## 2 Natural Neighbour Radial Point Interpolation Method  
Meshless methods are advanced discretisation techniques capable of discretising the problem domain using only a nodal or point distribution [21; 22]. Such a nodal distribution, which can be regular or irregular, does not form a mesh because there is no pre-established nodal connectivity. Since elasticity problems are addressed by solving a system of equations built based on integro-differential equations, it is necessary to numerically integrate such equations. The most common solution is the use of a background set of integration points capable of numerically integrate those integro-differential equations. Thus, after the nodal discretisation, in elasticity problems, meshless methods need to discretise the problem domains with a set of integration points, which can be dependent or independent of the initial nodal distributions [22]. Generally, if an independent set of integration points is constructed, the meshless method is called a non-truly meshless method because the set of integration points are obtained based on a structured grid of integration cells, leading to a background integration mesh. On the other hand, if the meshless method is capable of constructing a set of integration points dependent on the nodal distribution, the meshless method is called a truly meshless method [22]. The main advantage of non-truly meshless methods is their straightforward capability of being incorporated into any generic FEM software. In these methods, for example, the FEM's element meshes can be directly used as background integration cells, and the FEM's nodal meshes are naturally the meshless method's nodal distributions. The main advantage of dependent meshless methods is their ability to construct all the required mathematical entities (background integration points, nodal connectivity, shape functions, etc.) using only the information of the nodal distribution. After the construction of the background integration mesh, meshless methods must establish the nodal connectivity. For this, the most widespread technique is the influence domain concept, applied to various meshless formulations, such as the RPIM, EFGM, MLPG, and RKPM. Each integration point \(\mathbf{x}_{I}\) radially searches for its closed nodes, whose set will form its \"influence domain\" [22]. Afterwards, for each integration point \(\mathbf{x}_{I}\), shape functions are constructed. Once again, the literature offers several techniques to build shape functions [22], such as moving least squares, radial basis functions, polynomial basis functions, Taylor's expansion functions, Sibson functions, etc. These shape functions permit us to approximate/interpolate the variable field at an integration point \(\mathbf{x}_{I}\) with  
\[u(\mathbf{x}_{I})=\sum_{i=1}^{n}\varphi_{i}(\mathbf{x}_{I})\cdot u(\mathbf{x} _{i})=\Phi(\mathbf{x}_{I})^{T}\cdot\mathbf{u}_{s}=\left\{\varphi_{1}(\mathbf{ x}_{I}),\varphi_{2}(\mathbf{x}_{I}),\cdots,\varphi_{n}(\mathbf{x}_{I}) \right\}\cdot\begin{cases}u(\mathbf{x}_{1})\\ u(\mathbf{x}_{2})\\ \cdots\\ u(\mathbf{x}_{n})\end{cases} \tag{1}\]  
being the number of nodes within the influence domain of the integration point \(\mathbf{x}_{I}\) defined as \(n\). Vector \(\mathbf{u}_{s}\) contains the field variable components of each node \(\mathbf{x}_{i}\) inside the influence domain of \(\mathbf{x}_{I}\), and \(\varphi_{i}(\mathbf{x}_{I})\) is the \(i^{th}\) component of the shape function constructed for \(\mathbf{x}_{I}\). After constructing the shape functions and their partial derivatives, they are applied to the integro-differential equations of elasticity, leading to the system of algebraic equations. Then, the solution variable field across the domain is obtained by solving the system of algebraic equations.