# Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II: Mixed-hybrid finite element solution

Kamyar Malakpoor; Enrique F. Kaasschieter; Jacques M. Huyghe

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

- Volume: 41, Issue: 4, page 679-712
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topMalakpoor, Kamyar, Kaasschieter, Enrique F., and 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 41.4 (2007): 679-712. <http://eudml.org/doc/250064>.

@article{Malakpoor2007,

abstract = {
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: M2AN41 (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.
},

author = {Malakpoor, Kamyar, Kaasschieter, Enrique F., Huyghe, Jacques M.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Hydrated soft tissue; nonlinear parabolic partial differential equation; mixed hybrid finite element.},

language = {eng},

month = {10},

number = {4},

pages = {679-712},

publisher = {EDP Sciences},

title = {Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II: Mixed-hybrid finite element solution},

url = {http://eudml.org/doc/250064},

volume = {41},

year = {2007},

}

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: Mathematical Modelling and Numerical Analysis

DA - 2007/10//

PB - EDP Sciences

VL - 41

IS - 4

SP - 679

EP - 712

AB -
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: M2AN41 (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.

LA - eng

KW - Hydrated soft tissue; nonlinear parabolic partial differential equation; mixed hybrid finite element.

UR - http://eudml.org/doc/250064

ER -

## References

top- S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag, Berlin-Heidelberg- New York (2002).
- F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag, Berlin-Heidelberg-New York (1991).
- P.G. Ciarlet, The Finite Element Method for Elliptic Problems, Studies in Mathematics and Its Applications4. North Holland, Amsterdam (1978).
- S. Flügge, Handbuch der physik, Elastizität und plastizität. Springer-Verlag (1958).
- 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).
- B.X. Fraeijs de Veubeke, An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér.11 (1977) 341–354.
- A.J.H. Frijns, A four-component mixture theory applied to cartilaginous tissues. Ph.D. thesis, Eindhoven University of Technology (2001).
- J.M. Huyghe and J.D. Janssen, Quadriphasic mechanics of swelling incompressibleporous media. Int. J. Engng. Sci.35 (1997) 793–802.
- 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.
- 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.
- 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: M2AN41 (2007) 661–678.
- J.C. Nédélec, Mixed finite elements in ${\mathbb{R}}^{3}$. Numer. Math.35 (1980) 315.
- J.C. Nédélec, A new family of mixed finite elements in ${\mathbb{R}}^{3}$. Numer. Math.50 (1980) 57.
- 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 Mathematics606, I. Galligani and E. Magenes Eds., Springer, Berlin (1997) 292–315.
- 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.
- 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).
- 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.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.