The present paper proposes and analyzes a general locking free mixed strategy for computing the deformation of incompressible three dimensional structures placed inside flexible membranes. The model involves as in Chapelle and Ferent [Math. Models Methods Appl. Sci. 13 (2003) 573-595] a bending dominated shell envelope and a quasi incompressible elastic body. The present work extends an earlier work of Arnold and Brezzi [Math Comp. 66 (1997) 1-14] treating the shell part and proposes a global stable finite element approximation by coupling optimal mixed finite element formulations of the different subproblems by mortar techniques. Examples of adequate finite elements are proposed. Convergence results are derived in two steps. First a global inf-sup condition is proved, deduced from the local conditions to be satisfied by the finite elements used for the external shell problem, the internal incompressible 3D problem, and the mortar coupling, respectively. Second, the analysis of Arnold and Brezzi [Math. Comp. 66 (1997) 1-14] is extended to the present problem and least to convergence results for the full coupled problem, with constants independent of the problem's small parameters.
Mots clés : 3D coupling, mixed formulations, shells, incompressible elasticity, mortar elements, delinquent modes, inf-sup condition, locking free approximations
@article{M2AN_2007__41_1_129_0,
author = {Tallec, Patrick Le and Aouadi, Saloua Mani},
title = {Locking free matching of different three dimensional models in structural mechanics},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {129--145},
publisher = {EDP-Sciences},
volume = {41},
number = {1},
year = {2007},
doi = {10.1051/m2an:2007013},
mrnumber = {2323694},
zbl = {1223.74018},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007013/}
}
TY - JOUR AU - Tallec, Patrick Le AU - Aouadi, Saloua Mani TI - Locking free matching of different three dimensional models in structural mechanics JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 129 EP - 145 VL - 41 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007013/ DO - 10.1051/m2an:2007013 LA - en ID - M2AN_2007__41_1_129_0 ER -
%0 Journal Article %A Tallec, Patrick Le %A Aouadi, Saloua Mani %T Locking free matching of different three dimensional models in structural mechanics %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 129-145 %V 41 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007013/ %R 10.1051/m2an:2007013 %G en %F M2AN_2007__41_1_129_0
Tallec, Patrick Le; Aouadi, Saloua Mani. Locking free matching of different three dimensional models in structural mechanics. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 1, pp. 129-145. doi : 10.1051/m2an:2007013. http://www.numdam.org/articles/10.1051/m2an:2007013/
[1] and, Locking free finite element methods for shells. Math. Comp. 66 (1997) 1-14. | Zbl
[2] and, The Finite Element Analysis of Shells - fundamentals. Computational Fluid and Solid Mechanics, Springer Verlag, New York (2003). | MR | Zbl
[3] , and, An inf-sup test for shell finite elements. Comput. Structures 75 (2000) 439-456.
[4] and, The mortar element method for three dimensional finite element. RAIRO Modél. Math. Anal. Numér. 31 (1997) 289-303. | Numdam | Zbl
[5] and, Existence et unicité pour le modèle de Koiter pour une coque peu régulière. C.R. Acad. Sci. Paris 319 (1994) 1127-1132. | Zbl
[6] and, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR | Zbl
[7] and, Error estimates for the three-field formulation with bubble stabilization. Math. Comp. 70 (2000) 911-934. | Zbl
[8] and, Sur l'ellipticité du modèle linéaire de coque de Koiter. Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin (1976). | Zbl
[9] and, Modeling of the inclusion of a reinforcing sheet within a 3D medium. Math. Models Methods Appl. Sci. 13 (2003) 573-595. | Zbl
[10] and, Stabilized finite element formulations for shells in a bending dominated state. SIAM J. Numer. Anal. 36 (1999) 32-73. | Zbl
[11] and, Entrance of a bioartificial capsule in a pore. Comput. Modeling Engineering Sci. 3 (2002) 321-338. | Zbl
[12] , and, Mortar methods with curved interfaces. Technical report, Max Planck Institute (2004). | MR | Zbl
[13] , Méthodes numériques pour la dynamique des structures non-linéaires incompressibles à deux échelles. Ph.D. thesis, École polytechnique, France (2004).
[14] and, Numerical analysis of a linearized fluid-structure interaction problem. Numer. Math. 87 (2000) 317-354. | Zbl
[15] , A 3D mortar method for solid mechanic. Int. J. Num. Meth. Engr. 59 (2004) 315-336. | Zbl
[16] and, Finite element interpolation of non smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl
[17] , A technique for analysing finite element methods for viscous incompressible flow. Int. J. Num. Meth. Fluids 11 (1990) 935-948. | Zbl
[18] , Discretization methods and iterative solvers based on domain decomposition. Springer Verlag, New York (2001). | MR | Zbl
[19] , and, Error Analysis of mixed finite element for cylindrical shells, Centre de Recherche Mathématiques, Proceedings and Lecture Notes 21 (1999). | MR | Zbl
Cité par Sources :
