Finite Element Immersed Boundary Method

A major difficulty associated with the numerical approximation of fluid structure interaction problems is given by the fact that the natural frameworks for dealing with fluids and solids are different: it is customary and convenient to describe fluids in an Eulerian framework, where the unknowns are expressed at fixed points in space, while for solids the natural framework is Lagrangian, where the unknowns are associated with moving material particles. The usual solution approach based on the ALE technique consists in creating an hybrid Eulerian-Lagrangian framework to solve the Navier-Stokes equations in the fluid and a suitable solid model, coupled through transmission conditions on the interface between the two domains. The ALE formulation allows the solution of the fluid equations on a fixed reference domain, which is smoothly deformed through an arbitrary time dependent mapping which reflects the changes of the solid domain at the fluid-structure interface. The effects of this arbitrary and non physical motion are then taken into account in a modified partial differential equation. A known drawback of the ALE method is that the domain deformations must preserve the initial topological properties of the system. As such, the ALE method might not be particularly suitable for the solution of problems with high deformations, or where channels may be occluded. Moreover, it has been shown that when the fluid and the solid have similar densities, the time advancing scheme of the partitioned iterative procedure could produce unconditional instability unless fully implicit schemes are used (Causin et al. 2005). The Immersed Boundary Method (IBM) and Immersed Finite Element Method (IFEM) do not present these drawbacks, even though they possess inferior mass conservation properties. The basic idea of these methods consists in solving in the whole domain (solid and fluid) the fluid-dynamics equations and in considering the effect of the presence of the structure as a forcing term of the system. In this way the computational grid is fixed, while the motion of the solid is determined by the fluid velocity (Peskin 2002). In the first versions of the Immersed Boundary Method and of the Finite Element Immersed Boundary Method, the immersed structure was thought of as a co-dimension one structure (2004 and 2007). In this page I present my early works on the Finite Element Immersed Boundary Method.