Simplified Least Squares Shadowing sensitivity analysis for chaotic ODEs and PDEs

Chater, Mario; Ni, Angxiu; Wang, Qiqi

doi:10.1016/j.jcp.2016.10.035

Cited by 4 publications

(4 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…By deploying the shadowing lemma [120,121], the sensitivity of timeaveraged cost functionals was developed by Wang [122]. Such a technique was made more computationally feasible by the least-square shadowing method [123][124][125][126] and its improved versions [127][128][129]. The discussion of Larsson and Wang [130] is particularly relevant to fluid dynamics simulations.…”

Section: Non-reacting Flowsmentioning

confidence: 99%

Adjoint Methods as Design Tools in Thermoacoustics

Magri

2019

Applied Mechanics Reviews

View full text Add to dashboard Cite

In a thermoacoustic system, such as a flame in a combustor, heat release oscillations couple with acoustic pressure oscillations. If the heat release is sufficiently in phase with the pressure, these oscillations can grow, sometimes with catastrophic consequences. Thermoacoustic instabilities are still one of the most challenging problems faced by gas turbine and rocket motor manufacturers. Thermoacoustic systems are characterized by many parameters to which the stability may be extremely sensitive. However, often only few oscillation modes are unstable. Existing techniques examine how a change in one parameter affects all (calculated) oscillation modes, whether unstable or not. Adjoint techniques turn this around: They accurately and cheaply compute how each oscillation mode is affected by changes in all parameters. In a system with a million parameters, they calculate gradients a million times faster than finite difference methods. This review paper provides: (i) the methodology and theory of stability and adjoint analysis in thermoacoustics, which is characterized by degenerate and nondegenerate nonlinear eigenvalue problems; (ii) physical insight in the thermoacoustic spectrum, and its exceptional points; (iii) practical applications of adjoint sensitivity analysis to passive control of existing oscillations, and prevention of oscillations with ad hoc design modifications; (iv) accurate and efficient algorithms to perform uncertainty quantification of the stability calculations; (v) adjoint-based methods for optimization to suppress instabilities by placing acoustic dampers, and prevent instabilities by design modifications in the combustor's geometry; (vi) a methodology to gain physical insight in the stability mechanisms of thermoacoustic instability (intrinsic sensitivity); and (vii) in nonlinear periodic oscillations, the prediction of the amplitude of limit cycles with weakly nonlinear analysis, and the theoretical framework to calculate the sensitivity to design parameters of limit cycles with adjoint Floquet analysis. To show the robustness and versatility of adjoint methods, examples of applications are provided for different acoustic and flame models, both in longitudinal and annular combustors, with deterministic and probabilistic approaches. The successful application of adjoint sensitivity analysis to thermoacoustics opens up new possibilities for physical understanding, control and optimization to design safer, quieter, and cleaner aero-engines. The versatile methods proposed can be applied to other multiphysical and multiscale problems, such as fluid–structure interaction, with virtually no conceptual modification.

show abstract

Section: Non-reacting Flowsmentioning

confidence: 99%

Adjoint Methods as Design Tools in Thermoacoustics

Magri

2019

Applied Mechanics Reviews

View full text Add to dashboard Cite

show abstract

“…This is natural due to the fact that turbulence is a chaotic system so that any linear sensitivity evaluation (be it forward or backward) is bound to diverge. [20][21][22] This limits in practice the length of our optimal control window, which we take here to be 200 s.…”

Section: Appendix C: Gradient Verificationmentioning

confidence: 99%

Optimal dynamic induction control of a pair of inline wind turbines

Yılmaz

Meyers

2018

Physics of Fluids

View full text Add to dashboard Cite

We study dynamic induction control for mitigating the wake losses of a pair of inline wind turbines. In order to explore control strategies that account for unsteady interactions with the flow, we employ optimal control and adjoint-based optimization in combination with large-eddy simulations. The turbines are represented with an actuator line model. We consider a simple uniform inflow case with two NREL 5 MW turbines spaced 5 diameters apart and find that optimal control leads to 25% gains compared to standard Maximum-Power-Point Tracking (MPPT). It is further found that only the control dynamics of the first turbine are changed, improving wake mixing, while the second turbine controller remains close to the MPPT control. We further synthesize the optimal generator torque and blade pitch controls of the first turbine into a signal that can be periodically used as an open-loop controller, with a Strouhal number of 0.38, while realizing the same gains as the original optimal control signal. Further analysis of the improved wake mixing resulting from the open-loop signal reveals periodic shedding of a three-vortex ring system, which interacts and merges downstream of the first turbine, increasing entrainment of high-speed momentum into the wake. The sensitivity of the open-loop signal to inlet turbulence levels and turbine spacing is also investigated. At low to medium turbulence levels, the control remains effective, while at higher levels, the coherence of the vortex rings degrades too fast for them to remain effective.

show abstract

“…We estimate the sensitivity of x3 = lim T →∞ 1 T T 0 x 3 t dt with respect to θ at θ = 28, which is approximately 0.981, an average of 50 finite difference estimates using different initial conditions shown in [4]. Theorem 3.1 in chapter IV in [17] shows that the invariant measure π σ of the stochastic Lorenz equation weakly converges to π 0 as σ → 0 in discrete version, but the convergence speed is not known.…”

Section: Numerical Investigationmentioning

confidence: 99%

“…The black line shows the sensitivity of the ODE, a constant 0.981 shown in [4]. For the SDEs, we use the new PS estimator and similarly simulate longer T for smaller σ.…”

Section: Numerical Investigationmentioning

confidence: 99%

Importance Sampling for Pathwise Sensitivity of Stochastic Chaotic Systems

Fang

Giles

2020

Preprint

View full text Add to dashboard Cite

This paper proposes a new pathwise sensitivity estimator for chaotic SDEs. By introducing a spring term between the original and perturbated SDEs, we derive a new estimator by importance sampling. The variance of the new estimator increases only linearly in time T, compared with the exponential increase of the standard pathwise estimator. We compare our estimator with the Malliavin estimator and extend both of them to the Multilevel Monte Carlo method, which further improves the computational efficiency. Finally, we also consider using this estimator for the SDE with small volatility to approximate the sensitivities of the invariant measure of chaotic ODEs. Furthermore, Richardson-Romberg extrapolation on the volatility parameter gives a more accurate and efficient estimator. Numerical experiments support our analysis.

show abstract

Simplified Least Squares Shadowing sensitivity analysis for chaotic ODEs and PDEs

Cited by 4 publications

References 19 publications

Adjoint Methods as Design Tools in Thermoacoustics

Adjoint Methods as Design Tools in Thermoacoustics

Optimal dynamic induction control of a pair of inline wind turbines

Importance Sampling for Pathwise Sensitivity of Stochastic Chaotic Systems

Contact Info

Product

Resources

About