On the Partial Analytical Solution of the Kirchhoff Equation

Dominik L. Michels, Dmitry Lyakhov, Vladimir P. Gerdt, Gerrit A. Sobottka, Andreas G. Weber

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 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.
Original languageEnglish (US)
Title of host publicationComputer Algebra in Scientific Computing
PublisherSpringer Nature
Pages322-333
Number of pages12
ISBN (Print)978-3-319-24021-3
DOIs
StatePublished - Nov 12 2015
Externally publishedYes

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