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Wiener Chaos Expansion for an Inextensible Kirchhoff Beam Driven by Stochastic Forces

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Progress in Industrial Mathematics at ECMI 2016 (ECMI 2016)

Part of the book series: Mathematics in Industry ((TECMI,volume 26))

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Abstract

In this work we study the feasibility of the Wiener Chaos Expansion for the simulation of an inextensible elastic slender fiber driven by stochastic forces. The fiber is described as Kirchhoff beam with a 1d-parameterized, time-dependent curve, whose dynamics is given by a constrained stochastic partial differential equation. The stochastic forces due to a surrounding turbulent flow field are modeled by a space-time white noise with flow-dependent amplitude. Using the techniques of polynomial chaos, we derive a deterministic system which approximates the original stochastic equation. We explore the numerical performance of the approximation and compare the results with those obtained by Monte-Carlo simulations.

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References

  1. Cameron, R.H., Martin, W.T.: The orthogonal development of non-linear functionals in series of Fourier-Hermite functionals. Ann. Math. 48, 385–392 (1947)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ernst, O.G., Mugler, A., Starkloff, H.J., Ullmann, E.: On the convergence of generalized polynomial chaos expansions. ESAIM Math. Model. Numer. Anal. 46, 317–339 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  3. Hou, T.Y., Luo, W., Rozovskii, B., Zhou, H.M.: Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics. J. Comput. Phys. 216, 687–706 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  4. Janson S.: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997)

    Book  MATH  Google Scholar 

  5. Lindner, F., Marheineke, N., Stroot, H., Vibe, A., Wegener, R.: Stochastic dynamics for inextensible fibers in a spatially semi-discrete setting. Stochastics Dyn. (2016). doi:10.1142/S0219493717500162

    MATH  Google Scholar 

  6. Luo, W.: Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations. California Institute of Technology, Pasadena (2006)

    Google Scholar 

  7. Marheineke, N., Wegener, R.: Fiber dynamics in turbulent flows: general modeling framework. SIAM J. Appl. Math. 66, 1703–1726 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  8. Marheineke, N., Wegener, R.: Modeling and application of a stochastic drag for fibers in turbulent flows. Int. J. Multiphase Flow 37, 136–148 (2011)

    Article  Google Scholar 

  9. Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence, RI (1985)

    MATH  Google Scholar 

  10. Wong, E., Zakai, M.: Martingales and stochastic integrals for processes with a multi-dimensional parameter. Z. Wahr. Verw. Gebiete 29, 109–122 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

This work has been supported by German DFG, project 251706852, MA 4526/2-1 and by the German BMBF, project OPAL 05M13.

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Authors and Affiliations

  1. Department Mathematik, FAU Erlangen-Nürnberg, Cauerstr. 11, D-91058, Erlangen, Germany

    Alexander Vibe & Nicole Marheineke

Authors
  1. Alexander Vibe
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  2. Nicole Marheineke
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Corresponding author

Correspondence to Alexander Vibe .

Editor information

Editors and Affiliations

  1. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Peregrina Quintela

  2. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Patricia Barral

  3. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Dolores Gómez

  4. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Francisco J. Pena

  5. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Jerónimo Rodríguez

  6. Department of Applied Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain, Spain

    Pilar Salgado

  7. Department of Applied Mathematics, University of Santiago de Compostela, Lugo, Spain, Spain

    Miguel E. Vázquez-Méndez

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Vibe, A., Marheineke, N. (2017). Wiener Chaos Expansion for an Inextensible Kirchhoff Beam Driven by Stochastic Forces. In: Quintela, P., et al. Progress in Industrial Mathematics at ECMI 2016. ECMI 2016. Mathematics in Industry(), vol 26. Springer, Cham. https://doi.org/10.1007/978-3-319-63082-3_112

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