Solving transient heat transfer equations is required to understand the evolution of temperature and heat flux. This physics is highly dependent on the materials and environmental conditions. If these factors change with time and temperature, the process becomes nonlinear and numerical methods are required to predict the thermal response. Numerical tools are even more relevant when the number of parameters influencing the model is large, and it is necessary to isolate the most influential variables. In this regard, sensitivity analysis can be conducted to increase the process understanding and identify those variables. Here, we combine the complex-variable differentiation theory with the finite element formulation for transient heat transfer, allowing one to compute efficient and accurate first-order sensitivities. Although this approach takes advantage of complex algebra to calculate sensitivities, the method is implemented with real-variable solvers, facilitating the application within commercial software. We present this new methodology in a numerical example using the commercial software Abaqus. The calculation of sensitivities for the temperature and heat flux with respect to temperature-dependent material properties, boundary conditions, geometric parameters, and time are demonstrated. To highlight, the new sensitivity method showed step-size independence, mesh perturbation independence, and reduced computational time contrasting traditional sensitivity analysis methods such as finite differentiation.

A derivative-based uncertainty quantification (UQ) method called HYPAD-UQ that utilizes sensitivities from a computational model was developed to approximate the statistical moments and Sobol' indices of the model output. Hypercomplex automatic differentiation (HYPAD) was used as a means to obtain accurate high-order partial derivatives from computational models such as finite element analyses. These sensitivities are used to construct a surrogate model of the output using a Taylor series expansion and subsequently used to estimate statistical moments (mean, variance, skewness, and kurtosis) and Sobol' indices using algebraic expansions. The uncertainty in a transient linear heat transfer analysis was quantified with HYPAD-UQ using first-order through seventh-order partial derivatives with respect to seven random variables encompassing material properties, geometry, and boundary conditions. Random sampling of the analytical solution and the regression-based stochastic perturbation finite element method were also conducted to compare accuracy and computational cost. The results indicate that HYPAD-UQ has superior accuracy for the same computational effort compared to the regression-based stochastic perturbation finite element method. Sensitivities calculated with HYPAD can allow higher-order Taylor series expansions to be an effective and practical UQ method.

A complex-variable transient thermomechanical element was developed and used to compute highly accurate sensitivities of the thermal and stress time-dependent responses of a thick-walled cylinder with temperature-dependent material properties with respect to fifteen material, shape, and loading parameters using the complex Taylor series expansion method (CTSE). The analysis was conducted by employing a complex-variable user element within the Abaqus commercial finite element software, herein called ZFEM, using real and imaginary nodal degrees of freedom (DOF). The real coordinates denote the spatial coordinates, as typical, and the imaginary nodal DOF denote the perturbation of the geometry for shape sensitivities. Upon completion, the imaginary DOF hold the partial derivatives of the temperature and the displacements with respect to the parameter of interest. Sensitivities of stresses were obtained using the displacement sensitivities. The sensitivities were verified by a comparison with finite difference estimates using the built-in Abaqus elements. The numerical results demonstrate the superior accuracy, ease of use, and reduced run-time of ZFEM compared to the finite difference method. The conclusions show that ZFEM can be utilized to obtain time-dependent sensitivities with high accuracy and precision for transient thermomechanical analyses that are computationally more efficient in comparison to finite difference methods.