Arbitrary-order sensitivity analysis of frequency response functions using hypercomplex automatic differentiation and spectral finite elements

Accurately computing sensitivities of Frequency Response Functions (FRFs) is crucial for analyzing the dynamic behavior of structures by enabling quantifying the impact that variations in geometry, material properties, and boundary conditions have on their dynamic response. However, one of the primary challenges in calculating accurate sensitivities lies in the numerical differentiation required to estimate the sensitivities of the FRFs. This paper presents a new method called the Hypercomplex Spectral Finite Elements Method (HYPAD-SFEM). HYPAD-SFEM combines the HYPercomplex Automatic Differentiation method (HYPAD) with the Spectral Finite Elements Method (SFEM) to compute highly accurate arbitrary-order sensitivities of the FRFs. To demonstrate and verify the method's performance and accuracy, we analyzed a truss structure under a harmonic axial load and compared the results with analytical equations, Finite Differences (FD), and traditional Automatic Differentiation (AD). Excellent agreement was observed between the computed displacements and their sensitivities, considering material properties, geometry, and boundary conditions. The application of HYPAD-SFEM was then extended to a more complex problem by performing shape sensitivity analysis of the dynamic behavior of a phononic lattice. Again, excellent agreement was found between HYPAD, FD and AD. In general, the proposed HYPAD-SFEM ensures high accuracy independent of the perturbation step selection, alleviating FD’s fundamental issues. Moreover, HYPAD-SFEM delivers superior computational performance when compared with traditional AD. Hence, HYPAD-SFEM provides an effective approach for FRF sensitivity analysis, facilitating design optimization, parameter tuning, robustness analysis, and model updating and validation in structural dynamics.