Shim, Jay Jayne

A Multiphasic-Fluid-Structure Interaction Formulation Based on Mixture Theory and Its Finite Element Implementation in FEBio

[PhD thesis]. Columbia University

Ateshian, Gerard (supervisor)

2021

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- Abstract
Computational modeling is increasingly necessary to the biomechanics and biophysics communities in order to model the solid, fluid, and solute mechanics of biological systems of interest, spanning across the molecular, cellular, tissue, organ, and whole-body levels. However, most commercial and custom codes are not suitable for biomechanical applications. Many such problems require either the use of specialized boundary conditions and material models or multiphysics simulations to obtain sufficiently accurate results and predictions. Verifying and sharing results from available software can be problematic as they are typically compatible only within the same program and their specific implementation details are not well documented. In addition, the source codes of these software are generally unavailable. As a result, there is a need for a computational modeling tool for biomechanical applications that has robust multiphysics capabilities while also allowing for both customizability and accessibility.

To address these needs, FEBio, a free, open-source finite element software, was developed specifically for the computational modeling requirements of the biomechanics and biophysics communities. FEBio has a large collection of novel computational algorithms and methodologies with detailed documentation and new user tutorials, and source code that is freely available and customizable. However, prior to the studies presented in this dissertation, FEBio only had solvers for analyzing the structural mechanics of deformable and rigid solids, as well as standard multiphasic mixtures. It did not yet have a dedicated computational fluid dynamics (CFD) solver, which is a crucial tool for many applications in biomechanics. In this dissertation, a “standard multiphasic” material refers to the existing FEBio framework for modeling a mixture of solid, fluid, and solute constituents where dynamic effects and intrinsic solvent viscosity are neglected, and where all the constituents are assumed to be intrinsically incompressible. One of the main contributions of the studies presented here is the formulation and finite element implementation of a multiphasic-fluidstructure interaction (MFSI) domain where the dynamics of solid and fluid constituents, and the viscosity of the fluid are taken into account. The term “multiphasic” may be used as shorthand for “standard multiphasic mixture,” whereas the multiphasic-fluid-structure interaction mixture is always described explicitly, or abbreviated as MFSI.

Many biological fluids interact with surrounding tissues or contain solutes and other constituents, which would require such fluid solvers to be coupled with the structural mechanics and multiphasic mixture solvers. Fluid solvers that can account for interactions with deformable multiphasic mixtures, while also accommodating reactive processes, are currently unavailable in any finite element code, and their formulation and implementation would represent a major milestone that greatly expands modeling capabilities and problem configurations. Consequently, the objective of this thesis is to formulate and implement a novel and fully general MFSI finite element solver into FEBio that allows for dynamic fluid flow, dynamic and finite deformation solid mechanics, fluid-structure interactions (FSI), porous media mechanics, and reactive and charged solute transport simultaneously. To do so, for each of the components, we propose to systematically formulate the governing equations using the mixture framework and implement them as finite element code into FEBio before combining all of these features together into one solver. In particular: (1) We formulate and implement an isothermal and compressible CFD solver in FEBio that is combined with its solid mechanics solver to allow for FSI. (2) We extend biphasic theory by developing a biphasic-fluid-structure interaction (BFSI) formulation that allows for dynamic and viscous fluid flow and implement it in FEBio. (3) We formulate and implement a CFD-solute transport solver in FEBio that accommodates diffusion, convection, chemical reactions, osmotic effects, body forces, and frictional drag between the solvent and the solutes. (4) We finally combine all of the features from these solvers (CFD, FSI, BFSI, and CFD-solute) to develop the MFSI solver, in FEBio.

We develop the CFD solver using a solid mechanics approach to solve the governing equations in order to circumvent the need for stabilization methods. The fluid dilatation is used as a degree of freedom to represent the compressibility of the fluid, and as a kinematic quantity, may further serve as a state variable for functions of state such as the fluid pressure. Using the CFD solver, an FSI solver derived from mixture theory is developed to model the deformation of the fluid domain mesh and allow interactions with any of the hyperelastic materials available in FEBio. Here, the FSI material is a special case of a biphasic medium, where the fluid flows relative to the solid constituent, which is defined on the mesh.

We formulate a novel biphasic mixture that includes dynamics and models a viscous interstitial fluid that can interface with a dynamic viscous fluid domain, such that the fluid can flow across the interface and into or out of the biphasic mixture. The new biphasic material formulation employs a hybrid approach, where the porous solid skeleton is intrinsically incompressible but the interstitial fluid is compressible, such that the overall biphasic mixture is compressible due to changes in pore volume. The speed of sound in each medium (compressible porous solid and compressible fluid) remains finite. This framework is then implemented into FEBio as the BFSI solver.

Then, we develop a CFD-solute transport formulation derived from the mixture framework, as an extension of the CFD solver. In addition to modeling convection, diffusion, and reactions for solutes, the CFD-solute formulation can also consider body forces for the solutes, frictional drag between the fluid and solutes, and osmotic effects. Like the CFD solver, the CFD-solute solver can model both viscous Newtonian or non-Newtonian solvents. It can also include any number of reactive and charged solute species like the existing standard multiphasic solver in FEBio. The CFD-solute solver does not require any stabilization method, neither for the fluid nor the solute equations, unlike standard methods that assume intrinsic fluid incompressibility.

Finally we formulate and implement the MFSI solver by using the mixture framework to combine the capabilities of the previous solvers, namely the CFD, FSI, CFD-solute, standard biphasic, BFSI, and standard multiphasic solvers in FEBio, where solvent and solutes may be exchanged across the fluid-structure interfaces, while also allowing for chemical reactions and the presence of charged solute species. The MFSI solver forms the foundation of a fully general multiphysics finite element code that can, in principle, be used in almost any FEBio scenario to obtain the most physiological results possible for a wide range of biomechanics problems.

All of the solvers developed in this thesis encompass innovative formulations based on the framework of mixture theory, greatly expanding the modeling capabilities available to the biomechanics and biophysics communities. By being open-source, the FEBio project encourages verification and sharing of results through the availability of the source code and modular code structure. The MFSI finite element solver in particular can also potentially act as a foundation for further extension by being amenable the addition of new capabilities that are currently not considered here.

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