## 3 Methodology  
### RBF-FD Approximations  
A typical central finite difference scheme for a derivative of function \(u(x,y)\) with respect to \(x\) at any grid point \((i,j)\) can be written in the form as follows  
\[\frac{\partial u}{\partial x}\bigg{|}_{(i,j)}\approx\sum_{k\in\{i-1,i,i+1\}} \omega_{(k,j)}u_{(k,j)} \tag{17}\]  
where \(\omega_{(k,j)}\) are obtained by using polynomial interpolation or Taylor series expansion. Instead of using polynomial interpolation or Taylor series expansion, this study applied radial basis function interpolation to obtain the RBF-FD approximation [20; 21]. The RBF interpolation can be shown as Equation (18)  
\[u(x)\approx s(x)=\sum_{i=1}^{N}\lambda_{i}\phi(\|x-x_{i}\|)+\sum_{j=1}^{m} \beta_{j}p_{j}(x) \tag{18}\]  
where \(\|\cdot\|\) denotes the standard Euclidean 2-norm. \(\phi(r=\|\cdot\|)\) is the RBF that can be replaced by any other type in Table 1. \(p_{j}(x)\) is a basis for the space of all d-variate polynomials, and \(\varepsilon\) is the shape function, which can be tuned by the user. Furthermore, the expansion coefficients \(\lambda_{i}\) and \(\beta_{j}\) are determined by enforcing the conditions as Equations (19) and (20)  
\[s(x_{i})=u(x_{i})\quad i=1,2,\ldots N \tag{19}\]  
Figure 3: Thermal material properties of AISI 316L stainless steel. (**a**) thermal conductivity; (**b**) density; (**c**) heat capacity. \[\sum_{i=1}^{N}\lambda_{i}p_{j}(x_{i})=0\;\;j=1,2,\ldots m \tag{20}\]  
According to [18; 22], Equation (18) will be well posed without any polynomial augmentation when using some type of RBFs, such as GA, MQ, IMQ, and IQ. However, for the conditionally positive definite RBFs, such as PHS and MQ, polynomial augmentation can be used for Equation (18) to ensure the linear system is uniquely solvable and leads to more accurate results. Therefore, in this study, the RBF of type MQ was applied, and the Lagrange form of the RBF interpolation as Equation (21) was used for deriving the RBF-FD approximation  
\[u(x)\approx\sum_{i=1}^{N}\chi(\|x-x_{i}\|)u(x_{i}) \tag{21}\]  
where \(\chi(\|x-x_{i}\|)\) satisfies Equation (22),  
\[\chi(\|x-x_{i}\|)=\left\{ \begin{array}{ll}1,&if\;x=x_{i}\\ 0,&if\;x\
eq x_{i}\end{array} \;i=1\sim N\right. \tag{22}\]  
knowing that Equations (18) and (21) infer a global formulation if \(N\) is the total number of nodes located in the full domain. Therefore, a supporting region is defined first to derive the localized formulation, as shown in Figure 4. It illustrates that the unknown function \(u(x)\) at any node, say \(x_{1}\), is approximated by an RBF interpolation with the \"center node\", \(x_{1}\), placed on the node itself and the \(n-1\) closest surrounding nodes. The \(n\) nodes constitute the supporting region for the center node. In order to match the localized formulation, substitute \(N\) for \(n\) in Equation (22) with \(n\ll N\). Also, the linear differential operator \(\mathcal{L}\) is applied on both sides of Equation (21), and \(\mathcal{L}u(x_{1})\) can be described as Equation (23). \[\mathcal{L}u(x_{1})\approx\sum_{i=1}^{n}\mathcal{L}\chi(\|x_{1}-x_{i}\|)u(x_ {i}) \tag{23}\]  
According to the finite difference scheme mentioned in Equation (17), this approximation can be rewritten in the form of Equation (24)  
\[\mathcal{L}u(x_{1})\approx\sum_{i=1}^{n}\omega_{(1,i)}u(x_{i}) \tag{24}\]