On the general analytical solution of the kinematic cosserat equations

Dominik L. Michels; Dmitry A. Lyakhov; Vladimir P. Gerdt; Zahid Hossain; Ingmar H. Riedel-Kruse; Andreas G. Weber

doi:10.1007/978-3-319-45641-6_24

On the general analytical solution of the kinematic cosserat equations

Dominik L. Michels, Dmitry A. Lyakhov, Vladimir P. Gerdt, Zahid Hossain, Ingmar H. Riedel-Kruse, Andreas G. Weber

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

2 Scopus citations

Abstract

Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.

Original language	English (US)
Title of host publication	Computer Algebra in Scientific Computing - 18th International Workshop, CASC 2016, Proceedings
Editors	Evgenii V. Vorozhtsov, Vladimir P. Gerdt, Wolfram Koepf, Werner M. Seiler
Publisher	Springer-Verlag
Pages	367-380
Number of pages	14
ISBN (Print)	9783319456409
DOIs	https://doi.org/10.1007/978-3-319-45641-6_24
State	Published - 2016
Externally published	Yes
Event	18th International Workshop on Computer Algebra in Scientific Computing, CASC 2016 - Bucharest, Romania Duration: Sep 19 2016 → Sep 23 2016

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	9890 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	18th International Workshop on Computer Algebra in Scientific Computing, CASC 2016
Country/Territory	Romania
City	Bucharest
Period	9/19/16 → 9/23/16

Keywords

Cosserat rods
Differential thomas decomposition
Flagellated microswimmers
General analytical solution
Kinematic equations
Lie symmetry analysis
Stokes flow
Symbolic computation

ASJC Scopus subject areas

Theoretical Computer Science
Computer Science(all)

Access to Document

10.1007/978-3-319-45641-6_24

Cite this

Michels, D. L., Lyakhov, D. A., Gerdt, V. P., Hossain, Z., Riedel-Kruse, I. H., & Weber, A. G. (2016). On the general analytical solution of the kinematic cosserat equations. In E. V. Vorozhtsov, V. P. Gerdt, W. Koepf, & W. M. Seiler (Eds.), Computer Algebra in Scientific Computing - 18th International Workshop, CASC 2016, Proceedings (pp. 367-380). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9890 LNCS). Springer-Verlag. https://doi.org/10.1007/978-3-319-45641-6_24

On the general analytical solution of the kinematic cosserat equations. / Michels, Dominik L.; Lyakhov, Dmitry A.; Gerdt, Vladimir P. et al.
Computer Algebra in Scientific Computing - 18th International Workshop, CASC 2016, Proceedings. ed. / Evgenii V. Vorozhtsov; Vladimir P. Gerdt; Wolfram Koepf; Werner M. Seiler. Springer-Verlag, 2016. p. 367-380 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 9890 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

Michels, DL, Lyakhov, DA, Gerdt, VP, Hossain, Z, Riedel-Kruse, IH & Weber, AG 2016, On the general analytical solution of the kinematic cosserat equations. in EV Vorozhtsov, VP Gerdt, W Koepf & WM Seiler (eds), Computer Algebra in Scientific Computing - 18th International Workshop, CASC 2016, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 9890 LNCS, Springer-Verlag, pp. 367-380, 18th International Workshop on Computer Algebra in Scientific Computing, CASC 2016, Bucharest, Romania, 9/19/16. https://doi.org/10.1007/978-3-319-45641-6_24

Michels DL, Lyakhov DA, Gerdt VP, Hossain Z, Riedel-Kruse IH, Weber AG. On the general analytical solution of the kinematic cosserat equations. In Vorozhtsov EV, Gerdt VP, Koepf W, Seiler WM, editors, Computer Algebra in Scientific Computing - 18th International Workshop, CASC 2016, Proceedings. Springer-Verlag. 2016. p. 367-380. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-319-45641-6_24

Michels, Dominik L. ; Lyakhov, Dmitry A. ; Gerdt, Vladimir P. et al. / On the general analytical solution of the kinematic cosserat equations. Computer Algebra in Scientific Computing - 18th International Workshop, CASC 2016, Proceedings. editor / Evgenii V. Vorozhtsov ; Vladimir P. Gerdt ; Wolfram Koepf ; Werner M. Seiler. Springer-Verlag, 2016. pp. 367-380 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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abstract = "Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.",

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author = "Michels, {Dominik L.} and Lyakhov, {Dmitry A.} and Gerdt, {Vladimir P.} and Zahid Hossain and Riedel-Kruse, {Ingmar H.} and Weber, {Andreas G.}",

note = "Funding Information: This work has been partially supported by the Max Planck Society (FKZ-01IMC01/FKZ-01IM10001), the Russian Foundation for Basic Research (16-01-00080), and a BioX Stanford Interdisciplinary Graduate Fellowship. The reviewers{\textquoteright} valuable comments are gratefully acknowledged. Publisher Copyright: {\textcopyright} Springer International Publishing AG 2016.; 18th International Workshop on Computer Algebra in Scientific Computing, CASC 2016 ; Conference date: 19-09-2016 Through 23-09-2016",

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AU - Weber, Andreas G.

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N2 - Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.

AB - Based on a Lie symmetry analysis, we construct a closed form solution to the kinematic part of the (partial differential) Cosserat equations describing the mechanical behavior of elastic rods. The solution depends on two arbitrary analytical vector functions and is analytical everywhere except a certain domain of the independent variables in which one of the arbitrary vector functions satisfies a simple explicitly given algebraic relation. As our main theoretical result, in addition to the construction of the solution, we proof its generality. Based on this observation, a hybrid semi-analytical solver for highly viscous two-way coupled fluid-rod problems is developed which allows for the interactive high-fidelity simulations of flagellated microswimmers as a result of a substantial reduction of the numerical stiffness.

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On the general analytical solution of the kinematic cosserat equations

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