Generalized Robin–Neumann explicit coupling schemes for incompressible fluid‐structure interaction: Stability analysis and numerics

Abstract

SUMMARYWe introduce a new class of explicit coupling schemes for the numerical solution of fluid-structure interaction problems involving a viscous incompressible fluid and an elastic structure. These methods generalize the arguments reported in [Comput. Methods Appl. Mech. Engrg., 267:566-593, 2013, Numer. Math., 123(1):21-65, 2013] to the case of the coupling with thick-walled structures. The basic idea lies in the derivation of an intrinsic interface Robin consistency at the space semi-discrete level, using a lumpedmass approximation in the structure. The fluid-solid splitting is then performed through appropriate extrapolations of the solid velocity and stress on the interface. Based on these methods, a new, parameterfree, Robin-Neumann iterative procedure is also proposed for the partitioned solution of implicit coupling. A priori energy estimates, guaranteeing the stability of the schemes and the convergence of the iterative procedure, are established within a representative linear setting. The accuracy and performance of the methods are illustrated in several numerical examples.

Fully decoupled time-marching schemes for incompressible fluid/thin-walled structure interaction

“…Note that the latter includes higher-order through-the-thickness effects that overcome, in particular, the common modeling assumptions of Remark 2.2. An interesting step in this direction could be to combine the ideas recently reported in [71] with the present fully decoupled paradigm.…”

Explicit Robin–Neumann schemes for the coupling of incompressible fluids with thin-walled structures

“…An important setting, not covered by the present analysis, is the case of the coupling with thickwalled structures, for which the relation (5) is not valid anymore. A first attempt in this direction can be found in [25] (see also [24]), which generalizes the notion of interface Robin consistency using a mass lumping approximation in the solid. The resulting explicit coupling paradigm yields added-mass free stability (as in Theorem 1), but numerical evidence indicates that the Inria optimality of the accuracy (as in Theorem 3) is not necessarily preserved.…”

“…In summary, the relations (25) and (26) are nothing but an implicit time-discretization of (19) with the perturbed kinematic constraint (25). Therefore, in order to asses the stability and accuracy of Algorithm 3, we only need to investigate how this kinematic perturbation affects the stability and accuracy of the underlying implicit coupling scheme.…”

“…The last term of (37) represents the truncation error introduced by the Robin-Neumann splitting, that is, the time-consistency of the kinematic perturbation in (25). For the variant without extrapolation, the error estimate (37) predicts a sub-optimal O(τ 1 2 ) time-convergence rate in the energy-norm.…”

“…It should be noted that last two terms vanish due to definition (31) of the solid projection operators. Moreover, by combining (25) and (67) with (63), we have…”