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The immersed boundary method is both a general mathematical framework and a particular numerical approach to problems of fluid-structure 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 semi-rigid 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 fluid-structure coupling in the immersed boundary method, whereas in the interior of the fluid domain, we employ a standard four-point delta function which is frequently used with the immersed boundary method. The standard delta

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The Immersed Boundary method is one of the most useful computational methods in studying fluid structure interaction. On the other hand, the Immersed Boundary method is also known to require small time steps to maintain stability when solved with an explicit method. Many implicit or approximately implicit methods have been proposed in the literature to remove this severe time step stability constraint, but none of them give satisfactory performance. In this paper, we propose an efficient semiimplicit scheme to remove this stiffness from the Immersed Boundary method for the Navier-Stokes equations. The construction of our semi-implicit scheme consists of two steps. First, we obtain a semi-implicit discretization which is proved to be unconditionally stable. This unconditionally stable semi-implicit scheme is still quite expensive to implement in practice. Next, we apply the Small Scale Decomposition to the unconditionally stable semi-implicit scheme to construct our efficient semi-implicit scheme. Unlike other implicit or semi-implicit schemes proposed in the literature, our semi-implicit scheme can be solved explicitly in the spectral space. Thus the computational cost of our semi-implicit schemes is comparable to that of an explicit scheme. Our extensive numerical experiments show that our semi-implicit scheme has much better stability property than an explicit scheme. This offers a substantial computational saving in using the Immersed Boundary method.

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Abstract. The immersed boundary (IB) method is an approach to problems of fluid-struc-ture interaction in which an elastic structure is immersed in a viscous incompress-ible 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 ver-sions of the IB method can suffer from poor volume conservation. Methods have been introduced to improve the volume-conservation properties of the IB method, but they either have been fairly specialized, or have used complex, nonstandard Eulerian finite-difference discretizations. In this paper, we use quasi-static and dynamic bench-mark problems to investigate the effect of the choice of Eulerian discretization on the volume-conservation properties of a formally second-order accurate IB method. We consider both collocated and staggered-grid discretization methods. For the tests con-sidered herein, the staggered-grid IB scheme generally yields at least a modest im-provement in volume conservation when compared to cell-centered methods, and in many cases considered in this work, the spurious volume changes exhibited by the staggered-grid IB method are more than an order of magnitude smaller than those of

The immersed boundary method is an algorithm for simulating the interaction of immersed elastic bodies or boundaries with a viscous incompressible fluid. The immersed elastic material is represented in the fluid equations by a system or field of applied forces. The particular case of Stokes flow with applied forces on a periodic domain involves two related mathematical complications. One of these is that an arbitrary constant vector may be added to the fluid velocity, and the other is the constraint that the integral of the applied force must be zero. Typically, forces defined on a freely floating elastic immersed boundary or body satisfy this constraint, but there are many important classes of forces that do not. For example, the so-called tether forces that are used to prescribe the simulated configuration of an immersed boundary, possibly in a time-dependent manner, typically do not sum to zero. Another type of force that does not have zero integral is a uniform force density that may be used to simulate an overall pressure gradient driving flow through a system. We present a method for periodic Stokes flow that when used with tether points, admits the use of all forces irrespective of their integral over the domain. A byproduct of this method is that the additive constant velocity associated with periodic Stokes flow is uniquely determined. Indeed, the additive constant is chosen at each time step so that the sum of the tether forces balances the sum of any other forces that may be applied.

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The method of regularized Stokeslets is a numerical approach to approximating solutions of fluid-structure interaction problems in the Stokes regime. Regularized Stokeslets are fundamental solutions to the Stokes equations with a regularized point-force term that are used to represent forces generated by rigid or elastic object interacting with the fluid. Due to the linearity of the Stokes equations, the velocity at any point in the fluid can be computed by summing the contributions of regularized Stokeslets, and the time evolution of positions can be computed using standard methods for ordinary differential equations. Rigid or elastic objects in the flow are usually treated as immersed boundaries represented by a collection of regularized Stokeslets coupled together by virtual springs which determine the forces exerted by the boundary in the fluid. For problems with boundaries modeled by springs with large spring constants, the resulting ordinary differential equations become stiff, and hence the time step for explicit time integration methods is severely constrained. Unfortunately, the