Skip to main content
Springer Nature Link
Log in

On the General Analytical Solution of the Kinematic Cosserat Equations

  • Conference paper
  • First Online:
Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

Included in the following conference series:

Abstract

Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    The equalities in (11) are easily verifiable with Maple (cf. [16]), Sect. 3.5.

  2. 2.

    The computation time has been measured on a machine with an Intel(R) Xeon E5 with 3.5 GHz and 32 GB DDR-RAM.

  3. 3.

    It is easy to check with Maple that the right-hand sides of (19) satisfy (5) for arbitrary \({\varvec{q}}(s,t)\) if one takes (9a) and (9b) into account.

  4. 4.

    We do not explicitly write out the resulting equations here for brevity. A construction of a hybrid semi-analytical, semi-numerical solver is also described in our recent contribution [17].

  5. 5.

    Since at this point, the functions \(\phi _\epsilon \), \(G_\epsilon \), and \(B_\epsilon \) only depend on the norm of their arguments, we change the notation according to this.

  6. 6.

    The simulations illustrated in Figs. 2 and 3 can be carried out in real-time on a machine with an Intel(R) Xeon E5 with 3.5 GHz and 32 GB DDR-RAM.

References

  1. Ainley, J., Durkin, S., Embid, R., Boindala, P., Cortez, R.: The method of images for regularized stokeslets. J. Comput. Phys. 227, 4600–4616 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences, vol. 107. Springer, New York (1995)

    MATH  Google Scholar 

  3. Bächler, T., Gerdt, V., Langer-Hegermann, M., Robertz, D.: Algorithmic Thomas decomposition of algebraic and differential systems. J. Symbolic Comput. 47, 1233–1266 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  4. Blinkov, Y., Cid, C., Gerdt, V., Plesken, W., Robertz, D.: The Maple package Janet: II. linear partial differential equations. In: Ganzha, V., Mayr, E., Vorozhtsov, E. (eds.) Computer Algebra in Scientific Computing, CASC 2003, pp. 41–54. Springer, Heidelberg (2003)

    Google Scholar 

  5. 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)

    Article  Google Scholar 

  6. Butcher, J., Carminati, J., Vu, K.T.: A comparative study of some computer algebra packages which determine the Lie point symmetries of differential equations. Comput. Phys. Commun. 155, 92–114 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cao, D.Q., Tucker, R.W.: Nonlinear dynamics of elatic rods using the Cosserat theory: modelling and simulation. Int. J. Solids Struct. 45, 460–477 (2008)

    Article  MATH  Google Scholar 

  8. Carminati, J., Vu, K.T.: Symbolic computation and differential equations: Lie symmetries. J. Symb. Comput. 29, 95–116 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cortez, R.: The method of the regularized stokeslet. SIAM J. Sci. Comput. 23(4), 1204–1225 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  10. Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann, Paris (1909)

    MATH  Google Scholar 

  11. Elgeti, J., Winkler, R., Gompper, G.: Physics of microswimmers–single particle motion and collective behavior: a review. Rep. Prog. Phys. 78(5), 056601 (2015)

    Article  MathSciNet  Google Scholar 

  12. Goldstein, R.: Green algae as model organisms for biological fluid dynamics. Ann. Rev. Fluid Mech. 47(1), 343–375 (2015)

    Article  Google Scholar 

  13. Granger, R.: Fluid Mechanics. Dover Classics of Science and Mathematics. Courier Corporation, Mineola (1995)

    MATH  Google Scholar 

  14. Hereman, W.: Review of symbolic software for Lie symmetry analysis. CRC handbook of Lie group analysis of differential equations. In: Ibragimov, N.H. (ed.) New Trends in Theoretical Developments and Computational Methods, pp. 367–413. CRC Press, Boca Raton (1996)

    Google Scholar 

  15. Lang, H., Linn, J., Arnold, M.: Multibody dynamics simulation of geometrically exact Cosserat rods. In: Berichte des Fraunhofer ITWM, vol. 209 (2011)

    Google Scholar 

  16. 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, Heidelberg (2014)

    Google Scholar 

  17. Michels, D., Lyakhov, D., Gerdt, V., Sobottka, G., Weber, A.: On the partial analytical solution to the Kirchhoff equation. In: Gerdt, V., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing, CASC 2015, pp. 320–331. Springer, Heidelberg (2015)

    Google Scholar 

  18. Michels, D., Mueller, P., Sobottka, G.: A physically based approach to the accurate simulation of stiff fibers and stiff fiber meshes. Comput. Graph. 53B, 136–146 (2015)

    Article  Google Scholar 

  19. Oliveri, F.: Lie symmetries of differential equations: classical results and recent contributions. Symmetry 2, 658–706 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Riedel-Kruse, I., Hilfinger, A., Howard, J., Jülicher, F.: How molecular motors shape the flagellar beat. HFSP J. 1(3), 192–208 (2007)

    Article  Google Scholar 

  21. Robertz, D.: Formal Algorithmic Elimination for PDEs. Lecture Notes in Mathematics, vol. 2121. Springer, Heidelberg (2014)

    MATH  Google Scholar 

  22. Filho, R.T.M., Figueiredo, A.: [SADE] a Maple package for the symmetry analysis of differential equations. Comput. Phys. Commun. 182, 467–476 (2011)

    Article  MATH  Google Scholar 

  23. Seiler, W.M.: Involution: The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics. Springer, Heidelberg (2010)

    Book  MATH  Google Scholar 

  24. Thomas, J.M.: Riquier’s existence theorems. Ann. Math. 30, 285–310 (1929). 30, 306–311 (1934)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

This work has been partially supported by the Max Planck Society (FKZ-01IMC01/FKZ-01IM10001), the Russian Foundation for Basic Research (16-01-00080), and a BioX Stanford Interdisciplinary Graduate Fellowship. The reviewers’ valuable comments are gratefully acknowledged.

Author information

Authors and Affiliations

  1. Department of Computer Science, Stanford University, 353 Serra Mall, Stanford, CA, 94305, USA

    Dominik L. Michels & Zahid Hossain

  2. Visual Computing Center, King Abdullah University of Science and Technology, Al Khawarizmi Building, Thuwal, 23955-6900, Saudi Arabia

    Dmitry A. Lyakhov

  3. Group of Algebraic and Quantum Computations, Joint Institute for Nuclear Research, Joliot-Curie 6, 141980, Dubna, Moscow Region, Russia

    Vladimir P. Gerdt

  4. Department of Bioengineering, Stanford University, 318 Campus Drive, Stanford, CA, 94305, USA

    Zahid Hossain & Ingmar H. Riedel-Kruse

  5. Institute of Computer Science II, University of Bonn, Friedrich-Ebert-Allee 144, 53113, Bonn, Germany

    Andreas G. Weber

Authors
  1. Dominik L. Michels
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitry A. Lyakhov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vladimir P. Gerdt
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zahid Hossain
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Ingmar H. Riedel-Kruse
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Andreas G. Weber
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dominik L. Michels .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Michels, D.L., Lyakhov, D.A., Gerdt, V.P., Hossain, Z., Riedel-Kruse, I.H., Weber, A.G. (2016). On the General Analytical Solution of the Kinematic Cosserat Equations. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-45641-6_24

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-45640-9

  • Online ISBN: 978-3-319-45641-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation