# Fluid-structure interaction with embedded finite-element methods

Fluids are characterized by the fact that they cannot withstand shear stress, and hence they occupy all available space. As a consequence it is preferable to express their laws in a spatial framework, also called an Eulerian, that consider evaluating fields on fixed points. On the contrary solids have preferable shapes, and thus a material framework (also called a Lagrangian) to express their motions is more convenient since in such a framework fields are evaluated on points following the motion of the solid. In particular in a fluid-structure interaction problem at least a part of the boundary of the fluid is not fixed through time since it moves along with the solid and therefore expressing the fluid system of a fluid-structure interaction problem is a difficult task in an Eulerian framework. A common method to solve fluid-structure interaction problems is the so-called Arbitrarity Lagrangian Eulerian (ALE) method.

The ALE method was introduced to allow computations in a framework which is neither Eulerian or Lagrangian. In particular we may consider the fluid in an ALE setting such that computations are performed on a moving mesh following the motion of the solid. However, we can observe that if the solid undergoes large deformations the fluid mesh also undergoes large deformations. From a practical point of view such an issue may result in an important loss of accuracy of the method. As a consequence, we aim at finding a method in which the fluid can be computed on a mesh whose boundaries are not necessarily fitting those of the solid. Many methods of this type are commonly found under the name of fictitious, immersed, embedded, or even unfitted. We use the term embedded but this choice is arbitrary.

Moreover, the constraints to be imposed at the interface between the fluid and the solid are: continuity of all components of the velocity, since the fluid is viscous, and continuity of the normal stress to ensure conservation of momentum at the interface. It follows that if we consider a finite element method with a fluid mesh larger than the fluid itself we may have to take into account singularities inside fluid elements crossed by the solid. We may also have to impose essential constraints into some fluid elements. It is therefore crucial to develop finite element methods that are able to represent these singularities in order to obtain accurate and robust methods.

In this regard, our research focuses on:

1. suitable frameworks for fluid-structure interaction problems with embedded methods;
2. efficient and robust techniques for problems with embedded singularities and embedded essential constraints.

Poster presentation:

Main collaborations: