## 4 Numerical Example: Transient Fin Problem
### Sensitivity Analysis of the Heat Flux History: Defining the Fin's Performance
The other parameters, e.g., b, \(\rho,C_{p},T_{\infty}\), and \(T_{0}\) influence the effectiveness less than the 1% compared to the sum of all relative sensitivities. This means that an increment or reduction of these five variables will not produce a significant change in the effectiveness of the fin. Figure 10: Thermal effectiveness sensitivity history in fin tip to **(a)** thermal conductivity; **(b)** room temperature; **(c)** fin length. Solid black line, analytical sensitivity using analytical differentiation in Equation (26), 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 (26). **(d)** Steady-state influence of fin parameters in \(\varepsilon\) based on relative sensitivities. In the pie chart, the missing parameters \((b,\rho,C_{p},T_{\infty},T_{0})\) have an influence of less than 1% compared to the sum of all relative sensitivities. Moreover, the \((-)\) in the pie sections implies an inverse relationship between the output variable and the input parameter, while a \((+)\) parameter has a direct relationship with the output. Similarly, from Figure 11d, it is possible to identify that \(\delta\), \(b\), \(h_{U}\), and \(k\) are the most influential parameters to modify the thermal efficiency. Therefore, increments in \(\delta\) and \(k\) will increase this index. For the efficiency, the weights are not equally split as in the case of the effectiveness. For instance, the fin length is almost twice more influential than \(h_{U}\), and \(k\). Moreover, the fin thickness is approximately 2.4 times less influential in \(\eta\) than the length of the fin. This means that larger geometries will reduce the thermal efficiency of the fin while enhancing the heat transfer with more conductive materials larger heat transfer coefficients are beneficial for \(\eta\). From these results, some guidelines for the fin's design can be stated: To cool a heated wall, it is preferred to have a fin with a low aspect ratio made of a material with high conductivity and low heat transfer coefficient. This boundary condition requirement is counterintuitive because larger heat transfer coefficients typically improve the heat exchange from the external surfaces to the room. However, this also means that the heat will be transferred only at the beginning of the fin. Therefore, according to the first-order sensitivity information, a shorter and thicker fin made of more conductive materials is preferred to increase the dissipation performance. It is worth highlighting that although we show the Transient Thermal ZFEM methodology in a simple thermal problem, the methodology is general. For instance, we have  
Figure 11: Thermal efficiency sensitivity history in fin tip to **(a)** thermal conductivity; **(b)** room temperature; **(c)** fin length. Solid black line, analytical sensitivity using analytical differentiation in Equation (27), 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 (27). **(d)** Steady-state influence of fin parameters in \(\eta\) based on relative sensitivities. In the pie chart, the missing parameters \((\rho,C_{p},T_{\infty},T_{0})\) have an influence of less than 1% compared to the sum of all relative sensitivities. Moreover, the \((-)\) in the pie sections implies an inverse relationship between the output variable and the input parameter while a \((+)\) parameter has the opposite relationship with the output. included an additional verification of the capabilities of the Transient Thermal ZFEM in the Supplementary Material for a 2D problem corresponding to a rotating disk under nonconstant initial temperature.