Skip to main content
SpringerLink
Log in

Sensitivity analysis of deployable flexible space structures with a large number of design parameters

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

Abstract

The reliability and dynamic performance of deployable flexible space structures significantly depend on their key design parameters. Sensitivity analysis of these design parameters in the frame of multibody dynamics can serve as a powerful tool to evaluate and improve the dynamic performances of deployable space structures. Nevertheless, previous studies on the sensitivity analysis are mainly confined to planar multibody systems with a few design parameters. In this study, an efficient computational methodology is proposed to perform the sensitivity analysis of the complex deployable space structures with a large number of design parameters. Firstly, the analytical sensitivity analysis formulations of objective functions with the parameter-dependent integration bounds are deduced via the direct differentiation method and adjoint variable method. A checkpointing scheme is further introduced to assist the backward integration of the high-dimensional differential algebraic equations of the adjoint variables. The flexible beams in the deployable flexible space structure are described by the locking-free three-node spatial beam elements of absolute nodal coordinate formulation. Furthermore, a parallelized automatic differentiation algorithm is proposed to efficiently evaluate the complex partial derivatives in the sensitivity analysis formulations. Finally, four numerical examples are provided to validate the accuracy and efficiency of the proposed computational methodology.

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

Availability of data and materials

The datasets generated during the current study are available from the corresponding author on reasonable request.

References

  1. Mitsugi, J., Ando, K., Senbokuya, Y., Meguro, A.: Deployment analysis of large space antenna using flexible multibody dynamics simulation. Acta Astronaut. 47, 19–26 (2000)

    Article  Google Scholar 

  2. Li, P., Liu, C., Tian, Q., Hu, H., Song, Y.: Dynamics of a deployable mesh reflector of satellite antenna: parallel computation and deployment simulation1. J. Comput. Nonlinear Dyn. 11, 061005 (2016)

    Article  Google Scholar 

  3. Peng, Y., Zhao, Z., Zhou, M., He, J., Yang, J., Xiao, Y.: Flexible multibody model and the dynamics of the deployment of mesh antennas. J. Guid. Control Dyn. 40, 1499–1510 (2017)

    Article  Google Scholar 

  4. Li, K., Tian, Q., Shi, J., Liu, D.: Assembly dynamics of a large space modular satellite antenna. Mech. Mach. Theory. 142, 103601 (2019)

    Article  Google Scholar 

  5. Dopico, D., Sandu, A., Sandu, C., Zhu, Y.T.: Sensitivity analysis of multibody dynamic systems modeled by ODEs and DAEs. In: Proceedings of the ECCOMAS Thematic Conference on Multibody Dynamics. pp. 1–32. Zagreb, Croatia (2014)

  6. Dopico, D., González, F., Luaces, A., Saura, M., García-Vallejo, D.: Direct sensitivity analysis of multibody systems with holonomic and nonholonomic constraints via an index-3 augmented Lagrangian formulation with projections. Nonlinear Dyn. 93, 2039–2056 (2018)

    Article  Google Scholar 

  7. Dopico, D., Zhu, Y., Sandu, A., Sandu, C.: Direct and adjoint sensitivity analysis of ordinary differential equation multibody formulations. J. Comput. Nonlinear Dyn. 10, 011012 (2015)

    Article  Google Scholar 

  8. Bhalerao, K.D., Poursina, M., Anderson, K.S.: An efficient direct differentiation approach for sensitivity analysis of flexible multibody systems. Multibody Syst. Dyn. 23, 121–140 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Neto, M.A., Ambrósio, J.A.C., Leal, R.P.: Sensitivity analysis of flexible multibody systems using composite materials components. Int. J. Numer. Methods Eng. 77, 386–413 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Xiang, W., Yan, S., Wu, J., Niu, W.: Dynamic response and sensitivity analysis for mechanical systems with clearance joints and parameter uncertainties using Chebyshev polynomials method. Mech. Syst. Signal Process. 138, 106596 (2020)

    Article  Google Scholar 

  11. Tromme, E., Brüls, O., Emonds-Alt, J., Bruyneel, M., Virlez, G., Duysinx, P.: Discussion on the optimization problem formulation of flexible components in multibody systems. Struct. Multidiscip. Optim. 48, 1189–1206 (2013)

    Article  MathSciNet  Google Scholar 

  12. Ebrahimi, M., Butscher, A., Cheong, H., Iorio, F.: Design optimization of dynamic flexible multibody systems using the discrete adjoint variable method. Comput. Struct. 213, 82–99 (2019)

    Article  Google Scholar 

  13. Zhang, M., Peng, H., Song, N.: Semi-analytical sensitivity analysis approach for fully coupled optimization of flexible multibody systems. Mech. Mach. Theory. 159, 104256 (2021)

    Article  Google Scholar 

  14. Shourijeh, M.S.: Optimal Control and Multibody Dynamic Modelling of Human Musculoskeletal Systems. Ph.D. thesis, University of Waterloo (2013)

  15. Serban, R., Freeman, J.S.: Identification and identifiability of unknown parameters in multibody dynamic systems. Multibody Syst. Dyn. 5, 335–350 (2001)

    Article  MATH  Google Scholar 

  16. Ebrahimi, S., Kövecses, J.: Sensitivity analysis for estimation of inertial parameters of multibody mechanical systems. Mech. Syst. Signal Process. 24, 19–28 (2010)

    Article  MATH  Google Scholar 

  17. Burden, R.L., Faires, J.D.: Numerical Analysis. Brooks/Cole, Cengage Learning, Boston (2011)

  18. Schaffer, A.S.: On the adjoint formulation of design sensitivity analysis of multibody dynamics. Ph.D. thesis, University of Iowa (2005)

  19. Laflin, J.J., Anderson, K.S., Khan, I.M., Poursina, M.: Advances in the application of the divide-and-conquer algorithm to multibody system dynamics. J. Comput. Nonlinear Dyn. 9, 041003 (2014)

    Article  Google Scholar 

  20. Callejo, A., Dopico, D.: Direct sensitivity analysis of multibody systems: a vehicle dynamics benchmark. J. Comput. Nonlinear Dyn. 14, 021004 (2019)

    Article  Google Scholar 

  21. Nachbagauer, K., Oberpeilsteiner, S., Sherif, K., Steiner, W.: The use of the adjoint method for solving typical optimization problems in multibody dynamics. J. Comput. Nonlinear Dyn. 10, 061011 (2015)

    Article  Google Scholar 

  22. Lauß, T., Oberpeilsteiner, S., Steiner, W., Nachbagauer, K.: The discrete adjoint gradient computation for optimization problems in multibody dynamics. J. Comput. Nonlinear Dyn. 12, 031016 (2017)

    Article  MATH  Google Scholar 

  23. Serban, R., Hindmarsh, A.C.: CVODES: The Sensitivity-Enabled ODE Solver in SUNDIALS. In: Volume 6: 5th International Conference on Multibody Systems, Nonlinear Dynamics, and Control, Parts A, B, and C. pp. 257–269. ASMEDC, Long Beach, California, USA (2005)

  24. Hindmarsh, A.C., Brown, P.N., Grant, K.E., Lee, S.L., Serban, R., Shumaker, D.E., Woodward, C.S.: SUNDIALS: Suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31, 363–396 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sirkes, Z., Tziperman, E.: Finite difference of adjoint or adjoint of finite difference? Mon. Weather Rev. 125, 6 (1997)

    Article  Google Scholar 

  26. Pi, T., Zhang, Y., Chen, L.: First order sensitivity analysis of flexible multibody systems using absolute nodal coordinate formulation. Multibody Syst. Dyn. 27, 153–171 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Bauchau, O.A., Han, S., Mikkola, A., Matikainen, M.K., Gruber, P.: Experimental validation of flexible multibody dynamics beam formulations. Multibody Syst. Dyn. 34, 373–389 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Serban, R., Recuero, A.: Sensitivity analysis for hybrid systems and systems with memory. J. Comput. Nonlinear Dyn. 14, 091006 (2019)

    Article  Google Scholar 

  29. Corner, S., Sandu, C., Sandu, A.: Adjoint sensitivity analysis of hybrid multibody dynamical systems. Multibody Syst. Dyn. 49, 395–420 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sonneville, V., Brüls, O.: Sensitivity analysis for multibody systems formulated on a Lie group. Multibody Syst. Dyn. 31, 47–67 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Lauß, T., Oberpeilsteiner, S., Steiner, W., Nachbagauer, K.: The discrete adjoint method for parameter identification in multibody system dynamics. Multibody Syst. Dyn. 42, 397–410 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Azari Nejat, A., Moghadasi, A., Held, A.: Adjoint sensitivity analysis of flexible multibody systems in differential-algebraic form. Comput. Struct. 228, 106148 (2020)

    Article  Google Scholar 

  33. Wang, Q., Yu, W.: Sensitivity Analysis of Geometrically Exact Beam Theory (GEBT) Using the Adjoint Method with Hydra. In: 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference. American Institute of Aeronautics and Astronautics, Denver, Colorado (2011)

  34. Gutiérrez-López, M.D., Callejo, A., de Jalón, J.G.: Computation of independent sensitivities using Maggi’s formulation. In: Proceedings of the 2nd Joint International Conference on Multibody System Dynamics, Stuttgart, Germany (2012)

  35. Shabana, A.A., Xu, L.: Rotation-based finite elements: reference-configuration geometry and motion description. Acta Mech. Sin. 37, 105–126 (2021)

    Article  MathSciNet  Google Scholar 

  36. Martins, J.R.R.A., Hwang, J.T.: Review and unification of methods for computing derivatives of multidisciplinary computational models. AIAA J. 51, 2582–2599 (2013)

    Article  Google Scholar 

  37. Baydin, A.G., Pearlmutter, B.A., Radul, A.A., Siskind, J.M.: Automatic differentiation in machine learning: a survey. J. Mach. Learn. Res. 18, 5595–5637 (2017)

    MathSciNet  MATH  Google Scholar 

  38. Griewank, A., Walther, A.: Evaluating derivatives: principles and techniques of algorithmic differentiation. Society for Industrial and Applied Mathematics, Philadelphia, PA (2008)

  39. Yu, W., Blair, M.: DNAD, a simple tool for automatic differentiation of fortran codes using dual numbers. Comput. Phys. Commun. 184, 1446–1452 (2013)

    Article  MATH  Google Scholar 

  40. Straka, C.W.: ADF95: Tool for automatic differentiation of a FORTRAN code designed for large numbers of independent variables. Comput. Phys. Commun. 168, 123–139 (2005)

    Article  MATH  Google Scholar 

  41. Stamatiadis, S., Prosmiti, R., Farantos, S.C.: AUTO_DERIV: tool for automatic differentiation of a FORTRAN code. Comput. Phys. Commun. 127, 343–355 (2000)

    Article  MATH  Google Scholar 

  42. Bischof, C., Khademi, P., Mauer, A., Carle, A.: Adifor 2.0: automatic differentiation of Fortran 77 programs. IEEE Comput. Sci. Eng. 3, 18–32 (1996)

    Article  Google Scholar 

  43. Utke, J., Aachen, T.H., Fagan, M., Tallent, N., Strout, M., Hill, P.H.C., Fagan, M., Tallent, N., Univer, R.: OpenAD/F: a modular, open-source tool for automatic differentiation of Fortran codes. ACM Trans Math Softw. 34, 1–36 (2008)

    Article  MATH  Google Scholar 

  44. Callejo, A., Narayanan, S.H.K., García de Jalón, J., Norris, B.: Performance of automatic differentiation tools in the dynamic simulation of multibody systems. Adv. Eng. Softw. 73, 35–44 (2014)

    Article  Google Scholar 

  45. Kudruss, M., Manns, P., Kirches, C.: Efficient derivative evaluation for rigid-body dynamics based on recursive algorithms subject to kinematic and loop constraints. IEEE Control Syst. Lett. 3, 619–624 (2019)

    Article  MathSciNet  Google Scholar 

  46. Hoffait, S., Brüls, O., Granville, D., Cugnon, F., Kerschen, G.: Dynamic analysis of the self-locking phenomenon in tape-spring hinges. Acta Astronaut. 66, 1125–1132 (2010)

    Article  Google Scholar 

  47. Newmark, N.M.: A method of computation for structural dynamics. J. Eng. Mech. Div. 85, 67–94 (1959)

    Article  Google Scholar 

  48. Boopathy, K., Kennedy, G.J.: Parallel finite element framework for rotorcraft multibody dynamics and discrete adjoint sensitivities. AIAA J. 57, 3159–3172 (2019)

    Article  Google Scholar 

  49. Gavrea, B., Negrut, D., Potra, F.A.: The Newmark Integration Method for Simulation of Multibody Systems: Analytical Considerations. In: Design Engineering, Parts A and B. pp. 1079–1092. ASMEDC, Orlando, Florida, USA (2005)

  50. Nachbagauer, K., Gruber, P., Gerstmayr, J.: Structural and Continuum mechanics approaches for a 3D shear deformable ANCF beam finite element: application to static and linearized dynamic examples. J. Comput. Nonlinear Dyn. 8, 0004 (2013)

    Google Scholar 

  51. Gerstmayr, J., Shabana, A.A.: Efficient integration of the elastic forces and thin three-dimensional beam elements in the absolute nodal coordinate formulation. In: ECCOMAS Thematic Conference, Madrid, Spain, 21–24 June (2005)

  52. Tang, Y., Hu, H., Tian, Q.: A condensed algorithm for adaptive component mode synthesis of viscoelastic flexible multibody dynamics. Int. J. Numer. Methods Eng. 122, 609–637 (2021)

    Article  MathSciNet  Google Scholar 

  53. Lan, P., Tian, Q., Yu, Z.: A new absolute nodal coordinate formulation beam element with multilayer circular cross section. Acta Mech. Sin. 36, 82–96 (2020)

    Article  MathSciNet  Google Scholar 

  54. Maqueda, L.G., Shabana, A.A.: Poisson modes and general nonlinear constitutive models in the large displacement analysis of beams. Multibody Syst. Dyn. 18, 375–396 (2007)

    Article  MATH  Google Scholar 

  55. Shabana, A.A.: ANCF reference node for multibody system analysis. Proc. Inst. Mech. Eng. Part K J. Multi-Body Dyn. 229, 109–112 (2015)

    Google Scholar 

Download references

Acknowledgements

This research was supported in part by the National Natural Science Foundations of China under Grants 11722216 and 11832005. The research was also supported in part by the 111 China Project (B16003). This work was supported in part by the project from the Beijing Institute of Technology-Shanghai Academy of Spaceflight Technology joint laboratory.

Author information

Authors and Affiliations

  1. MOE Key Laboratory of Dynamics and Control of Flight Vehicle, School of Aerospace Engineering, Beijing Institute of Technology, Beijing, 100081, China

    Shuai Wang, Qiang Tian & Haiyan Hu

  2. Shanghai Institute of Aerospace System Engineering, Shanghai, 201108, China

    Junwei Shi & Lingbin Zeng

Authors
  1. Shuai Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qiang Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Haiyan Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Junwei Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Lingbin Zeng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qiang Tian.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, S., Tian, Q., Hu, H. et al. Sensitivity analysis of deployable flexible space structures with a large number of design parameters. Nonlinear Dyn 105, 2055–2079 (2021). https://doi.org/10.1007/s11071-021-06741-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-021-06741-4

Keywords

Search

Navigation