Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T19:42:24.530Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation of the arch problem by residual-free bubbles

Published online by Cambridge University Press:  15 April 2002

A. Agouzal  and
M. El Alami El Ferricha
A. Agouzal
Affiliation:
Université Lyon 1, L.A.N., bâtiment 101, boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France.
M. El Alami El Ferricha
Affiliation:
École Normale Supérieure, Fès, Département de Mathématiques et Informatique, BP 5206, Bensouda, Fès, Morocco. ([email protected])
Get access

Abstract

We consider a general loaded arch problem with a small thickness. Toapproximate the solution of this problem, a conforming mixed finite elementmethod which takes into account an approximation of the middle line of thearch is given. But for a very small thickness such a method gives poor errorbounds. the conforming Galerkin method is then enriched with residual-freebubble functions.

Keywords

Mixed methodLagrange multipliersconforming approximationsresidual-free-bubble.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 2 , March 2001 , pp. 271 - 293
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnold, D.N. and Falk, R.S., A uniformly accurate finite element method for the Reissner Mindlin plate. SIAM J. Numer. Anal 26 (1989) 1276-1250. CrossRef
Babuska, I., The finite element method with Lagrangian multipliers. Numer. Math 20 (1973) 179-192. CrossRef
Babuska, I. and Suri, M., On the locking and robustness in the finite element method. SIAM J. Numer. Anal. 29 (1992) 1276-1290.
Baiocchi, C., Brezzi, F. and Franca, L., Virtual bubbles and the Galerkin-Least-squares method. Comput. Methods Appl. Mech. Engrg. 105 (1993) 125-141. CrossRef
Bernadou, M. and Ducatel, Y., Approximation of a general arch problems by straight beam elements. Numer. Math. 40 (1982) 1-29. CrossRef
F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO-Anal. Numér. (1974) 129-151.
Brezzi, F. and Douglas, I., Stabilized mixed methods for the stokes problem. Numér. Math. 53 (1988) 225-236. CrossRef
F. Brezzi and M. Fortin, Mixed and hybrid finite Element Methods. Springer-Verlag, Berlin, New-York, Springer Ser. Comput. Math. 15 (1991).
F. Brezzi and A. Russo, Choosing bubbles for advection-diffusion problems. Math. Models Methods Appl. Sci. 4 (1994) 571-578.
B. Budiansky and J.L. Sanders, On the best first order linear shell theory. Progr. Appl. Mech., Mac Millan, New-York, 129-140.
Chenais, D., 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. CrossRef
D. Chenais and J.-C. Paumier, On the locking phenomenon for a class of elliptic problems. Numer. Math. 67 (1994) 427-440
P.G. Ciarlet, The finite element method for elliptic problems. North Holland, Amsterdam (1978).
Destuyender, Ph., Some numerical aspects of mixed finite elements for bending plates. Comput. Methods. Appl. Mech. Engrg. 78 (1990) 73-87. CrossRef
Franca, L.P. and Hughes, T.J.R., Two classes of mixed finite element methods. Comput. Methods Appl. Mech. Engrg. 69 (1986) 89-129. CrossRef
L.P. Franca and A. Russo, Unlocking with residual-free bubbles. Comput. Methods Appl. Mech. Engrg. 142 (1997) 361-364 CrossRef
Habbal, A. and Chenais, D., Deterioration of a finite element method for arch structures when thickness goes to zero. Numer. Math. 62 (1992) 321-341. CrossRef
Lods, V., A new formulation for arch structures. Application to optimization problems. RAIRO-Modél. Math. Anal. Numér. 28 (1994) 873-902. CrossRef
Loula, A.F.D., Franca, L.P., Hughes, T.J.R. and Miranda, I., Stability Convergence and accuracy of a New finite element method for the circular arch problem. Comput. Methods Appl. Mech. Engrg. 63 (1987) 281-303. CrossRef
Z. Ould Zeidane, Contributions théoriques en Optimisation et Modélisation des structures. Thèse Université de Nice Sophia-Antipolis, Nice (1995).
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.
Russo, A., Bubble Stabilization of finite element methods for the linearized incompressible Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 132 (1996) 333-343. CrossRef