Advertisement

Bending Problems

  • Sören BartelsEmail author
Chapter
Part of the Springer Series in Computational Mathematics book series (SSCM, volume 47)

Abstract

Deforming thin elastic objects within the bending regime is a classical problem in continuum mechanics and efficient mathematical descriptions lead to fourth-order problems. In the case of small displacements, a linear partial differential equation provides accurate approximations, but finite element methods have to be carefully developed to avoid locking phenomena. In the case of large deformations, a pointwise isometry constraint has to be incorporated which can be treated with techniques developed for approximating harmonic maps. Related problems arise in modeling of fluid membranes, but require different numerical methods. Iterative algorithms, together with short implementations, are motivated and analyzed.

References

  1. 1.
    Arnold, D.N., Falk, R.S.: A uniformly accurate finite element method for the Reissner-Mindlin plate. SIAM J. Numer. Anal. 26(6), 1276–1290 (1989). http://dx.doi.org/10.1137/0726074
  2. 2.
    Barrett, J.W., Garcke, H., Nürnberg, R.: Parametric approximation of Willmore flow and related geometric evolution equations. SIAM J. Sci. Comput. 31(1), 225–253 (2008). http://dx.doi.org/10.1137/070700231
  3. 3.
    Bartels, S.: Approximation of large bending isometries with discrete Kirchhoff triangles. SIAM J. Numer. Anal. 51(1), 516–525 (2013). http://dx.doi.org/10.1137/110855405
  4. 4.
    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol. 44. Springer, Heidelberg (2013)Google Scholar
  5. 5.
    Braess, D.: Finite Elements, 3rd edn. Cambridge University Press, Cambridge (2007)Google Scholar
  6. 6.
    Ciarlet, P.G.: Mathematical Elasticity. Vol. II: Theory of Plates, Studies in Mathematics and Its Applications, vol. 27. North-Holland Publishing, Amsterdam (1997)Google Scholar
  7. 7.
    Conti, S.: Derivation of nonlinear plate models (2009). personal communicationGoogle Scholar
  8. 8.
    Dziuk, G.: Finite elements for the Beltrami operator on arbitrary surfaces. In: Partial Differential Equations and Calculus of Variations. Lecture Notes in Math., vol. 1357, pp. 142–155. Springer, Berlin (1988). http://dx.doi.org/10.1007/BFb0082865
  9. 9.
    Dziuk, G.: Computational parametric Willmore flow. Numer. Math. 111(1), 55–80 (2008). http://dx.doi.org/10.1007/s00211-008-0179-1
  10. 10.
    Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Commun. Pure Appl. Math. 55(11), 1461–1506 (2002). http://dx.doi.org/10.1002/cpa.10048
  11. 11.
    Girault, V., Raviart, P.A.: Finite Element Methods for Navier-Stokes Equations, Springer Series in Computational Mathematics, vol. 5. Springer, Berlin (1986)Google Scholar
  12. 12.
    Hornung, P.: Approximating \(W^{2,2}\) isometric immersions. C. R. Math. Acad. Sci. Paris 346(3–4), 189–192 (2008). http://dx.doi.org/10.1016/j.crma.2008.01.001
  13. 13.
    Kühnel, W.: Differential Geometry. Student Mathematical Library, vol. 16. American Mathematical Society, Providence (2002)Google Scholar
  14. 14.
    Willmore, T.J.: Riemannian geometry. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York (1993)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Abteilung für Angewandte MathematikAlbert-Ludwigs-Universität FreiburgFreiburgGermany

Personalised recommendations