On the General Analytical Solution of the Kinematic Cosserat Equations

Dominik L. Michels; Dmitry 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 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
Publisher	Springer Nature
Pages	367-380
Number of pages	14
ISBN (Print)	9783319456409
DOIs	https://doi.org/10.1007/978-3-319-45641-6_24
State	Published - Sep 9 2016

Bibliographical note

KAUST Repository Item: Exported on 2020-10-01
Acknowledgements: 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’ valuable comments are gratefully acknowledged.

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

https://repository.kaust.edu.sa/bitstream/10754/621970/1/GeneralAnalyticalSolutionOfKinematicCosseratEquations.pdf

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title = "On the General Analytical Solution of the Kinematic Cosserat Equations",

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.",

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

note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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.",

year = "2016",

month = sep,

day = "9",

doi = "10.1007/978-3-319-45641-6_24",

language = "English (US)",

isbn = "9783319456409",

pages = "367--380",

booktitle = "Computer Algebra in Scientific Computing",

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T1 - On the General Analytical Solution of the Kinematic Cosserat Equations

AU - Michels, Dominik L.

AU - Lyakhov, Dmitry

AU - Gerdt, Vladimir P.

AU - Hossain, Zahid

AU - Riedel-Kruse, Ingmar H.

AU - Weber, Andreas G.

N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledgements: 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’ valuable comments are gratefully acknowledged.

PY - 2016/9/9

Y1 - 2016/9/9

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.

UR - http://hdl.handle.net/10754/621970

UR - http://link.springer.com/chapter/10.1007%2F978-3-319-45641-6_24

UR - http://www.scopus.com/inward/record.url?scp=84988432391&partnerID=8YFLogxK

U2 - 10.1007/978-3-319-45641-6_24

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

M3 - Conference contribution

SN - 9783319456409

SP - 367

EP - 380

BT - Computer Algebra in Scientific Computing

PB - Springer Nature

ER -

On the General Analytical Solution of the Kinematic Cosserat Equations

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