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.

The calculation of accurate arbitrary-order sensitivities of eigenvalues and eigenvectors is crucial for structural analysis applications, including topology optimization, system identification, finite element model updating, damage detection, and fault diagnosis. Current approaches to obtaining sensitivities for eigenvalues and eigenvectors lack generality, are complicated to implement, prone to numerical errors, and are computationally expensive. In this work, a novel methodology is introduced that uses hypercomplex automatic differentiation (HYPAD) and semi-analytical expressions to obtain arbitrary-order sensitivities for eigenfrequency problems. The new methodology exhibits no sign of truncation nor subtractive cancellation errors regardless of the order of the sensitivity, it is general, and can obtain any high-order sensitivities with the simplicity of first-order computations. A numerical example is presented to verify the accuracy of the method, where the free vibration of a homogeneous cantilever beam is studied. For this problem, up to third-order sensitivities of the eigenvalues and eigenvectors with respect to the material and geometrical parameters were obtained, considering the cases of close and distinct eigenvalues. The results were verified using analytical equations, showing excellent agreement for the eigenvalues and the eigenvectors. The new method promises to facilitate the computation of sensitivities for eigenfrequency problems into routine practice and commercial software.

Derivative enhanced Gaussian Processes (DEGPs) can significantly improve surrogate model accuracy over standard Gaussian process (GP) formulations by incorporating derivative information. However, standard implementations scale poorly with dimension, limiting their use in high dimensional engineering problems. JetGP is a Python framework that unifies existing derivative enhanced GP methodologies into a single library and extends them to support arbitrary order derivative information. The library implements four complementary formulations: standard derivative enhanced Gaussian processes (DEGP), directional DEGP (DDEGP), generalized directional DEGP (GDDEGP), and weighted DEGP (WDEGP). By unifying these approaches in a consistent interface with robust numerical implementations, JetGP enables practitioners to balance predictive accuracy and computational efficiency for high dimensional optimization, uncertainty quantification, and sensitivity analysis in engineering design.