An immersogeometric variational framework for fluid-structure interaction: application to bioprosthetic heart valves.

In this paper, we develop a geometrically flexible technique for computational fluid-structure interaction (FSI). The motivating application is the simulation of tri-leaflet bioprosthetic heart valve function over the complete cardiac cycle. Due to the complex motion of the heart valve leaflets, the fluid domain undergoes large deformations, including changes of topology. The proposed method directly analyzes a spline-based surface representation of the structure by immersing it into a non-boundary-fitted discretization of the surrounding fluid domain. This places our method within an emerging class of computational techniques that aim to capture geometry on non-boundary-fitted analysis meshes. We introduce the term "immersogeometric analysis" to identify this paradigm. The framework starts with an augmented Lagrangian formulation for FSI that enforces kinematic constraints with a combination of Lagrange multipliers and penalty forces. For immersed volumetric objects, we formally eliminate the multiplier field by substituting a fluid-structure interface traction, arriving at Nitsche's method for enforcing Dirichlet boundary conditions on object surfaces. For immersed thin shell structures modeled geometrically as surfaces, the tractions from opposite sides cancel due to the continuity of the background fluid solution space, leaving a penalty method. Application to a bioprosthetic heart valve, where there is a large pressure jump across the leaflets, reveals shortcomings of the penalty approach. To counteract steep pressure gradients through the structure without the conditioning problems that accompany strong penalty forces, we resurrect the Lagrange multiplier field. Further, since the fluid discretization is not tailored to the structure geometry, there is a significant error in the approximation of pressure discontinuities across the shell. This error becomes especially troublesome in residual-based stabilized methods for incompressible flow, leading to problematic compressibility at practical levels of refinement. We modify existing stabilized methods to improve performance. To evaluate the accuracy of the proposed methods, we test them on benchmark problems and compare the results with those of established boundary-fitted techniques. Finally, we simulate the coupling of the bioprosthetic heart valve and the surrounding blood flow under physiological conditions, demonstrating the effectiveness of the proposed techniques in practical computations.

[1]  G. Burch,et al.  The second heart sound. , 1968, American heart journal.

[2]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[3]  B. Bellhouse,et al.  Mechanism of Closure of the Aortic Valve , 1968, Nature.

[4]  Hwa Liang Leo,et al.  Fluid Dynamic Assessment of Three Polymeric Heart Valves Using Particle Image Velocimetry , 2006, Annals of Biomedical Engineering.

[5]  Tayfun E. Tezduyar,et al.  Modeling of fluid–structure interactions with the space–time finite elements: contact problems , 2008 .

[6]  Ajit P. Yoganathan,et al.  A Comparison of Flow Field Structures of Two Tri-Leaflet Polymeric Heart Valves , 2005, Annals of Biomedical Engineering.

[7]  Thomas J. R. Hughes,et al.  Multiscale and Stabilized Methods , 2007 .

[8]  Jack Lemmon,et al.  A numerical simulation of mechanical heart valve closure fluid dynamics. , 2002, Journal of biomechanics.

[9]  Victor M. Calo,et al.  The role of continuity in residual-based variational multiscale modeling of turbulence , 2007 .

[10]  Yuri Bazilevs,et al.  Space–Time and ALE-VMS Techniques for Patient-Specific Cardiovascular Fluid–Structure Interaction Modeling , 2012 .

[11]  P. Wriggers,et al.  Isogeometric large deformation frictionless contact using T-splines , 2014 .

[12]  M. J. D. Powell,et al.  A method for nonlinear constraints in minimization problems , 1969 .

[13]  A. Chorin A Numerical Method for Solving Incompressible Viscous Flow Problems , 1997 .

[14]  Tayfun E. Tezduyar,et al.  Automatic mesh update with the solid-extension mesh moving technique , 2004 .

[15]  C. Peskin The immersed boundary method , 2002, Acta Numerica.

[16]  Victor M. Calo,et al.  Weak Dirichlet Boundary Conditions for Wall-Bounded Turbulent Flows , 2007 .

[17]  F. Hartmann,et al.  The discrete Babuška-Brezzi condition , 1986 .

[18]  Yuri Bazilevs,et al.  Isogeometric fluid–structure interaction analysis with emphasis on non-matching discretizations, and with application to wind turbines , 2012 .

[19]  Steven A Bogen,et al.  Clinical laboratory measurement of serum, plasma, and blood viscosity. , 2006, American journal of clinical pathology.

[20]  T. Hughes,et al.  Large Eddy Simulation and the variational multiscale method , 2000 .

[21]  Victor M. Calo,et al.  Improving stability of stabilized and multiscale formulations in flow simulations at small time steps , 2010 .

[22]  Tayfun E. Tezduyar,et al.  PARALLEL COMPUTATION OF INCOMPRESSIBLE FLOWS WITH COMPLEX GEOMETRIES , 1997 .

[23]  M. Hestenes Multiplier and gradient methods , 1969 .

[24]  C. Peskin Flow patterns around heart valves: A numerical method , 1972 .

[25]  Fotis Sotiropoulos,et al.  Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies , 2008, J. Comput. Phys..

[26]  Toshio Kobayashi,et al.  Fluid-structure interaction modeling of blood flow and cerebral aneurysm: Significance of artery and aneurysm shapes , 2009 .

[27]  Tayfun E. Tezduyar,et al.  Fluid–structure interaction modeling and performance analysis of the Orion spacecraft parachutes , 2011 .

[28]  David J. Benson,et al.  Sliding interfaces with contact-impact in large-scale Lagrangian computations , 1985 .

[29]  Thomas J. R. Hughes,et al.  What are C and h ?: inequalities for the analysis and design of finite element methods , 1992 .

[30]  W. Shyy,et al.  Regular Article: An Accurate Cartesian Grid Method for Viscous Incompressible Flows with Complex Immersed Boundaries , 1999 .

[31]  T. Tezduyar,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. I: The concept and the preliminary numerical tests , 1992 .

[32]  T. E. TezduyarAerospace,et al.  3d Simulation of Fluid-particle Interactions with the Number of Particles Reaching 100 , 1996 .

[33]  Y. Bazilevs,et al.  Small and large deformation analysis with the p- and B-spline versions of the Finite Cell Method , 2012 .

[34]  Tayfun E. Tezduyar,et al.  Space–time finite element computation of complex fluid–structure interactions , 2010 .

[35]  G. Tallini,et al.  ON THE EXISTENCE OF , 1996 .

[36]  J. Butany,et al.  Bioprosthetic heart valves: modes of failure , 2009, Histopathology.

[37]  Yuri Bazilevs,et al.  Fluid–structure interaction modeling of wind turbines: simulating the full machine , 2012, Computational Mechanics.

[38]  G. Hulbert,et al.  A generalized-α method for integrating the filtered Navier–Stokes equations with a stabilized finite element method , 2000 .

[39]  T. Hughes,et al.  Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes , 2010 .

[40]  Tayfun E. Tezduyar,et al.  Fluid–structure interaction modeling of parachute clusters , 2011 .

[41]  Gianluca Iaccarino,et al.  IMMERSED BOUNDARY METHODS , 2005 .

[42]  I. Akkerman,et al.  Large eddy simulation of turbulent Taylor-Couette flow using isogeometric analysis and the residual-based variational multiscale method , 2010, J. Comput. Phys..

[43]  D. Goldstein,et al.  Secondary flow induced by riblets , 1998, Journal of Fluid Mechanics.

[44]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[45]  P. Pibarot,et al.  Prosthetic Heart Valves Selection of the Optimal Prosthesis and Long-Term Management , 2009 .

[46]  Antonio J. Gil,et al.  An enhanced Immersed Structural Potential Method for fluid-structure interaction , 2013, J. Comput. Phys..

[47]  Tayfun E. Tezduyar,et al.  Computation of Inviscid Supersonic Flows Around Cylinders and Spheres With the V-SGS Stabilization and YZβ Shock-Capturing , 2009 .

[48]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[49]  Tayfun E. Tezduyar,et al.  Interface projection techniques for fluid–structure interaction modeling with moving-mesh methods , 2008 .

[50]  F. Baaijens A fictitious domain/mortar element method for fluid-structure interaction , 2001 .

[51]  Fehmi Cirak,et al.  A fixed‐grid b‐spline finite element technique for fluid–structure interaction , 2014 .

[52]  J. Guermond,et al.  Theory and practice of finite elements , 2004 .

[53]  H N Sabbah,et al.  Relation of the second sound to diastolic vibration of the closed aortic valve. , 1978, The American journal of physiology.

[54]  Tayfun E. Tezduyar,et al.  Space–time FSI modeling and dynamical analysis of spacecraft parachutes and parachute clusters , 2011 .

[55]  Marek Behr,et al.  Massively parallel finite element computation of 3d flows - mesh update strategies in computation of moving boundaries and interfaces° , 1995 .

[56]  Tayfun E. Tezduyar,et al.  Interface-tracking and interface-capturing techniques for finite element computation of moving boundaries and interfaces , 2006 .

[57]  S. Mittal,et al.  International Journal for Numerical Methods in Fluids Control of Vortex Shedding behind Circular Cylinder for Flows at Low Reynolds Numbers , 2022 .

[58]  Thomas J. R. Hughes,et al.  Fluid–structure interaction analysis of bioprosthetic heart valves: significance of arterial wall deformation , 2014, Computational Mechanics.

[59]  V. Brummelen Added Mass Effects of Compressible and Incompressible Flows in Fluid-Structure Interaction , 2009 .

[60]  Tayfun E. Tezduyar,et al.  Space–time fluid mechanics computation of heart valve models , 2014 .

[61]  Helio J. C. Barbosa,et al.  The finite element method with Lagrange multiplier on the boundary: circumventing the Babuscka-Brezzi condition , 1991 .

[62]  Peter Wriggers,et al.  A large deformation frictional contact formulation using NURBS‐based isogeometric analysis , 2011 .

[63]  Kazufumi Ito,et al.  Augmented Lagrangian methods for nonsmooth, convex optimization in Hilbert spaces , 2000 .

[64]  J B Uther,et al.  Measurement of ascending aortic flow patterns in man. , 1973, Journal of applied physiology.

[65]  I. Borazjani Fluid–structure interaction, immersed boundary-finite element method simulations of bio-prosthetic heart valves , 2013 .

[66]  M. Ulbrich,et al.  The Nitsche Method of the Navier-Stokes Equations for Immersed and Moving Boundaries , 2011 .

[67]  Tayfun E. Tezduyar,et al.  Modelling of fluid–structure interactions with the space–time finite elements: Solution techniques , 2007 .

[68]  Roger A. Sauer,et al.  A computational contact formulation based on surface potentials , 2013 .

[69]  W. Wall,et al.  An eXtended Finite Element Method/Lagrange multiplier based approach for fluid-structure interaction , 2008 .

[70]  T. Hughes,et al.  Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows , 2007 .

[71]  Yuri Bazilevs,et al.  The bending strip method for isogeometric analysis of Kirchhoff–Love shell structures comprised of multiple patches , 2010 .

[72]  Ahmad H. Nasri,et al.  T-splines and T-NURCCs , 2003, ACM Trans. Graph..

[73]  T. Wick Flapping and contact FSI computations with the fluid–solid interface-tracking/interface-capturing technique and mesh adaptivity , 2014 .

[74]  John A. Evans,et al.  An Isogeometric design-through-analysis methodology based on adaptive hierarchical refinement of NURBS, immersed boundary methods, and T-spline CAD surfaces , 2012 .

[75]  J. Damasceno,et al.  Numerical simulation of two-dimensional flows over a circular cylinder using the immersed boundary method , 2003 .

[76]  R. Verzicco,et al.  Combined Immersed-Boundary Finite-Difference Methods for Three-Dimensional Complex Flow Simulations , 2000 .

[77]  T. Tezduyar,et al.  Computation of inviscid compressible flows with the V‐SGS stabilization and YZβ shock‐capturing , 2007 .

[78]  Alessandro Corsini,et al.  Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD) , 2007 .

[79]  Berend E. Westerhof,et al.  The arterial Windkessel , 2009, Medical & Biological Engineering & Computing.

[80]  Fotis Sotiropoulos,et al.  A numerical method for solving the 3D unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries , 2007, J. Comput. Phys..

[81]  T. Hughes,et al.  The variational multiscale method—a paradigm for computational mechanics , 1998 .

[82]  H N Sabbah,et al.  Frequency spectrum of the aortic component of the second heart sound in patients with normal valves, aortic stenosis and aortic porcine xenografts. Potential for detection of porcine xenograft degeneration. , 1980, The American journal of cardiology.

[83]  Jintai Chung,et al.  A Time Integration Algorithm for Structural Dynamics With Improved Numerical Dissipation: The Generalized-α Method , 1993 .

[84]  Chih-Yung Wen,et al.  Experimental and numerical study of the separation angle for flow around a circular cylinder at low Reynolds number , 2004, Journal of Fluid Mechanics.

[85]  Charles A. Taylor,et al.  A coupled momentum method for modeling blood flow in three-dimensional deformable arteries , 2006 .

[86]  Tayfun E. Tezduyar,et al.  Mesh update strategies in parallel finite element computations of flow problems with moving boundaries and interfaces , 1994 .

[87]  M. Sacks,et al.  Simulated bioprosthetic heart valve deformation under quasi-static loading. , 2005, Journal of biomechanical engineering.

[88]  Lucy T. Zhang,et al.  Immersed finite element method , 2004 .

[89]  Tayfun E. Tezduyar,et al.  Advanced mesh generation and update methods for 3D flow simulations , 1999 .

[90]  A. Korobenko,et al.  Aerodynamic Simulation of Vertical-Axis Wind Turbines , 2014 .

[91]  T. Laursen Computational Contact and Impact Mechanics: Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis , 2002 .

[92]  A. Marsden,et al.  A comparison of outlet boundary treatments for prevention of backflow divergence with relevance to blood flow simulations , 2011 .

[93]  Ernst Rank,et al.  Finite cell method , 2007 .

[94]  Tayfun E. Tezduyar,et al.  Space-time finite element techniques for computation of fluid-structure interactions , 2005 .

[95]  Toshio Kobayashi,et al.  Computer modeling of cardiovascular fluid-structure interactions with the deforming-spatial-domain/stabilized space-time formulation , 2006 .

[96]  Kenji Takizawa,et al.  Space–time interface-tracking with topology change (ST-TC) , 2014 .

[97]  K. Höllig Finite element methods with B-splines , 1987 .

[98]  Roland Wüchner,et al.  Isogeometric shell analysis with Kirchhoff–Love elements , 2009 .

[99]  Tony W. H. Sheu,et al.  A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier-Stokes equations in time-varying complex geometries , 2010, J. Comput. Phys..

[100]  Charles A. Taylor,et al.  Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries , 2006 .

[101]  I. Babuska Error-bounds for finite element method , 1971 .

[102]  Tayfan E. Tezduyar,et al.  Stabilized Finite Element Formulations for Incompressible Flow Computations , 1991 .

[103]  Yuri Bazilevs,et al.  High-performance computing of wind turbine aerodynamics using isogeometric analysis , 2011 .

[104]  P. Wriggers Finite element algorithms for contact problems , 1995 .

[105]  T. Hughes,et al.  Isogeometric fluid-structure interaction: theory, algorithms, and computations , 2008 .

[106]  Kazufumi Ito,et al.  4. Augmented Lagrangian Methods for Nonsmooth, Convex Optimization , 2008 .

[107]  Constantin Bacuta,et al.  A Unified Approach for Uzawa Algorithms , 2006, SIAM J. Numer. Anal..

[108]  Marek Behr,et al.  Parallel finite-element computation of 3D flows , 1993, Computer.

[109]  Thomas J. R. Hughes,et al.  Finite element modeling of blood flow in arteries , 1998 .

[110]  Ernst Rank,et al.  Weak coupling for isogeometric analysis of non-matching and trimmed multi-patch geometries , 2014 .

[111]  Philippe Angot,et al.  A penalization method to take into account obstacles in incompressible viscous flows , 1999, Numerische Mathematik.

[112]  Gerhard A. Holzapfel,et al.  Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science , 2000 .

[113]  T. Kenner,et al.  The measurement of blood density and its meaning , 1989, Basic Research in Cardiology.

[114]  Yuri Bazilevs,et al.  Computational Fluid-Structure Interaction: Methods and Applications , 2013 .

[115]  C. C. Long,et al.  Fluid–structure interaction simulations of the Fontan procedure using variable wall properties , 2012, International journal for numerical methods in biomedical engineering.

[116]  Yuri Bazilevs,et al.  Wind turbine aerodynamics using ALE–VMS: validation and the role of weakly enforced boundary conditions , 2012, Computational Mechanics.

[117]  Thomas J. R. Hughes,et al.  Isogeometric divergence-conforming B-splines for the unsteady Navier-Stokes equations , 2013, J. Comput. Phys..

[118]  A. Whittemore,et al.  The Gibbs phenomenon. , 1990, AJR. American journal of roentgenology.

[119]  Wing Kam Liu,et al.  Lagrangian-Eulerian finite element formulation for incompressible viscous flows☆ , 1981 .

[120]  Frederick J Schoen,et al.  Calcification of tissue heart valve substitutes: progress toward understanding and prevention. , 2005, The Annals of thoracic surgery.

[121]  Georg Stadler,et al.  Path-following and augmented Lagrangian methods for contact problems in linear elasticity , 2007 .

[122]  H N Sabbah,et al.  Turbulent Blood Flow in the Ascending Aorta of Humans with Normal and Diseased Aortic Valves , 1976, Circulation research.

[123]  Paul R. Stay The Definition and Ray-Tracing of B-Spline Objects in a Combinatorial Solid Geometric Modeling System , 2013 .

[124]  Thomas J. R. Hughes,et al.  The multiscale formulation of large eddy simulation: Decay of homogeneous isotropic turbulence , 2001 .

[125]  Brummelen van Eh,et al.  Flux evaluation in primal and dual boundary-coupled problems , 2011 .

[126]  D. J. Benson,et al.  Patient-specific isogeometric structural analysis of aortic valve closure , 2015 .

[127]  Michael S Sacks,et al.  Effects of Leaflet Stiffness on In Vitro Dynamic Bioprosthetic Heart Valve Leaflet Shape , 2013, Cardiovascular engineering and technology.

[128]  S. Mittal,et al.  A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space-time procedure. II: Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders , 1992 .

[129]  M. Daniels,et al.  Architectural Trends in the Human Normal and Bicuspid Aortic Valve Leaflet and Its Relevance to Valve Disease , 2014, Annals of Biomedical Engineering.

[130]  Fehmi Cirak,et al.  Subdivision-stabilised immersed b-spline finite elements for moving boundary flows , 2012 .

[131]  Yuri Bazilevs,et al.  ALE-VMS AND ST-VMS METHODS FOR COMPUTER MODELING OF WIND-TURBINE ROTOR AERODYNAMICS AND FLUID–STRUCTURE INTERACTION , 2012 .

[132]  Thomas Wick,et al.  Fully Eulerian fluid-structure interaction for time-dependent problems , 2013 .

[133]  Alexei Lozinski,et al.  A fictitious domain approach for the Stokes problem based on the extended finite element method , 2013, 1303.6850.

[134]  Thomas J. R. Hughes,et al.  Explicit trace inequalities for isogeometric analysis and parametric hexahedral finite elements , 2013, Numerische Mathematik.

[135]  Thomas J. R. Hughes,et al.  Weak imposition of Dirichlet boundary conditions in fluid mechanics , 2007 .

[136]  Josef Kiendl,et al.  Isogeometric Analysis and Shape Optimal Design of Shell Structures , 2011 .

[137]  T. Hughes,et al.  Isogeometric analysis : CAD, finite elements, NURBS, exact geometry and mesh refinement , 2005 .

[138]  Tayfun E. Tezduyar,et al.  Finite element stabilization parameters computed from element matrices and vectors , 2000 .

[139]  Zhaosheng Yu A DLM/FD method for fluid/flexible-body interactions , 2005 .

[140]  Rolf Stenberg,et al.  Nitsche's method for general boundary conditions , 2009, Math. Comput..

[141]  J. Halleux,et al.  An arbitrary lagrangian-eulerian finite element method for transient dynamic fluid-structure interactions , 1982 .

[142]  S. Mittal,et al.  Incompressible flow past a circular cylinder: dependence of the computed flow field on the location of the lateral boundaries , 1995 .

[143]  Y. Bazilevs,et al.  Weakly enforced essential boundary conditions for NURBS‐embedded and trimmed NURBS geometries on the basis of the finite cell method , 2013 .

[144]  R. Cheng,et al.  Three-Dimensional Fluid-Structure Interaction Simulation of Bileaflet Mechanical Heart Valve Flow Dynamics , 2004, Annals of Biomedical Engineering.

[145]  D. J. Hart Fluid-structure interaction in the aortic heart valve : a three-dimensional computational analysis , 2002 .

[146]  L. Sirovich,et al.  Modeling a no-slip flow boundary with an external force field , 1993 .

[147]  T. Hughes,et al.  Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations , 1990 .

[148]  Dominik Schillinger,et al.  The Finite Cell Method: A Review in the Context of Higher-Order Structural Analysis of CAD and Image-Based Geometric Models , 2015 .

[149]  Tayfun E. Tezduyar,et al.  Computational Methods for Parachute Fluid–Structure Interactions , 2012 .

[150]  Ming-Chen Hsu,et al.  Computational vascular fluid–structure interaction: methodology and application to cerebral aneurysms , 2010, Biomechanics and modeling in mechanobiology.

[151]  Jia Lu,et al.  Dynamic Simulation of Bioprosthetic Heart Valves Using a Stress Resultant Shell Model , 2008, Annals of Biomedical Engineering.

[152]  T. Tezduyar,et al.  Mesh Moving Techniques for Fluid-Structure Interactions With Large Displacements , 2003 .

[153]  Ernst Rank,et al.  The finite cell method for three-dimensional problems of solid mechanics , 2008 .

[154]  F Auricchio,et al.  Patient-specific simulation of a stentless aortic valve implant: the impact of fibres on leaflet performance , 2014, Computer methods in biomechanics and biomedical engineering.

[155]  Z. J. Wang,et al.  A cartesian grid method for modeling multiple moving objects in 2D incompressible viscous flow , 2003 .

[156]  J. Nitsche Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind , 1971 .

[157]  Claes Johnson Numerical solution of partial differential equations by the finite element method , 1988 .

[158]  F. Brezzi On the existence, uniqueness and approximation of saddle-point problems arising from lagrangian multipliers , 1974 .

[159]  Thomas J. R. Hughes,et al.  Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device , 2009 .

[160]  Yuri Bazilevs,et al.  3D simulation of wind turbine rotors at full scale. Part I: Geometry modeling and aerodynamics , 2011 .

[161]  H. Uzawa,et al.  Preference, production, and capital: Iterative methods for concave programming , 1989 .

[162]  A. Huerta,et al.  Arbitrary Lagrangian–Eulerian Methods , 2004 .

[163]  P. Wriggers Computational contact mechanics , 2012 .

[164]  Antonio J. Gil,et al.  On continuum immersed strategies for Fluid-Structure Interaction , 2012 .

[165]  J. Dolbow,et al.  Imposing Dirichlet boundary conditions with Nitsche's method and spline‐based finite elements , 2010 .

[166]  F. Sotiropoulos,et al.  Immersed boundary methods for simulating fluid-structure interaction , 2014 .