On the Partial Analytical Solution of the Kirchhoff Equation

Dominik L. Michels,Stanford University,
Dmitry A. Lyakhov,National Academy of Sciences of Belarus,
Vladimir P. Gerdt,Joint Institute for Nuclear Research,
Gerrit A. Sobottka,University of Bonn,
Andreas G. Weber,University of Bonn.
In Proceedings of Computer Algebra in Scientific Computing, CASC 2015, Pages 320-331,
Lecture Notes in Computer Science, Springer, Sept. 2015.


author = {Dominik L.~Michels and Dmitry A.~Lyakhov and Vladimir P.~Gerdt and Gerrit A.~Sobottka and Andreas G.~Weber},
title = {On the Partial Analytical Solution to the Kirchhoff Equation},
pages = {320--331},
booktitle = {Computer Algebra in Scientific Computing – CASC 2015},
series = {Lecture Notes in Computer Science},
volume = {9301},
year= {2015},
publisher= {Springer},
address = {Berlin, Heidelberg},
isbn = {978-3-319-24020-6}


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.

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