On the robustness of Dirichlet-Neumann coupling schemes for fluid-structure-interaction problems with nearly-closed fluid domains

Partitioned methods for fluid-structure interaction (FSI) involve solving the structural and flow problems sequentially. These methods allow for separate settings for the fluid and solid subsystems and thus modularity, enabling reuse of advanced commercial and open-source software. Most partitioned FSI schemes apply a Dirichlet-Neumann (DN) split of the interface conditions. The DN scheme is adequate in a wide range of applications, but it is sensitive to the added-mass effect, and it is susceptible to the incompressibility dilemma, i.e. it completely fails for FSI problems with an incompressible fluid furnished with Dirichlet boundary conditions on the part of its boundary complementary to the interface. In this paper, we show that if the fluid is incompressible and the fluid domain is nearly-closed, i.e. it carries Dirichlet conditions except for a permeable part of the boundary carrying a Robin condition, then the DN partitioned approach is sensitive to the flow resistance at the permeable part, and convergence of the partitioned approach deteriorates as the flow resistance increases. The DN scheme then becomes unstable in the limit as the flow resistance passes to infinity. Based on a simple model problem, we show that in the nearly-closed case, the convergence rate of the DN partitioned method depends on a so-called added-damping effect. The analysis gives insights that can aid to improve robustness and efficiency of partitioned method for FSI problems with contact, e.g. valve applications. In addition, the results elucidate the incompressibility dilemma as a limit of the added-damping effect passing to infinity, and the corresponding challenges related to FSI problems with nearly closed fluid-domain configurations. Via numerical experiments, we consider the generalization of the results of the simple model problem to more complex nearly-closed FSI problems.