Abstract
A direct differentiation method and an adjoint variable method are proposed for the efficient semi-analytical evaluation of the sensitivities of multibody systems formulated in a matrix Lie group framework. These methods rely on the linearization of the equations of motion and/or of the time integration procedure. The simpler structure of the equations of motion in the Lie group formalism appears as an advantage for that purpose. Lie bracket contributions and the non-linearity of the exponential map need to be taken into account in the sensitivity algorithms. Nevertheless, essential characteristics of formulations of the direct differentiation method and the adjoint variable method on linear spaces are recovered. Some implementation issues are discussed and two relevant examples illustrate the properties of these methods.
Similar content being viewed by others
References
Brüls, O., Lemaire, E., Duysinx, P., Eberhard, P.: Optimization of multibody systems and their structural components. In: Blajer, W., Arczewski, J., Fraczek, K., Wojtyra, M. (eds.) Multibody Dynamics: Computational Methods and Applications. Computational Methods in Applied Sciences, vol. 23, pp. 49–68. Springer, Berlin (2011)
Bottasso, C.L., Croce, A., Ghezzi, L., Faure, P.: On the solution of inverse dynamics and trajectory optimization problems for multibody systems. Multibody Syst. Dyn. 11, 1–22 (2004)
Eberhard, P., Bischof, C.: Automatic differentiation of numerical integration algorithms. Math. Comput. 68, 717–731 (1999)
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)
Ambrósio, J.A.C., Neto, M.A., Leal, R.P.: Optimization of a complex flexible multibody systems with composite materials. Multibody Syst. Dyn. 18, 117–144 (2007)
Géradin, M., Cardona, A.: Flexible Multibody Dynamics: A Finite Element Approach. Wiley, Chichester (2001)
Brüls, O., Cardona, A.: On the use of Lie group time integrators in multibody dynamics. J. Comput. Nonlinear Dyn. 5(3), 031002 (2010)
Brüls, O., Cardona, A., Arnold, M.: Lie group generalized-α time integration of constrained flexible multibody systems. Mech. Mach. Theory 48, 121–137 (2012)
Paraskevopoulos, E., Natsiavas, S.: A new look into the kinematics and dynamics of finite rigid body rotations using Lie group theory. Int. J. Solids Struct. 50, 57–72 (2013)
Brüls, O., Eberhard, P.: Sensitivity analysis for dynamic mechanical systems with finite rotations. Int. J. Numer. Methods Eng. 74(13), 1897–1927 (2008)
Haftka, R.T., Adelman, H.M.: Recent developments in structural sensitivity analysis. Struct. Optim. 1, 137–151 (1989)
Haug, E.J., Choi, K.K., Komkov, V.: Design Sensitivity Analysis of Structural Systems. Academic Press, Orlando (1986)
Kowalczyk, P.: Design sensitivity analysis in large deformation elasto-plastic and elasto-viscoplastic problems. Int. J. Numer. Methods Eng. 66, 1234–1270 (2006)
Michaleris, P., Tortorelli, D.A., Vidal, C.A.: Tangent operators and design sensitivity formulations for transient non-linear coupled problems with applications to elastoplasticity. Int. J. Numer. Methods Eng. 37, 2471–2499 (1994)
Tortorelli, D.A.: Sensitivity analysis for non-linear constrained elastoplastic systems. Int. J. Numer. Methods Eng. 33, 1643–1660 (1992)
Bestle, D., Eberhard, P.: Analyzing and optimizing multibody systems. Mech. Struct. Mach. 20, 67–92 (1992)
Bestle, D., Seybold, J.: Sensitivity analysis of constrained multibody systems. Arch. Appl. Mech. 62, 181–190 (1992)
Dias, J.M.P., Pereira, M.S.: Sensitivity analysis of rigid-flexible multibody systems. Multibody Syst. Dyn. 1(3), 303–322 (1997)
Wasfy, T.M., Noor, K.: Modeling and sensitivity analysis of multibody systems using new solid, shell and beam elements. Comput. Methods Appl. Mech. Eng. 138, 187–211 (1996)
Holm, D.D., Schmah, T., Stoica, C., Ellis, D.C.P.: Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions. Oxford Texts in Applied and Engineering Mathematics. Oxford University Press, London (2009)
Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. Academic Press, San Diego (2003)
Brüls, O., Arnold, M., Cardona, A.: Two Lie group formulations for dynamic multibody systems with large rotations. In: Proceedings of the IDETC/MSNDC Conference, Washington DC, USA, August 2011
Udwadia, F.E., Phohomsiri, P.: Explicit equations of motion for constrained mechanical systems with singular mass matrices and applications to multi-body dynamics. Proc. R. Soc. Lond. Ser. A 462, 2097–2117 (2006)
Schaffer, A.S.: On the adjoint formulation of design sensitivity analysis of multibody dynamics. PhD thesis, The University of Iowa (2005)
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)
Acknowledgements
Valentin Sonneville would like to acknowledge the Belgian National Fund for Scientific research (FRIA) for its financial support. The authors also thank Martin Arnold for useful suggestions about the treatment of an acceleration-dependent functional using the adjoint variable method.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sonneville, V., Brüls, O. Sensitivity analysis for multibody systems formulated on a Lie group. Multibody Syst Dyn 31, 47–67 (2014). https://doi.org/10.1007/s11044-013-9345-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11044-013-9345-z