  In this work, we present a computational formulation based on continuum
mechanics to study the interaction of fluid membranes embedded with
semiflexible filaments. This is motivated by systems in membrane biology, such
as cytoskeletal networks and protein filaments aiding the cell fission process.
We model the membrane as a fluid shell via the Helfrich-Canham energy and the
filament as a one-dimensional Cosserat continuum. We assume the filament to be
tethered to the surface of the membrane in a way that it is allowed to float on
the surface freely. The novel filament-membrane coupling, which is anticipated
to yield interesting physics, also gives rise to unique computational
challenges, which we address in this work. We present validation results and
apply the formulation to certain problems inspired by cellular biology.
