Skip to main content
SpringerLink
Log in

Convergence of Finite Element Approximations for Generalized Marguerre–von Kármán Equations

  • Conference paper
  • First Online:
Advances in Applied Mathematics

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 87))

Abstract

In this work, we establish the convergence of a conforming finite element approximations to the generalized Marguerre–von Kármán equations. More precisely, we consider here the generalized Marguerre–von Kármán equations, which constitute a mathematical model for a nonlinearly elastic shallow shell subjected to boundary conditions of von Kármán’s type only on a portion of its lateral face, the remaining portion being free. We first reduce the discrete problem of these equations to a single discrete cubic operator equation, whose unknown is the approximate of vertical displacement of the shallow shell. We next solve this discrete operator equation, by adapting a compactness method due to J.L. Lions and on Brouwer’s fixed point theorem (Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris, 1969). Then we establish the convergence of a conforming finite element approximations to these equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Marguerre, K.: Zur Theorie der gekrummten Platte grosser Formanderung. In: Proceedings of the 5th International Congress for Applied Mechanics, pp. 93–101 (1938)

    Google Scholar 

  2. von Kármán, T., Tsien, H.S.: The buckling of spherical shells by external pressure. J. Aero. Sci. 7, 43–50 (1939)

    Article  Google Scholar 

  3. von Kármán, T.: Festigkeitesprobleme in Maschinenbau. Encyklopadie der Mathematischen Wissenschaften, vol. IV/4, pp. 311–385. Taubner, Leipzig (1910)

    Google Scholar 

  4. Ciarlet, P.G., Paumier, J.C.: A justification of the Marguerre-von Kármán equations. Comput. Mech. 1, 177–202 (1986)

    Article  MATH  Google Scholar 

  5. Gratie, L.: Generalized Marguerre-von Kármán equations of a nonlinearly elastic shallow shell. Appl. Anal. 81, 1107–1126 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  6. Ciarlet, P.G., Gratie, L.: On the existence of solutions to the generalized Marguerre-von Kármán equations. Math. Mech. Solids 11, 83–100 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chacha, D.A., Ghezal, A., Bensayah, A.: Modélisation asymptotique d’une coque peu-profonde de Marguerre-von Kármán généralisée dans le cas dynamique. J. ARIMA 13, 63–76 (2010)

    MathSciNet  Google Scholar 

  8. Chacha, D.A., Ghezal, A., Bensayah, A.: Existence result for a dynamical equations of generalized Marguerre-von Kármán shallow shells. J. Elast. 111, 265–283 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  9. Lions, J.L.: Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Dunod, Paris (1969)

    MATH  Google Scholar 

  10. Bensayah, A., Chacha, D.A., Ghezal, A.: Asymptotic modelling of a signorini problem of generalized Marguerre-von Kármán shallow shells. Appl. Anal. 92(9), 1848–1862 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  11. Miyoshi, T.: A mixed finite element method for the solution of von Kármán equations. Numer. Math. 26, 255–269 (1976)

    Article  MathSciNet  Google Scholar 

  12. Kesavan, S.: La méthode de Kikuchi appliquée aux équations de von Kármán. Numer. Math. 32, 209–232 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kesavan, S.: Une méthode d’éléments finis mixte pour les équations de von Kármán. RAIRO Anal. Numér. 14(2), 149–173 (1980)

    MATH  MathSciNet  Google Scholar 

  14. Brezzi, F.: Finite element approximations of the von Kármán equations. RAIRO Anal. Numér. 12(4), 303–312 (1978)

    MATH  MathSciNet  Google Scholar 

  15. Brezzi, F., Rappaz, J., Raviart, P.A.: Finite dimensional approximation of nonlinear problems, Part I: branches of nonsingular solutions. Numer. Math. 36, 1–25 (1980)

    MATH  MathSciNet  Google Scholar 

  16. Brezzi, F., Rappaz, J., Raviart, P.A.: Finite dimensional approximation of nonlinear problems, Part III: simple bifurcation points. Numer. Math. 38, 1–30 (1981)

    MATH  MathSciNet  Google Scholar 

  17. Reinhart, L.: On the numerical analysis of the von Kármán equations: mixed finite element approximation and continuation techniques. Numer. Math. 39, 371–404 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  18. Ciarlet, P.G., Gratie, L., Kesavan, S.: On the generalized von Kármán equations and their approximation. J. Math. Models Methods Appl. Sci. 4(17), 617–633 (2007)

    Article  MathSciNet  Google Scholar 

  19. Berger, M.S.: On von Kármán equations and the buckling of a thin elastic plate. I. The clamped plate. Commun. Pure Appl. Math. 20, 687–719 (1967)

    Article  MATH  Google Scholar 

  20. Berger, M.S., Fife, P.C.: On von Kármán equations and the buckling of a thin elastic plate. II. Plate with general edge conditions. Commun. Pure Appl. Math. 21, 227–241, (1968)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ciarlet, P.G., Gratie, L., Sabu, N.: An existence theorem for generalized von Kármán equations. J. Elast. 62, 239–248 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  23. Ciarlet, P.G.: Mathematical Elasticity, Theory of Plates, vol. II. North-Holland, Amsterdam (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Laboratoire de mathématiques appliquées, Université Kasdi Merbah, B.P 511, Ouargla, 30000, Algérie

    A. Ghezal & D. A. Chacha

Authors
  1. A. Ghezal
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. D. A. Chacha
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Ghezal .

Editor information

Ali R. Ansari

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this paper

Cite this paper

Ghezal, A., Chacha, D.A. (2014). Convergence of Finite Element Approximations for Generalized Marguerre–von Kármán Equations. In: Ansari, A. (eds) Advances in Applied Mathematics. Springer Proceedings in Mathematics & Statistics, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-06923-4_9

Download citation

Publish with us

Policies and ethics

Search

Navigation