International Workshop on Computer Algebra in Scientific Computing

Computer Algebra in Scientific Computing pp 322-333 | Cite as

On the Partial Analytical Solution of the Kirchhoff Equation

  • Dominik L. Michels
  • 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)

Abstract

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.

Keywords

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

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Dominik L. Michels
    • 1
  • 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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