Approximation of the arch problem by residual-free bubbles

A. Agouzal; M. El Alami El Ferricha

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2001)

  • Volume: 35, Issue: 2, page 271-293
  • ISSN: 0764-583X

Abstract

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We consider a general loaded arch problem with a small thickness. To approximate the solution of this problem, a conforming mixed finite element method which takes into account an approximation of the middle line of the arch is given. But for a very small thickness such a method gives poor error bounds. the conforming Galerkin method is then enriched with residual-free bubble functions.

How to cite

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Agouzal, A., and El Alami El Ferricha, M.. "Approximation of the arch problem by residual-free bubbles." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 35.2 (2001): 271-293. <http://eudml.org/doc/194050>.

@article{Agouzal2001,
abstract = {We consider a general loaded arch problem with a small thickness. To approximate the solution of this problem, a conforming mixed finite element method which takes into account an approximation of the middle line of the arch is given. But for a very small thickness such a method gives poor error bounds. the conforming Galerkin method is then enriched with residual-free bubble functions.},
author = {Agouzal, A., El Alami El Ferricha, M.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {mixed method; Lagrange multipliers; conforming approximations; residual-free-bubble; loaded arch; conforming Galerkin mixed finite element method; residual-free bubble functions; small thickness; approximation of middle line},
language = {eng},
number = {2},
pages = {271-293},
publisher = {EDP-Sciences},
title = {Approximation of the arch problem by residual-free bubbles},
url = {http://eudml.org/doc/194050},
volume = {35},
year = {2001},
}

TY - JOUR
AU - Agouzal, A.
AU - El Alami El Ferricha, M.
TI - Approximation of the arch problem by residual-free bubbles
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2001
PB - EDP-Sciences
VL - 35
IS - 2
SP - 271
EP - 293
AB - We consider a general loaded arch problem with a small thickness. To approximate the solution of this problem, a conforming mixed finite element method which takes into account an approximation of the middle line of the arch is given. But for a very small thickness such a method gives poor error bounds. the conforming Galerkin method is then enriched with residual-free bubble functions.
LA - eng
KW - mixed method; Lagrange multipliers; conforming approximations; residual-free-bubble; loaded arch; conforming Galerkin mixed finite element method; residual-free bubble functions; small thickness; approximation of middle line
UR - http://eudml.org/doc/194050
ER -

References

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  1. [1] D.N. Arnold and R.S. Falk, A uniformly accurate finite element method for the Reissner Mindlin plate. SIAM J. Numer. Anal 26 (1989) 1276–1250. Zbl0696.73040
  2. [2] I. Babuska, The finite element method with Lagrangian multipliers. Numer. Math 20 (1973) 179–192. Zbl0258.65108
  3. [3] I. Babuska and M. Suri, On the locking and robustness in the finite element method. SIAM J. Numer. Anal. 29 (1992) 1276–1290. Zbl0763.65085
  4. [4] C. Baiocchi, F. Brezzi and L. Franca, Virtual bubbles and the Galerkin-Least-squares method. Comput. Methods Appl. Mech. Engrg. 105 (1993) 125–141. Zbl0772.76033
  5. [5] M. Bernadou and Y. Ducatel, Approximation of a general arch problems by straight beam elements. Numer. Math. 40 (1982) 1–29. Zbl0508.73069
  6. [6] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO-Anal. Numér. (1974) 129–151. Zbl0338.90047
  7. [7] F. Brezzi and I. Douglas, Stabilized mixed methods for the stokes problem. Numér. Math. 53 (1988) 225–236. Zbl0669.76052
  8. [8] F. Brezzi and M. Fortin, Mixed and hybrid finite Element Methods. Springer-Verlag, Berlin, New-York, Springer Ser. Comput. Math. 15 (1991). Zbl0788.73002MR1115205
  9. [9] F. Brezzi and A. Russo, Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4 (1994) 571–578. Zbl0819.65128
  10. [10] B. Budiansky and J.L. Sanders, On the best first order linear shell theory. Progr. Appl. Mech., Mac Millan, New-York, 129–140. 
  11. [11] D. Chenais, Rousselet and B. Benedict, Design sensibivity for arch structures with respect to midsurface shape under static loading. J. Optim. Theory Appl. 58 (1988) 225–239. Zbl0631.49015
  12. [12] D. Chenais and J.-C. Paumier, On the locking phenomenon for a class of elliptic problems. Numer. Math. 67 (1994) 427–440 Zbl0798.73054
  13. [13] P.G. Ciarlet, The finite element method for elliptic problems. North Holland, Amsterdam (1978). Zbl0383.65058MR520174
  14. [14] Ph. Destuyender, Some numerical aspects of mixed finite elements for bending plates. Comput. Methods. Appl. Mech. Engrg. 78 (1990) 73–87. Zbl0707.73074
  15. [15] L.P. Franca and T.J.R. Hughes, Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Engrg. 69 (1986) 89–129. Zbl0629.73053
  16. [16] L.P. Franca and A. Russo, Unlocking with residual-free bubbles. Comput. Methods Appl. Mech. Engrg. 142 (1997) 361–364 Zbl0890.73064
  17. [17] A. Habbal and D. Chenais, Deterioration of a finite element method for arch structures when thickness goes to zero. Numer. Math. 62 (1992) 321–341. Zbl0756.73088
  18. [18] V. Lods, A new formulation for arch structures. Application to optimization problems. RAIRO-Modél. Math. Anal. Numér. 28 (1994) 873–902. Zbl0817.73041
  19. [19] A.F.D. Loula, L.P. Franca, T.J.R. Hughes and I. Miranda, Stability Convergence and accuracy of a New finite element method for the circular arch problem. Comput. Methods Appl. Mech. Engrg. 63 (1987) 281–303. Zbl0607.73077
  20. [20] Z. Ould Zeidane, Contributions théoriques en Optimisation et Modélisation des structures. Thèse Université de Nice Sophia-Antipolis, Nice (1995). 
  21. [21] A. Russo, Residual-free bubbles and Stabilized methods, in Proc. of the ninth International Conference on finite Elements in Fluids-New Trends and Applications, M.M. Cacchi, K. Morgan, J. Pariaux, B.A. Schreffer, O.C. Zienkiewicz, Eds., Venice (1995) 377–386. 
  22. [22] A. Russo, Bubble Stabilization of finite element methods for the linearized incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 132 (1996) 333–343. Zbl0887.76038

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