  In this paper we propose a novel and general approach to design semi-implicit
methods for the simulation of fluid-structure interaction problems in a fully
Eulerian framework. In order to properly present the new method, we focus on
the two-dimensional version of the general model developed to describe full
membrane elasticity. The approach consists in treating the elastic source term
by writing an evolution equation on the structure stress tensor, even if it is
nonlinear. Then, it is possible to show that its semi-implicit discretization
allows us to add to the linear system of the Navier-Stokes equations some
consistent dissipation terms that depend on the local deformation and stiffness
of the membrane. Due to the linearly implicit discretization, the approach does
not need iterative solvers and can be easily applied to any Eulerian framework
for fluid-structure interaction. Its stability properties are studied by
performing a Von Neumann analysis on a simplified one-dimensional model and
proving that, thanks to the additional dissipation, the discretized coupled
system is unconditionally stable. Several numerical experiments are shown for
two-dimensional problems by comparing the new method to the original explicit
scheme and studying the effect of structure stiffness and mesh refinement on
the membrane dynamics. The newly designed scheme is able to relax the time step
restrictions that affect the explicit method and reduce crucially the
computational costs, especially when very stiff membranes are under
consideration.
