551 research outputs found
Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations
The adjoint method, among other sensitivity analysis methods, can fail in
chaotic dynamical systems. The result from these methods can be too large,
often by orders of magnitude, when the result is the derivative of a long time
averaged quantity. This failure is known to be caused by ill-conditioned
initial value problems. This paper overcomes this failure by replacing the
initial value problem with the well-conditioned "least squares shadowing (LSS)
problem". The LSS problem is then linearized in our sensitivity analysis
algorithm, which computes a derivative that converges to the derivative of the
infinitely long time average. We demonstrate our algorithm in several dynamical
systems exhibiting both periodic and chaotic oscillations.Comment: submitted to JCP in revised for
Least Squares Shadowing Sensitivity Analysis of a Modified Kuramoto-Sivashinsky Equation
Computational methods for sensitivity analysis are invaluable tools for
scientists and engineers investigating a wide range of physical phenomena.
However, many of these methods fail when applied to chaotic systems, such as
the Kuramoto-Sivashinsky (K-S) equation, which models a number of different
chaotic systems found in nature. The following paper discusses the application
of a new sensitivity analysis method developed by the authors to a modified K-S
equation. We find that least squares shadowing sensitivity analysis computes
accurate gradients for solutions corresponding to a wide range of system
parameters.Comment: 23 pages, 14 figures. Submitted to Chaos, Solitons and Fractals, in
revie
Stability, sensitivity and optimisation of chaotic acoustic oscillations
In an acoustic cavity with a heat source, such as a flame in a gas turbine,
the thermal energy of the heat source can be converted into acoustic energy,
which may generate a loud oscillation. If uncontrolled, these nonlinear
acoustic oscillations, also known as thermoacoustic instabilities, can cause
large vibrations up to structural failure. Numerical and experimental studies
showed that thermoacoustic oscillations can be chaotic. It is not yet known,
however, how to minimise such chaotic oscillations. We propose a strategy to
analyse and minimise chaotic acoustic oscillations, for which traditional
stability and sensitivity methods break down. We investigate the acoustics of a
nonlinear heat source in an acoustic resonator. First, we propose covariant
Lyapunov analysis as a tool to calculate the stability of chaotic acoustics
making connections with eigenvalue and Floquet analyses. We show that covariant
Lyapunov analysis is the most general flow stability tool. Second, covariant
Lyapunov vector analysis is applied to a chaotic system. The time-averaged
acoustic energy is investigated by varying the heat-source parameters.
Thermoacoustic systems can display both hyperbolic and non-hyperbolic chaos, as
well as discontinuities in the time-averaged acoustic energy. Third, we embed
sensitivities of the time-averaged acoustic energy in an optimisation routine.
This procedure achieves a significant reduction in acoustic energy and
identifies the bifurcations to chaos.
The analysis and methods proposed enable the reduction of chaotic
oscillations in thermoacoustic systems by optimal passive control. The
techniques presented can be used in other unsteady fluid-dynamics problems with
virtually no modification
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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
Least Squares Shadowing method for sensitivity analysis of differential equations
For a parameterized hyperbolic system the derivative
of the ergodic average to the parameter can be computed via
the Least Squares Shadowing algorithm (LSS). We assume that the sytem is
ergodic which means that depends only on (not on the
initial condition of the hyperbolic system). After discretizing this continuous
system using a fixed timestep, the algorithm solves a constrained least squares
problem and, from the solution to this problem, computes the desired derivative
. The purpose of this paper is to prove that the
value given by the LSS algorithm approaches the exact derivative when the
discretization timestep goes to and the timespan used to formulate the
least squares problem grows to infinity.Comment: 21 pages, this article complements arXiv:1304.3635 and analyzes LSS
for the case of continuous hyperbolic system
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