Advertisement

Symbolic-Numeric Integration of the Dynamical Cosserat Equations

  • Dmitry A. Lyakhov
  • Vladimir P. Gerdt
  • Andreas G. Weber
  • Dominik L. Michels
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10490)

Abstract

We devise a symbolic-numeric approach to the integration of the dynamical part of the Cosserat equations, a system of nonlinear partial differential equations describing the mechanical behavior of slender structures, like fibers and rods. This is based on our previous results on the construction of a closed form general solution to the kinematic part of the Cosserat system. Our approach combines methods of numerical exponential integration and symbolic integration of the intermediate system of nonlinear ordinary differential equations describing the dynamics of one of the arbitrary vector-functions in the general solution of the kinematic part in terms of the module of the twist vector-function. We present an experimental comparison with the well-established generalized \(\alpha \)-method illustrating the computational efficiency of our approach for problems in structural mechanics.

Keywords

Analytical solution Cosserat rods Dynamic equations Exponential integration Generalized \(\alpha \)-method Kinematic equations Symbolic computation 

Notes

Acknowledgements

The authors appreciate the insightful comments of the anonymous referees. This work has been partially supported by the King Abdullah University of Science and Technology (KAUST baseline funding), the Max Planck Center for Visual Computing and Communication (MPC-VCC) funded by Stanford University and the Federal Ministry of Education and Research of the Federal Republic of Germany (BMBF grants FKZ-01IMC01 and FKZ-01IM10001), the Russian Foundation for Basic Research (grant 16-01-00080) and the Ministry of Education and Science of the Russian Federation (agreement 02.a03.21.0008).

References

  1. 1.
    Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences, vol. 107. Springer, Heidelberg (1995). doi: 10.1007/0-387-27649-1 MATHGoogle Scholar
  2. 2.
    Boyer, F., De Nayer, G., Leroyer, A., Visonneau, M.: Geometrically exact Kirchhoff beam theory: application to cable dynamics. J. Comput. Nonlinear Dyn. 6(4), 041004 (2011)CrossRefGoogle Scholar
  3. 3.
    Cao, D.Q., Tucker, R.W.: Nonlinear dynamics of elastic rods using the Cosserat theory: modelling and simulation. Int. J. Solids Struct. 45, 460–477 (2008)CrossRefMATHGoogle Scholar
  4. 4.
    Chung, J., Hulbert, G.M.: A time integration algorithm for structural dynamics with improved numerical dissipation: the generalized-\(\alpha \) method. J. Appl. Mech. 60(2), 371–375 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann, Paris (1909)MATHGoogle Scholar
  6. 6.
    Hilber, H.M., Hughes, T.J.R., Taylor, R.L.: Improved numerical dissipation for time integration algorithms in structural dynamics. Earthq. Eng. Struct. Dyn. 5, 283–292 (1977)CrossRefGoogle Scholar
  7. 7.
    Hilfinger, A.: Dynamics of cilia and flagella. Ph.D. thesis, Technische Universität Dresden (2006)Google Scholar
  8. 8.
    Hochbruck, M., Ostermann, A.: Exponential integrators. Acta Numerica 19, 209–286 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Ibragimov, N.H.: A Practical Course in Differential Equations and Mathematical Modelling. Classical and New Methods. Nonlinear Mathematical Models. Symmetry and Invariance Principles. Higher Education Press/World Scientific, Beijing (2009)CrossRefGoogle Scholar
  10. 10.
    Lang, H., Linn, J., Arnold, M.: Multibody Dynamics Simulation of Geometrically Exact Cosserat Rods. Berichte des Fraunhofer ITWM, vol. 209. Fraunhofer, Munich (2011)MATHGoogle Scholar
  11. 11.
    Luo, A.C.J.: Nonlinear Deformable-Body Dynamics. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-12136-4 CrossRefMATHGoogle Scholar
  12. 12.
    Michels, D.L., Desbrun, M.: A semi-analytical approach to molecular dynamics. J. Comput. Phys. 303, 336–354 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Michels, D.L., Lyakhov, D.A., Gerdt, V.P., Sobottka, G.A., Weber, A.G.: Lie symmetry analysis for cosserat rods. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2014. LNCS, vol. 8660, pp. 324–334. Springer, Cham (2014). doi: 10.1007/978-3-319-10515-4_23 Google Scholar
  14. 14.
    Michels, D.L., Lyakhov, D.A., Gerdt, V.P., Hossain, Z., Riedel-Kruse, I.H., Weber, A.G.: On the general analytical solution of the Kinematic Cosserat equations. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 367–380. Springer, Cham (2016). doi: 10.1007/978-3-319-45641-6_24 CrossRefGoogle Scholar
  15. 15.
    Michels, D.L., Mueller, J.P.T., Sobottka, G.: A physically based approach to the accurate simulation of stiff fibers and stiff fiber meshes. Comput. Graph. 53B, 136–146 (2015)CrossRefGoogle Scholar
  16. 16.
    Michels, D.L., Sobottka, G.A., Weber, A.G.: Exponential integrators for stiff elastodynamic problems. ACM Trans. Graph. 33, 7:1–7:20 (2014)CrossRefMATHGoogle Scholar
  17. 17.
    Olver, P.J.: Applications of Lie Groups to Differential Equations. Graduate Texts in Mathematics, vol. 107, 2nd edn. Springer, Heidelberg (1993). doi: 10.1007/978-1-4684-0274-2 CrossRefMATHGoogle Scholar
  18. 18.
    Rubin, M.B.: Cosserat Theories: Shells, Rods and Points. Kluwer Academic Publishers, Dordrecht (2000)CrossRefMATHGoogle Scholar
  19. 19.
    Seiler, W.M.: Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics, vol. 24. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-01287-7 CrossRefMATHGoogle Scholar
  20. 20.
    Sobottka, G.A., Lay, T., Weber, A.G.: Stable integration of the dynamic Cosserat equations with application to hair modeling. J. WSCG 16, 73–80 (2008)Google Scholar
  21. 21.
    Wood, W.L., Bossak, M., Zienkiewicz, O.C.: An alpha modification of Newmarks method. Int. J. Numer. Methods Eng. 15, 1562–1566 (1981)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Dmitry A. Lyakhov
    • 1
  • Vladimir P. Gerdt
    • 3
    • 4
  • Andreas G. Weber
    • 5
  • Dominik L. Michels
    • 1
    • 2
  1. 1.Visual Computing CenterKing Abdullah University of Science and TechnologyThuwalKingdom of Saudi Arabia
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA
  3. 3.Laboratory of Information TechnologiesJoint Institute for Nuclear ResearchDubnaRussian Federation
  4. 4.Peoples’ Friendship University of RussiaMoscowRussian Federation
  5. 5.Institute of Computer Science IIUniversity of BonnBonnGermany

Personalised recommendations