Locking free matching of different three dimensional models in structural mechanics

Patrick Le Tallec; Saloua Mani Aouadi

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 1, page 129-145
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Le Tallec, Patrick, and Mani Aouadi, Saloua. "Locking free matching of different three dimensional models in structural mechanics." ESAIM: Mathematical Modelling and Numerical Analysis 41.1 (2007): 129-145. <http://eudml.org/doc/249943>.

@article{LeTallec2007,
abstract = { 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. },
author = {Le Tallec, Patrick, Mani Aouadi, Saloua},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {3D coupling; mixed formulations; shells; incompressible elasticity; mortar elements; delinquent modes; inf-sup condition; locking free approximations.; locking free approximations},
language = {eng},
month = {4},
number = {1},
pages = {129-145},
publisher = {EDP Sciences},
title = {Locking free matching of different three dimensional models in structural mechanics},
url = {http://eudml.org/doc/249943},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Le Tallec, Patrick
AU - Mani Aouadi, Saloua
TI - Locking free matching of different three dimensional models in structural mechanics
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/4//
PB - EDP Sciences
VL - 41
IS - 1
SP - 129
EP - 145
AB - 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.
LA - eng
KW - 3D coupling; mixed formulations; shells; incompressible elasticity; mortar elements; delinquent modes; inf-sup condition; locking free approximations.; locking free approximations
UR - http://eudml.org/doc/249943
ER -

References

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  2. K.J. Bathe and D. Chapelle, The Finite Element Analysis of Shells - fundamentals. Computational Fluid and Solid Mechanics, Springer Verlag, New York (2003).  Zbl1103.74003
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  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.  Zbl1057.74021
  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.  Zbl0940.74059
  11. A. Diaz and D. Barthes-Biesel, Entrance of a bioartificial capsule in a pore. Comput. Modeling Engineering Sci.3 (2002) 321–338.  Zbl1039.74034
  12. B. Flemisch, J.M. Melenk and B. Wohlmuth, Mortar methods with curved interfaces. Technical report, Max Planck Institute (2004).  Zbl1078.65119
  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.  Zbl0998.76050
  15. M.A. Puso, A 3D mortar method for solid mechanic. Int. J. Num. Meth. Engr.59 (2004) 315–336.  Zbl1047.74065
  16. L.R. Scott and S. Zhang, Finite element interpolation of non smooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493.  Zbl0696.65007
  17. R. Stenberg, A technique for analysing finite element methods for viscous incompressible flow. Int. J. Num. Meth. Fluids11 (1990) 935–948.  Zbl0704.76017
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  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 Notes21 (1999).  Zbl0958.74075

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