## 3 Sensitivity Analysis in Transient Heat Transfer Problems Using Transient Thermal ZFEM  
To obtain sensitivity information for transient heat transfer problems, we propose a methodology that combines the solution of the transient heat transfer equation using the finite element method with the Complex Taylor Series Expansion. We call this methodology Transient Thermal ZFEM, and the main concepts are presented in Figure 1b. To begin, all of the components in the FEM formulation, such as nodal coordinates, material properties, and boundary conditions, become complex-valued variables [79]. Then, to obtain sensitivities of \(T\) or any other output of the Transient Thermal ZFEM model with respect to the input parameters \(\phi\), different perturbation schemes for \(\phi\) should be followed depending on the kind of input, as follows:  
* **Constant inputs:** to obtain sensitivities with respect to the input variables in the model that are constant, the procedure presented in Equation (13) can be applied with only a small imaginary perturbation \(h\) applied to the variable of interest. For instance, to obtain sensitivities with respect to \(T_{\infty}\), we evaluated the model is evaluated with a value of \(T_{\infty}^{*}=T_{\infty}(1+hi)\). * **Geometry sensitivities:** to obtain sensitivities with respect to geometric shapes, perturbations to the geometry are required. In Transient Thermal ZFEM, the domain shape is represented by the nodal coordinates. As previously mentioned, the nodal coordinates are converted to complex-variable. For reference, Figure 2a shows a mesh where the nodal coordinates are in the plane \(xy\), and the out of plane represents the imaginary component \(Im\). Moreover, Figure 2b shows the representation of a planar element with real and imaginary coordinates. To obtain sensitivities with respect to the geometry, the imaginary component of the nodal coordinates should be perturbed according to the following guidelines: First, the direction of the perturbation should be selected. For instance, considering sensitivities with respect to the length \(b[\text{m}]\) in the fin problem (see Figure 2a), as \(b\) is parallel to the \(x\)-axis and its boundaries normal to that axis, the perturbation should be applied by translating the imaginary nodal coordinates along the direction of positive \(Im\) axis. Second, a fraction of the nodes (\(\lambda\)) should be selected in the mesh to be perturbed. As presented in Figure 2a, \(\lambda\) can take values from zero to one. The selected value for this fraction depends on the balance between precision and computational time. This parameter is problem-dependent and could be determined with convergence analysis. However, previous works have shown that considering a value of \(\lambda\) large enough to perturb at least two elements normal to the boundary is enough for most problems [60; 71; 86; 87]. Next, a perturbation function \(P(x)\) is selected. In Figure 2a a linear \(P(x)\) is considered. This function has a minimum value of zero and a maximum value of one. The maximum and minimum values of \(P(x)\) are located along the geometric boundaries of dimension \(\lambda b\). Finally, the fraction of the imaginary coordinates of the nodes are shifted in the selected imaginary direction, a magnitude \(h\times P(x)\), and the sensitivity is obtained following Equation (13). An example of this perturbation scheme is presented in Figure 2a, where the wireframe mesh presents the real components of the nodal coordinates and surface pointing out of the plane, representing the perturbation on the imaginary axis. * **Temperature-dependent properties:** to obtain sensitivities with respect to temperature-dependent properties, the properties should be defined for any given temperature; therefore, fitting or interpolating from experimental values is necessary. After fitting, the perturbation is applied by shifting \(h\) units in the imaginary axis of the resulting temperature-dependent curve, and the perturbed function is used to perform the calculations as usual.