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                <forename type="first">Matteo</forename>
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            <idno type="stamp" n="UNIV-PARIS7" corresp="UNIV-PARIS">Université Denis Diderot - Paris VII</idno>
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                <title xml:lang="en">Fast numerical methods for fluid-structure interaction problems arising in blood flows</title>
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                <title level="m">Proceedings of the Third Chilean Workshop on Numerical Analysis of Partial Differential Equations, WONAPDE 2010</title>
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              <p>Numerical computations from patient-specific medical data have become a key issue now in biomedical engineering. The complexity of the geometry and of the physical problem are generally such that simulation time can be problematically elevated. As a result, numerical achievements which aim at accelerating the computations are of very high interest in this context. We will present in this talk two numerical strategies to speed-up the numerical resolution of fluid-structure interaction problems involving a viscous incompressible fluid and that arise typically in blood flows. These strategies are both based on a partitioned algorithm to solve the coupled problem. In a first part, we will describe a semi-implicit coupling scheme [1], stable irrespectively of the so-called added-mass effect [4] and which allows for conservative time-stepping within the structure. Its efficiency is based on the explicit splitting of the viscous effects and geometrical/convective non-linearities, through the use of the Chorin-Temam projection scheme within the fluid [6]. Stability relies on the implicit treatment of the pressure stresses and on a Robin-based treatment of the viscous coupling, inspired from Nitsche's method [2, 3]. The numerical stability of the scheme is proved theoretically through a priori energy estimation. Numerical results in two and three dimensions illustrate the stability and efficiency of the scheme as well as its potentiality in the context of blood flow simulations. In a second part, we will present a parallel time-marching scheme for coupled parabolic- hyperbolic problems [5], as a prototype of fluid-structure interaction problems involving a linear structure and a viscous fluid. This scheme should allow further speed-up in the numerical simulations. It is obtained through application of the classical Parareal scheme [9] to the parabolic part, while the modified algorithm proposed by Farhat et al. [7, 8] is applied to the hyperbolic part. This hybrid Parareal treatment takes advantage of the partitioned for- mulation of the coupled propagator. No linearity assumption is made on the parabolic side. Numerical evidence shows that the resulting scheme is stable for a wider range of physical and discretization parameters than the classical Parareal.</p>
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