary rupture lines at first, and then the central rupture and, in the end, the diagonal lines. Such results indicate that the nonlinear solution algorithm was successfully combined with the linear RPIM formulation.  
The main advantage of RPIM is its potential to be directly integrated in existing FEM software. For instance, the RPIM can assume the finite elements as integration cells, distributing inside them the required integration points (following the most appropriated scheme). Moreover, the element nodes can be utilised to form the global nodal discretization. Then, by using the influence-domain concept, the nodal connectivity can be established. Afterward, the process moves on, constructing the RPI functions, establishing the system of equations, obtaining the displacement, strain and stress fields and following the elastoplastic analysis. The final results can then be depicted in existent FEM software. Moreover, because RPIM is an interpolating numerical method, it can be combined with FEM. Allowing to use meshless methods in parts of the domain in which higher accuracy is required (for instance, the vicinity of cracks or other locations in which stress concentration is expected) and the FEM can be applied in the remaining part of the solid (to speed-up the analysis).  
Conceptualisation, J.B., methodology; J.B., software; J.B., validation; M.A. and J.B., formal analysis; J.B., investigation, J.B., resources; J.B., data curation; M.A. and J.B., writing--original draft preparation; J.B. and M.A., writing--review and editing; J.B., visualization; J.B., supervision; J.B., project administration; J.B., funding acquisition. All authors have read and agreed to the published version of the manuscript.  
The authors acknowledge the funding provided by Ministerio da Ciencia, Tecnologia e Ensino Superior--Fundacao para a Ciencia e a Tecnologia (Portugal) and LAETA, under internal project UIDB/50022/2020.  
Not applicable.  
The authors declare no conflict of interest.