# Elastoplastic Analysis of Frame Structures Using Radial Point Interpolation Meshless Methods
## 2 Radial Point Interpolation Meshless Methods
The Euclidean distance between the point \(\mathbf{x}_{I}\) and a node \(\mathbf{x}_{i}\) within the same influence-domain can be calculated for 2D problems using:  
\[r_{ij}=\sqrt{(\mathbf{x}_{I}-\mathbf{x}_{i})^{2}+(\mathbf{y}_{I}-\mathbf{y}_ {i})^{2}} \tag{3}\]  
Thus, Equation (1) can be rewritten in a matrix form as:  
\[\mathbf{u}(\mathbf{x}_{I})=\{\mathbf{R}^{T}(\mathbf{x}_{I}),\mathbf{p}^{T}( \mathbf{x}_{I}\}\mathbf{G}^{-1}\left\{ \begin{array}{l}\mathbf{u}_{s}\\ 0\end{array} \right\}=\varphi(\mathbf{x}_{I})\mathbf{u}_{s} \tag{4}\]  
where \(\varphi(\mathbf{x}_{I})\) is the matrix containing the values of the interpolation functions of every node composing the influence-domain calculated at the interest point \(\mathbf{x}_{I}\), and matrix \(\mathbf{R}\), of  
Figure 3: Sub-cell obtained through the overlapping of the Voronoi diagram, the Delaunay tessellation, and how quadrature points following the Gauss–Legendre integration scheme are established. size \([n\times n]\), contains the values of the MQ-RBFs calculated by Equation (2) at the interest point \(\mathbf{x}_{I}\):  
\[\mathbf{R}= \begin{bmatrix}R(r_{11})&R(r_{12})&\ldots&R(r_{1n})\\ R(r_{21})&R(r_{22})&\ldots&R(r_{2n})\\ \vdots&\vdots&\ddots&\vdots\\ R(r_{n1})&R(r_{n2})&\ldots&R(r_{nn})\end{bmatrix}  \tag{5}\]  
Matrix \(\mathbf{p}\), of size \([n\times m]\), is the matrix containing the polynomial basis (m is the monomial number, i.e., the number of terms of the considered polynomial basis):  
\[\mathbf{p}= \begin{bmatrix}p_{1}(\mathbf{x}_{1})&p_{2}(\mathbf{x}_{1})&\ldots &p_{m}(\mathbf{x}_{1})\\ p_{1}(\mathbf{x}_{2})&p_{2}(\mathbf{x}_{2})&\ldots&p_{m}(\mathbf{x}_{2})\\ \vdots&\vdots&\ddots&\vdots\\ p_{1}(\mathbf{x}_{n})&p_{2}(\mathbf{x}_{n})&\ldots&p_{m}(\mathbf{x}_{n})\end{bmatrix}  \tag{6}\]  
\(\mathbf{u}_{s}\) is the nodal displacement vector of the influence-domain of \(\mathbf{x}_{I}\), and \(\mathbf{G}\) is given as:  
\[ \begin{bmatrix}\mathbf{R}&\mathbf{p}\\ \mathbf{p}^{T}&0\end{bmatrix} =\mathbf{G} \tag{7}\]  
For more detailed mathematical information on the interpolation functions used in the RPIM and NNPRIM, the book by [PERSON] et al. [5] is well documented. With the shape functions established, the discrete system of equations can be obtained using the Galerkin Weak Form. This is a rather standard procedure, well documented in the literature [5], and therefore it will not be presented in this paper. In comparison with the FEM, the RPIM and the NNRIPM possess higher nodal connectivity (due to the overlap rule of influence-domains), which usually leads to more accurate solutions, even for non-transitory geometry problems. The accuracy of the meshless methods used in this work will be proven in Section 4. Additionally, the interpolation functions cited here have virtually a higher order, allowing a higher continuity and reproducibility. Due to the preprocessing of the Voronoi diagram and the construction of the node-dependent integration mesh, higher computational costs are expected in the NNRIPM, this being its main disadvantage.