On the Partial Analytical Solution of the Kirchhoff Equation

  • Dominik L. MichelsEmail author
  • Dmitry A. Lyakhov
  • Vladimir P. Gerdt
  • Gerrit A. Sobottka
  • Andreas G. Weber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9301)


We derive a combined analytical and numerical scheme to solve the (1+1)-dimensional differential Kirchhoff system. Here the object is to obtain an accurate as well as an efficient solution process. Purely numerical algorithms typically have the disadvantage that the quality of the solutions decreases enormously with increasing temporal step sizes, which results from the numerical stiffness of the underlying partial differential equations. To prevent that, we apply a differential Thomas decomposition and a Lie symmetry analysis to derive explicit analytical solutions to specific parts of the Kirchhoff system. These solutions are general and depend on arbitrary functions, which we set up according to the numerical solution of the remaining parts. In contrast to a purely numerical handling, this reduces the numerical solution space and prevents the system from becoming unstable. The differential Kirchhoff equation describes the dynamic equilibrium of one-dimensional continua, i.e. slender structures like fibers. We evaluate the advantage of our method by simulating a cilia carpet.


Differential Thomas Decomposition Kirchhoff Rods Lie Symmetry Analysis Partial Analytical Solutions Partial Differential Equations Semi-analytical Integration 


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  1. 1.
    Antman, S.S.: Nonlinear Problems of Elasticity. Applied Mathematical Sciences, vol. 107. Springer-Verlag, Berlin (1995)Google Scholar
  2. 2.
    Bächler, T., Gerdt, V.P., Lange-Hegermann, M., Robertz, D.: Algorithmic Thomas decomposition of algebraic and differential systems. J. Symb. Comput. 47, 1233–1266 (2012)Google Scholar
  3. 3.
    Blinkov, Y.A., Cid, C.F., Gerdt, V.P., Plesken, W., Robertz, D.: The maple package Janet: II. linear partial differential equations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2003, pp. 41–54. Institut für Informatik, Technische Universität München, Garching (2003)Google Scholar
  4. 4.
    Carminati, J., Vu, K.: Symbolic computation and differential equations: Lie symmetries. J. Symb. Comput. 29, 95–116 (2000)Google Scholar
  5. 5.
    Fikhtengol’ts, G.M.: A Course of Differential and Integral Calculus [in Russian], vol. 3. Nauka, Moscow (1966)Google Scholar
  6. 6.
    Fulford, G.R., Blake, J.R.: Muco-ciliary transport in the lung. J. Theor. Biol. 121(4), 381–402 (1986)Google Scholar
  7. 7.
    Gerdt, V.P.: Algebraically simple involutive differential systems and the Cauchy problem. J. Math. Sci. 168(3), 362–367 (2010)Google Scholar
  8. 8.
    Hartman, P., Nirenberg, L.: On spherical image maps whose Jacobians do not change sign. Am. J. Math. 81(4), 901–920 (1959)Google Scholar
  9. 9.
    Ibragimov, N.H. (ed.): CRS Handbook of Lie Group Analysis of Differential Equations. New Trends in Theoretical Developments and Computational Methods, vol. 3. CRC Press, Boca Raton (1996)Google Scholar
  10. 10.
    Krivoshapko, S., Ivanov, V.N.: Encyclopedia of Analytical Surfaces. Springer, Cham (2015)Google Scholar
  11. 11.
    Lange-Hegermann, M.: The Differential Dimension Polynomial for Characterizable Differential Ideals. arXiv:1401.5959 (2014)Google Scholar
  12. 12.
    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
  13. 13.
    Robertz, D.: Formal Algorithmic Elimination for PDEs. Lecture Notes in Mathematics, vol. 2121. Springer, Cham (2014)Google Scholar
  14. 14.
    Seiler, W.M.: Involution - The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics, vol. 24. Springer, Heildelberg (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Dominik L. Michels
    • 1
    Email author
  • Dmitry A. Lyakhov
    • 2
  • Vladimir P. Gerdt
    • 3
  • Gerrit A. Sobottka
    • 4
  • Andreas G. Weber
    • 4
  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA
  2. 2.Radiation Gaseous Dynamics Lab, A. V. Luikov Heat and Mass Transfer Institute of the National Academy of Sciences of BelarusMinskBelarus
  3. 3.Group of Algebraic and Quantum Computations, Joint Institute for Nuclear ResearchDubnaRussia
  4. 4.Multimedia, Simulation and Virtual Reality Group, Institute of Computer Science IIUniversity of BonnBonnGermany

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