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SIMULATING THE FLUID DYNAMICS OF NATURAL AND PROSTHETIC HEART VALVES USING THE IMMERSED BOUNDARY METHOD
, 2009
"... The immersed boundary method is both a general mathematical framework and a particular numerical approach to problems of fluidstructure interaction. In the present work, we describe the application of the immersed boundary method to the simulation of the fluid dynamics of heart valves, including a ..."
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Cited by 19 (6 self)
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The immersed boundary method is both a general mathematical framework and a particular numerical approach to problems of fluidstructure interaction. In the present work, we describe the application of the immersed boundary method to the simulation of the fluid dynamics of heart valves, including a model of a natural aortic valve and a model of a chorded prosthetic mitral valve. Each valve is mounted in a semirigid flow chamber. In the case of the mitral valve, the flow chamber is a circular pipe, and in the case of the aortic valve, the flow chamber is a model of the aortic root. The model valves and flow chambers are immersed in a viscous incompressible fluid, and realistic fluid boundary conditions are prescribed at the upstream and downstream ends of the chambers. To connect the immersed boundary models to the boundaries of the fluid domain, we introduce a novel modification of the standard immersed boundary scheme. In particular, near the outer boundaries of the fluid domain, we modify the construction of the regularized delta function which mediates fluidstructure coupling in the immersed boundary method, whereas in the interior of the fluid domain, we employ a standard fourpoint delta function which is frequently used with the immersed boundary method. The standard delta
On the Volume Conservation of the Immersed Boundary Method
, 2012
"... Abstract. The immersed boundary (IB) method is an approach to problems of fluidstructure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the f ..."
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Cited by 12 (4 self)
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Abstract. The immersed boundary (IB) method is an approach to problems of fluidstructure interaction in which an elastic structure is immersed in a viscous incompressible fluid. The IB formulation of such problems uses a Lagrangian description of the structure and an Eulerian description of the fluid. It is well known that some versions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volumeconservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finitedifference discretizations. In this paper, we use quasistatic and dynamic benchmark problems to investigate the effect of the choice of Eulerian discretization on the volumeconservation properties of a formally secondorder accurate IB method. We consider both collocated and staggeredgrid discretization methods. For the tests considered herein, the staggeredgrid IB scheme generally yields at least a modest improvement in volume conservation when compared to cellcentered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggeredgrid IB method are more than an order of magnitude smaller than those of
2009] Parallel and adaptive simulation of cardiac fluid dynamics
 Advanced Computational Infrastructures for Parallel and Distributed Adaptive Applications (John Wiley and Sons), expected publication in
, 2009
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Fluidstructure interaction and multibody contact. Application to the aortic valves
 Comput. Methods Appl. Mech. Eng
, 2009
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Locally corrected semiLagrangian methods for Stokes flow with moving elastic interfaces
, 2007
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Modelling and simulation of porous immersed boundaries
 Comput. Struct
, 2009
"... The immersed boundary method has been used to simulate a wide range of fluidstructure interaction problems from biology and engineering, wherein flexible solid structures deform in response to a surrounding incompressible fluid flow. We generalize the IB method to handle porous membranes by incorp ..."
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Cited by 4 (1 self)
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The immersed boundary method has been used to simulate a wide range of fluidstructure interaction problems from biology and engineering, wherein flexible solid structures deform in response to a surrounding incompressible fluid flow. We generalize the IB method to handle porous membranes by incorporating an additional transmembrane flux that obeys Darcy’s law. An approximate analytical solution is derived that clearly illustrates the effect of porosity on the immersed boundary motion. Numerical simulations in two dimensions are used to validate the analytical results and to illustrate the motion of more general porous membrane dynamics.
A Velocity Decomposition Approach for Moving Interfaces in Viscous Fluids
"... We present a secondorder accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the NavierStokes equations, with a singular force due to the stretching of the moving interface. We decompose the veloc ..."
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Cited by 3 (2 self)
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We present a secondorder accurate method for computing the coupled motion of a viscous fluid and an elastic material interface with zero thickness. The fluid flow is described by the NavierStokes equations, with a singular force due to the stretching of the moving interface. We decompose the velocity into a “Stokes ” part and a “regular ” part. The first part is determined by the Stokes equations and the singular interfacial force. The Stokes solution is obtained using the immersed interface method, which gives secondorder accurate values by incorporating known jumps for the solution and its derivatives into a finite difference method. The regular part of the velocity is given by the NavierStokes equations with a body force resulting from the Stokes part. The regular velocity is obtained using a timestepping method that combines the semiLagrangian method with the backward difference formula. Because the body force is continuous, jump conditions are not necessary. For problems with stiff boundary forces, the decomposition approach can be combined with fractional timestepping, using a smaller time step to advance the interface quickly by Stokes flow, with the velocity computed using boundary integrals. The small time steps maintain numerical stability, while the overall solution is updated on a larger time step to reduce computational cost.
A Second Order Virtual Node Algorithm for Stokes Flow Problems with Interfacial Forces and Discontinuous Material Properties
"... We present a numerical method for the solution of the Stokes equations that handles interfacial discontinuities due to both singular forces and discontinuous fluid properties such as viscosity and density. The discretization couples a Lagrangian representation of the material interface with an Euler ..."
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We present a numerical method for the solution of the Stokes equations that handles interfacial discontinuities due to both singular forces and discontinuous fluid properties such as viscosity and density. The discretization couples a Lagrangian representation of the material interface with an Eulerian representation of the fluid velocity and pressure. The method is efficient, easy to implement and yields discretely divergencefree velocities that are second order accurate. No knowledge of the jumps on the fluid variables and their derivatives is required along the interface. We discretize the equations using an embedded approach on a uniform MAC grid employing virtual nodes and duplicated cells at the interfaces. These additional degrees of freedom allow for accurate resolution of discontinuities in the fluid stress at the material interface but require a Lagrange multiplier term to enforce continuity of the fluid velocity. We provide a novel discretization of this term that accurately resolves the constant pressure null modes. We show that the accurate resolution of these modes accelerates the overall speed of our simulations. Interfaces are represented with a hybrid Lagrangian/level set method. The discrete coupled equations for the velocity, pressure and Lagrange multipliers are in the form of a symmetric KKT system. Numerical results indicate second order accuracy for the velocities and first order accuracy for the pressure (in L ∞).
Spatially adaptive stochastic methods for fluidstructure interactions subject to thermal fluctuations in domains with complex geometries
 Journal of Computational Physics
, 2014
"... Abstract. We develop stochastic mixed finite element methods for spatially adaptive simulations of fluidstructure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuationdissipation balance condition to develop compatible stochas ..."
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Abstract. We develop stochastic mixed finite element methods for spatially adaptive simulations of fluidstructure interactions when subject to thermal fluctuations. To account for thermal fluctuations, we introduce a discrete fluctuationdissipation balance condition to develop compatible stochastic driving fields for our discretization. We perform analysis that shows our condition is sufficient to ensure results consistent with statistical mechanics. We show the GibbsBoltzmann distribution is invariant under the stochastic dynamics of the semidiscretization. To generate efficiently the required stochastic driving fields, we develop a Gibbs sampler based on iterative methods and multigrid to generate fields with O(N) computational complexity. Our stochastic methods provide an alternative to uniform discretizations on periodic domains that rely on Fast Fourier Transforms. To demonstrate in practice our stochastic computational methods, we investigate within channel geometries having internal obstacles and noslip walls how the mobility/diffusivity of particles depends on location. Our methods extend the applicability of fluctuating hydrodynamic approaches by allowing for spatially adaptive resolution of the mechanics and for domains that have complex geometries relevant in many applications.
An Efficient Parallel Immersed Boundary Algorithm using a PseudoCompressible Fluid Solver
, 2013
"... We propose an efficient algorithm for the immersed boundary method on distributedmemory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudocompressibility method recently proposed by Guermond and Minev t ..."
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Cited by 3 (2 self)
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We propose an efficient algorithm for the immersed boundary method on distributedmemory architectures, with the computational complexity of a completely explicit method and excellent parallel scaling. The algorithm utilizes the pseudocompressibility method recently proposed by Guermond and Minev that uses a directional splitting strategy to discretize the incompressible NavierStokes equations, thereby reducing the linear systems to a series of onedimensional tridiagonal systems. We perform numerical simulations of several fluidstructure interaction problems in two and three dimensions and study the accuracy and convergence rates of the proposed algorithm. For these problems, we compare the proposed algorithm against other secondorder projectionbased fluid solvers. Lastly, the strong and weak scaling properties of the proposed algorithm are investigated.