A Mixed Finite Element Method for Nearly Incompressible Multiple-Network Poroelasticity

Abstract

In this paper, we present and analyze a new mixed finite element formulation of a general family of quasi-static multiple-network poroelasticity (MPET) equations. The MPET equations describe flow and deformation in an elastic porous medium that is permeated by multiple fluid networks of differing characteristics. As such, the MPET equations represent a generalization of Biot's equations, and numerical discretizations of the MPET equations face similar challenges. Here, we focus on the nearly incompressible case for which standard mixed finite element discretizations of the MPET equations perform poorly. Instead, we propose a new mixed finite element formulation based on introducing an additional total pressure variable. By presenting energy estimates for the continuous solutions and a priori error estimates for a family of compatible semi-discretizations, we show that this formulation is robust in the limits of incompressibility, vanishing storage coefficients, and vanishing transfer between networks. These theoretical results are corroborated by numerical experiments. Our primary interest in the MPET equations stems from the use of these equations in modelling interactions between biological fluids and tissues in physiological settings. So, we additionally present physiologically realistic numerical results for blood and tissue fluid flow interactions in the human brain.

On the optimization of the fixed‐stress splitting for Biot's equations

“…We mention that the fixed‐stress splitting scheme also can be applied to more involved extensions of Biot's equations, for example, including nonlinear water compressibility, unsaturated poroelasticity, the multiple‐network poroelasticity theory, finite‐strain poroplasticity, fractured porous media, and fracture propagation . For nonlinear problems, one combines a linearization technique, eg, the L ‐scheme, with the splitting algorithm; the convergence of the resulting scheme can be proved rigorously .…”

Experimental and Numerical Study of the Mechanical Behavior of Bio-Loaded PVC Subjected to Aging

“…After a tensile test carried out by the EZ-20 tensile machine (figure 4) and a numerical analysis using the finite element method, which is known for its strong ability to predict the behavior of more complex materials [10][11][12][13][14][15][16][17][18][19], on the flexible and rigid PVC samples after and before aging, the results obtained from the test are grouped together in the figures 5 and 6 and table 2. We note that the stiffness of rigid PVC has changed slightly after aging and this is due to the inclination of the slope towards the elastic region.…”

Iterative solvers for Biot model under small and large deformations

“…In terms of modelling, Biot's model has been extended to unsaturated flow [14,37], multiphase flow [27,28,34,36,47], thermo-poroelasticity [20], and reactive transport in porous media [33,48], where nonlinearities arise in the flow model, specifically in the diffusion term, the time derivative term, and/or in Biot's coupling term. The mechanics model can also be extended to the elastoplastic [3,56], the fracture propagation [35], and the hyperelasticity [21,22], where the nonlinearities appear in the constitutive law of the material, in the compatibility condition and/or the conservation of momentum equation.…”