Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-06-12T13:02:03.870Z Has data issue: false hasContentIssue false
  • English
  • Français

Stability and Conservation Properties of Collocated Constraints in Immersogeometric Fluid-Thin Structure Interaction Analysis

Part of: Coupling of solid mechanics with other effects Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  15 October 2015

David Kamensky ,
John A. Evans  and
Ming-Chen Hsu
David Kamensky
Affiliation:
Center for Cardiovascular Simulation, Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th St, Stop C0200, Austin, TX 78712, USA
John A. Evans
Affiliation:
Center for Cardiovascular Simulation, Institute for Computational Engineering and Sciences, The University of Texas at Austin, 201 East 24th St, Stop C0200, Austin, TX 78712, USA
Ming-Chen Hsu*
Affiliation:
Department of Mechanical Engineering, Iowa State University, 2025 Black Engineering, Ames, IA 50011, USA
*
*Corresponding author. Email addresses: kamensky@ices.utexas.edu (D. Kamensky), john.a.evans@colorado.edu (J. A. Evans), jmchsu@iastate.edu (M.-C. Hsu)
Get access

Abstract

The purpose of this study is to enhance the stability properties of our recently-developed numerical method [D. Kamensky, M.-C. Hsu, D. Schillinger, J.A. Evans, A. Aggarwal, Y. Bazilevs, M.S. Sacks, T.J.R. Hughes, “An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves”, Comput. Methods Appl. Mech. Engrg., 284 (2015) 1005–1053] for immersing spline-based representations of shell structures into unsteady viscous incompressible flows. In the cited work, we formulated the fluid-structure interaction (FSI) problem using an augmented Lagrangian to enforce kinematic constraints. We discretized this Lagrangian as a set of collocated constraints, at quadrature points of the surface integration rule for the immersed interface. Because the density of quadrature points is not controlled relative to the fluid discretization, the resulting semi-discrete problem may be over-constrained. Semi-implicit time integration circumvents this difficulty in the fully-discrete scheme. If this time-stepping algorithm is applied to fluid-structure systems that approach steady solutions, though, we find that spatially-oscillating modes of the Lagrange multiplier field can grow over time. In the present work, we stabilize the semi-implicit integration scheme to prevent potential divergence of the multiplier field as time goes to infinity. This stabilized time integration may also be applied in pseudo-time within each time step, giving rise to a fully implicit solution method. We discuss the theoretical implications of this stabilization scheme for several simplified model problems, then demonstrate its practical efficacy through numerical examples.

Keywords

Immersogeometric analysisfluid-structure interactionisogeometric analysiscollocationstabilized methodsaugmented Lagrangian

MSC classification

Secondary: 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65M85: Fictitious domain methods 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 4 , October 2015 , pp. 1147 - 1180
Copyright
Copyright © Global-Science Press 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Stein, K., Benney, R., Kalro, V., Tezduyar, T. E., Leonard, J., and Accorsi, M.. Parachute fluid-structure interactions: 3-D Computation. Computer Methods in Applied Mechanics and Engineering, 190:373–386, 2000.CrossRefGoogle Scholar
[2]Tezduyar, T. and Osawa, Y.. Fluid-structure interactions of a parachute crossing the far wake of an aircraft. Computer Methods in Applied Mechanics and Engineering, 191:717–726, 2001.Google Scholar
[3]Stein, K., Tezduyar, T. E., Sathe, S., Benney, R., and Charles, R.. Fluid-structure interaction modeling of parachute soft-landing dynamics. International Journal for Numerical Methods in Fluids, 47:619–631, 2005.Google Scholar
[4]Tezduyar, T. E., Sathe, S., Pausewang, J., Schwaab, M., Christopher, J., and Crabtree, J.. Fluid-structure interaction modeling of ringsail parachutes. Computational Mechanics, 43:133–142, 2008.Google Scholar
[5]Takizawa, K., Wright, S., Moorman, C., and Tezduyar, T. E.. Fluid-structure interaction modeling of parachute clusters. International Journal for Numerical Methods in Fluids, 65:286–307, 2011.Google Scholar
[6]Takizawa, K., Spielman, T., and Tezduyar, T. E.. Space-time FSI modeling and dynamical analysis of spacecraft parachutes and parachute clusters. Computational Mechanics, 48:345–364, 2011.Google Scholar
[7]Takizawa, K. and Tezduyar, T. E.. Computational methods for parachute fluid-structure interactions. Archives of Computational Methods in Engineering, 19:125–169, 2012.CrossRefGoogle Scholar
[8]Takizawa, K., Tezduyar, T. E., Boswell, R. Kolesar C., and Montel, K.. FSI modeling of the reefed stages and disreefing of the Orion spacecraft parachutes. Computational Mechanics, 54:1203–1220, 2014.CrossRefGoogle Scholar
[9]Takizawa, K., Henicke, B., Puntel, A., Kostov, N., and Tezduyar, T. E.. Space-time techniques for computational aerodynamics modeling of flapping wings of an actual locust. Computational Mechanics, 50:743–760, 2012.CrossRefGoogle Scholar
[10]Takizawa, K., Henicke, B., Puntel, A., Kostov, N., and Tezduyar, T. E.. Computer modeling techniques for flapping-wing aerodynamics of a locust. Computers & Fluids, 85:125–134, 2013.Google Scholar
[11]Takizawa, K., Tezduyar, T. E., and Kostov, N.. Sequentially-coupled space-time FSI analysis of bio-inspired flapping-wing aerodynamics of an MAV. Computational Mechanics, 54:213–233, 2014.Google Scholar
[12]Makhijani, V. B., Yang, H. Q., Dionne, P. J., and Thubrikar, M. J.. Three-dimensional coupled fluid-structure simulation of pericardial bioprosthetic aortic valve function. ASAIO Journal, 43:M387–M392, 1997.Google Scholar
[13]Hart, J. De, Peters, G. W. M., Schreurs, P. J. G., and Baaijens, F. P. T.. A three-dimensional computational analysis of fluid-structure interaction in the aortic valve. Journal of Biomechanics, 36:103–112, 2003.Google Scholar
[14]Hart, J. De, Baaijens, F. P. T., Peters, G. W. M., and Schreurs, P. J. G.. A computational fluid-structure interaction analysis of a fiber-reinforced stentless aortic valve. Journal of Biomechanics, 36:699–712, 2003.Google Scholar
[15]Cheng, R., Lai, Y. G., and Chandran, K. B.. Three-dimensional fluid-structure interaction simulation of bileaflet mechanical heart valve flow dynamics. Annals of Biomedical Engineering, 32(11):1471–1483, 2004.CrossRefGoogle ScholarPubMed
[16]Carmody, C. J., Burriesci, G., Howard, I. C., and Patterson, E. A.. An approach to the simulation of fluid-structure interaction in the aortic valve. Journal of Biomechanics, 39:158–169, 2006.CrossRefGoogle Scholar
[17]Loon, R. van, Anderson, P. D., and Vosse, F. N. van de. A fluid-structure interaction method with solid-rigid contact for heart valve dynamics. Journal of Computational Physics, 217:806–823, 2006.CrossRefGoogle Scholar
[18]Astorino, M., Gerbeau, J.-F., Pantz, O., and Traoré, K.-F.. Fluid-structure interaction and multi-body contact: Application to aortic valves. Computer Methods in Applied Mechanics and Engineering, 198:3603–3612, 2009.CrossRefGoogle Scholar
[19]Borazjani, I., Ge, L., and Sotiropoulos, F.. High-resolution fluid-structure interaction simulations of flow through a bi-leaflet mechanical heart valve in an anatomic aorta. Annals of Biomedical Engineering, 38(2):326–344, 2010.Google Scholar
[20]Loon, R. van. Towards computational modelling of aortic stenosis. International Journal for Numerical Methods in Biomedical Engineering, 26:405–420, 2010.CrossRefGoogle Scholar
[21]Shadden, S. C., Astorino, M., and Gerbeau, J.-F.. Computational analysis of an aortic valve jet with Lagrangian coherent structures. Chaos, 20:017512, 2010.CrossRefGoogle ScholarPubMed
[22]Griffith, B. E.. Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions. International Journal for Numerical Methods in Biomedical Engineering, 28(3):317–345, 2012.Google Scholar
[23]Sturla, F., Votta, E., Stevanella, M., Conti, C. A., and Redaelli, A.. Impact of modeling fluid-structure interaction in the computational analysis of aortic root biomechanics. Medical Engineering and Physics, 35:1721–1730, 2013.Google Scholar
[24]Borazjani, I.. Fluid-structure interaction, immersed boundary-finite element method simulations of bio-prosthetic heart valves. Computer Methods in Applied Mechanics and Engineering, 257:103–116, 2013.Google Scholar
[25]Votta, E., Le, T. B., Stevanella, M., Fusini, L., Caiani, E. G., Redaelli, A., and Sotiropoulos, F.. Toward patient-specific simulations of cardiac valves State-of-the-art and future directions. Journal of Biomechanics, 46:217–228, 2013.Google Scholar
[26]Hsu, M.-C., Kamensky, D., Bazilevs, Y., Sacks, M. S., and Hughes, T. J. R.. Fluid-structure interaction analysis of bioprosthetic heart valves: significance of arterial wall deformation. Computational Mechanics, 54:1055–1071, 2014.Google Scholar
[27]Kamensky, D., Hsu, M.-C., Schillinger, D., Evans, J. A., Aggarwal, A., Bazilevs, Y., Sacks, M. S., and Hughes, T. J. R.. An immersogeometric variational framework for fluid-structure interaction: Application to bioprosthetic heart valves. Computer Methods in Applied Mechanics and Engineering, 284:1005–1053, 2015.Google Scholar
[28]Schoen, F. J. and Levy, R. J.. Calcification of tissue heart valve substitutes: progress toward understanding and prevention. Ann. Thorac. Surg., 79(3):1072–1080, 2005.Google Scholar
[29]Pibarot, P. and Dumesnil, J. G.. Prosthetic heart valves: selection of the optimal prosthesis and long-term management. Circulation, 119(7):1034–1048, 2009.CrossRefGoogle ScholarPubMed
[30]Siddiqui, R. F., Abraham, J. R., and Butany, J.. Bioprosthetic heart valves: modes of failure. Histopathology, 55:135–144, 2009.Google Scholar
[31]Hamid, M. S., Sabbah, H. N., and Stein, P. D.. Finite element evaluation of stresses on closed leaflets of bioprosthetic heart valves with flexible stents. Finite Elements in Analysis and Design, 1(3):213–225, 1985.Google Scholar
[32]Hamid, M. S., Sabbah, H. N., and Stein, P. D.. Influence of stent height upon stresses on the cusps of closed bioprosthetic valves. Journal of Biomechanics, 19(9):759–769, 1986.Google Scholar
[33]Rousseau, E. P., Steenhoven, A. A. van, and Janssen, J. D.. A mechanical analysis of the closed Hancock heart valve prosthesis. Journal of Biomechanics, 21(7):545–562, 1988.CrossRefGoogle ScholarPubMed
[34]Chandran, K. B., Kim, S. H., and Han, G.. Stress distribution on the cusps of a polyurethane trileaflet heart valve prosthesis in the closed position. Journal of Biomechanics, 24(6):385–395, 1991.CrossRefGoogle ScholarPubMed
[35]Black, M. M., Howard, I. C., Huang, X., and Patterson, E. A.. A three-dimensional analysis of a bioprosthetic heart valve. Journal of Biomechanics, 24(9):793–801, 1991.Google Scholar
[36]Patterson, E. A., Howard, I. C., and Thornton, M. A.. A comparative study of linear and nonlinear simulations of the leaflets in a bioprosthetic heart valve during the cardiac cycle. J Med Eng Technol, 20(3):95–108, 1996.Google Scholar
[37]Li, J., Luo, X. Y., and Kuang, Z. B.. A nonlinear anisotropic model for porcine aortic heart valves. Journal of Biomechanics, 34(10):1279–1289, 2001.Google Scholar
[38]Sripathi, V. C., Kumar, R. K., and Balakrishnan, K. R.. Further insights into normal aortic valve function: role of a compliant aortic root on leaflet opening and valve orifice area. Ann. Thorac. Surg., 77(3):844–851, 2004.CrossRefGoogle ScholarPubMed
[39]Sun, W., Abad, A., and Sacks, M. S.. Simulated bioprosthetic heart valve deformation under quasi-static loading. Journal of Biomechanical Engineering, 127(6):905–914, 2005.Google Scholar
[40]Kim, H., Lu, J., Sacks, M. S., and Chandran, K. B.. Dynamic simulation pericardial bioprosthetic heart valve function. Journal of Biomechanical Engineering, 128:717–724, 2006.Google Scholar
[41]Kim, H., Lu, J., Sacks, M. S., and Chandran, K. B.. Dynamic simulation of bioprosthetic heart valves using a stress resultant shell model. Annals of Biomedical Engineering, 36(2):262–275, 2008.Google Scholar
[42]Auricchio, F., Conti, M., Morganti, S., and Totaro, P.. A computational tool to support pre-operative planning of stentless aortic valve implant. Medical Engineering and Physics, 33(10):1183–1192, 2011.CrossRefGoogle ScholarPubMed
[43]Hammer, P. E., Sacks, M. S., Nido, P. J. Del, and Howe, R. D.. Mass-spring model for simulation of heart valve tissue mechanical behavior. Annals of Biomedical Engineering, 39(6):1668–1679, 2011.Google Scholar
[44]Auricchio, F., Conti, M., Morganti, S., and Reali, A.. Simulation of transcatheter aortic valve implantation: a patient-specific finite element approach. Computer Methods in Biomechanics and Biomedical Engineering, 17(12):1347–1357, 2013.CrossRefGoogle ScholarPubMed
[45]Fan, R., Bayoumi, A. S., Chen, P., Hobson, C. M., Wagner, W. R., Mayer, J. E. Jr., and Sacks, M. S.. Optimal elastomeric scaffold leaflet shape for pulmonary heart valve leaflet replacement. Journal of Biomechanics, 46:662–669, 2013.CrossRefGoogle ScholarPubMed
[46]Auricchio, F., Conti, M., Ferrara, A., Morganti, S., and Reali, A.. Patient-specific simulation of a stentless aortic valve implant: the impact of fibres on leaflet performance. Computer Methods in Biomechanics and Biomedical Engineering, 17(3):277–285, 2014.CrossRefGoogle ScholarPubMed
[47]Morganti, S., Auricchio, F., Benson, D. J., Gambarin, F. I., Hartmann, S., Hughes, T. J. R., and Reali, A.. Patient-specific isogeometric structural analysis of aortic valve closure. Computer Methods in Applied Mechanics and Engineering, 284:508–520, 2015.Google Scholar
[48]Hughes, T. J. R., Liu, W. K., and Zimmermann, T. K.. Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Computer Methods in Applied Mechanics and Engineering, 29:329–349, 1981.Google Scholar
[49]Donea, J., Giuliani, S., and Halleux, J. P.. An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Computer Methods in Applied Mechanics and Engineering, 33:689–723, 1982.Google Scholar
[50]Donea, J., Huerta, A., Ponthot, J.-P., and Rodriguez-Ferran, A.. Arbitrary Lagrangian-Eulerian methods. In Encyclopedia of Computational Mechanics, Volume 3: Fluids, chapter 14. John Wiley & Sons, 2004.Google Scholar
[51]Tezduyar, T. E., Takizawa, K., Moorman, C., Wright, S., and Christopher, J.. Space-time finite element computation of complex fluid-structure interactions. International Journal for Numerical Methods in Fluids, 64:1201–1218, 2010.Google Scholar
[52]Wick, T.. Flapping and contact FSI computations with the fluid-solid interface-tracking/interface-capturing technique and mesh adaptivity. Computational Mechanics, 53(1):29–43, 2014.CrossRefGoogle Scholar
[53]Piegl, L. and Tiller, W.. The NURBS Book (Monographs in Visual Communication), 2nd ed. Springer-Verlag, New York, 1997.CrossRefGoogle Scholar
[54]Hughes, T. J. R., Cottrell, J. A., and Bazilevs, Y.. Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement. Computer Methods in Applied Mechanics and Engineering, 194:4135–4195, 2005.Google Scholar
[55]Schillinger, D., Dedè, L., Scott, M. A., Evans, J. A., Borden, M. J., Rank, E., and Hughes, T. J. R.. An isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces. Computer Methods in Applied Mechanics and Engineering, 249–252:116–150, 2012.Google Scholar
[56]Breitenberger, M., Apostolatos, A., Philipp, B., Wüchner, R., and Bletzinger, K.-U.. Analysis in computer aided design: Nonlinear isogeometric B-Rep analysis of shell structures. Computer Methods in Applied Mechanics and Engineering, 284:401–457, 2015.Google Scholar
[57]Cottrell, J. A., Reali, A., Bazilevs, Y., and Hughes, T. J. R.. Isogeometric analysis of structural vibrations. Computer Methods in Applied Mechanics and Engineering, 195:5257–5297, 2006.Google Scholar
[58]Cottrell, J. A., Hughes, T. J. R., and Reali, A.. Studies of refinement and continuity in isogeometric structural analysis. Computer Methods in Applied Mechanics and Engineering, 196:4160–4183, 2007.Google Scholar
[59]Cottrell, J. A., Hughes, T. J. R., and Bazilevs, Y.. Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester, 2009.Google Scholar
[60]Benson, D. J., Bazilevs, Y., Hsu, M.-C., and Hughes, T. J. R.. Isogeometric shell analysis: The Reissner-Mindlin shell. Computer Methods in Applied Mechanics and Engineering, 199:276–289, 2010.Google Scholar
[61]Bazilevs, Y. and Hughes, T. J. R.. NURBS-based isogeometric analysis for the computation of flows about rotating components. Computational Mechanics, 43:143–150, 2008.Google Scholar
[62]Bazilevs, Y. and Akkerman, I.. Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and the residual-based variational multiscale method. Journal of Computational Physics, 229:3402–3414, 2010.Google Scholar
[63]Evans, J. A. and Hughes, T. J. R.. Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations. Journal of Computational Physics, 241:141–167, 2013.Google Scholar
[64]Lorenzis, L. De, Temizer, İ., Wriggers, P., and Zavarise, G.. A large deformation frictional contact formulation using NURBS-based isogeometric analysis. International Journal for Numerical Methods in Engineering, 87:1278–1300, 2011.CrossRefGoogle Scholar
[65]Temizer, İ., Wriggers, P., and Hughes, T. J. R.. Three-dimensional mortar-based frictional contact treatment in isogeometric analysis with NURBS. Computer Methods in Applied Mechanics and Engineering, 209-212:115–128, 2012.Google Scholar
[66]Lorenzis, L. De, Wriggers, P., and Hughes, T. J. R.. Isogeometric contact: A review. GAMM-Mitteilungen, 37(1):85–123, 2014.CrossRefGoogle Scholar
[67]Kiendl, J., Bletzinger, K.-U., Linhard, J., and Wüchner, R.. Isogeometric shell analysis with Kirchhoff-Love elements. Computer Methods in Applied Mechanics and Engineering, 198:3902– 3914, 2009.Google Scholar
[68]Kiendl, J., Bazilevs, Y., Hsu, M.-C., Wüchner, R., and Bletzinger, K.-U.. The bending strip method for isogeometric analysis of Kirchhoff-Love shell structures comprised of multiple patches. Computer Methods in Applied Mechanics and Engineering, 199:2403–2416, 2010.Google Scholar
[69]Kiendl, J.. Isogeometric Analysis and Shape Optimal Design of Shell Structures. PhD thesis, Lehrstuhl für Statik, Technische Universität München, 2011.Google Scholar
[70]Benson, D. J., Bazilevs, Y., Hsu, M.-C., and Hughes, T. J. R.. A large deformation, rotation-free, isogeometric shell. Computer Methods in Applied Mechanics and Engineering, 200:1367–1378, 2011.Google Scholar
[71]Rank, E., Ruess, M., Kollmannsberger, S., Schillinger, D., and Düster, A.. Geometric modeling, isogeometric analysis and the finite cell method. Computer Methods in Applied Mechanics and Engineering, 249-252:104–115, 2012.Google Scholar
[72]Ruess, M., Schillinger, D., Bazilevs, Y., Varduhn, V., and Rank, E.. Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method. International Journal for Numerical Methods in Engineering, 95:811–846, 2013.Google Scholar
[73]Schillinger, D. and Ruess, M.. The Finite Cell Method: A review in the context of higher-order structural analysis of CAD and image-based geometric models. Archives of Computational Methods in Engineering, 22(3):391–455, 2015.Google Scholar
[74]Barbosa, H. J. C. and Hughes, T. J. R.. The finite element method with Lagrange multipliers on the boundary: circumventing the Babuška-Brezzi condition. Computer Methods in Applied Mechanics and Engineering, 85(1):109–128, 1991.Google Scholar
[75]Bazilevs, Y., Hsu, M.-C., and Scott, M. A.. Isogeometric fluid-structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines. Computer Methods in Applied Mechanics and Engineering, 249–252:28–41, 2012.Google Scholar
[76]Nitsche, J.. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 36:9–15, 1971.Google Scholar
[77]Hansbo, A. and Hansbo, P.. An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Computer Methods in Applied Mechanics and Engineering, 191:5537–5552, 2002.Google Scholar
[78]Juntunen, J. and Stenberg, R.. Nitsche’s method for general boundary conditions. Mathematics of Computation, 78:1353–1374, 2009.Google Scholar
[79]Embar, A., Dolbow, J., and Harari, I.. Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements. International Journal for Numerical Methods in Engineering, 83:877–898, 2010.Google Scholar
[80]Bazilevs, Y. and Hughes, T. J. R.. Weak imposition of Dirichlet boundary conditions in fluid mechanics. Computers and Fluids, 36:12–26, 2007.Google Scholar
[81]Bazilevs, Y., Michler, C., Calo, V. M., and Hughes, T. J. R.. Weak Dirichlet boundary conditions for wall-bounded turbulent flows. Computer Methods in Applied Mechanics and Engineering, 196:4853–4862, 2007.Google Scholar
[82]Bazilevs, Y., Michler, C., Calo, V. M., and Hughes, T. J. R.. Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes. Computer Methods in Applied Mechanics and Engineering, 199:780–790, 2010.Google Scholar
[83]Hsu, M.-C., Akkerman, I., and Bazilevs, Y.. Wind turbine aerodynamics using ALE-VMS: Validation and the role of weakly enforced boundary conditions. Computational Mechanics, 50:499–511, 2012.Google Scholar
[84]Brenner, S. C. and Scott, L. R.. The Mathematical Theory of Finite Element Methods, 3rd ed. Springer, 2008.Google Scholar
[85]Johansson, A. and Larson, M. G.. A high order discontinuous Galerkin Nitsche method for elliptic problems with fictitious boundary. Numerische Mathematik, 123:607–628, 2012.Google Scholar
[86]Evans, J. A. and Hughes, T. J. R.. Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements. Numerische Mathematik, 123:259–290, 2013.Google Scholar
[87]Brooks, A. N. and Hughes, T. J. R.. Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Computer Methods in Applied Mechanics and Engineering, 32:199–259, 1982.CrossRefGoogle Scholar
[88]Franca, L. P., Frey, S. L., and Hughes, T. J. R.. Stabilized finite element methods: I. Application to the advective-diffusive model. Computer Methods in Applied Mechanics and Engineering, 95:253–276, 1992.Google Scholar
[89]Hartmann, F.. The discrete Babuška-Brezzi condition. Ingenieur-Archiv, 56(3):221–228, 1986.Google Scholar
[90]Hughes, T. J. R., Feijóo, G. R., Mazzei, L., and Quincy, J.B.. The variational multiscale method–A paradigm for computational mechanics. Computer Methods in Applied Mechanics and Engineering, 166:3–24, 1998.CrossRefGoogle Scholar
[91]Bazilevs, Y., Calo, V. M., Cottrel, J. A., Hughes, T. J. R., Reali, A., and Scovazzi, G.. Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Computer Methods in Applied Mechanics and Engineering, 197:173–201, 2007.Google Scholar
[92]Taylor, C. A., Hughes, T. J. R., and Zarins, C. K.. Finite element modeling of blood flow in arteries. Computer Methods in Applied Mechanics and Engineering, 158:155–196, 1998.Google Scholar
[93]Holzapfel, G. A.. Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Wiley, Chichester, 2000.Google Scholar
[94]Chung, J. and Hulbert, G. M.. A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized-α method. Journal of Applied Mechanics, 60:371–75, 1993.Google Scholar
[95]Bazilevs, Y., Calo, V. M., Hughes, T. J. R., and Zhang, Y.. Isogeometric fluid-structure interaction: theory, algorithms, and computations. Computational Mechanics, 43:3–37, 2008.Google Scholar
[96]Jansen, K. E., Whiting, C. H., and Hulbert, G. M.. A generalized-α method for integrating the filtered Navier-Stokes equations with a stabilized finite element method. Computer Methods in Applied Mechanics and Engineering, 190:305–319, 2000.Google Scholar
[97]Hestenes, M. R.. Multiplier and gradient methods. Journal of Optimization Theory and Applications, 4(5):303–320, 1969.Google Scholar
[98]Powell, M. J. D.. A method for nonlinear constraints in minimization problems. In Fletcher, R., editor, Optimization, pages 283–298. Academic Press, New York, 1969.Google Scholar
[99]Uzawa, H. and Arrow, K. J.. Iterative methods for concave programming. In Preference, production, and capital, pages 135–148. Cambridge University Press, 1989. Cambridge Books Online.Google Scholar
[100]Bacuta, C.. A unified approach for Uzawa algorithms. SIAM Journal on Numerical Analysis, 44(6):2633–2649, 2006.Google Scholar
[101]Court, S., Fournié, M., and Lozinski, A.. A fictitious domain approach for the Stokes problem based on the extended finite element method. International Journal for Numerical Methods in Fluids, 74(2):73–99, 2014.Google Scholar
[102]Bazilevs, Y., Takizawa, K., and Tezduyar, T. E.. Computational Fluid-Structure Interaction: Methods and Applications. Wiley, Chichester, 2013.Google Scholar
[103]Brummelen, E. H. van. Added mass effects of compressible and incompressible flows in fluid-structure interaction. Journal of Applied Mechanics, 76:021206, 2009.Google Scholar
[104]Roos, H.-G., Stynes, M., and Tobiska, L.. Numerical Methods for Singularly Perturbed Differential Equations. Convection-Diffusion and Flow Problems. Springer, 1996.CrossRefGoogle Scholar
[105]Hesch, C., Gil, A. J., Carreño, A. Arranz, and Bonet, J.. On continuum immersed strategies for fluid-structure interaction. Computer Methods in Applied Mechanics and Engineering, 247-248:51–64, 2012.Google Scholar
[106]Gil, A.J., Carreño, A. Arranz, Bonet, J., and Hassan, O.. An enhanced immersed structural potential method for fluid-structure interaction. Journal of Computational Physics, 250:178–205, 2013.CrossRefGoogle Scholar
[107]Kenner, T.. The measurement of blood density and its meaning. Basic Research in Cardiology, 84(2):111–124, 1989.Google Scholar
[108]Rosencranz, R. and Bogen, S. A.. Clinical laboratory measurement of serum, plasma, and blood viscosity. American Journal of Clinical Pathology, 125:S78–S86, 2006.Google Scholar
[109]Hasimoto, H.. On the flow of a viscous fluid past a thin screen at small Reynolds numbers. Journal of the Physical Society of Japan, 13(6):633–639, 1958.Google Scholar
[110]Yap, C. H., Saikrishnan, N., Tamilselvan, G., and Yoganathan, A. P.. Experimental technique of measuring dynamic fluid shear stress on the aortic surface of the aortic valve leaflet. Journal of Biomechanical Engineering, 133(6):061007, 2011.Google Scholar
[111]Bazilevs, Y., Gohean, J. R., Hughes, T. J. R., Moser, R. D., and Zhang, Y.. Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device. Computer Methods in Applied Mechanics and Engineering, 198:3534–3550, 2009.CrossRefGoogle Scholar
[112]Esmaily-Moghadam, M., Bazilevs, Y., Hsia, T.-Y., Vignon-Clementel, I. E., Marsden, A. L., and Modeling of Congenital Hearts Alliance (MOCHA). A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations. Computational Mechanics, 48:277–291, 2011.Google Scholar
[113]Westerhof, N., Lankhaar, J.-W., and Westerhof, B. E.. The arterial Windkessel. Medical & Biological Engineering & Computing, 47:131–141, 2009.Google Scholar
[114]Hughes, T. J. R., Liu, W. K., and Brooks, A.. Finite element analysis of incompressible viscous flows by the penalty function formulation. Journal of Computational Physics, 30(1):1–60, 1979.CrossRefGoogle Scholar