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Research Papers

A Formulation for Fluid–Structure Interactions in febio Using Mixture Theory

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

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

1febio.org.

2febio.org/febio/febio-downloads.

The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health or the National Science Foundation.

Manuscript received July 1, 2018; final manuscript received February 25, 2019; published online March 27, 2019. Assoc. Editor: Sarah Kieweg.

J Biomech Eng 141(5), 051010 (Mar 27, 2019) (15 pages) Paper No: BIO-18-1306; doi: 10.1115/1.4043031 History: Received July 01, 2018; Revised February 25, 2019

Many physiological systems involve strong interactions between fluids and solids, posing a significant challenge when modeling biomechanics. The objective of this study was to implement a fluid–structure interaction (FSI) solver in the free, open-source finite element code FEBio, that combined the existing solid mechanics and rigid body dynamics solver with a recently developed computational fluid dynamics (CFD) solver. A novel Galerkin-based finite element FSI formulation was introduced based on mixture theory, where the FSI domain was described as a mixture of fluid and solid constituents that have distinct motions. The mesh was defined on the solid domain, specialized to have zero mass, negligible stiffness, and zero frictional interactions with the fluid, whereas the fluid was modeled as isothermal and compressible. The mixture framework provided the foundation for evaluating material time derivatives in a material frame for the solid and in a spatial frame for the fluid. Similar to our recently reported CFD solver, our FSI formulation did not require stabilization methods to achieve good convergence, producing a compact set of equations and code implementation. The code was successfully verified against benchmark problems from the FSI literature and an analytical solution for squeeze-film lubrication. It was validated against experimental measurements of the flow rate in a peristaltic pump and illustrated using non-Newtonian blood flow through a bifurcated carotid artery with a thick arterial wall. The successful formulation and implementation of this FSI solver enhance the multiphysics modeling capabilities in febio relevant to the biomechanics and biophysics communities.

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Figures

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Fig. 1

One-dimensional slip piston analysis, showing the mesh at times t = 0, 3, 7, and 10 s. A constant velocity was prescribed on the left boundary of Ωs and the fluid pressure was set to zero at the right boundary of Ωf. The side walls of Ωf were set to slip.

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Fig. 2

One-dimensional slip piston analysis, comparing febio results with those of Bathe and Ledezma [58] for the displacement, velocity, and normal traction at the fluid–solid interfaceΓfs

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Fig. 3

Moving boundary piston-cylinder problem, showing the mesh of the quarter model at times t = 0, 100, and 195 s. The fluid pressure was set to zero at the rightmost boundary of the outlet tube. The piston face, represented by the moving boundary, was imparted a constant velocity. All internal faces of the piston cylinder and tube were set to no-slip.

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Fig. 4

Moving boundary piston-cylinder problem, comparing febio results with those of Bathe and Ledezma [58] for (a) the fluid pressure and (b) the axial fluid velocity, along the centerline of the piston cylinder and tube at times t = 40, 100, and 160

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Fig. 5

Bubble inflation problem, showing mesh in reference configuration (left) and final configuration at time t = 0.25 (right). The pressure was prescribed as a linearly increasing function of time p0(t) at the bottom face of Ωbotf.

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Fig. 6

Bubble inflation problem, comparing febio results to those of Bathe and Ledezma [58] for (a) the vertical displacement and (b) the first principal stretch of the top of the bubble

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Fig. 15

Inflation of the bifurcated artery for model C, where the inlet nodes are fixed, at different time points up until the time it failed (t =0.168 s)

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Fig. 16

Results from bifurcated artery model A, showing the mass flow rates at inlet and outlet boundaries, the time rate of change m˙ of fluid mass in the fluid domain, and the sum of these measures (Total m˙), which should be zero according to the axiom of mass balance. The error observed in this analysis was less than 1.2% of the peak mass flow rate over the entire analysis.

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Fig. 14

Fluid velocity magnitude and first principal stress on the outer surface of the arterial wall, at systole in the second cycle (t =0.87 s). The arterial wall has heterogeneous properties in the media and adventitia, described by the fiber-reinforced constitutive model of [60]. In (a), the arterial wall is surrounded by subcutaneous adipose tissue, whose outer boundaries are fixed. In (b), the inlet and outlets are fixed and there is no surrounding external matrix.

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Fig. 7

Free surface wave problem using an inviscid fluid, showing (a) the mesh at t =0 and color contours of the normalized vertical displacement at the midpoint (t =143.1 s) and endpoint (t =286.2 s) of the analysis; and (b) a comparison of febio results and those of Hughes et al. [33] for the normalized vertical displacement versus normalized horizontal positions

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Fig. 8

Free surface wave problem, showing the sum of potential and kinetic energies of the fluid over the entire domain Ωf for various values of ρ. Increasing values of ρ demonstrate improved energy conservation, with ideal results achieved at ρ = 1. The total energy is normalized to the steady-state value achieved under ideal conditions.

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Fig. 9

Squeeze-film lubrication between flat parallel plates: Comparison of febio results against the analytical solution of the Reynolds equation for (a) peak fluid pressure p(0, t) at the center of the plate, over the entire duration of the analysis (0 ≤ t 10 s), and (b) shear stress τ(x, 10) along the entire length of the plate at the final time

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Fig. 10

Squeeze-film lubrication of flat parallel plates: Comparison of febio fluid pressure results against the analytical solution of the Reynolds equation, as a function of normalized length X, at selected time points

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Fig. 11

Mesh convergence study of squeeze-film lubrication between flat parallel plates, showing relative error of peak pressure at center of plate at t =10 s as a function of the number of elements in the mesh

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Fig. 12

Prescribed average inlet velocity of bifurcated artery. Note that inlet is set to be parabolic, fully developed flow and the pulsatile flow runs for three cycles.

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Fig. 13

(a) Mesh of bifurcated carotid artery, showing fluid domain Ωf in red, tunica media Ωmeds in cyan, and tunica adventitia Ωadvs in yellow. (b) For model A, the artery was embedded in a rectangular box representing subcutaneous adipose tissue Ωadts, with a relatively coarser mesh. The interface Γs between Ωadvs and Ωadts was modeled as a tied contact interface.

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Fig. 17

Mesh of the overall peristaltic pump (top). The pump mesh at t = 0 s (bottom left), pump with velocity arrows at t = 0.25 s (bottom center), and pump with velocity arrows at t = 0.5 s (bottom right).

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Fig. 18

Comparison of the experimental flow rates and the flow rates obtained from febio for different amounts of occlusion

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