Skip to main content
SpringerLink
Log in

Adjoint-based limit cycle oscillation instability sensitivity and suppression

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Dynamical systems often exhibit limit cycle oscillations (LCOs), self-sustaining oscillations of limited amplitude. LCOs can be supercritical or subcritical. The supercritical response is benign, while the subcritical response can be bi-stable and exhibit a hysteretic response. Subcritical responses can be avoided in design optimization by enforcing LCO stability. However, many high-fidelity system models are computationally expensive to evaluate. Thus, there is a need for an efficient computational approach that can model instability and handle hundreds or thousands of design variables. To address this need, we propose a simple metric to determine the LCO stability using a fitted bifurcation diagram slope. We develop an adjoint-based formula to efficiently compute the stability derivative with respect to many design variables. To evaluate the stability derivative, we only need to compute the time-spectral adjoint equation three times, regardless of the number of design variables. The proposed adjoint method is verified with finite differences, achieving a five-digit agreement between the two approaches. We consider a stability-constrained LCO parameter optimization problem using an analytic model to demonstrate that the optimizer can suppress the instability. We also consider a more realistic LCO speed and stability-constrained airfoil problem that minimizes the normalized mass and stiffness. The proposed method could be extended to optimization problems with a partial differential equation (PDE)-based model, opening the door to other applications where high-fidelity models are needed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data Availability Statement

The numerical implementation of algorithms and results discussed in this work are available in the form of open-source Python software at DOI:10.5281/zenodo.5992080.

Notes

  1. https://github.com/SichengHe/LCO_stability_fit_data.

References

  1. Strogatz, S.H.: Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Perseus books, Reading, Massachusetts (1994)

    MATH  Google Scholar 

  2. He, S., Jonsson, E., Mader, C.A., Martins, J.R.R.A.: Aerodynamic shape optimization with time spectral flutter adjoint. In: 2019 AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference. American Institute of Aeronautics and Astronautics, San Diego, CA (2019a)

  3. Huang, D., Rokita, T., Friedmann, P.P.: Integrated aerothermoelastic analysis framework with application to skin panels. AIAA J. 56(11), 4562–4581 (2018)

    Article  Google Scholar 

  4. Jonsson, E., Riso, C., Lupp, C.A., Cesnik, C.E.S., Martins, J.R.R.A., Epureanu, B.I.: Flutter and post-flutter constraints in aircraft design optimization. Progress Aerospace Sci. 109(100), 537 (2019)

    Google Scholar 

  5. Li, H., Ekici, K.: Aeroelastic modeling of the AGARD 445.6 wing using the harmonic-balance-based one-shot method. AIAA J. 57(11), 4885–4902 (2019)

    Article  Google Scholar 

  6. Riso, C., Ghadami, A., Cesnik, C.E.S., Epureanu, B.I.: Data-driven forecasting of postflutter responses of geometrically nonlinear wings. AIAA J. 58(6), 2726–2736 (2020)

    Article  Google Scholar 

  7. Thomas, J.P., Dowell, E.H., Hall, K.C.: Nonlinear inviscid aerodynamic effects on transonic divergence, flutter, and limit-cycle oscillations. AIAA J. 40(4), 638–646 (2002)

    Article  Google Scholar 

  8. Grizzle, J., Abba, G., Plestan, F.: Asymptotically stable walking for biped robots: analysis via systems with impulse effects. IEEE Trans. Autom. Control 46(1), 51–64 (2001)

    Article  MATH  Google Scholar 

  9. Manchester, I.R., Tobenkin, M.M., Levashov, M., Tedrake, R.: Regions of attraction for hybrid limit cycles of walking robots. IFAC Proc. Volume. 44(1), 5801–5806 (2011)

    Article  Google Scholar 

  10. Shiriaev, A., Freidovich, L., Manchester, I.: Can we make a robot ballerina perform a pirouette? orbital stabilization of periodic motions of underactuated mechanical systems. Ann. Rev. Control 32(2), 200–211 (2008)

    Article  Google Scholar 

  11. Waugh, I.C., Kashinath, K., Juniper, M.P.: Matrix-free continuation of limit cycles and their bifurcations for a ducted premixed flame. J. Fluid Mech. 759, 1–27 (2014)

    Article  Google Scholar 

  12. Xu, M., Song, S., Sun, X., Chen, W., Zhang, W.: Machine learning for adjoint vector in aerodynamic shape optimization. (2020) arXiv preprint arXiv:2012.15730

  13. He, S., Jonsson, E., Martins, J.R.R.A.: Wing aerostructural optimization with time spectral limit-cycle oscillation adjoint. In: AIAA AVIATION 2022 Forum (2022)

  14. Lyu, Z., Xu, Z., Martins, J.R.R.A.: Benchmarking optimization algorithms for wing aerodynamic design optimization. In: Proceedings of the 8th International Conference on Computational Fluid Dynamics, Chengdu, Sichuan, China, iCCFD8-2014-0203 (2014)

  15. Martins, J.R.R.A., Ning, A.: Engineering Design Optimization. Cambridge University Press, Cambridge (2021)

  16. Jameson, A.: Aerodynamic design via control theory. J. Sci. Comput. 3(3), 233–260 (1988)

    Article  MATH  Google Scholar 

  17. Kenway, G.K.W., Kennedy, G.J., Martins, J.R.R.A.: Scalable parallel approach for high-fidelity steady-state aeroelastic analysis and adjoint derivative computations. AIAA J. 52(5), 935–951 (2014)

    Article  Google Scholar 

  18. Kenway, G.K.W., Mader, C.A., He, P., Martins, J.R.R.A.: Effective adjoint approaches for computational fluid dynamics. Progress Aeros. Sci. 110(100), 542 (2019)

    Google Scholar 

  19. Shi, Y., Mader, C.A., He, S., Halila, G.L.O., Martins, J.R.R.A.: Natural laminar-flow airfoil optimization design using a discrete adjoint approach. AIAA J. 58(11), 4702–4722 (2020)

    Article  Google Scholar 

  20. Thomas, J.P., Dowell, E.H.: Discrete adjoint approach for nonlinear unsteady aeroelastic design optimization. AIAA J. 57, 4368–4376 (2019)

    Article  Google Scholar 

  21. Lasagna, D.: Sensitivity analysis of chaotic systems using unstable periodic orbits. SIAM J. Appl. Dyn. Syst. 17(1), 547–580 (2018)

    Article  MATH  Google Scholar 

  22. Shimizu, YS., Fidkowski, K.: Output error estimation for chaotic flows. In: 46th AIAA Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics (2016)

  23. Wang, Q.: Forward and adjoint sensitivity computation of chaotic dynamical systems. J. Comput. Phys. 235, 1–13 (2013)

    Article  MATH  Google Scholar 

  24. Wilkins, A.K., Tidor, B., White, J., Barton, P.I.: Sensitivity analysis for oscillating dynamical systems. SIAM J. Sci. Comput. 31(4), 2706–2732 (2009)

    Article  MATH  Google Scholar 

  25. Chicone, C.: Ordinary Differential Equations with Applications. Springer, New York (2006)

    MATH  Google Scholar 

  26. Floquet, G.: Sur les équations différentielles linéaires à coefficients périodiques. Ann. Scientifiques de l’École normale supérieure 12, 47–88 (1883)

    Article  MATH  Google Scholar 

  27. Friedmann, P., Hammond, C.E., Woo, T.H.: Efficient numerical treatment of periodic systems with application to stability problems. Int. J. Numerical Methods Eng. 11(7), 1117–1136 (1977)

    Article  MATH  Google Scholar 

  28. Dimitriadis, G.: Continuation of higher-order harmonic balance solutions for nonlinear aeroelastic systems. J. Aircr. 45(2), 523–537 (2008)

    Article  Google Scholar 

  29. Shukla, H., Patil, M.J.: Nonlinear state feedback control design to eliminate subcritical limit cycle oscillations in aeroelastic systems. Nonlinear Dyn. 88(3), 1599–1614 (2017)

    Article  Google Scholar 

  30. Gai, G., Timme, S.: Nonlinear reduced-order modelling for limit-cycle oscillation analysis. Nonlinear Dyn. 84(2), 991–1009 (2016)

    Article  Google Scholar 

  31. Habib, G., Kerschen, G.: Suppression of limit cycle oscillations using the nonlinear tuned vibration absorber. Proc. Royal Soc. A Math. Phys. Eng. Sci. 471(2176), 20140976 (2015)

    MATH  Google Scholar 

  32. Kuznetsov, Y.A.: Elements of applied bifurcation theory. Applied mathematical sciences, Springer, New York, NY (2010)

    Google Scholar 

  33. Malher, A., Touzé, C., Doaré, O., Habib, G., Kerschen, G.: Flutter control of a two-degrees-of-freedom airfoil using a nonlinear tuned vibration absorber. J. Comput. Nonlinear Dyn. 12(5), 016–051 (2017)

    Google Scholar 

  34. Stanford, B., Beran, P.: Direct flutter and limit cycle computations of highly flexible wings for efficient analysis and optimization. J. Fluids Struct. 36, 111–123 (2013)

    Article  Google Scholar 

  35. Bauchau, O.A., Nikishkov, Y.G.: An implicit Floquet analysis for rotorcraft stability evaluation. J. Am. Helicopter Soc. 46(3), 200 (2001)

    Article  Google Scholar 

  36. Cao, Y., Li, S., Petzold, L., Serban, R.: Adjoint sensitivity analysis for differential-algebraic equations: the adjoint DAE system and its numerical solution. SIAM J. Sci. Comput. 24(3), 1076–1089 (2003)

    Article  MATH  Google Scholar 

  37. Seyranian, A., Solem, F., Pedersen, P.: Sensitivity analysis for the floquet multipliers. In: 2000 2nd International Conference. Control of Oscillations and Chaos. Proceedings (Cat. No.00TH8521), IEEE (2000)

  38. Shukla, H., Patil, M.J.: Controlling limit cycle oscillation amplitudes in nonlinear aeroelastic systems. J. Aircraft 54(5), 1921–1932 (2017)

    Article  Google Scholar 

  39. Riso, C., Cesnik, CES., Epureanu, BI., Teufel, P.: A post-flutter response constraint for gradient-based aircraft design optimization. In: AIAA Aviation 2021 Forum, American Institute of Aeronautics and Astronautics (2021)

  40. Stanford, B., Beran, P., Kurdi, M.: Adjoint sensitivities of time-periodic nonlinear structural dynamics via model reduction. Comput. Struct. 88(19–20), 1110–1123 (2010)

    Article  Google Scholar 

  41. He, S., Jonsson, E., Mader, CA., Martins, J.R.R.A.: A coupled Newton–Krylov time-spectral solver for wing flutter and LCO prediction. In: AIAA Aviation Forum, Dallas, TX (2019b)

  42. He, S., Jonsson, E., Mader, C.A., Martins, J.R.R.A.: Coupled Newton-Krylov time-spectral solver for flutter and limit cycle oscillation prediction. AIAA J. 59(6), 2214–2232 (2021)

    Article  Google Scholar 

  43. Virtanen, P., Gommers, R., Oliphant, T.E., Haberland, M., Reddy, T., Cournapeau, D., Burovski, E., Peterson, P., Weckesser, W., Bright, J., van der Walt, S.J., Brett, M., Wilson, J., Millman, K.J., Mayorov, N., Nelson, A.R.J., Jones, E., Kern, R., Larson, E., Carey, C.J., Polat, I., Feng, Y., Moore, E.W., VanderPlas, J., Laxalde, D., Perktold, J., Cimrman, R., Henriksen, I., Quintero, E.A., Harris, C.R., Archibald, A.M., Ribeiro, A.H., Pedregosa, F., van Mulbregt, P.: SciPy 1.0: fundamental algorithms for scientific computing in python. Nat. Methods 17(3), 261–272 (2020)

    Article  Google Scholar 

  44. Knoll, D.A., Keyes, D.E.: Jacobian-free Newton-Krylov methods: a survey of approaches and applications. J. Comput. Phys. 193(2), 357–397 (2004)

  45. Yildirim, A., Kenway, G.K.W., Mader, C.A., Martins, J.R.R.A.: A Jacobian-free approximate Newton-Krylov startup strategy for RANS simulations. J. Comput. Phys. 397(108), 741 (2019)

  46. Gill, P.E., Murray, W., Saunders, M.A.: SNOPT: an SQP algorithm for large-scale constrained optimization. SIAM J. Opt. 12(4), 979–1006 (2002)

    Article  MATH  Google Scholar 

  47. Wu, N., Kenway, G., Mader, C.A., Jasa, J., Martins, J.R.R.A.: PyOptSparse: a Python framework for large-scale constrained nonlinear optimization of sparse systems. J. Open Source Softw. 5(54), 2564 (2020)

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank Cristina Riso for the helpful technical discussions of the work.

Author information

Authors and Affiliations

  1. Aerospace Engineering, University of Michigan, François-Xavier Bagnoud Aerospace Building, 1320 Beal Avenue, Ann Arbor, MI, 48109-2140, United States

    Sicheng He, Eirikur Jonsson & Joaquim R. R. A. Martins

Authors
  1. Sicheng He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eirikur Jonsson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Joaquim R. R. A. Martins
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sicheng He.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, S., Jonsson, E. & Martins, J.R.R.A. Adjoint-based limit cycle oscillation instability sensitivity and suppression. Nonlinear Dyn 111, 3191–3205 (2023). https://doi.org/10.1007/s11071-022-07989-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07989-0

Keywords

Search

Navigation