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

Published online by Cambridge University Press:  04 October 2007

Kamyar Malakpoor ,
Enrique F. Kaasschieter  and
Jacques M. Huyghe
Kamyar Malakpoor
Affiliation:
Korteweg-de Vries Institute for Mathematics (Faculty NWI), University of Amsterdam, Plantage Muidergracht 24, 1018 TV, Amsterdam, The Netherlands.
Enrique F. Kaasschieter
Affiliation:
Departement of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. e.f.kaasschieter@tue.nl
Jacques M. Huyghe
Affiliation:
Faculty of Mechanical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. j.m.r.huyghe@tue.nl
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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.

Keywords

Hydrated soft tissuenonlinear parabolic partial differential equationmixed hybrid finite element.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 4 , July 2007 , pp. 679 - 712
Copyright
© EDP Sciences, SMAI, 2007

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