  Every animal cell is filled with a cytoskeleton, a dynamic gel made of
inextensible fibers, such as microtubules, actin fibers, and intermediate
filaments, all suspended in a viscous fluid. Numerical simulation of this gel
is challenging because the fiber aspect ratios can be as large as $10^4$. We
describe a new method for rapidly computing the dynamics of inextensible
slender filaments in periodically-sheared Stokes flow. The dynamics of the
filaments are governed by a nonlocal slender body theory which we partially
reformulate in terms of the Rotne-Prager-Yamakawa hydrodynamic tensor. To
enforce inextensibility, we parameterize the space of inextensible fiber
motions and strictly confine the dynamics to the manifold of inextensible
configurations. To do this, we introduce a set of Lagrange multipliers for the
tensile force densities on the filaments and impose the constraint of no
virtual work in an $L^2$ weak sense. We augment this approach with a spectral
discretization of the local and nonlocal slender body theory operators which is
linear in the number of unknowns and gives improved spatial accuracy over
approaches based on solving a line tension equation. For dynamics, we develop a
second-order semi-implicit temporal integrator which requires at most a few
evaluations of nonlocal hydrodynamics and a few block diagonal linear solves
per time step. After demonstrating the improved accuracy and robustness of our
approach through numerical examples, we apply our formulation to a permanently
cross-linked actin mesh in a background oscillatory shear flow. We observe a
characteristic frequency at which the network transitions from quasi-static,
primarily elastic behavior to dynamic, primarily viscous behavior. We find that
nonlocal hydrodynamics increases the viscous modulus by as much as 25%, even
for semi-dilute fiber suspensions.
