In bioengineering applications problems of flow interacting with
elastic solid are very common. We formulate the problem of interaction
for an incompressible fluid and an incompressible elastic material in
a fully coupled arbitrary Lagrangian-Eulerian formulation. The
mathematical description and the numerical schemes are designed in
such a way that more complicated constitutive relations (and more
realistic for bioengineering applications) can be incorporated easily.
The whole domain of interest is treated as one continuum and the same
discretization in space (Q2/P1 FEM) and time (Crank-Nicholson) is used
for both, solid and fluid, parts. The resulting nonlinear algebraic
system is solved by an approximate Newton method. The combination of
second order discretization and fully coupled solution method gives a
method with high accuracy and robustness. To demonstrate the
flexibility of this numerical approach we apply the same method to a
mixture based model of elastic material with perfusion which also
falls into the category of fluid structure interactions. A few simple
2D example calculations with simple material models and a large
deformations of the solid part are presented.