Combined field formulation and a simple stable explicit interface advancing scheme for fluid structure interaction

TL;DR

This paper introduces a combined field formulation and a simple explicit interface advancing scheme for fluid-structure interaction, enabling decoupled computation, energy stability, and reduced unknowns, with validation through numerical tests.

Contribution

The paper presents a novel combined field formulation with an explicit interface scheme that improves computational efficiency and stability in fluid-structure interaction problems.

Findings

Scheme is energy stable for convex strain energy solids.

Only a linear system needs to be solved each time step.

Numerical tests confirm stability and first-order accuracy.

Abstract

We develop a combined field formulation for the fluid structure (FS) interaction problem. The unknowns are (u;p;v), being the fluid velocity, fluid pressure and solid velocity. This combined field formulation uses Arbitrary Lagrangian Eulerian (ALE) description for the fluid and Lagrangian description for the solid. It automatically enforces the simultaneous continuities of both velocity and traction along the FS interface. We present a first order in time fully discrete scheme when the flow is incompressible Navier-Stokes and when the solid is elastic. The interface position is determined by first order extrapolation so that the generation of the fluid mesh and the computation of (u;p;v) are decoupled. This explicit interface advancing enables us to save half of the unknowns comparing with traditional monolithic schemes. When the solid has convex strain energy (e.g. linear elastic), we prove that the total energy of the fluid and the solid at time t^{n} is bounded by the total energy at time t^{n-1}. Like in the continuous case, the fluid mesh velocity which is used in ALE description does not enter into the stability bound. Surprisingly, the nonlinear convection term in the Navier-Stokes equations plays a crucial role to stabilize the scheme and the stability result does not apply to Stokes flow. As the nonlinear convection term is treated semi-implicitly, in each time step, we only need to solve a linear system (and only once) which involves merely (u;p;v) if the solid is linear elastic. Two numerical tests including the benchmark test of Navier-Stokes flow past a Saint Venant-Kirchhoff elastic bar are performed. In addition to the stability, we also confirm the first order temporal accuracy of our explicit interface advancing scheme.