Locking free matching of different three dimensional models in structural mechanics
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, p. 129-145

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.

DOI : https://doi.org/10.1051/m2an:2007013
Classification:  65N30,  65N55,  74G15,  74K25,  74S05
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {1},
     year = {2007},
     pages = {129-145},
     doi = {10.1051/m2an:2007013},
     zbl = {1223.74018},
     zbl = {pre05178753},
     mrnumber = {2323694},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2007__41_1_129_0}
}
Tallec, Patrick Le; Aouadi, Saloua Mani. Locking free matching of different three dimensional models in structural mechanics. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 1, pp. 129-145. doi : 10.1051/m2an:2007013. http://www.numdam.org/item/M2AN_2007__41_1_129_0/

[1] A. Arnold and F. Brezzi, Locking free finite element methods for shells. Math. Comp. 66 (1997) 1-14. | Zbl 0854.65095

[2] K.J. Bathe and D. Chapelle, The Finite Element Analysis of Shells - fundamentals. Computational Fluid and Solid Mechanics, Springer Verlag, New York (2003). | MR 2143259 | Zbl 1103.74003

[3] J. Bathe, D. Chapelle and A. Iosilevich, An inf-sup test for shell finite elements. Comput. Structures 75 (2000) 439-456.

[4] F. Ben Belgacem and Y. Maday, The mortar element method for three dimensional finite element. RAIRO Modél. Math. Anal. Numér. 31 (1997) 289-303. | Numdam | Zbl 0868.65082

[5] A. Blouza and H. Le Dret, Existence et unicité pour le modèle de Koiter pour une coque peu régulière. C.R. Acad. Sci. Paris 319 (1994) 1127-1132. | Zbl 0813.73039

[6] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, New York (1991). | MR 1115205 | Zbl 0788.73002

[7] F. Brezzi and D. Marini, Error estimates for the three-field formulation with bubble stabilization. Math. Comp. 70 (2000) 911-934. | Zbl 0970.65118

[8] M. Bernadou and P.G. Ciarlet, Sur l'ellipticité du modèle linéaire de coque de Koiter. Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin (1976). | Zbl 0356.73066

[9] D. Chapelle and A. Ferent, Modeling of the inclusion of a reinforcing sheet within a 3D medium. Math. Models Methods Appl. Sci. 13 (2003) 573-595. | Zbl 1057.74021

[10] D. Chapelle and R. Stenberg, Stabilized finite element formulations for shells in a bending dominated state. SIAM J. Numer. Anal. 36 (1999) 32-73. | Zbl 0940.74059

[11] A. Diaz and D. Barthes-Biesel, Entrance of a bioartificial capsule in a pore. Comput. Modeling Engineering Sci. 3 (2002) 321-338. | Zbl 1039.74034

[12] B. Flemisch, J.M. Melenk and B. Wohlmuth, Mortar methods with curved interfaces. Technical report, Max Planck Institute (2004). | MR 2149357 | Zbl 1078.65119

[13] P. Hauret, Méthodes numériques pour la dynamique des structures non-linéaires incompressibles à deux échelles. Ph.D. thesis, École polytechnique, France (2004).

[14] P. Le Tallec and S. Mani, Numerical analysis of a linearized fluid-structure interaction problem. Numer. Math. 87 (2000) 317-354. | Zbl 0998.76050

[15] M.A. Puso, A 3D mortar method for solid mechanic. Int. J. Num. Meth. Engr. 59 (2004) 315-336. | Zbl 1047.74065

[16] L.R. Scott and S. Zhang, Finite element interpolation of non smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl 0696.65007

[17] R. Stenberg, A technique for analysing finite element methods for viscous incompressible flow. Int. J. Num. Meth. Fluids 11 (1990) 935-948. | Zbl 0704.76017

[18] B.I. Wohlmuth, Discretization methods and iterative solvers based on domain decomposition. Springer Verlag, New York (2001). | MR 1820470 | Zbl 0966.65097

[19] G. Yang, M.C. Delfour and M. Fortin, Error Analysis of mixed finite element for cylindrical shells, Centre de Recherche Mathématiques, Proceedings and Lecture Notes 21 (1999). | MR 1696490 | Zbl 0958.74075