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.

SUMMARYWe consider a Leray model with a deconvolution-based indicator function for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of few thousand) with underresolved meshes. For the implementation of the model, we adopt a three-step algorithm called evolve-filterrelax (EFR) that requires (i) the solution of a Navier-Stokes problem, (ii) the solution of a Stokes-like problem to filter the Navier-Stokes velocity field, and (iii) a final relaxation step. We take advantage of a reformulation of the EFR algorithm as an operator splitting method to analyze the impact of the filter on the final solution vs a direct simulation of the Navier-Stokes equations. In addition, we provide some direction for tuning the parameters involved in the model based on physical and numerical arguments. Our approach is validated against experimental data for fluid flow in an idealized medical device (consisting of a conical convergent, a narrow throat, and a sudden expansion, as recommended by the U.S. Food and Drug Administration). Numerical results are in good quantitative agreement with the measured axial components of the velocity and pressures for two different flow rates corresponding to turbulent regimes, even for meshes with a mesh size more than 40 times larger than the smallest turbulent scale. After several numerical experiments, we perform a preliminary sensitivity analysis of the computed solution to the parameters involved in the model.

We consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in the model. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data. * mgirfogl@sissa.it †

We consider a Stokes problem posed on a 2D surface embedded in a 3D domain. The equations describe an equilibrium, area-preserving tangential flow of a viscous surface fluid and serve as a model problem in the dynamics of material interfaces. In this paper, we develop and analyze a Trace finite element method (TraceFEM) for such a surface Stokes problem. TraceFEM relies on finite element spaces defined on a fixed, surface-independent background mesh which consists of shape-regular tetrahedra. Thus, there is no need for surface parametrization or surface fitting with the mesh. The TraceFEM treated here is based on P 1 bulk finite elements for both the velocity and the pressure. In order to enforce the velocity vector field to be tangential to the surface we introduce a penalty term. The method is straightforward to implement and has an O(h 2 ) geometric consistency error, which is of the same order as the approximation error due to the P 1 -P 1 pair for velocity and pressure. We prove stability and optimal order discretization error bounds in the surface H 1 and L 2 norms. A series of numerical experiments is presented to illustrate certain features of the proposed TraceFEM.