Abstract. We discuss in this paper the numerical approximation of fluid-structure interaction (FSI) problems dealing with strong added-mass effect. We propose new semi-implicit algorithms based on inexact block-LU factorization of the linear system obtained after the space-time discretization and linearization of the FSI problem. As a result, the fluid velocity is computed separately from the coupled pressure-structure velocity system at each iteration, reducing the computational cost. We investigate explicit-implicit decomposition through algebraic splitting techniques originally designed for the FSI problem. This approach leads to two different families of methods which extend to FSI the algebraic pressure correction method and the Yosida method, two schemes that were previously adopted for pure fluid problems. Furthermore, we have considered the inexact factorization of the fluid-structure system as a preconditioner. The numerical properties of these methods have been tested on a model problem representing a blood-vessel system. Key words. fluid-structure interaction, semi-implicit coupling, inexact factorization, algebraic splitting methods, added-mass effect, blood flow AMS subject classifications. 74F10, 65N30, 76D05 DOI. 10.1137/070680497 1. Introduction. We are interested in the numerical approximation of the heterogeneous mechanical system which couples the equations governing a fluid flow and the deformation of a structure; this situation arises in many engineering problems.A great variety of strategies have been proposed to solve fluid-structure interaction (FSI) problems. A first issue is how to deal with the nonlinearity of the problem. In fact, not only are the fluid (and in some cases the structure) equations nonlinear, but the structure displacement also modifies the fluid domain generating geometrical nonlinearities. The fixed point technique (e.g., [3,20,21,10]) is the simplest to linearize the FSI problem; however, Newton (e.g., [13]) and quasi-Newton (e.g., [30,15,19,17]) methods have also been considered.A classical restriction for fluid-structure algorithms is modularity. Most of the time the codes for the pure fluid problem and for the pure structure problem already exist, and they are optimized for the specific mathematical features of the two different problems. Then the best way to solve the FSI problem would be to design algorithms

We present a new model and a novel loosely coupled partitioned numerical scheme modeling fluid-structure interaction (FSI) in blood flow allowing non-zero longitudinal displacement. Arterial walls are modeled by a linearly viscoelastic, cylindrical Koiter shell model capturing both radial and longitudinal displacement. Fluid flow is modeled by the Navier-Stokes equations for an incompressible, viscous fluid. The two are fully coupled via kinematic and dynamic coupling conditions. Our numerical scheme is based on a new modified Lie operator splitting that decouples the fluid and structure sub-problems in a way that leads to a loosely coupled scheme which is unconditionally stable. This was achieved by a clever use of the kinematic coupling condition at the fluid and structure sub-problems, leading to an implicit coupling between the fluid and structure velocities. The proposed scheme is a modification of the recently introduced "kinematically coupled scheme" for which the newly proposed modified Lie splitting significantly increases the accuracy. The performance and accuracy of the scheme were studied on a couple of instructive examples including a comparison with a monolithic scheme. It was shown that the accuracy of our scheme was comparable to that of the monolithic scheme, while our scheme retains all the main ad-$ vantages of partitioned schemes, such as modularity, simple implementation, and low computational costs.

In this article we address the numerical simulation of fluid-structure interaction (FSI) problems featuring large added-mass effect. We analyze different preconditioners for the coupled system matrix obtained after space-time discretization and linearization of the FSI problem. The classical Dirichlet-Neumann preconditioner has the advantage of "modularity" because it allows to reuse existing fluid and structure codes with minimum effort (simple interface communication). Unfortunately, its performance is very poor in case of large added-mass effects. Alternatively, we consider two non-modular approaches. The first one consists in preconditioning the coupled system with a suitable diagonal scaling combined with an ILUT 1 preconditioner. The system is then solved by a Krylov method. The drawback of this procedure is that the combination of fluid and structure codes to solve the coupled system is not straightforward. The second non-modular approach we consider is a splitting technique based on an inexact block-LU factorization of the linear FSI system. The resulting algorithm computes the fluid velocity separately from the coupled pressure-structure system at each iteration, reducing the computational cost. Independently of the preconditioner, the efficiency of semi-implicit algorithms (i.e., those that treat geometric and fluid nonlinearities in an explicit way) is highlighted and their performance compared to the one of implicit algorithms. All the methods are tested on three-dimensional blood-vessel systems. The algorithm combining the non-modular ILUT preconditioner with Krylov methods proved to be the fastest.

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We address the numerical simulation of fluid-structure interaction problems dealing with an incompressible fluid whose density is close to the structure density. We propose a semi-implicit coupling scheme based on an algebraic fractional-step method. The basic idea of a semi-implicit scheme consists in coupling implicitly the added-mass effect, while the other terms (dissipation, convection and geometrical non-linearities) are treated explicitly. Thanks to this kind of explicit-implicit splitting, computational costs can be reduced (in comparison to fully implicit coupling algorithms) and the scheme remains stable for a wide range of discretization parameters. In this paper we propose to derive this kind of splitting from the algebraic formulation of the coupled fluid-structure problem (after finite-element space discretization). This approach extends for the first time to fluid-structure problems the algebraic fractional step methodology that was previously advocated to treat the pure fluid problem in a fixed domain. More particularly, for the specific semi-implicit method presented in this report we adapt the Yosida scheme to the case of a coupled fluid-structure problem. 1This scheme relies on an approximate LU block factorization of the matrix obtained after the discretization in time and space of the fluid-structure system. We analyze the numerical performances of this scheme on 2D fluid-structure simulations performed with a simple 1D structure model.