0
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).

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Updegrove, A. , Wilson, N. M. , Merkow, J. , Lan, H. , Marsden, A. L. , and Shadden, S. C. , 2017, “ SimVascular: An Open Source Pipeline for Cardiovascular Simulation,” Ann. Biomed. Eng., 45(3), pp. 525–541. [CrossRef] [PubMed]
Taylor, C. A. , Hughes, T. J. , and Zarins, C. K. , 1998, “ Finite Element Modeling of Blood Flow in Arteries,” Comput. Methods Appl. Mech. Eng., 158(1–2), pp. 155–196. [CrossRef]
Whiting, C. H. , and Jansen, K. E. , 2001, “ A Stabilized Finite Element Method for the Incompressible Navier-Stokes Equations Using a Hierarchical Basis,” Int. J. Numer. Methods Eng., 35(1), pp. 93–116. [CrossRef]
Maas, S. A. , Ellis, B. J. , Ateshian, G. A. , and Weiss, J. A. , 2012, “ FEBio: Finite Elements for Biomechanics,” ASME J. Biomech. Eng., 134(1), p. 011005. [CrossRef]
Ateshian, G. A. , Albro, M. B. , Maas, S. , and Weiss, J. A. , 2011, “ Finite Element Implementation of Mechanochemical Phenomena in Neutral Deformable Porous Media Under Finite Deformation,” ASME J. Biomech. Eng., 133(8), p. 081005. [CrossRef]
Ateshian, G. A. , Maas, S. , and Weiss, J. A. , 2013, “ Multiphasic Finite Element Framework for Modeling Hydrated Mixtures With Multiple Neutral and Charged Solutes,” ASME J. Biomech. Eng., 135(11), p. 111001. [CrossRef]
Ateshian, G. A. , Maas, S. , and Weiss, J. A. , 2010, “ Finite Element Algorithm for Frictionless Contact of Porous Permeable Media Under Finite Deformation and Sliding,” ASME J. Biomech. Eng., 132(6), p. 061006. [CrossRef]
Ateshian, G. A. , Maas, S. , and Weiss, J. A. , 2012, “ Solute Transport Across a Contact Interface in Deformable Porous Media,” J. Biomech., 45(6), pp. 1023–1027. [CrossRef] [PubMed]
Ateshian, G. A. , Nims, R. J. , Maas, S. , and Weiss, J. A. , 2014, “ Computational Modeling of Chemical Reactions and Interstitial Growth and Remodeling Involving Charged Solutes and Solid-Bound Molecules,” Biomech. Model. Mechanobiol., 13(5), pp. 1105–1120. [CrossRef] [PubMed]
Ateshian, G. A. , 2015, “ Viscoelasticity Using Reactive Constrained Solid Mixtures,” J. Biomech., 48(6), pp. 941–947. [CrossRef] [PubMed]
Nims, R. J. , Durney, K. M. , Cigan, A. D. , Dusséaux, A. , Hung, C. T. , and Ateshian, G. A. , 2016, “ Continuum Theory of Fibrous Tissue Damage Mechanics Using Bond Kinetics: Application to Cartilage Tissue Engineering,” Interface Focus, 6(1), p. 20150063. [CrossRef] [PubMed]
Krittian, S. , Janoske, U. , Oertel, H. , and Böhlke, T. , 2010, “ Partitioned Fluid-Solid Coupling for Cardiovascular Blood Flow: Left-Ventricular Fluid Mechanics,” Ann. Biomed. Eng., 38(4), pp. 1426–1441. [CrossRef] [PubMed]
Wolters, B. J. B. M. , Rutten, M. C. M. , Schurink, G. W. H. , Kose, U. , de Hart, J. , and van de Vosse, F. N. , 2005, “ A Patient-Specific Computational Model of Fluid-Structure Interaction in Abdominal Aortic Aneurysms,” Med. Eng. Phys., 27(10), pp. 871–883. [CrossRef] [PubMed]
Zhang, W. , Herrera, C. , Atluri, S. N. , and Kassab, G. S. , 2004, “ Effect of Surrounding Tissue on Vessel Fluid and Solid Mechanics,” ASME J. Biomech. Eng., 126(6), pp. 760–769. [CrossRef]
Moghani, T. , Butler, J. P. , Lin, J. L.-W. , and Loring, S. H. , 2007, “ Finite Element Simulation of Elastohydrodynamic Lubrication of Soft Biological Tissues,” Comput. Struct., 85(11), pp. 1114–1120. [CrossRef] [PubMed]
Masoumi, N. , Framanzad, F. , Zamanian, B. , Seddighi, A. S. , Moosavi, M. H. , Najarian, S. , and Bastani, D. , 2013, “ 2D Computational Fluid Dynamic Modeling of Human Ventricle System Based on Fluid-Solid Interaction and Pulsatile Flow,” Basic Clin. Neurosci., 4(1), pp. 64–75. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4202551/ [PubMed]
Sweetman, B. , and Linninger, A. A. , 2011, “ Cerebrospinal Fluid Flow Dynamics in the Central Nervous System,” Ann. Biomed. Eng., 39(1), pp. 484–496. [CrossRef] [PubMed]
Sweetman, B. , Xenos, M. , Zitella, L. , and Linninger, A. A. , 2011, “ Three-Dimensional Computational Prediction of Cerebrospinal Fluid Flow in the Human Brain,” Comput. Biol. Med., 41(2), pp. 67–75. [CrossRef] [PubMed]
Luo, H. , Mittal, R. , Zheng, X. , Bielamowicz, S. A. , Walsh, R. J. , and Hahn, J. K. , 2008, “ An Immersed-Boundary Method for Flow–Structure Interaction in Biological Systems With Application to Phonation,” J. Comput. Phys., 227(22), pp. 9303–9332. [CrossRef] [PubMed]
Malvè, M. , Pérez del Palomar, A. , Chandra, S. , López-Villalobos, J. L. , Mena, A. , Finol, E. A. , Ginel, A. , and Doblaré, M. , 2011, “ FSI Analysis of a Healthy and a Stenotic Human Trachea Under Impedance-Based Boundary Conditions,” ASME J. Biomech. Eng., 133(2), p. 021001. [CrossRef]
Barakat, A. I. , 2001, “ A Model for Shear Stress-Induced Deformation of a Flow Sensor on the Surface of Vascular Endothelial Cells,” J. Theor. Biol., 210(2), pp. 221–236. [CrossRef] [PubMed]
Fung, Y. C. , and Liu, S. Q. , 1993, “ Elementary Mechanics of the Endothelium of Blood Vessels,” ASME J. Biomech. Eng., 115(1), pp. 1–12. [CrossRef]
Fritton, S. P. , and Weinbaum, S. , 2009, “ Fluid and Solute Transport in Bone: Flow-Induced Mechanotransduction,” Annu. Rev. Fluid Mech., 41, pp. 347–374. [CrossRef] [PubMed]
Kwon, R. Y. , and Frangos, J. A. , 2010, “ Quantification of Lacunar-Canalicular Interstitial Fluid Flow Through Computational Modeling of Fluorescence Recovery After Photobleaching,” Cell Mol. Bioeng., 3(3), pp. 296–306. [CrossRef] [PubMed]
Scheiner, S. , Pivonka, P. , and Hellmich, C. , 2016, “ Poromicromechanics Reveals That Physiological Bone Strains Induce Osteocyte-Stimulating Lacunar Pressure,” Biomech. Model. Mechanobiol., 15(1), pp. 9–28. [CrossRef] [PubMed]
Avanzini, A. , and Battini, D. , 2014, “ Structural Analysis of a Stented Pericardial Heart Valve With Leaflets Mounted Externally,” Proc. Inst. Mech. Eng. H, 228(10), pp. 985–995. [CrossRef] [PubMed]
Fiore, G. B. , Redaelli, A. , Guadagni, G. , Inzoli, F. , and Fumero, R. , 2002, “ Development of a New Disposable Pulsatile Pump for Cardiopulmonary Bypass: Computational Fluid-Dynamic Design and In Vitro Tests,” ASAIO J., 48(3), pp. 260–267. [CrossRef] [PubMed]
Malvè, M. , Del Palomar, A. P. , Chandra, S. , López-Villalobos, J. L. , Finol, E. A. , Ginel, A. , and Doblaré, M. , 2011, “ FSI Analysis of a Human Trachea Before and After Prosthesis Implantation,” ASME J. Biomech. Eng., 133(7), p. 071003. [CrossRef]
Malkus, D. S. , and Hughes, T. J. , 1978, “ Mixed Finite Element Methods—Reduced and Selective Integration Techniques: A Unification of Concepts,” Comput. Methods Appl. Mech. Eng., 15(1), pp. 63–81. [CrossRef]
Hughes, T. J. , Liu, W. K. , and Brooks, A. , 1979, “ Finite Element Analysis of Incompressible Viscous Flows by the Penalty Function Formulation,” J. Comput. Phys., 30(1), pp. 1–60. [CrossRef]
Bercovier, M. , and Engelman, M. , 1979, “ A Finite Element for the Numerical Solution of Viscous Incompressible Flows,” J. Comput. Phys., 30(2), pp. 181–201. [CrossRef]
Reddy, J. N. , and Gartling, D. K. , 2001, The Finite Element Method in Heat Transfer and Fluid Dynamics, 2nd ed., CRC Press, Boca Raton, FL.
Brezzi, F. , and Bathe, K.-J. , 1986, “ The Inf-Sup Condition, Equivalent Forms and Applications,” International Conference Reliability of Methods for Engineering Analysis, Swansea, UK, July 9–11, pp. 197–219 http://web.mit.edu/kjb/www/Publications_Prior_to_1998/Studies_of_Finite_Element_Procedures_The_Inf-Sup_Condition_Equivalent_Forms_and_Applications.pdf.
Brezzi, F. , and Bathe, K.-J. , 1990, “ A Discourse on the Stability Conditions for Mixed Finite Element Formulations,” Comput. Methods Appl. Mech. Eng., 82(1–3), pp. 27–57. [CrossRef]
Brooks, A. N. , and Hughes, T. J. , 1982, “ Streamline Upwind/Petrov-Galerkin Formulations for Convection Dominated Flows With Particular Emphasis on the Incompressible Navier-Stokes Equations,” Comput. Methods Appl. Mech. Eng., 32(1), pp. 199–259. [CrossRef]
Hughes, T. J. , Franca, L. P. , and Balestra, M. , 1986, “ A New Finite Element Formulation for Computational Fluid Dynamics: V. Circumventing the Babuška-Brezzi Condition: A Stable Petrov-Galerkin Formulation of the Stokes Problem Accommodating Equal-Order Interpolations,” Comput. Methods Appl. Mech. Eng., 59(1), pp. 85–99. [CrossRef]
Hughes, T. J. , Franca, L. P. , and Hulbert, G. M. , 1989, “ A New Finite Element Formulation for Computational Fluid Dynamics—VIII: The Galerkin/Least-Squares Method for Advective-Diffusive Equations,” Comput. Methods Appl. Mech. Eng., 73(2), pp. 173–189. [CrossRef]
Tezduyar, T. E. , 1991, “ Stabilized Finite Element Formulations for Incompressible Flow Computations,” Adv. Appl. Mech., 28, pp. 1–44. [CrossRef]
Franca, L. P. , and Frey, S. L. , 1992, “ Stabilized Finite Element Methods—II: The Incompressible Navier-Stokes Equations,” Comput. Methods Appl. Mech. Eng., 99(2–3), pp. 209–233. [CrossRef]
Tezduyar, T. E. , Mittal, S. , Ray, S. , and Shih, R. , 1992, “ Incompressible Flow Computations With Stabilized Bilinear and Linear Equal-Order-Interpolation Velocity-Pressure Elements,” Comput. Methods Appl. Mech. Eng., 95(2), pp. 221–242. [CrossRef]
Tezduyar, T. E. , and Osawa, Y. , 2000, “ Finite Element Stabilization Parameters Computed From Element Matrices and Vectors,” Comput. Methods Appl. Mech. Eng., 190(3), pp. 411–430. [CrossRef]
Akin, J. E. , and Tezduyar, T. E. , 2004, “ Calculation of the Advective Limit of the SUPG Stabilization Parameter for Linear and Higher-Order Elements,” Comput. Methods Appl. Mech. Eng., 193(21), pp. 1909–1922. [CrossRef]
Bonet, J. , and Wood, R. D. , 1997, Nonlinear Continuum Mechanics for Finite Element Analysis, Cambridge University Press, New York.
Jansen, K. E. , Whiting, C. H. , and Hulbert, G. M. , 2000, “ A Generalized-α Method for Integrating the Filtered Navier–Stokes Equations With a Stabilized Finite Element Method,” Comput. Methods Appl. Mech. Eng., 190(3), pp. 305–319. [CrossRef]
Reddy, J. N. , 2008, An Introduction to Continuum Mechanics: With Applications, Cambridge University Press, New York.
Panton, R. L. , 2006, Incompressible Flow, Wiley, New York.
Cho, Y. I. , and Kensey, K. R. , 1991, “ Effects of the Non-Newtonian Viscosity of Blood on Flows in a Diseased Arterial Vessel—Part 1: Steady Flows,” Biorheology, 28(3–4), pp. 241–262. [CrossRef] [PubMed]
Ryzhakov, P. B. , Rossi, R. , Idelsohn, S. R. , and Onate, E. , 2010, “ A Monolithic Lagrangian Approach for Fluid-Structure Interaction Problems,” Comput. Mech., 46(6), pp. 883–899. [CrossRef]
Zienkiewicz, O. C. , and Codina, R. , 1995, “ A General Algorithm for Compressible and Incompressible Flow—Part I: The Split, Characteristic-Based Scheme,” Int. J. Numer. Methods Eng., 20(8–9), pp. 869–885. [CrossRef]
Zienkiewicz, O. C. , Taylor, R. L. , and Nithiarasu, P. , 2014, The Finite Element Method for Fluid Dynamics, 7th ed., Butterworth-Heinemann, Oxford, UK..
Bazilevs, Y. , Gohean, J. , Hughes, T. , Moser, R. , and Zhang, Y. , 2009, “ Patient-Specific Isogeometric Fluid–Structure Interaction Analysis of Thoracic Aortic Blood Flow Due to Implantation of the Jarvik 2000 Left Ventricular Assist Device,” Comput. Methods Appl. Mech. Eng., 198(45), pp. 3534–3550. [CrossRef]
Esmaily Moghadam, M. , Bazilevs, Y. , Hsia, T.-Y. , Vignon-Clementel, I. E. , and Marsden, A. L. , 2011, “ A Comparison of Outlet Boundary Treatments for Prevention of Backflow Divergence With Relevance to Blood Flow Simulations,” Comput. Mech., 48(3), pp. 277–291. [CrossRef]
Vignon-Clementel, I. E. , Figueroa, C. A. , Jansen, K. E. , and Taylor, C. A. , 2006, “ Outflow Boundary Conditions for Three-Dimensional Finite Element Modeling of Blood Flow and Pressure in Arteries,” Comput. Methods Appl. Mech. Eng., 195(29), pp. 3776–3796. [CrossRef]
Engelman, M. , Strang, G. , and Bathe, K.-J. , 1981, “ The Application of Quasi-Newton Methods in Fluid Mechanics,” Int. J. Numer. Methods Eng., 17(5), pp. 707–718. [CrossRef]
Schenk, O. , and Gärtner, K. , 2004, “ Solving Unsymmetric Sparse Systems of Linear Equations With PARDISO,” Future Gener. Comput. Syst., 20(3), pp. 475–487. [CrossRef]
Ghia, U. , Ghia, K. N. , and Shin, C. , 1982, “ High-Re Solutions for Incompressible Flow Using the Navier-Stokes Equations and a Multigrid Method,” J. Comput. Phys., 48(3), pp. 387–411. [CrossRef]
Denham, M. , and Patrick, M. , 1974, “ Laminar Flow Over a Downstream-Facing Step in a Two-Dimensional Flow Channel,” Trans. Inst. Chem. Eng., 52(4), pp. 361–367. http://archive.icheme.org/cgi-bin/somsid.cgi?session=924344F&page=52ap0361-001&type=framedpdf
Seeley, B. D. , and Young, D. F. , 1976, “ Effect of Geometry on Pressure Losses Across Models of Arterial Stenoses,” J. Biomech., 9(7), pp. 439–448. [CrossRef] [PubMed]
Tezduyar, T. E. , 1992, Stabilized Finite Element Formulations for Incompressible Flow Computations, Academic Press, New York.
Pelletier, D. , Fortin, A. , and Camarero, R. , 1989, “ Are Fem Solutions of Incompressible Flows Really Incompressible? (or How Simple Flows Can Cause Headaches!),” Int. J. Numer. Methods Fluids, 9(1), pp. 99–112. [CrossRef]

Figures

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).

Grahic Jump Location
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.

Grahic Jump Location
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].

Grahic Jump Location
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∞.

Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
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.

Grahic Jump Location
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

Grahic Jump Location
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.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In