On the General Analytical Solution of the Kinematic Cosserat Equations

  • Dominik L. Michels
  • Dmitry A. Lyakhov
  • Vladimir P. Gerdt
  • Zahid Hossain
  • Ingmar H. Riedel-Kruse
  • Andreas G. Weber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9890)

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.

Keywords

Cosserat rods Differential thomas decomposition Flagellated microswimmers General analytical solution Kinematic equations Lie symmetry analysis Stokes flow Symbolic computation 

References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences, vol. 107. Springer, New York (1995)MATHGoogle Scholar
  3. 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 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. 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)CrossRefGoogle Scholar
  6. 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)MathSciNetCrossRefMATHGoogle Scholar
  7. 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)CrossRefMATHGoogle Scholar
  8. 8.
    Carminati, J., Vu, K.T.: Symbolic computation and differential equations: Lie symmetries. J. Symb. Comput. 29, 95–116 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Cortez, R.: The method of the regularized stokeslet. SIAM J. Sci. Comput. 23(4), 1204–1225 (2001)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Cosserat, E., Cosserat, F.: Théorie des corps déformables. Hermann, Paris (1909)MATHGoogle Scholar
  11. 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)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Goldstein, R.: Green algae as model organisms for biological fluid dynamics. Ann. Rev. Fluid Mech. 47(1), 343–375 (2015)CrossRefGoogle Scholar
  13. 13.
    Granger, R.: Fluid Mechanics. Dover Classics of Science and Mathematics. Courier Corporation, Mineola (1995)MATHGoogle Scholar
  14. 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. 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. 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. 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. 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)CrossRefGoogle Scholar
  19. 19.
    Oliveri, F.: Lie symmetries of differential equations: classical results and recent contributions. Symmetry 2, 658–706 (2010)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Riedel-Kruse, I., Hilfinger, A., Howard, J., Jülicher, F.: How molecular motors shape the flagellar beat. HFSP J. 1(3), 192–208 (2007)CrossRefGoogle Scholar
  21. 21.
    Robertz, D.: Formal Algorithmic Elimination for PDEs. Lecture Notes in Mathematics, vol. 2121. Springer, Heidelberg (2014)MATHGoogle Scholar
  22. 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)CrossRefMATHGoogle Scholar
  23. 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)CrossRefMATHGoogle Scholar
  24. 24.
    Thomas, J.M.: Riquier’s existence theorems. Ann. Math. 30, 285–310 (1929). 30, 306–311 (1934)CrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  • Dominik L. Michels
    • 1
  • Dmitry A. Lyakhov
    • 2
  • Vladimir P. Gerdt
    • 3
  • Zahid Hossain
    • 1
    • 4
  • Ingmar H. Riedel-Kruse
    • 4
  • Andreas G. Weber
    • 5
  1. 1.Department of Computer ScienceStanford UniversityStanfordUSA
  2. 2.Visual Computing CenterKing Abdullah University of Science and TechnologyThuwalSaudi Arabia
  3. 3.Group of Algebraic and Quantum ComputationsJoint Institute for Nuclear ResearchDubnaRussia
  4. 4.Department of BioengineeringStanford UniversityStanfordUSA
  5. 5.Institute of Computer Science IIUniversity of BonnBonnGermany

Personalised recommendations