A mixed stabilized finite element formulation for finite deformation of a poroelastic solid saturated with a compressible fluid

Finite element methods for large-strain poroelasticity/chemotaxis models simulating the formation of myocardial oedema

“…Adopting its parabolic formulation solely in terms of pressure head (which condenses momentum and mass conservation equations for the fluid) and combining it with the conservation of linear momentum in the solid, we obtain the extension of Biot consolidation equations to the regime of large strains. This can be carried out using the descriptions already available in, e.g., [15,50,77]. In this paper we choose to write them using the first Piola-Kirchhoff stress tensor P = P eff − αpJF −t and the effective (hyperelastic) stress P eff = ∂Ψs ∂F .…”

“…Note that the specific form of the strain energy density Ψ s is considered irrespective of the fluid saturating the solid. It can also be derived from subtracting the volumetric free energy of the fluid from the total Helmholtz free energy of the mixture Ψ s = Ψ −φ f Ψ f , where φ f is the nominal (Lagrangian) porosity measuring the current fluid volume per unit reference total volume [56,77]. Note also that rearrangement of tissue components as a result of change in fluid content will imply an additional solid deformation as well as a stress modification [26,48,58].…”

“…where κ 0 = d 2 m φ 3 0 /180(1−φ 0 ) 2 , d m is the typical diameter of pores or of solid grains, and M 0 is the slope of the normalised permeability curve [7,14,77]. As porosity is a volumetric quantity, these forms of permeability are isotropic and therefore indifferent to rotation and shear effects.…”

“…The setup of this validation example proceeds similarly as in [47,Sect. 4.2] (see also [77,Sect. 4.2] and [49,62]).…”

“…Even if many numerical methods for the approximation of linear poroelasticity and their convergence analysis can be found in the recent literature (see [12,[16][17][18]52] and the references therein), much less attention has been given to formulations addressing large-strain poromechanics. In this regard, recent works include an enriched Galerkin framework [26], stabilised finite elements [15,77] and hybrid finite elements as well [76]. Although most contributions address the problem in a Lagrangian frame of reference, some alternative formulations include descriptions in Eulerian [62] and ALE [21] coordinates.…”