The swelling and shrinkage of biological tissues are modelled by a four-component mixture theory [J.M. Huyghe and J.D. Janssen, Int. J. Engng. Sci. 35 (1997) 793-802; K. Malakpoor, E.F. Kaasschieter and J.M. Huyghe, Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modeling of incompressible charged porous media. ESAIM: M2AN 41 (2007) 661-678]. This theory results in a coupled system of nonlinear parabolic differential equations together with an algebraic constraint for electroneutrality. In this model, it is desirable to obtain accurate approximations of the fluid flow and ions flow. Such accurate approximations can be determined by the mixed finite element method. The solid displacement, fluid and ions flow and electro-chemical potentials are taken as degrees of freedom. In this article the lowest-order mixed method is discussed. This results into a first-order nonlinear algebraic equation with an indefinite coefficient matrix. The hybridization technique is then used to reduce the list of degrees of freedom and to speed up the numerical computation. The mixed hybrid finite element method is then validated for small deformations using the analytical solutions for one-dimensional confined consolidation and swelling. Two-dimensional results are shown in a swelling cylindrical hydrogel sample.
Mots-clés : hydrated soft tissue, nonlinear parabolic partial differential equation, mixed hybrid finite element
@article{M2AN_2007__41_4_679_0,
author = {Malakpoor, Kamyar and Kaasschieter, Enrique F. and Huyghe, Jacques M.},
title = {Mathematical modelling and numerical solution of swelling of cartilaginous tissues. {Part} {II} : mixed-hybrid finite element solution},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {679--712},
publisher = {EDP-Sciences},
volume = {41},
number = {4},
year = {2007},
doi = {10.1051/m2an:2007037},
mrnumber = {2362911},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2007037/}
}
TY - JOUR AU - Malakpoor, Kamyar AU - Kaasschieter, Enrique F. AU - Huyghe, Jacques M. TI - Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II : mixed-hybrid finite element solution JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2007 SP - 679 EP - 712 VL - 41 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2007037/ DO - 10.1051/m2an:2007037 LA - en ID - M2AN_2007__41_4_679_0 ER -
%0 Journal Article %A Malakpoor, Kamyar %A Kaasschieter, Enrique F. %A Huyghe, Jacques M. %T Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II : mixed-hybrid finite element solution %J ESAIM: Modélisation mathématique et analyse numérique %D 2007 %P 679-712 %V 41 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2007037/ %R 10.1051/m2an:2007037 %G en %F M2AN_2007__41_4_679_0
Malakpoor, Kamyar; Kaasschieter, Enrique F.; Huyghe, Jacques M. Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II : mixed-hybrid finite element solution. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 4, pp. 679-712. doi : 10.1051/m2an:2007037. http://www.numdam.org/articles/10.1051/m2an:2007037/
[1] and, The Mathematical Theory of Finite Element Methods. Springer-Verlag, Berlin-Heidelberg- New York (2002). | MR | Zbl
[2] and, Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin-Heidelberg-New York (1991). | MR | Zbl
[3] , The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications 4. North Holland, Amsterdam (1978). | MR | Zbl
[4] , Handbuch der physik, Elastizität und plastizität. Springer-Verlag (1958). | Zbl
[5] , Displacement and equilibrium models in the finite element method, in Stress Analysis, O.C. Zienkiewicz and G. Holister Eds., John Wiley, New York (1965).
[6] , An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11 (1977) 341-354. | Numdam | Zbl
[7] , A four-component mixture theory applied to cartilaginous tissues. Ph.D. thesis, Eindhoven University of Technology (2001). | Zbl
[8] and, Quadriphasic mechanics of swelling incompressibleporous media. Int. J. Engng. Sci. 35 (1997) 793-802. | Zbl
[9] and, Mixed-hybrid finite elements and streamline computation for the potential flow problem. Numer. Methods Partial Differ. Equat. 8 (1992) 221-266. | Zbl
[10] , and, An analytical solution of incompressible charged porous media. Z. Angew. Math. Mech. 86 (2006) 667-681. | MR | Zbl
[11] , and, Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part I: Modeling of incompressible charged porous media. ESAIM: M2AN 41 (2007) 661-678. | Numdam
[12] , Mixed finite elements in . Numer. Math. 35 (1980) 315. | MR | Zbl
[13] , A new family of mixed finite elements in . Numer. Math. 50 (1980) 57. | MR | Zbl
[14] and, A mixed finite element method for 2nd-order elliptic problems, in Mathematical Aspects of Finite Element Methods, Lecture Note in Mathematics 606, I. Galligani and E. Magenes Eds., Springer, Berlin (1997) 292-315. | Zbl
[15] and, Mixed and hybrid finite element methods, in Handbook of Numerical Analysis, Volume II: Finite Element Methods, P.G. Ciarlet and J.L. Lions Eds., North Holland, Amsterdam (1991) 523-639. | Zbl
[16] , Sur l'analyse numérique des méthodes d'éléments finis hybrides et mixtes. Ph.D. thesis, University Pierre et Marie Curie, Paris (1977).
[17] ,, and, 3D FE implementation of an incompressible quadriphasic mixture model. Inter. J. Numer. Meth. Eng. 57 (2003) 1243-1258. | Zbl
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