Adjoint-based sensitivity analysis of periodic orbits by the Fourier-Galerkin method

Sensitivity of periodic solutions of time-dependent partial differential equations is commonly computed using time-consuming direct and adjoint time integrations. Particular attention must be provided to the periodicity condition in order to obtain accurate results. Furthermore, stabilization techniques are required if the orbit is unstable. The present article aims to propose an alternative methodology to evaluate the sensitivity of periodic flows via the Fourier-Galerkin method. Unstable periodic orbits are directly computed and continued without any stabilizing technique. The stability of the periodic state is determined via Hill's method: the frequency-domain counterpart of Floquet analysis. Sensitivity maps, used for open-loop control and physical instability identification, are directly evaluated using the adjoint of the projected operator. Furthermore, we propose an efficient and robust iterative algorithm for the resolution of underlying linear systems. First of all, the new approach is applied on the Feigenbaum route to chaos in the Lorenz system. Second, the transition to a three-dimensional state in the periodic vortex-shedding past a circular cylinder is investigated. Such a flow case allows the validation of the sensitivity approach by a systematic comparison with previous results presented in the literature. Finally, the transition to a quasi-periodic state past two side-by-side cylinders is considered. These last two cases also served to test the performance of the proposed iterative algorithm. (C) 2021 Elsevier Inc. All rights reserved.

Automated summary

In this study, an alternative methodology using the Fourier-Galerkin method is proposed for evaluating the sensitivity of periodic flows, with the stability of periodic states determined via Hill's method. The approach is applied to the Feigenbaum route to chaos in the Lorenz system and the transition to a three-dimensional state in the periodic vortex-shedding past a circular cylinder, with systematic comparisons to validate the sensitivity approach. The proposed iterative algorithm is also tested in the transition to a quasi-periodic state past two side-by-side cylinders.