Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II : mixed-hybrid finite element solution
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 4, p. 679-712

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.

DOI : https://doi.org/10.1051/m2an:2007037
Classification:  35K55,  35M10,  65F10,  65M60
Keywords: 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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {4},
     year = {2007},
     pages = {679-712},
     doi = {10.1051/m2an:2007037},
     zbl = {pre05289493},
     mrnumber = {2362911},
     language = {en},
     url = {http://www.numdam.org/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 4, pp. 679-712. doi : 10.1051/m2an:2007037. http://www.numdam.org/item/M2AN_2007__41_4_679_0/

[1] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, Berlin-Heidelberg- New York (2002). | MR 1894376 | Zbl 0804.65101

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

[3] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications 4. North Holland, Amsterdam (1978). | MR 520174 | Zbl 0383.65058

[4] S. Flügge, Handbuch der physik, Elastizität und plastizität. Springer-Verlag (1958). | Zbl 0103.16403

[5] B.X. Fraeijs De Veubeke, 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] B.X. Fraeijs De Veubeke, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11 (1977) 341-354. | Numdam | Zbl 0373.65055

[7] A.J.H. Frijns, A four-component mixture theory applied to cartilaginous tissues. Ph.D. thesis, Eindhoven University of Technology (2001). | Zbl 0966.92002

[8] J.M. Huyghe and J.D. Janssen, Quadriphasic mechanics of swelling incompressibleporous media. Int. J. Engng. Sci. 35 (1997) 793-802. | Zbl 0903.73004

[9] E.F. Kaasschieter and A.J.M. Huijben, Mixed-hybrid finite elements and streamline computation for the potential flow problem. Numer. Methods Partial Differ. Equat. 8 (1992) 221-266. | Zbl 0767.76029

[10] K. Malakpoor, E.F. Kaasschieter and J.M. Huyghe, An analytical solution of incompressible charged porous media. Z. Angew. Math. Mech. 86 (2006) 667-681. | MR 2254775 | Zbl 1106.76069

[11] 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. | Numdam | Zbl pre05289492

[12] J.C. Nédélec, Mixed finite elements in 3 . Numer. Math. 35 (1980) 315. | MR 592160 | Zbl 0419.65069

[13] J.C. Nédélec, A new family of mixed finite elements in 3 . Numer. Math. 50 (1980) 57. | MR 864305 | Zbl 0625.65107

[14] P.A. Raviart and J.M. Thomas, 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 0362.65089

[15] J.E. Roberts and J.M. Thomas, 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 0875.65090

[16] J.M. Thomas, Sur l'analyse numérique des méthodes d'éléments finis hybrides et mixtes. Ph.D. thesis, University Pierre et Marie Curie, Paris (1977).

[17] R. Van Loon, J.M. Huyghe, M.W. Wijlaars and F.P.T. Baaijens, 3D FE implementation of an incompressible quadriphasic mixture model. Inter. J. Numer. Meth. Eng. 57 (2003) 1243-1258. | Zbl 1062.74634