## 2 Natural Neighbour Radial Point Interpolation Method
### Nodal Connectivity and Numerical Integration  
Since the NNRPIM uses the mathematical concept of natural neighbours to construct the background set of integration points and to establish the nodal connectivity, it is necessary first to present it briefly. The natural neighbour concept requires the construction of the Voronoi diagram of the problem's nodal discretisation [44]. Thus, assuming a 2D domain, \(\Omega\in\mathbb{R}^{2}\) discretised with \(N\) nodes: \(\mathbf{X}=\left\{\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N}\right\} \in\mathbb{R}^{2}\), the Voronoi diagram, \(\mathbf{V}\), of \(\mathbf{X}\) is formed by a set of sub-regions \(V_{i}\), closed and convex, defined in (and defining) the same sub-space \(\Omega\in\mathbb{R}^{2}\). Each node \(\mathbf{x}_{i}\) possesses its own Voronoi cell \(V_{i}\), which is defined as the geometric place where all points in its interior are closer to \(\mathbf{x}_{i}\) than any other\(\mathbf{x}_{i}\in\mathbf{X}\), being \(j\
eq i\). Thus, the Voronoi diagram of \(\mathbf{X}\) is defined by \(\mathbf{V}=\{V_{1},V_{2},\ldots,V_{N}\}\). The distinct Voronoi cells \(V_{i}\) do not overlap or leave gaps between each other, allowing us to verify: \(\Omega=\sum_{i=1}^{N}V_{i}\therefore\mathbf{V}=\Omega\). Figure 1 shows the Voronoi diagram of a given nodal discretisation. The natural neighbours of node \(\mathbf{x}_{i}\) are all the nodes whose Voronoi cells share an edge with the Voronoi cell of node \(\mathbf{x}_{i}\) (light grey cells in Figure 1a surrounding the dark grey cell in node \(\mathbf{x}_{i}\)). Using the natural neighbour concept, it is possible to automatically establish influence domains, i.e., the nodal connectivity. Thus, in the NNRPIM, integration points inside Voronoi cell \(V_{i}\) will inherit the nodal connectivity of node \(\mathbf{x}_{i}\). In the NNRPIM, the \"influence domain\" concept is substituted by the similar \"influence cell\" concept. Instead of searching for the closest nodes, the influence domain of node \(\mathbf{x}_{i}\) is formed by the natural neighbours of node \(\mathbf{x}_{i}\), which were already defined by the Voronoi diagram. Thus, two kinds of influence cells are possible: first-degree influence cells, Figure 1a, and second-degree influence cells, Figure 1b. The first-degree influence cells of node \(\mathbf{x}_{i}\) are formed by the natural neighbours of node \(\mathbf{x}_{i}\) and node \(\mathbf{x}_{i}\) themselves. The second-degree influence cells of node \(\mathbf{x}_{i}\) are formed by node \(\mathbf{x}_{i}\) itself, plus the natural neighbours of node \(\mathbf{x}_{i}\) (as a first-degree influence cell) and also the natural neighbour of those first natural neighbours. In the NNRPIM, the construction of the background set of integration points also uses the information of the Voronoi diagram. As Figure 2 represents, for a generic irregular nodal distribution, the Voronoi cell \(V_{i}\) of node \(\mathbf{x}_{i}\) can be sub-divided in quadrilaterals. Then, using the Gauss-Legendre quadrature integration scheme, it is possible to define the position and weight of each integration point inside the quadrilateral. Figure 1: Voronoi diagram of a given nodal discretisation. (**a**) First-degree influence cell. (**b**) second-degree influence cell. Figure 2: Generic procedure for the construction of the background set of integration points based on the Voronoi diagram. The procedure follows the following steps. First, each quadrilateral of a given Voronoi cell \(V_{i}\) is transformed into a unit isoparametric square, allowing for the distribution of integration points within the isoparametric square respecting the Gauss-Legendre integration scheme [22]. Afterwards, as Figure 2 shows, the Cartesian coordinates of the integration points are defined from their isoparametric coordinates with Equation (2). \[\mathbf{x}_{l}=\left\{ \begin{array}{l}x_{l}\\ y_{l}\end{array} \right\}=\left[ \begin{array}{l}\mathbf{N}(\xi_{l},\eta_{l})^ {T}\\ \mathbf{N}(\xi_{l},\eta_{l})^{T}\end{array} \right]\cdot\left[