A stable added-mass partitioned (AMP) algorithm for elastic solids and incompressible flow

Abstract

A stable added-mass partitioned (AMP) algorithm is developed for fluid-structure interaction (FSI) problems coupling viscous incompressible flow and compressible elastic solids. The hyperbolic system governing the solid is integrated using an explicit upwind (Godunov) scheme. The equations for the fluid are integrated using an implicit-explicit (IMEX) fractional-step scheme whereby the velocity is advanced in one step, treating the viscous terms implicitly, and the pressure is computed in a second step. The AMP interface conditions for the fluid arise from the outgoing characteristic variables in the solid, and they are partitioned into a Robin (mixed) interface condition for the pressure and interface conditions for the velocity. The latter conditions include an impedance-weighted average between fluid and solid velocities using a fluid impedance of a special form. A similar impedance-weighted average is used to define interface values for the solid. Deforming composite grids are used to effectively handle the evolving geometry and large deformations. The new algorithm is verified for accuracy and stability on a number of useful benchmark problems in two space dimensions, including a radial-piston problem for which exact solutions for radial and azimuthal motions are found and tested. Exact traveling wave solutions for a solid disk surrounded by an annular region of fluid are also derived and used to verify the AMP scheme. Fluid flow in a channel past a deformable solid annulus is computed, and the errors are estimated from a self-convergence grid refinement study. The AMP scheme is found to be stable and second-order accurate, without sub-time-step iterations, even for very difficult cases of very light solids when added-mass and added-damping effects are large.