## 4 Numerical Example: Transient Fin Problem
### Sensitivity Analysis of Temperature History
This means that all the imaginary coordinates of the mesh have non-zero values as shown in the bottom right inset in Figure 5b. The sensitivities history of \(\theta\) concerning the thermal conductivity, \(S_{k}^{\theta}\), the room temperature, \(S_{T_{\infty}}^{\theta}\), and the fin's height, \(S_{b}^{\theta}\), are shown in Figure 6a-c, respectively. A negative relative sensitivity predicts a decreasing trend for the temperature with an increment of the input parameter and vice versa. Overall, a good agreement between analytical differentiation, FD, and Transient Thermal ZFEM is observed. The solutions from the simulations using Transient Thermal ZFEM had a better match than built-in Abaqus functions with FD match when compared with the analytical values, as could be seen by error measurements in Table 2. These errors are on the same order as the NRMSE for \(\theta\); therefore, we attribute the good agreement among methodologies to the enhanced dissipation implemented to counter the lack of heat transfer in the out-of-plane direction in the 2D FEM and Transient Thermal ZFEM elements. Figure 5: (**a**) Step size convergence analysis comparing Transient Thermal ZFEM and FD for the UEL and the selected analyzed variables. (**b**) Convergence study for mesh perturbation fraction \(\lambda\). The insets show \(P(x)\) for the rule of thumb of the literature and \(\lambda=1\) in the top left and bottom right corners, respectively. From the time-dependent sensitivity analysis in Figure 6, it can be stated that \(T_{\infty}\) is only influential in the transient state, since \(S_{T_{\infty}}^{\theta}\approx 0\) at steady state while \(k\) and \(b\) are not influential at the beginning, but they are influential in steady-state given that \(S_{k}^{\theta}\
eq 0\) and \(S_{b}^{\theta}\
eq 0\) at 425. Based on the information provided by Figure 6a, a less conductive fin will reduce \(\theta\) given that \(S_{k}^{\theta}(t)\geq 0\). On the other hand, Figure 6b,c show that longer fins exposed to higher room temperatures will reduce the fin's tip temperature. This is explained by the inverse relation of \(\theta\), \(S_{b}^{\theta}(t)\), and \(S_{T_{\infty}}^{\theta}(t)\). In the current application, reducing the fin's tip is desired because it means that more energy can be dissipated in the system before reaching the tip. \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & & \multicolumn{2}{c}{**NRMSE**} & \\  & \(\theta\) & \(S_{k}^{\theta}\) & \(T_{\infty}\) & \(S_{b}^{\theta}\) \\ \hline ZFEM & 0.003 & 0.011 & 0.002 & 0.003 \\ FD & 0.003 & 0.009 & 0.009 & 0.019 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: NRMSE of \(\theta\) and the selected sensitivities compared to the analytical solution. Figure 6: Dimensionless temperature sensitivity history at the fin tip to (**a**) thermal conductivity; (**b**) room temperature; (**c**) fin length. Solid black line, analytical sensitivity using analytical differentiation in Equation (23), yellow star, sensitivity using Transient Thermal ZFEM, blue square, sensitivity using 1% FD and Abaqus built-in element solutions, thin red dashed line, sensitivity using analytical differentiation for steady-state using Equation (24). Computational Efficiency of Transient Thermal ZFEM  
For the numerical solutions to the transient fin problem, the computational system used consisted of an Intel Xeon W-2265 at 3.50 GHz, 128 Gb of RAM, Abaqus 2021 Hot Fix 5, and Ubuntu 20.04 Long Term Support. In this specific case, the total CPU time of three pairs of sequential and independent simulation runs made with ZFEM and FD were conducted and averaged. The results showed that on average, ZFEM required 45% less total CPU time to solve the Transient Thermal ZFEM model and obtain sensitivities compared to using built-in Abaqus functions and FD. This contrasts with that reported in the ZFEM literature for more complicated physics, where 2.5 to 4 times longer computational times were obtained for equivalent real variable runs [71; 74; 87].