Sensitivity Analysis in Structural Dynamics Using Hypercomplex Automatic Differentiation and Spectral Finite Elements

The dynamic behavior of structural systems is significantly influenced by variations in structural parameters and service conditions, making it crucial to understand their impact for ensuring structural integrity and safety. Sensitivity analysis provides a means to quantify the influence of parameter variations on the dynamic response of structures. This dissertation presents a novel methodology that combines the Hypercomplex Automatic Differentiation method (HYPAD) with the Spectral Finite Element Method (SFEM), known as the Hypercomplex Spectral Finite Element Method (HYPAD-SFEM). The objective is to compute highly accurate arbitrary-order sensitivities in structural dynamics, enabling a deeper understanding of how variations in structural parameters and service conditions impact the dynamic response of structures.

The dissertation is structured as a compendium of four publications focused on three specific aims. Firstly, the influence of parameter variations in the free vibration of structures is investigated, where the natural modes and frequencies of vibration are analyzed using generalized eigenvalue problems. Secondly, the influence of parameter variations in the forced vibration of structures is examined by obtaining sensitivities of the frequency response function (FRF). Lastly, the influence of parameter variations in the transient dynamic response of structures is explored through explicit dynamic simulations. Subsequently, the methodology is used to assess the detection capabilities of waveguide Structural Health Monitoring (SHM), providing an efficient and accurate framework for estimating the probability of detection (POD) of SHM systems.

Overall, this dissertation extends the application of HYPAD to complex-valued problems, eigenvalue problems, frequency-domain solutions, explicit time integration schemes, and spectral elements.