# A stable added-mass partitioned (AMP) algorithm for elastic solids and incompressible flow: model problem analysis

## TL;DR

This paper introduces a stable, second-order accurate partitioned algorithm for fluid-structure interaction problems involving incompressible flow and elastic solids, effectively handling large added-mass effects.

## Contribution

A novel AMP scheme that remains stable and accurate for large added-mass effects in FSI problems, with detailed stability analysis and verification.

## Key findings

- The AMP scheme is stable for any solid-to-fluid density ratio.
- The algorithm achieves second-order accuracy in complex FSI scenarios.
- Validated with exact benchmark solutions including large interface deformations.

## Abstract

A stable added-mass partitioned (AMP) algorithm is developed for fluid-structure interaction (FSI) problems involving viscous incompressible flow and compressible elastic-solids. The AMP scheme remains stable and second-order accurate even when added-mass and added-damping effects are large. The fluid is updated with 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 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. The fluid impedance is defined using material and discretization parameters and follows from a careful analysis of the discretization of the governing equations and coupling conditions near the interface. A normal mode analysis is performed to show that the AMP scheme is stable for a few carefully-selected model problems. Two extensions of the analysis in Banks et al. are considered, including a first-order accurate discretization of a viscous model problem and a second-order accurate discretization of an inviscid model problem. The AMP algorithm is shown to be stable for any ratio of solid and fluid densities, including when added-mass effects are large. The algorithm is verified for accuracy and stability for a set of new exact benchmark solutions where finite interface deformations are permitted. The AMP scheme is found to be stable and second-order accurate even for very difficult cases of very light solids.