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

Conference paper
Part of the Mathematics in Industry book series (MATHINDUSTRY, volume 26)

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.

Notes

Acknowledgements

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

References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 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)MathSciNetCrossRefMATHGoogle Scholar
  3. 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)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Janson S.: Gaussian Hilbert Spaces. Cambridge University Press, Cambridge (1997)CrossRefMATHGoogle Scholar
  5. 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/S0219493717500162MATHGoogle Scholar
  6. 6.
    Luo, W.: Wiener Chaos Expansion and Numerical Solutions of Stochastic Partial Differential Equations. California Institute of Technology, Pasadena (2006)Google Scholar
  7. 7.
    Marheineke, N., Wegener, R.: Fiber dynamics in turbulent flows: general modeling framework. SIAM J. Appl. Math. 66, 1703–1726 (2006)MathSciNetCrossRefMATHGoogle Scholar
  8. 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)CrossRefGoogle Scholar
  9. 9.
    Szegö, G.: Orthogonal Polynomials, 4th edn. American Mathematical Society, Providence, RI (1985)MATHGoogle Scholar
  10. 10.
    Wong, E., Zakai, M.: Martingales and stochastic integrals for processes with a multi-dimensional parameter. Z. Wahr. Verw. Gebiete 29, 109–122 (1974)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department MathematikFAU Erlangen-NürnbergErlangenGermany

Personalised recommendations