An analysis of a new stable partitioned algorithm for FSI problems. Part I: Incompressible flow and elastic solids

Stable partitioned algorithms for fluid-structure interaction (FSI) problems are developed and analyzed in this two-part paper. Part I describes an algorithm for incompressible flow coupled with compressible elastic solids, while Part II discusses an algorithm for incompressible flow coupled with structural shells. Importantly, these new added-mass partitioned (AMP) schemes are stable and retain full accuracy with no sub-iterations per time step, even in the presence of strong added-mass effects (e.g. for light solids). The numerical approach described here for bulk compressible solids extends the scheme of Banks et al. [1, 2] for inviscid compressible flow, and uses Robin (mixed) boundary conditions with the fluid and solid solvers at the interface. The basic AMP Robin conditions, involving a linear combination of velocity and stress, are determined from the outgoing solid characteristic relation normal to the fluid-solid interface combined with the matching conditions on the velocity and traction. Two alternative forms of the AMP conditions are then derived depending on whether the fluid equations are advanced with a fractional-step method or not. The stability and accuracy of the AMP algorithm is evaluated for linearized FSI model problems; the full nonlinear case being left for future consideration. A normal mode analysis is performed to show that the new AMP algorithm is stable for any ratio of the solid and fluid densities, including the case of very light solids when added-mass effects are large. In contrast, it is shown that a traditional partitioned algorithm involving a Dirichlet-Neumann coupling for the same FSI problem is formally unconditionally unstable for any ratio of densities. Exact traveling wave solutions are derived for the FSI model problems, and these solutions are used to verify the stability and accuracy of the corresponding numerical results obtained from the AMP algorithm for the cases of light, medium and heavy solids.