A Stable Added-Mass Partitioned (AMP) Algorithm for Elastic Solids and Incompressible Flow: Model Problem Analysis

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) fractionalstep 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 [1] 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. On the other hand, the traditional algorithm involving a Dirichlet-Neumann coupling is shown to be unconditionally unstable as added-mass effects become large with grid refinement. The algorithm is verified for accuracy and stability for set of new exact benchmark solutions. These new solutions are elastic piston problems, 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.

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

“…The interface between the fluid and solid, in physical space given by x ∈ Γ(t), is determined by the mapping in (8) for the boundary of the solid given byx ∈Γ 0 . The interface is assumed to be smooth so that a well-defined normal to the interface exists.…”

“…Once the fluid velocity is advanced to the next time step, a Poisson problem is solved to update the fluid pressure. Following the analysis in [1,8], the interface condition in (15a) is used with the momentum equation in (1b) to derive a Robin condition for the pressure-Poisson problem that takes the form…”

“…In particular, the normal and tangential components of the velocity on the interface x ∈ Γ(t) are given by not result in a stable scheme. The analysis of a model problem that accounts for viscous effects performed in the companion paper [8], provides the suitable form for z f which is a key result that leads to a stable scheme.…”

“…The incorporation of the IMEX scheme has necesitated important changes to the original AMP interface conditions introduced in [1] to handle viscous added-damping effects that arise when the viscous CFL number becomes large. Modifications to the AMP interface conditions are guided by the analysis of carefully chosen FSI model problems, as discussed in the companion manuscript [8]. In particular, the analysis reveals a stable definition of the fluid impedance, which is a critical ingredient in the present AMP algorithm.…”

A split-step finite-element method for incompressible Navier-Stokes equations with high-order accuracy up-to the boundary

“…The AMP condition, which is a non-standard Robin-type boundary condition involving the fluid stress tensor, requires no adjustable parameters and in principle is applicable at the discrete level to couple the fluid and structure solvers of any accuracy and of any approximation methods (finite difference, finite element, finite volume, spectral element methods, etc). Within the finite-difference framework, the AMP algorithms have been developed and implemented to solve FSI problems involving the interaction of incompressible flows with a wide range of structures, such as elastic beams/shells [25,26], bulk solids [27][28][29] and rigid bodies [30][31][32]. It has been shown in these works that the AMP schemes are second-order accurate and stable without sub-time-step iterations, even for very light structures when added-mass effects are strong.…”