## 4 Numerical Example: Transient Fin Problem
### Sensitivity Analysis of Temperature History  
As stated in Section 3, the first step to obtain sensitivities using Transient Thermal ZFEM is to apply a perturbation, \(h\), to the variable of interest. While many other references have shown the step size independence for CTSE when applied in other fields, a step size analysis was conducted for transient thermal analysis to verify previous findings. In addition, the optimal step size for FD was also computed for comparison purposes. Considering the analytical equations as a reference for the calculation of the error, the NRMSE for the responses as a function of the step size for Transient Thermal ZFEM and FD calculated with built-in Abaqus functions is presented in Figure 5a, using continuous and dashed lines, respectively. All the FD results were obtained by using a forward differentiation scheme and first-order accuracy, which is analogous to applying a positive imaginary perturbation in the CTSE method integrated into Transient Thermal ZFEM. In agreement \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter Type** & **Parameter of Interest (\(\phi\))** & **Value** & **Units** \\ \hline \multirow{3}{*}{Material Properties} & \(k\) & 7.1 & \(\mathrm{W}/(\mathrm{m}\cdot\mathrm{K})\) \\  & \(C_{P}\) & 580 & \(\mathrm{J}/(\mathrm{kg}\cdot\mathrm{K})\) \\  & \(\rho\) & 4430 & \(\mathrm{kg}/\mathrm{m}^{3}\) \\  & \(h_{\mathrm{H}\mathrm{I}}\) & 114 & \(\mathrm{W}/(\mathrm{m}^{2}\cdot\mathrm{K})\) \\ \multirow{3}{*}{Boundary Conditions} & \(T_{\infty}\) & 283 & \(\mathrm{K}\) \\  & \(T_{w}\) & 389 & \(\mathrm{K}\) \\ \multirow{3}{*}{Initial Conditions} & \(T_{0}\) & 283 & \(\mathrm{K}\) \\  & \(b\) & 51 & mm \\ \multirow{3}{*}{Geometry} & \(L\) & 1 & \(\mathrm{m}\) \\  & \(\delta\) & 4.75 & mm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters involved in the transient thermal fin problem. Figure 4: (**a**) Convergence analysis for the heat transfer coefficient per unit of thickness where the optimal value that minimizes the NRMSE is \(735(W/(\mathrm{m}^{2}\mathrm{K}))/\mathrm{m}\). (**b**) The dimensionless temperature history of fin tip: solid black line, Equation (23), yellow star, the real part of Transient Thermal ZFEM solution, blue square, Abaqus built-in element solution, thin red dashed line, steady-state solution (Equation (24)). with previous literature about numerical sensitivities [89], for values above \(h\times\phi\) with \(h=10^{-4}\), the truncation error completely cancels the precision of the sensitivity response in the FD method. However, the Transient Thermal ZFEM is completely insensitive to the step size for the analyzed input parameters. The change in the trend for the dashed red and dashed green lines in FD is related to a shift in the dominance of the truncation error to round-off error. This analysis highlights one of the main advantages of the Transient Thermal ZFEM over the regular FD in that CTSE is dramatically less prone to step size inaccuracies. Based on the perturbation step analysis study, the rest of the results presented herein were calculated using \(h=10^{-10}\phi\) for the Transient Thermal ZFEM and \(h=1\%\phi\) for FD. For geometrical perturbations, it is required to perturb a fraction of nodes in the mesh. Besides using the rule of thumb for this parameter, a convergence for study for \(\lambda\) was performed. Figure 5b, shows the NRMSE for the sensitivity with respect to \(b\), \(S_{b}^{\theta}\), as a function of \(\lambda\). In this figure, the top insert shows the rule of thumb suggested \(\lambda\), and the bottom inset shows a completely perturbed mesh. For the transient thermal fin problem, the precision of \(S_{b}^{\theta}\) is insensitive to \(\lambda\), as shown by the almost flat line of Figure 5b. Therefore, the rule of thumb and the case when \(\lambda=1\) have almost the same precision. For illustration purposes in the fin problem, we picked \(\lambda=1\) for the sensitivity analysis of the fin length.