We present a formulation for fluid-structure interaction problems in Fully Eulerian coordinates for both sub-problems, fluid and solid. By the use of Eulerian coordinates, we prevent the introduction of artificial coordinate system transformations as present in the common "Arbitrary Lagrangian Eulerian" (ALE) approach. Such mappings can prevent very large deformation, motion of the structure or even contact in a strictly monolithic setting.
The Fully Eulerian formulation, which is based on a variational monolithic coupling of the FSI problem is of front-capturing type, as the moving Eulerian domains are discretized on a fixed background mesh. The interface between fluid and solid must be captured by the discretization. For this, we introduce the "Initial Point Set" (IPS) function, that transports the initial coordinate system. The IPS is afterwards used for two purposes: First, to decide about the domain affiliation, and second to express the solid's stresses with respect to the deformation.
The major advantage of the Fully Eulerian formulation is its ability to handle problems with large deformation and contact in a strictly monolithic way, such that implicit discretization and solution techniques as well as gradient based methods for error estimation and adaptivity are available. Further, the method is of striking simplicity in a mathematical sense, as it can be regarded as one single momentum equation using material parameters and material laws, that change within the common fluid-structure domain.
The Fully Eulerian formulation is similar to multiphase-flows or general interface problems. This gives rise to several difficulties connected to the numerical approximation of the coupled systems: First, the solution may be smooth, but its derivatives have jumps across the interface. Second, the interface is moving, such that the design of time-stepping schemes is complicated, as a certain point may change its affiliation from fluid to solid within one time-step.
We will present the Fully Eulerian formulation in detail and comment on various issues connected to the numerical approximation. Further, we show numerical results that will both validate the formulation by comparison to traditional techniques, and that will show the potentials of the Fully Eulerian method when dealing with large deformation and contact.