## 2 Radial Point Interpolation Method  
Similar to most numerical node-dependent discretisation methods, the majority of meshless methods follow a common procedure. The main differences between meshless methods and the standard and well-known FEM can be found at the pre-processing phase [7]. In opposition to FEM, which discretizes the problem domain with elements (containing nodes and integration points), meshless methods discretize the problem domain using only a nodal set. In meshless methods there are no pre-established connections between the discretization nodes. Such connection is enforced by some mathematical concept (radial search, natural neighbours, etc.) [7]. Also, the background integration points of meshless formulations can be constructed using node-independent or node-dependent rules, allowing a higher flexibility in their distribution along the problem domain [7]. Regarding the shape function construction, contrasting with FEM, meshless methods are able to apply distinct functions to build their shape functions, such as Moving Least Squares, Point Interpolation, Polynomial Interpolation, Sibson Interpolation, etc. [7]. Thus, next, in order to understand the general meshless procedure, a general overview is provided. In meshless formulations, first, the solid domain is discretised with a nodal set, and the essential and natural boundaries are established. Following discretisation, it is necessary to construct a cloud of background integration points, covering the complete solid domain. The integration points can be obtained using the same procedure as FEM. First, integration cells are constructed, discretising the complete solid domain, and then the integration points are obtained following the Gauss-Legendre quadrature integration technique [7]. Each cell is subsequently transformed into a unit square using a parametric transformation (Figure 1). This transformation allows for the distribution of integration points within the resulting isoparametric square [7; 17]. Once the integration mesh is established, the next step involves defining the nodal connectivity. In FEM, element definitions dictate nodal connectivity. However, meshless methods, as mentioned earlier, rely on influence domains rather than elements. Nodal connectivity is established by overlapping these influence domains. Finding these domains involves searching for nodes within a specific area or volume (depending on the problem's spatial dimension). This approach, due to its simplicity and low computational cost, is widely used in various meshless methods like RPIM, EFGM, MLPG, and RKPM. It is important to consider the impact of domain size and shape on the method's performance to achieve accurate and satisfactory solutions. While the literature suggests a single optimal size for influence domains is not feasible, a fixed reference dimension is often adopted. Each integration point \(\mathbf{x_{I}}\) searches for its closest nodes, forming its own influence domain. For the RPIM, the literature recommends between nine and sixteen nodes inside each influence domain [7]. In order to establish the system of equations, it is necessary to construct the shape functions. To obtain them, RPIM uses the radial point interpolating technique [7; 17]. Considering a function, \(\mu(\mathbf{x})\), defined within the domain \(\Omega\in\mathbb{R}^{2}\). The domain is discretised  
Figure 1: General RPIM procedure. (**a**) Domain’s nodal discretisation, (**b**) background integration grid, (**c**) integration points filling on integration cell, and (**d**) final background integration mesh. by a set of \(N\) nodes, denoted by \(\mathbf{X}=\{\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{N}\}\).