On the Partial Analytical Solution of the Kirchhoff Equation

Handle URI:
http://hdl.handle.net/10754/621975
Title:
On the Partial Analytical Solution of the Kirchhoff Equation
Authors:
Michels, Dominik L.; Lyakhov, Dmitry A. ( 0000-0001-8034-9568 ) ; Gerdt, Vladimir P.; Sobottka, Gerrit A.; Weber, Andreas G.
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.
KAUST Department:
Visual Computing Center (VCC)
Publisher:
Springer International Publishing
Journal:
Computer Algebra in Scientific Computing
Conference/Event name:
International Workshop on Computer Algebra in Scientific Computing
Issue Date:
Sep-2015
DOI:
10.1007/978-3-319-24021-3_24
ARXIV:
arXiv:1510.00521
Type:
Conference Paper
ISBN:
978-3-319-24021-3
Sponsors:
This work was partially supported by the Max Planck Center for Visual Computing and Communication (D.L.M.) as well as by the grant No. 13-01-00668 from the Russian Foundation for Basic Research (V.P.G.). The authors thank Robert Bryant for useful comments on the solution of PDE system (23)-(25) and the anonymous referees for their remarks and suggestions.
Additional Links:
http://link.springer.com/chapter/10.1007/978-3-319-24021-3_24; https://arxiv.org/abs/1510.00521
Appears in Collections:
Conference Papers

Full metadata record

DC FieldValue Language
dc.contributor.authorMichels, Dominik L.en
dc.contributor.authorLyakhov, Dmitry A.en
dc.contributor.authorGerdt, Vladimir P.en
dc.contributor.authorSobottka, Gerrit A.en
dc.contributor.authorWeber, Andreas G.en
dc.date.accessioned2016-12-08T08:50:17Z-
dc.date.available2016-12-08T08:50:17Z-
dc.date.issued2015-09-
dc.identifier.isbn978-3-319-24021-3-
dc.identifier.doi10.1007/978-3-319-24021-3_24-
dc.identifier.urihttp://hdl.handle.net/10754/621975-
dc.description.abstractWe 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.en
dc.description.sponsorshipThis work was partially supported by the Max Planck Center for Visual Computing and Communication (D.L.M.) as well as by the grant No. 13-01-00668 from the Russian Foundation for Basic Research (V.P.G.). The authors thank Robert Bryant for useful comments on the solution of PDE system (23)-(25) and the anonymous referees for their remarks and suggestions.en
dc.publisherSpringer International Publishingen
dc.relation.urlhttp://link.springer.com/chapter/10.1007/978-3-319-24021-3_24en
dc.relation.urlhttps://arxiv.org/abs/1510.00521en
dc.rightsThe final version of this paper is available from Springer at: http://link.springer.com/chapter/10.1007/978-3-319-24021-3_24en
dc.subjectDifferential Thomas Decomposition, Kirchhoff Rods, Lie Symmetry Analysis, Partial Analytical Solutions, Partial Differential Equations, Semi-analytical Integrationen
dc.titleOn the Partial Analytical Solution of the Kirchhoff Equationen
dc.typeConference Paperen
dc.contributor.departmentVisual Computing Center (VCC)en
dc.identifier.journalComputer Algebra in Scientific Computingen
dc.conference.dateSeptember 14 - 18, 2015en
dc.conference.nameInternational Workshop on Computer Algebra in Scientific Computingen
dc.conference.locationAachen, Germanyen
dc.eprint.versionPost-printen
dc.identifier.arxividarXiv:1510.00521-
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