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Optimisation of chaotically perturbed acoustic limit cycles
In an acoustic cavity with a heat source, the thermal energy of the heat source can be converted into acoustic energy, which may generate a loud oscillation.
If uncontrolled, these acoustic oscillations, also known as thermoacoustic instabilities, can cause mechanical vibrations, fatigue and structural failure.
The objective of manufacturers is to design stable thermoacoustic configurations.
In this paper, we propose a method to optimise a chaotically perturbed limit cycle in the bistable region of a subcritical bifurcation.
In this situation, traditional stability and sensitivity methods, such as eigenvalue and Floquet analysis, break down.
First, we propose covariant Lyapunov analysis and shadowing methods as tools to calculate the stability and sensitivity of chaotically perturbed acoustic limit cycles.
Second, covariant Lyapunov vector analysis is applied to an acoustic system with a heat source. The acoustic velocity at the heat source is chaotically perturbed to qualitatively mimic the effect of the turbulent hydrodynamic field. It is shown that the tangent space of the acoustic attractor is hyperbolic, which has a practical implication: the sensitivities of time--averaged cost functionals exist and can be robustly calculated by a shadowing method.
Third, we calculate the sensitivities of the time--averaged acoustic energy and Rayleigh index to small changes to the heat--source intensity and time delay. By embedding the sensitivities into a gradient--update routine, we suppress an existing chaotic acoustic oscillation by optimal design of the heat source.
The analysis and methods proposed enable the reduction of chaotic oscillations in thermoacoustic systems by optimal passive control. Because the theoretical framework is general, the techniques presented can be used in other unsteady deterministic multi-physics problems with virtually no modification
Sensitivity analysis of chaotic systems using a frequency-domain shadowing approach
We present a frequency-domain method for computing the sensitivities of
time-averaged quantities of chaotic systems with respect to input parameters.
Such sensitivities cannot be computed by conventional adjoint analysis tools,
because the presence of positive Lyapunov exponents leads to exponential growth
of the adjoint variables. The proposed method is based on the least-square
shadowing (LSS) approach [1], that formulates the evaluation of sensitivities
as an optimisation problem, thereby avoiding the exponential growth of the
solution. However, all existing formulations of LSS (and its variants) are in
the time domain and the computational cost scales with the number of positive
Lyapunov exponents. In the present paper, we reformulate the LSS method in the
Fourier space using harmonic balancing. The new method is tested on the
Kuramoto-Sivashinski system and the results match with those obtained using the
standard time-domain formulation. Although the cost of the direct solution is
independent of the number of positive Lyapunov exponents, storage and computing
requirements grow rapidly with the size of the system. To mitigate these
requirements, we propose a resolvent-based iterative approach that needs much
less storage. Application to the Kuramoto-Sivashinski system gave accurate
results with very low computational cost. The method is applicable to large
systems and paves the way for application of the resolvent-based shadowing
approach to turbulent flows. Further work is needed to assess its performance
and scalability
Ergodic Sensitivity Analysis of One-Dimensional Chaotic Maps
Sensitivity analysis in chaotic dynamical systems is a challenging task from
a computational point of view. In this work, we present a numerical
investigation of a novel approach, known as the space-split sensitivity or S3
algorithm. The S3 algorithm is an ergodic-averaging method to differentiate
statistics in ergodic, chaotic systems, rigorously based on the theory of
hyperbolic dynamics. We illustrate S3 on one-dimensional chaotic maps,
revealing its computational advantage over naive finite difference computations
of the same statistical response. In addition, we provide an intuitive
explanation of the key components of the S3 algorithm, including the density
gradient function.Comment: 20 pages, 17 figure