Robin-Robin preconditioned Krylov methods for fluid-structure interaction problems

Revision as of 10:50, 31 March 2020

Abstract

In this work, we propose a Robin–Robin preconditioner combined with Krylov iterations for the solution of the interface system arising in fluid–structure interaction (FSI) problems. It can be seen as a partitioned FSI procedure and in this respect it generalizes the ideas introduced in [S. Badia, F. Nobile, C. Vergara, J. Comput. Phys. 227 (2008) 7027–7051]. We analyze the convergence of GMRES iterations with the Robin–Robin preconditioner on a model problem and compare its efficiency with some existing algorithms. The method is shown to be very efficient for many challenging fluid–structure interaction problems, such as those characterized by a large added-mass effect or by enclosed fluids. In particular, the possibility to solve balloon-type problems without any special treatment makes this algorithm very appealing compared to the computationally intensive existing approaches.

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 == Abstract ==
-We are interested in those FSI problems where the added-mass effect is high, that is when the ratio between the fluid and structure densities is close to one (or larger). This typically appears in hemodynamic applications. It has been reported in the literature [41], [9], [18], [29] that the solution of the FSI system using explicit partitioned approaches (also called loosely coupled) is problematic in this situation. We refer to [44] for a discussion about the added-mass effect for compressible flows. In general, explicit algorithms that solve only once (or just few times) per time step the fluid and structure sub-problems are unstable, unlike for low added-mass problems such as those arising in aeroelasticity.
+In this work, we propose a Robin–Robin preconditioner combined with Krylov iterations for the solution of the interface system arising in fluid–structure interaction (FSI) problems. It can be seen as a partitioned FSI procedure and in this respect it generalizes the ideas introduced in [S. Badia, F. Nobile, C. Vergara, J. Comput. Phys. 227 (2008) 7027–7051]. We analyze the convergence of GMRES iterations with the Robin–Robin preconditioner on a model problem and compare its efficiency with some existing algorithms. The method is shown to be very efficient for many challenging fluid–structure interaction problems, such as those characterized by a large added-mass effect or by enclosed fluids. In particular, the possibility to solve balloon-type problems without any special treatment makes this algorithm very appealing compared to the computationally intensive existing approaches.
-To obtain stable numerical schemes to solve the monolithic system, one has then to consider algorithms that enforce exactly at each time step the continuity of the velocity and normal stresses at the fluid–structure interface. Among them, we consider modular algorithms that involve separate fluid and structure evaluations and that interact through the exchange of suitable transmission conditions on the interface. These algorithms are based on domain decomposition preconditioners (see [11]), which are applied to the interface equation (Schur complement) related to the whole FSI system. Then, the preconditioned system is solved using an iterative solver. Let us note that any approach employing domain decomposition preconditioners is monolithic, in the sense that it provides the monolithic solution. Thus, at convergence, they guarantee the continuity of the velocity and the normal stress at the interface (strong coupling).
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