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Direct Differentiation of the Floating Frame of Reference Formulation via Invariants for Gradient-Based Design Optimization

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Optimal Design and Control of Multibody Systems (IUTAM 2022)

Part of the book series: IUTAM Bookseries ((IUTAMBOOK,volume 42))

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Abstract

This paper applies the analytical differentiation of the equations of motion of flexible multibody systems modeled with the floating frame of reference formulation based on invariants. This leads to an efficient sensitivity analysis with the direct differentiation method and enables efficient design optimization of flexible multibody systems. The main results are the analytical derivatives of the equation terms of the floating frame of reference formulation in terms of inertia shape integrals or invariants. The introduced sensitivity analysis is applied and verified with a slider–crank mechanism modeled with beam elements. After numerical studies to assess the speedup, design optimization is carried out using the lightweight design formulation.

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References

  1. Baumgarte, J.: Stabilization of constraints and integrals of motion in dynamical systems. Comput. Methods Appl. Mech. Eng. 1(1), 1–16 (1972). https://doi.org/10.1016/0045-7825(72)90018-7

    Article  MathSciNet  Google Scholar 

  2. Bestle, D.: Analyse und Optimierung von Mehrkörpersystemen. Springer, Berlin (1994). https://doi.org/10.1007/978-3-642-52352-6

  3. Bestle, D., Eberhard, P.: Analyzing and optimizing multibody systems. Mech. Struct. Mach. 20(1), 67–92 (1992). https://doi.org/10.1080/08905459208905161

    Article  Google Scholar 

  4. Boopathy, K.: Adjoint based design optimization of systems with time dependent physics and probabilistically modeled uncertainties. Ph.D. thesis, Georgia Institute of Technology (2020). https://smartech.gatech.edu/handle/1853/63658

  5. Chung, J., Hulbert, G.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(\alpha \) method. J. Appl. Mech. 60(2), 371–375 (1993). https://doi.org/10.1115/1.2900803

    Article  MathSciNet  Google Scholar 

  6. Dai, Y.H., Schittkowski, K.: A sequential quadratic programming algorithm with non-monotone line search. Pacific J. Optimiz. 4(2), 335–351 (2008)

    MathSciNet  Google Scholar 

  7. 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). https://doi.org/10.1016/j.compstruc.2018.12.007

    Article  Google Scholar 

  8. Etman, L., Van Campen, D., Schoofs, A.: Optimization of multibody systems using approximation concepts. In: IUTAM Symposium on Optimization of Mechanical Systems, Solid Mechanics and its Applications, vol. 43, pp. 81–88. Springer, Dordrecht (1996). https://doi.org/10.1007/978-94-009-0153-7_11

  9. Etman, L.F.P., van Campen, D.H., Schoofs, A.J.G.: Design optimization of multibody systems by sequential approximation. Multibody Sys.Dyn. 2(4), 393–415 (1998). https://doi.org/10.1023/A:1009780119839

    Article  MathSciNet  Google Scholar 

  10. Gufler, V., Wehrle, E., Achleitner, J., Vidoni, R.: Flexible multibody dynamics and sensitivity analysis in the design of a morphing leading edge for high-performance sailplanes. In: ECCOMAS Thematic Conference on Multibody Dynamics (2021). https://doi.org/10.3311/ECCOMASMBD2021-203. Budapest, Hungary (online)

  11. Gufler, V., Wehrle, E., Achleitner, J., Vidoni, R.: A semi-analytical approach to sensitivity analysis with flexible multibody dynamics of a morphing forward wing section. Multibody Sys. Dyn. 58(1), 1–20 (2023). https://doi.org/10.1007/s11044-023-09886-9

    Article  MathSciNet  Google Scholar 

  12. Gufler, V., Wehrle, E., Vidoni, R.: Multiphysical design optimization of multibody systems: application to a Tyrolean Weir cleaning mechanism. In: Niola, V., Gasparetto, A. (eds.) IFToMM ITALY 2020. MMS, vol. 91, pp. 459–467. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-55807-9_52

  13. Gufler, V., Wehrle, E., Vidoni, R.: Sensitivity analysis of flexible multibody dynamics with generalized-\(\alpha \) time integration and Baumgarte stabilization. In: Advances in Italian Mechanism Science, Mechanisms and Machine Science, vol. 122, pp. 147–155. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-10776-4_18

  14. Gufler, V., Wehrle, E., Vidoni, R.: Analytical sensitivity analysis of flexible multibody dynamics with index-1 differential-algebraic equations and Baumgarte stabilization. Int. J. Mech. Control 24(1), 3–14 (2023)

    Google Scholar 

  15. Gufler, V., Wehrle, E., Zwölfer, A.: A review of flexible multibody dynamics for gradient-based design optimization. Multibody Sys. Dyn. 53(4), 379–409 (2021). https://doi.org/10.1007/s11044-021-09802-z

    Article  MathSciNet  Google Scholar 

  16. Gufler, V., Zwölfer, A., Wehrle, E.: Analytical derivatives of flexible multibody dynamics with the floating frame of reference formulation. Multibody Sys. Dyn. (2022). https://doi.org/10.1007/s11044-022-09858-5

    Article  Google Scholar 

  17. Haug, E.J., Arora, J.S.: Design sensitivity analysis of elastic mechanical systems. Comput. Methods Appl. Mech. Eng. 15(1), 35–62 (1978). https://doi.org/10.1016/0045-7825(78)90004-X

    Article  Google Scholar 

  18. Held, A.: On design sensitivities in the structural analysis and optimization of flexible multibody systems. Multibody Sys. Dyn. 54(1), 53–74 (2021). https://doi.org/10.1007/s11044-021-09800-1

    Article  MathSciNet  Google Scholar 

  19. Orzechowski, G., Matikainen, M.K., Mikkola, A.M.: Inertia forces and shape integrals in the floating frame of reference formulation. Nonlinear Dyn. 88(3), 1953–1968 (2017). https://doi.org/10.1007/s11071-017-3355-y

    Article  Google Scholar 

  20. Perez, R., Jansen, P., Martins, J.: pyOpt: a Python-based object-oriented framework for nonlinear constrained optimization. Struct. Multidiscip. Optim. 45, 101–118 (2012). https://doi.org/10.1007/s00158-011-0666-3

    Article  MathSciNet  Google Scholar 

  21. Schittkowski, K.: NLPQLP: A Fortran Implementation of a Sequential Quadratic Programming Algorithm with Distributed and Non-Monotone Line Search, 4.2 edn. (2015). http://klaus-schittkowski.de/NLPQLP.pdf

  22. Shabana, A.A.: Flexible multibody dynamics: review of past and recent developments. Multibody Sys. Dyn. 1(2), 189–222 (1997). https://doi.org/10.1023/A:1009773505418

    Article  MathSciNet  Google Scholar 

  23. Shabana, A.A.: Dynamics of Multibody Systems, 5 edn. Cambridge University Press (2020). https://doi.org/10.1017/9781108757553

  24. Wehrle, E.: Design optimization of lightweight space-frame structures considering crashworthiness and parameter uncertainty. Ph.D. thesis, Lehrstuhl für Leichtbau, Technische Universität München (2015). https://mediatum.ub.tum.de/doc/1244214/1244214.pdf

  25. Wehrle, E.: DesOptPy: design optimization in python (2022). https://doi.org/10.5281/ZENODO.5908099. https://github.com/e-dub/DesOptPy

  26. Wehrle, E.: Structural design optimization of dynamic systems: optimal lightweight engineering design via a three-block solver scheme for mechanical analysis. In: Optimal Design and Control of Multibody Systems, IUTAM 2022, IUTAM Bookseries. Springer (2022). https://doi.org/10.1007/978-3-031-50000-8_2

  27. Wehrle, E., Gufler, V.: Lightweight engineering design of nonlinear dynamic systems with gradient-based structural design optimization. In: Proceedings of the Munich Symposium on Lightweight Design 2020, pp. 44–57. Springer, Berlin (2021). https://doi.org/10.1007/978-3-662-63143-0_5

  28. Wehrle, E., Gufler, V.: Analytical sensitivity analysis of dynamic problems with direct differentiation of generalized-\(\alpha \) time integration. (Submitted). https://doi.org/10.31224/osf.io/2mb6y

  29. 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). https://doi.org/10.1016/j.mechmachtheory.2021.104256

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Acknowledgments

This work is supported by the project CRC 2017 TN2091 doloMULTI Design of Lightweight Optimized structures and systems under MULTIdisciplinary considerations through integration of MULTIbody dynamics in a MULTIphysics framework funded by the Free University of Bozen-Bolzano.

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Authors and Affiliations

  1. Faculty of Engineering, Free University of Bozen-Bolzano, Universitätsplatz 1, 39100, Bozen, South Tyrol, Italy

    Veit Gufler & Erich Wehrle

  2. Applied Research and Technology, Collins Aerospace, Munich, Germany

    Erich Wehrle

  3. Department of Mechanical Engineering, Chair of Applied Mechanics, Technical University of Munich, TUM School of Engineering and Design, 85748, Garching bei München, Germany

    Veit Gufler & Andreas Zwölfer

Authors
  1. Veit Gufler
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  2. Erich Wehrle
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  3. Andreas Zwölfer
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Correspondence to Veit Gufler .

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Editors and Affiliations

  1. Institute of Mechanics and Ocean Engineering, Hamburg University of Technology, Hamburg, Germany

    Karin Nachbagauer

  2. Technical University of Munich, Institute for Advanced Studies TUM–IAS, Garching, Germany

    Alexander Held

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Gufler, V., Wehrle, E., Zwölfer, A. (2024). Direct Differentiation of the Floating Frame of Reference Formulation via Invariants for Gradient-Based Design Optimization. In: Nachbagauer, K., Held, A. (eds) Optimal Design and Control of Multibody Systems. IUTAM 2022. IUTAM Bookseries, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-031-50000-8_4

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  • DOI: https://doi.org/10.1007/978-3-031-50000-8_4

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