Research Papers

Finite Element Framework for Computational Fluid Dynamics in FEBio

[+] Author and Article Information
Gerard A. Ateshian, Jay J. Shim

Department of Mechanical Engineering,
Columbia University,
New York, NY 10027

Steve A. Maas, Jeffrey A. Weiss

Department of Bioengineering,
University of Utah,
Salt Lake City, UT 84112

Manuscript received May 4, 2017; final manuscript received December 5, 2017; published online January 12, 2018. Editor: Victor H. Barocas.

J Biomech Eng 140(2), 021001 (Jan 12, 2018) Paper No: BIO-17-1193; doi: 10.1115/1.4038716 History: Received May 04, 2017; Revised December 05, 2017

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 properties. This study describes the formulation and implementation of a finite element framework for computational fluid dynamics (CFD) in FEBio, a free software designed to meet the computational needs of the biomechanics and biophysics communities. This formulation models nearly incompressible flow with a compressible isothermal formulation that uses a physically realistic value for the fluid bulk modulus. It employs fluid velocity and dilatation as essential variables: The virtual work integral enforces the balance of linear momentum and the kinematic constraint between fluid velocity and dilatation, while fluid density varies with dilatation as prescribed by the axiom of mass balance. Using this approach, equal-order interpolations may be used for both essential variables over each element, contrary to traditional mixed formulations that must explicitly satisfy the inf-sup condition. The formulation accommodates Newtonian and non-Newtonian viscous responses as well as inviscid fluids. The efficiency of numerical solutions is enhanced using Broyden's quasi-Newton method. The results of finite element simulations were verified using well-documented benchmark problems as well as comparisons with 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).

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Grahic Jump Location
Fig. 1

Lid-driven cavity flow: Unstructured tetrahedral mesh (left column) and structured hexahedral mesh (right column). Vorticity contours are shown for Re=400 (middle row) and Re=5000 (bottom row).

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Fig. 2

Velocity profiles in lid-driven cavity flow, for hexahedral and tetrahedral meshes, compared to Ghia et al. [56]: (a) v1 versus x2 along vertical centerline (x1=0.5); and (b) v2 versus x1 along horizontal centerline (x2=0.5)

Grahic Jump Location
Fig. 3

Flow past backward facing step with Re=229. Top panel shows the geometry, tetrahedral mesh, and boundary conditions. Middle panel shows contours of horizontal velocity component v1; bottom panel shows pressure contours.

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Fig. 4

Velocity profiles v1(x1,x2) in the flow past a backward facing step. Solid curves represent the finite element solution and symbols represent experimental data [57].

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Fig. 5

1D wave propagation in an inviscid fluid: (a) pressure profile p(x,t) as wave propagates rightward starting from x=−0.5, reflects at right wall (x=0.5), and reverses direction; finite element result using Euler integration is shown in orange, generalized α-method with ρ∞=1 is shown in blue and (b) fluid internal + kinetic energy E(t), evaluated over entire domain −0.5≤x≤0.5, as it varies with time t, using Euler method and generalized α-method with various values of ρ∞.

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Fig. 6

Flow past block in a narrow channel. Top panel shows geometry, mesh, and boundary conditions. Remaining panels show vorticity contour plots and velocity vector plots (ρ∞=1) at four time points within a cycle of the periodic response achieved after t≈30.

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Fig. 7

(a) Model of bifurcated carotid artery using coarse mesh (top) and finer mesh (bottom) of four-node tetrahedral elements. Boundary conditions are shown on the coarse mesh; in addition, v=0 on the arterial wall. The prescribed inlet velocity has a parabolic profile, with average value v0 and (b) time history of the average inlet velocity v0

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Fig. 8

FEBio solution for bifurcated carotid artery model (coarse mesh) at time t = 0.2 s, showing the WSS distribution (top) and the velocity vector field in a sectioned transparent model (bottom); the largest velocity magnitude at this time point was 1.05 m/s.

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Fig. 9

Bifurcated carotid artery results at point P1: (a) FEBio mesh convergence analysis for fluid pressure p. Comparison of SimVascular, Fluent and FEBio results for (b) p and (c) WSS, using finite element model with finer mesh

Grahic Jump Location
Fig. 10

Bifurcated carotid artery results at point P2: Mesh convergence analyses for WSS with (a) SimVascular, (b) Fluent, (c) FEBio, and (d) comparison of WSS results using finite element model with finer mesh

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Fig. 11

Experimental and CFD results for simulated stenosis, plotting the Euler number against the Reynolds number: Experimental measurements using water and water + glycerol mixtures are from [58]. CFD results from this study include FEBio analyses at 18 increasing values of Re (from 25 to 1100), and Fluent analyses at a subset of 15 values of Re.




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