## Fluid-Structure Interaction

### Introduction

Fluid-Structure Interaction(FSI) is basically concerned with the interaction of materials of two different types, e.g. fluid and solid.

One aspect of the coupling of different materials lies in the displacement of interface, which may cause trouble due to the fact that fluid and solid are modeled based on different type of coordinates.

Due to the constitutive laws, mathematical equations for solids are usually in Lagrangian discription while those for fluids are in Eulerian discription. The difference in discription causes trouble in both theoretical analysis and numerical computation.

### Model

The computational domain consists of fluid domain and structure domain, with interface depending on time.

• The fluid domain is governed by a parabolic equation in Eulerian discription.
• The structure domain is governed by a hyperbolic equation in Lagrangian discription.
• On the interface of fluid and solid, we have boundary conditions about continuity of velocity and stresses.

To resolve the inconsistancy in fluid and structure domains, an intermediate discription, Arbitrary Lagrangian-Eulerian (ALE), is introduced. ALE is built on fluid domain to track the displacement of interface and provide a relatively stable reference domain for fluids. With ALE, both the cost in remeshing and error in interpolation can be reduced.

### Numerical Methods

There are two different approaches in numerical methods.
• Monolithic Approach: the degrees of freedom of fluid and structure are included in a single linear system and solved by a single solver.
• Partitioned Approach: the degrees of freedom of fluid and structure are solved separately, with preexisting solvers.

While partitioned approach is proved to be unconditionally unstable under some circumstances, monolithic approach is considered to be more stable.