## 1 Introduction
However, in the hand-code differentiation, the sensitivities are obtained by hand procedures and typed directly in the code. These make the method error-prone, time-consuming to develop and code, and challenging for implicit models. In contrast, symbolic differentiation generates explicit symbolic expressions of explicit analytical models and uses computational symbolic packages to obtain the sensitivities automatically [53]. Automatic differentiation or algorithmic differentiation takes advantage of the chain rule and function composition to sequentially decompose a function in the root variables and its operations in the function. Then the recurrent evaluation of the decomposition in the chain rule of the function allows the algorithm to recover the sensitivity using forward, backward, or adjoint schemes. The formulation of the method makes the algorithm slow and memory demanding [52; 53; 54]. Another traditional method for obtaining the sensitivities of a model is the numerical FD. This technique has been widely applied in sensitivity analysis in many fields because it is easy to implement as it only requires multiple model evaluations with a small step size for the analyzed variable. Nevertheless, when the model becomes nonlinear, obtaining high precision sensitivities using FD is challenging because the method is subject to computational subtraction cancellation errors and truncation errors [55; 56]. In addition, the step size for each input to the model is problem dependent and usually requires a convergence analysis to select it for each of the inputs. Moreover, multiple evaluations of the model are required to calculate each parameter sensitivity of interest, making the technique disk memory intensive and time demanding [54]. An alternative approach for the numerical calculation of sensitivities is the Complex Taylor Series Expansion Method (CTSE) [57]. Unlike finite differencing, CTSE does not require the difference of two analyses. Instead, the derivatives are computed using a small perturbation along the imaginary axis of the input parameters, evaluating the function, extracting the imaginary part of the outputs, and dividing by the perturbation magnitude (more details shown in Section 2.2). In this context, since no subtraction operations are needed, no subtractive error is generated, and the computed sensitivity is highly accurate [56]. CTSE can be incorporated in partial differential equation solvers based on techniques such as the finite-differences method [58], the boundary element method [59], and the FEM [60; 61]. The combination between CTSE and analytical modeling has been successfully applied to transient heat transfer analysis. For instance, [PERSON] et al. [53] showed a sensitivity study for a 1D transient heat conduction problem. The authors compared analytical differentiation, FD, and CTSE methodologies for calculating sensitivities and concluded that CTSE provides certain advantages over other methods, such as near analytical accuracy, step size independency, reduction of computing time, and elimination of the subtractive cancelation problem. Similarly, CTSE has been applied to calculate sensitivities to solve inverse heat transfer problems using the finite-differences method. In this application, the sensitivities calculated using CTSE were used to guide the optimization algorithms to recover the values of temperature-dependent thermal properties and thermal boundary conditions [62; 63; 64]. Moreover, some authors have reported improved computational times and convergence rates in the minimization algorithms when CTSE is used over other numerical differentiation algorithms [65; 66; 67]. This work presents and verifies a methodology that integrates CTSE and the transient heat transfer FEM. We call this method the transient heat transfer complex-variable finite element method or Transient Thermal ZFEM.