Arbitrary-Order Sensitivity Analysis in Phononic Metamaterials Using the Multicomplex Taylor Series Expansion Method Coupled With Bloch’s Theorem

Abstract

Phononic metamaterials (PMs) exhibit frequency ranges at which elastic waves are attenuated called band gaps. However, this phenomenon is highly sensitive to geometrical variations and the unit cell’s mechanical properties. It is useful to have accurate sensitivity information to identify the variables that produce the highest impact on band gaps and guide the design of PMs with a desired wave propagation behavior. Current methodologies for sensitivity analysis in PMs, such as the finite difference method (FDM), are computationally inefficient, subjected to subtraction cancelation errors, and their accuracy is highly dependent on the magnitude of the perturbation step size. In this study, we introduce a new computational methodology to perform parameter sensitivity in the dynamic behavior of PMs using the multicomplex Taylor series expansion (ZTSE) coupled with Bloch’s theorem. The methodology allows one to obtain arbitrary-order sensitivities with high accuracy. In contrast to FDM, this methodology is computationally more efficient, eliminates the step size selection issue, and is not subjected to subtractive cancelation errors. Also, we show how the method can be applied using real algebra solvers. We limit our analysis to linear undamped PMs. The methodology using ZTSE with Bloch’s theorem is presented in numerical examples for the diatomic lattice and a 2D square lattice, where we compute up to third-order sensitivities. The results show a maximum normalized root-mean-squared deviation in the order of 10−9 for the diatomic lattice and in the order of 10−8 for the 2D square lattice when compared to the analytical solutions.