## 5 Conclusions  
In this work, the complex Taylor series expansion method was integrated with the transient heat transfer FEM to create the Transient Thermal ZFEM methodology. This methodology enables the calculation of high precision sensitivities regarding geometrical parameters, boundary conditions, time, and material properties. Although the method takes advantage of complex algebra to obtain the sensitivities, we do not require complex variable solvers. Instead, the system of equations is solved using the Cauchy-Reimann matrix representation of a complex number. This transformation allows the solution of the FEM equations in any real-variable solver facilitating the implementation of the new method within commercial finite element software. We verified the Transient Thermal ZFEM method against the analytical solution for a fin with an insulated tip. Our methodology accurately computed temperature and temperature sensitivity history regarding specific heat, thermal conductivity, ambient temperature, and fin geometry. Good agreement with the reference solutions was obtained for the numerical example. Similarly, the thermal performance of the fin based on the fin's heat flux presented a good agreement with the analytical solution. The sensitivities using Transient Thermal ZFEM were compared with FD using built-in Abaqus elements. Overall, we found a reduction of 45% in the computational time compared with FD. Moreover, we demonstrated that the Transient Thermal ZFEM is step-size independent and mesh perturbation independent for the transient thermal linear problem. On the other hand, the time derivative obtained by Transient Thermal ZFEM is highly dependent on the number of increments of time but only limited by the computing hardware. Implementing the Transient Thermal ZFEM in finite element analysis programs will enable the sensitivity analysis of heat transfer models with a fraction of the computational cost and higher precision than sensitivity analysis using the traditional finite differentiation method. Moreover, Transient Thermal ZFEM can improve the convergence rates and computational times on gradient-based optimizations, especially in inverse heat transfer problems. These improvements are possible because the Transient Thermal ZFEM methodology allows one to obtain a more precise and computationally efficient calculation of partial derivatives than the traditional methods. Given the advantages of Transient Thermal ZFEM over the traditional sensitivity analysis methodologies, our methodology is especially attractive for complicated systems in alternative energies, petroleum engineering, chemical, agriculture, materials, and metallurgical industries. Accurate sensitivity information in transient thermal problems allows one to understand the system's thermal behavior better. For instance, in the numerical example, using Transient Thermal ZFEM, the variables can be sorted by their influence in the transient state and steady states. Then, it was possible to effectively filter the variables that affect the processes after reaching the steady-state, even if the parameters of the systems are temperature dependent. As future work, this method will be extended to calculate higher-order sensitivities and to explore other multiphysics problems such as thermo-mechanical, microstructural evolution, crystal plasticity, heat treatments, and thermo-electrical. Moreover, we will explore the use of sensitivities to perform a derivative-based approximation of functions that enable predictions in complicated problems or to evaluate models with output uncertainty related to the input parameters for uncertainty quantification. In addition, these sensitivities could guide semiempirical techniques that rely on simulations to recover basic parameters of the process as the Nusselt numbers, Biot numbers, or Prandtl numbers.