  The dynamics of membranes, shells and capsules in fluid flow has become an
active research area in computational physics and computational biology. The
small thickness of these elastic materials enables their efficient
approximation as a hypersurface which exhibits an elastic response to in-plane
stretching and out-of-plane bending, possibly accompanied by a surface tension
force. In this work, we present a novel ALE method to simulate such elastic
surfaces immersed in Navier-Stokes fluids. The method combines high accuracy
with computational efficiency, since the grid is matched to the elastic surface
and can therefore be resolved with relatively few grid points. The focus of
this work is on axisymmetric shapes and flow conditions which are present in a
wide range of biophysical problems. We formulate axisymmetric elastic surface
forces and propose a discretization with surface finite-differences coupled to
evolving finite elements. We further develop an implicit coupling strategy to
reduce time step restrictions. We show in several numerical test cases that
accurate results can be achieved at computational times on the order of minutes
on a single core CPU. We demonstrate two state-of-the-art applications which to
our knowledge cannot be simulated with any other numerical method so far: We
present first simulations of the observed shape oscillations of novel
microswimming shells and the uniaxial compression of the cortex of a biological
cell during an AFM experiment.
