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

Research output: Contribution to journalArticleAcademicpeer-review

11 Citations (Scopus)

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: 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.
Original languageEnglish
Pages (from-to)679-712
JournalESAIM : Mathematical Modelling and Numerical Analysis
Volume41
Issue number4
DOIs
Publication statusPublished - 2007

Fingerprint

Swelling
Finite Element Solution
Mathematical Modeling
Numerical Solution
Tissue
Degree of freedom
Mixture Theory
Hydrogel
Finite element method
Parabolic Differential Equations
Nonlinear Algebraic Equations
Biological Tissue
Mixed Methods
Mixed Finite Element Method
Chemical potential
Consolidation
Ions
Degrees of freedom (mechanics)
Chemical Potential
Approximation

Cite this

@article{5b6c4ccee7ce4a06a7be1db909582c88,
title = "Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II: Mixed hybrid finite element solution",
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: 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.",
author = "K. Malakpoor and E.F. Kaasschieter and J.M.R.J. Huyghe",
year = "2007",
doi = "10.1051/m2an:2007037",
language = "English",
volume = "41",
pages = "679--712",
journal = "ESAIM : Mathematical Modelling and Numerical Analysis",
issn = "0764-583X",
publisher = "EDP Sciences",
number = "4",

}

TY - JOUR

T1 - Mathematical modelling and numerical solution of swelling of cartilaginous tissues. Part II: Mixed hybrid finite element solution

AU - Malakpoor, K.

AU - Kaasschieter, E.F.

AU - Huyghe, J.M.R.J.

PY - 2007

Y1 - 2007

N2 - 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.

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: 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.

U2 - 10.1051/m2an:2007037

DO - 10.1051/m2an:2007037

M3 - Article

VL - 41

SP - 679

EP - 712

JO - ESAIM : Mathematical Modelling and Numerical Analysis

JF - ESAIM : Mathematical Modelling and Numerical Analysis

SN - 0764-583X

IS - 4

ER -