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Abstract
We construct a stabilized finite-element method to compute flow and finite-strain deformations in an incompressible poroelastic medium. We employ a three-field mixed formulation to calculate displacement, fluid flux and pressure directly and introduce a Lagrange multiplier to enforce flux boundary conditions. We use a low order approximation, namely, continuous piecewise-linear approximation for the displacements and fluid flux, and piecewise-constant approximation for the pressure. This results in a simple matrix structure with low bandwidth. The method is stable in both the limiting cases of small and large permeability. Moreover, the discontinuous pressure space enables efficient approximation of steep gradients such as those occurring due to rapidly changing material coefficients or boundary conditions, both of which are commonly seen in physical and biological applications.
Highlights
Poroelasticity theory assumes a superposition of solid and fluid components to capture complex interactions between a deformable porous medium and the fluid flow within it, and was originally developed to study geophysical applications such as reservoir geomechanics [26,28,41]
Biological examples include the coupling of flow in coronary vessels with the mechanical deformation of myocardial tissue to create a poroelastic model of coronary perfusion [13,15]
It has been suggested that this problem is caused by the saddle point structure in the coupled equations resulting in a violation of the famous Ladyzhenskaya– Babuska–Brezzi (LBB) condition, highlighting the need for a stable combination of mixed finite elements [23]
Summary
Poroelasticity theory assumes a superposition of solid and fluid components to capture complex interactions between a deformable porous medium and the fluid flow within it, and was originally developed to study geophysical applications such as reservoir geomechanics [26,28,41]. Biological examples include the coupling of flow in coronary vessels with the mechanical deformation of myocardial tissue to create a poroelastic model of coronary perfusion [13,15]. There is a need for methods that do not give rise to localised pressure oscillations when seeking to approximate steep pressure gradients in the solution. When modelling the diseased lung, abrupt changes in tissue properties and heterogeneous airway narrowing are possible. This can result in a patchy ventilation and pressure distribution [51]. In this situation methods that solve the poroelastic equations using a continuous pressure approximation strug-. The method presented here is able to overcome these types of pressure instability
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