A General Fluid-Porous-Structure Interactions Formulation Implemented in Febio

Fluid-structure interactions (FSI) and poroelasticity are useful continuum mechanics frameworks for modeling a variety of materials and engineering processes. The interaction of these two types of material domains, which we call fluid-porous-structure interactions (FPSI), has not been studied in detail. Currently, it is not possible to computationally model porous-permeable materials that can simultaneously incorporate finite deformation of the porous solid matrix, dynamics, and a viscous interstitial fluid, or any fluid exchange between such a porous domain and an adjoining viscous fluid domain. Such computational tools would be useful for of variety of applications, most notably in biomechanics of soft biological tissues and their interactions with biological fluids. Currently, researchers cannot model the interstitial fluid exchanged between synovial fluid and cartilage, blood and the arterial wall, or cerebrospinal fluid and the brain. Moreover, few computational tools exist that can solve independently for FSI and large deformation porous materials with viscous interstitial fluid, and even fewer can address FPSI.  Among these, none are available as open-source software, thus lacking the benefits of easy access, and the ability to replicate and disseminate models and results.  The objective of this study was to formulate and implement a FPSI solver in the free, open-source finite element code FEBio (febio.org). FEBio currently offers solvers for solid mechanics, computational fluid dynamics, and porous deformable materials [1,2]. The FPSI solver presented here is a generalization of the recently developed FSI solver that defined interactions at the interface between a deformable (non-porous) solid domain and a viscous fluid domain [3]. Previously, the deformable fluid domain in an FSI analysis was formulated as a specialized solid-fluid mixture where the mesh was defined on the solid material, with zero mass density and negligible stiffness to regularize its deformation. Here, this formulation is generalized to FPSI materials where the viscous fluid flows through a porous solid matrix that has non-zero mass density. In addition, to account for Darcy-Brinkman-like behavior, frictional interactions between the fluid and porous solid are incorporated using a hydraulic permeability tensor. Constitutive relations for the solid include hyperelastic and viscoelastic models, and those for the fluid include Newtonian and non-Newtonian models. The fluid constituent is assumed to be compressible, using its dilatation as a state variable and a suitable function of state to evaluate the pressure. For nearly incompressible responses, the fluid pressure is calculated from the negated product of its bulk modulus with the dilatation. Here, the mass balance equation is replaced with the equivalent kinematic constraint between the fluid dilatation and its velocity. Therefore, like the FSI solver, our FPSI formulation does not require stabilization methods to achieve good convergence. Governing equations for the fluid and solid constituents were rearranged to satisfy the jump conditions related to momentum balance across interfaces automatically, and as a result, two FPSI domains can be interfaced without requiring additional user-specified boundary conditions. A FPSI domain can be easily reduced to a standard poroelastic domain, or a deformable viscous fluid domain, as was verified using a variety of test problems. The finite element code was also successfully verified against various analytical solutions from the literature, such as Couette and Poiseuille-like flow where rigid-impermeable walls are replaced with porous deformable domains. A newly-derived analytical solution for 1D ultrafiltration of a viscous fluid through a poroelastic domain was used to further validate FEBio results. Finally, the FPSI solver was used to examine poro-elastohydrodynamic squeeze-film lubrication, non-Newtonian viscous blood flow within and through a porous deformable arterial wall, and a flow constriction problem where the deformable impermeable walls of a fluid domain can be squeezed shut. The successful formulation and implementation of this FPSI solver offers enhanced multiphysics modeling capabilities that are accessible via an open source software platform.

References: [1] Maas, SA et al., J Biomech Eng, 2012.

[2] Ateshian, GA et al., J Biomech Eng, 2018.

[3] Shim, JJ et al., J Biomech Eng, 2019.