Finite Element Framework for Computational Fluid Dynamics In

Finite Element Framework for Computational Fluid Gerard A. Ateshian FEBIO Department of Mechanical Engineering, Dynamics in Columbia University, New York, NY 10027 The mechanics of biological fluids is an important topic in biomechanics, often requiring the use of computational tools to analyze problems with realistic geometries and material Jay J. Shim properties. This study describes the formulation and implementation of a finite element Department of Mechanical Engineering, framework for computational fluid dynamics (CFD) in FEBIO, a free software designed to Columbia University, meet the computational needs of the biomechanics and biophysics communities. This for- New York, NY 10027 mulation models nearly incompressible flow with a compressible isothermal formulation Downloaded from http://asmedigitalcollection.asme.org/biomechanical/article-pdf/140/2/021001/5989031/bio_140_02_021001.pdf by guest on 26 September 2021 that uses a physically realistic value for the fluid bulk modulus. It employs fluid velocity Steve A. Maas and dilatation as essential variables: The virtual work integral enforces the balance of linear momentum and the kinematic constraint between fluid velocity and dilatation, Department of Bioengineering, while fluid density varies with dilatation as prescribed by the axiom of mass balance. University of Utah, Using this approach, equal-order interpolations may be used for both essential variables Salt Lake City, UT 84112 over each element, contrary to traditional mixed formulations that must explicitly satisfy the inf-sup condition. The formulation accommodates Newtonian and non-Newtonian vis- Jeffrey A. Weiss cous responses as well as inviscid fluids. The efficiency of numerical solutions is Department of Bioengineering, enhanced using Broyden’s quasi-Newton method. The results of finite element simulations University of Utah, were verified using well-documented benchmark problems as well as comparisons with Salt Lake City, UT 84112 other free and commercial codes. These analyses demonstrated that the novel formulation introduced in FEBIO could successfully reproduce the results of other codes. The analogy between this CFD formulation and standard finite element formulations for solid mechanics makes it suitable for future extension to fluid–structure interactions (FSIs). [DOI: 10.1115/1.4038716] 8 9 1 Introduction SOLVER uses an octree finite volume discretization, and COOLFLUID uses either a spectral finite difference solver or a finite element The mechanics of biological fluids is an important topic in bio- solver. Currently, the CFD code most relevant to biomechanics is mechanics, most notably for the study of blood flow through the SIMVASCULAR, a finite element code specifically designed for cardi- cardiovascular system and related biomedical devices, cerebrospi- ovascular fluid mechanics, “providing a complete pipeline from nal fluid flow, airflow through the respiratory system, biotribology medical image data segmentation to patient-specific blood flow by fluid film lubrication, and flow through microfluidic biomedical simulation and analysis.”10 This open-source code [1] is based on devices. Therefore, the application domain of multiphysics com- the original work of Taylor [2]; it uses the flow solver from the putational frameworks geared toward biomechanics and biophy- 11 1 PHASTA project [3]. sics, such as the free finite element software FEBIO, can be In contrast, FEBIO was originally developed with a focus on the expanded by incorporating solvers for computational fluid dynam- biomechanics of soft tissues, providing the ability to model non- ics (CFD). Biological fluids are generally modeled as incompres- linear anisotropic tissue responses under finite deformation. It sible materials, though compressible flow may be needed to accommodates hyperelastic, viscoelastic, biphasic (poroelastic), and analyze wave propagation, for example, in acoustics (airflow multiphasic material responses which can combine solid mechanics through vocal folds) and the analysis of ultrasound propagation and mass (solvent and solute) transport, with a wide range of constitu- for imaging blood flow. tive models relevant to biological tissues and cells [4–6]. It also pro- Many open-source CFD codes are currently available in the vides robust algorithms to model tied or sliding contact under large public domain, though few are geared specifically for applications 2 deformations between elastic, biphasic, or multiphasic materials [7,8]. in biomechanics. OpenFOAM has many solvers applicable to a very More recently, FEBIO has incorporated reactive mechanisms, including broad range of fluid analyses, using the finite volume method. chemical reactions in multiphasic media, growth mechanics, reactive Other open-source CFD codes are typically more specialized: 3 4 viscoelasticity, and reactive damage mechanics [9–11]. Many of these SU2 is geared toward aerospace design, FLUIDITY is well suited 5 features are specifically geared toward applications in biomechanics, for geophysical fluid dynamics, and REEF3D is focused on marine including growth and remodeling and mechanobiology. applications. Some CFD codes explore alternative solution meth- 6 7 Our medium-term goal is to expand the capabilities of FEBIO by ods: PALABOS uses the lattice Boltzmann method, LIGGGHTS uses a including robust fluid–structure interactions (FSIs). Such interac- discrete-element method particle simulation, the GERRIS FLOW tions occur commonly in biomechanics and biophysics, most notably in cardiovascular mechanics where blood flows through the deforming heart and vasculature [12–14], diarthrodial joint lubrication where pressurized synovial fluid flows between deforming 1 febio.org articular layers [15], cerebrospinal mechanics where fluid flow 2openfoam.org 3su2.stanford.edu through ventricular cavities may cause significant deformation of 4fluidityproject.github.io 5reef3d.wordpress.com 6freecode.com/projects/xflows-2 8gfs.sourceforge.net/wiki/ 7www.cfdem.com 9coolfluid.github.io Manuscript received May 4, 2017; final manuscript received December 5, 2017; 10simvascular.org published online January 12, 2018. Editor: Victor H. Barocas. 11secure.cci.rpi.edu/wiki/index.php/PHASTA Journal of Biomechanical Engineering Copyright VC 2018 by ASME FEBRUARY 2018, Vol. 140 / 021001-1 surrounding soft tissues [16–18], vocal fold and upper airway compressibility of the fluid, using the dilatation as the only mechanics [19,20], viscous flow over endothelial cells [21,22], required component of the strain tensor, since fluids are isotropic canalicular and lacunar flow around osteocytes resulting from and cannot sustain shear stresses under steady shear strains. In this bone deformation [23–25], and many applications in biomedical approach, fluid velocity and dilatation may be obtained by satisfy- device design [26–28]. Some FSI capabilities already exist in sev- ing the momentum balance equation as well as the fundamental eral of the open-source CFD codes referenced previously; these kinematic relation between the material time derivative of the dil- implementations were developed as addendums to sophisticated atation and the divergence of the velocity. The finite element fluid analyses, often employing a limited range of solid material implementation, using a standard Galerkin residual formulation, responses, such as small strain analyses of linear isotropic elastic produces accurate numerical solutions even when using equal- solids. By formulating a CFD code in FEBIO, we expect to capital- order interpolation for the velocity and dilatation. ize on the much broader library of solid and multiphasic material models already available in that framework to considerably 2 Finite Element Implementation expand the range of FSI analyses in biomechanics. Our long-term goal is to expand these FSI features to incorporate chemical 2.1 Governing Equations. In a spatial (Eulerian) frame, the reactions within the fluid and at the fluid–solid interface, growth, momentum balance equation for a continuum is Downloaded from http://asmedigitalcollection.asme.org/biomechanical/article-pdf/140/2/021001/5989031/bio_140_02_021001.pdf by guest on 26 September 2021 and remodeling of the solid matrix of multiphasic domains inter- faced with the fluid, active transport of solutes across cell mem- qa ¼ div r þ qb (2.1) branes as triggered by mechanical, electrical, or chemical signals at fluid–membrane interfaces, cellular and muscle contraction where q is the density, r is the Cauchy stress, b is the body force mechanisms, and a host of other mechanisms relevant to biome- per mass, and a is the acceleration, given by the material time chanics, biophysics, and mechanobiology, complementing many derivative of the velocity v in the spatial frame of the similar features already implemented in FEBIO’s elastic, biphasic, and multiphasic domain frameworks. The FEBIO CFD @v a ¼ v_ ¼ þ L Á v (2.2) formulation proposed in this study represents a significant mile- @t stone toward that long-term goal. Since FEBIO uses the finite element method, this study formu- where L ¼ grad v is the spatial velocity gradient. The mass bal- lates a finite element CFD code. Finite element analyses of incom- ance equation is pressible flow typically enforce the incompressibility condition using either mixed formulations [29], which solve for the fluid q_ þ q div v ¼ 0 (2.3) pressure by enforcing zero divergence of the velocity, or the pen- alty method [30,31], where the fluid pressure is proportional to the where the material time derivative of the density in the spatial divergence

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