# Elastoplastic Analysis of Frame Structures Using Radial Point Interpolation Meshless Methods
## 2 Radial Point Interpolation Meshless Methods  
The standard meshless procedure starts with the discretization of the problem domain into a nodal mesh, which can arbitrarily be created (Figure 1). A regular or irregular nodal distribution can be assigned to the domain. A higher nodal density generally leads to more accurate results and, also in a general case scenario, irregular distributions have a lower accuracy when compared to regular distributions. The second step in a meshless algorithm is the construction of the integration mesh. The NNRPIM uses a nodal-dependent integration mesh to integrate the integro-differential equations from the Galerkin Weak form. On the other hand, the RPIM uses the GaussFigure 1: (**a**) Solid Domain. (**b**) Regular discretization. (**c**) Irregular discretization. After that, the nodal connectivity is imposed using the influence-domain concept: for a certain interest point \(\mathbf{x}_{l}\), the domain of influence of such a point will be an area or volume centered at said point. Thus, the nodes inside these areas or volumes will be used to interpolate the field variables at the interest point, and the overlap of the domains will ensure nodal connectivity. The procedure is similar in both RPIM and NNPRIM, although the shape of the influence-domains is different for both methods. Figure 2a shows some influence-domains with different sizes and shapes. According to [16], influence-domains with a fixed number of nodes (between \(n=9\) and \(n=16\)) allow the construction of shape-functions with the same level of complexity. Figure 2: (**a**) Fixed rectangular influence-domains. (**b**) Fixed circular influence-domain. (**c**) Flexible circular shaped influence-domains. natural neighbors. For mathematical completeness, [PERSON]. [5] provides a detailed explanation of how these geometrical constructions are established. As a consequence of the creation of the Voronoi diagram, a nodal-dependent integration mesh is constructed, as stated before. The Delaunay tessellation is the geometrical dual of the Voronoi diagram, and it is constructed by connecting the nodes whose Voronoi cells have common boundaries. Using this geometrical concept, [PERSON] et al. [5] then proposed an integration scheme: the Delaunay tessellation divides the Voronoi cells into subcells, and integration points are placed at the center of the sub-cell, the integration weight of that point being related with the area occupied by such integration point--Figure 3.  
The remaining RPIM and NNRPIM formulations are similar. As stated before, the field variables are interpolated within each influence-domain. For instance, consider the displacement field \(\mathbf{u}\). The displacement components \(\mathbf{u}_{I}=(u,v,w)\) at an interest point \(\mathbf{x}_{I}\) are computed using the nodal displacement of the nodes inside the influence-domain of said interest point:  
\[\mathbf{u}(\mathbf{x}_{I})=\sum_{j=1}^{n}\varphi_{j}(\mathbf{x}_{I})\mathbf{u }(\mathbf{x}_{i}) \tag{1}\]  
where \(n\) is the number of nodes inside the influence-domain or influence-cell of the interest point \(\mathbf{x}_{I}\), \(\mathbf{u}(\mathbf{x}_{i})\) is a vector containing the nodal displacements of the nodes within the influence-domain, and \(\varphi_{j}(\mathbf{x}_{I})\) is the interpolation function value at the \(j\)th node. Both RPIM and NNRPIM use the same interpolation functions, which are a combination of polynomial basis functions and the Multiquadratic Radial Basis Function (MQ-RBF)--proposed initially by [PERSON] [25]:  
\[R(r_{ij})=(r_{ij}^{2}+c^{2})^{p} \tag{2}\]  
with \(c\) and \(p\) being shape parameters, whose optimal values are \(c\) = 1.42 and \(p\) = 1.03, according to [16; 26]. Nevertheless, [PERSON] et al. [5] performed an optimization study for these shape parameters and concluded that \(c\) = 0.0001 and \(p\) = 0.9999 are more suitable values. These last shape parameter values were used in this work.