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On the General Analytical Solution of the Kinematic Cosserat Equations

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Computer Algebra in Scientific Computing (CASC 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9890))

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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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Notes

  1. 1.

    The equalities in (11) are easily verifiable with Maple (cf. [16]), Sect. 3.5.

  2. 2.

    The computation time has been measured on a machine with an Intel(R) Xeon E5 with 3.5 GHz and 32 GB DDR-RAM.

  3. 3.

    It is easy to check with Maple that the right-hand sides of (19) satisfy (5) for arbitrary \({\varvec{q}}(s,t)\) if one takes (9a) and (9b) into account.

  4. 4.

    We do not explicitly write out the resulting equations here for brevity. A construction of a hybrid semi-analytical, semi-numerical solver is also described in our recent contribution [17].

  5. 5.

    Since at this point, the functions \(\phi _\epsilon \), \(G_\epsilon \), and \(B_\epsilon \) only depend on the norm of their arguments, we change the notation according to this.

  6. 6.

    The simulations illustrated in Figs. 2 and 3 can be carried out in real-time on a machine with an Intel(R) Xeon E5 with 3.5 GHz and 32 GB DDR-RAM.

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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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Authors and Affiliations

  1. Department of Computer Science, Stanford University, 353 Serra Mall, Stanford, CA, 94305, USA

    Dominik L. Michels & Zahid Hossain

  2. Visual Computing Center, King Abdullah University of Science and Technology, Al Khawarizmi Building, Thuwal, 23955-6900, Saudi Arabia

    Dmitry A. Lyakhov

  3. Group of Algebraic and Quantum Computations, Joint Institute for Nuclear Research, Joliot-Curie 6, 141980, Dubna, Moscow Region, Russia

    Vladimir P. Gerdt

  4. Department of Bioengineering, Stanford University, 318 Campus Drive, Stanford, CA, 94305, USA

    Zahid Hossain & Ingmar H. Riedel-Kruse

  5. Institute of Computer Science II, University of Bonn, Friedrich-Ebert-Allee 144, 53113, Bonn, Germany

    Andreas G. Weber

Authors
  1. Dominik L. Michels
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  2. Dmitry A. Lyakhov
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  3. Vladimir P. Gerdt
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  4. Zahid Hossain
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  5. Ingmar H. Riedel-Kruse
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  6. Andreas G. Weber
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Correspondence to Dominik L. Michels .

Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies, Joint Institute of Nuclear Research, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Wolfram Koepf

  3. Institut für Mathematik, Universität Kassel, Kassel, Germany

    Werner M. Seiler

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, Russia

    Evgenii V. Vorozhtsov

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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: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2016. Lecture Notes in Computer Science(), vol 9890. Springer, Cham. https://doi.org/10.1007/978-3-319-45641-6_24

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  • DOI: https://doi.org/10.1007/978-3-319-45641-6_24

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